Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.01Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equationsZhangXinqiu
School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China
LiuLishan
School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China;Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia
WuYonghong
Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia

In this article, we study the existence and the uniqueness of iterative positive solutions for a class of nonlinear singular integral equations in which the nonlinear terms may be singular in both time and space variables. By using the fixed point theorem of mixed monotone operators in cones, we establish the conditions for the existence and uniqueness of positive solutions to the problem. Moreover, we derive various properties of the positive solutions to the equation and establish their dependence on the model parameter. The theorem obtained in this paper is more general and complements many previous known results including singular and nonlinear cases. Application of the results to the study of differential equations are also given in the article.

34B1634B18Mixed monotone operatorfixed point theoremiterative positive solutionsingular integral equationsboundary value problemcone.
R. P.AgarwalOn fourth order boundary value problems arising in beam analysisDifferential Integral Equations1989291110 R. P.Agarwal Y. M.ChowIterative methods for a fourth order boundary value problem10.1016/0377-0427(84)90058-X J. Comput. Appl. Math.198410203217A.Cabada G.-T.WangPositive solutions of nonlinear fractional differential equations with integral boundary value conditions10.1016/j.jmaa.2011.11.065J. Math. Anal. Appl.2012389403411 Z.-W.Cao D.-Q.Jiang C.-J.Yuan D.O’ReganExistence and uniqueness of solutions for singular integral equation10.1007/s11117-008-2209-8Positivity200812725732 Y.-J.Cui L.-S.Liu X.-Q.ZhangUniqueness and existence of positive solutions for singular differential systems with coupled integral boundary value problems10.1155/2013/340487Abstr. Appl. Anal.2013201319D.-J.Guo Y. J.Cho J.ZhuPartial ordering methods in nonlinear problemsNova Science Publishers, Inc., Hauppauge, NY2004 D.-J.Guo V.LakshmikanthamNonlinear problems in abstract conesNotes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., , Boston, MA1988 X.-A.Hao L.-S.Liu Y.-H.Wu Q.SunPositive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions10.1016/j.na.2010.04.074Nonlinear Anal.20107316531662H. H. G.HashemOn successive approximation method for coupled systems of Chandrasekhar quadratic integral equations10.1016/j.joems.2014.03.004 J. Egyptian Math. Soc.201523108112W.-H.Jiang J.-L.ZhangPositive solutions for (k, n-k) conjugate boundary value problems in Banach spaces10.1016/j.na.2008.10.104 Nonlinear Anal.200971723729 M.Jleli B.Samet Existence of positive solutions to an arbitrary order fractional differential equation via a mixed monotone operator method10.15388/NA.2015.3.4Nonlinear Anal. Model. Control201520367376A. A.Kilbas H. M.Srivastava J. J.TrujilloTheory and applications of fractional differential equationsNorth-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam2006 K. Q.LanMultiple positive solutions of conjugate boundary value problems with singularities10.1016/S0096-3003(02)00739-7 Appl. Math. Comput.2004147461474 K.Latrach M. A.TaoudiExistence results for a generalized nonlinear Hammerstein equation on $$L_1$$ spaces10.1016/j.na.2006.03.022 Nonlinear Anal.20076623252333P.-D.Lei X.-N.Lin D.-Q.JiangExistence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems10.1016/j.na.2007.08.049 Nonlinear Anal.20086927732779 F.-Y.Li Y.-H.Li Z.-P.LiangExistence of solutions to nonlinear Hammerstein integral equations and applications10.1016/j.jmaa.2005.10.014 J. Math. Anal. Appl.2006323209227H.-D.Li L.-S.Liu Y.-H.WuPositive solutions for singular nonlinear fractional differential equation with integral boundary conditions10.1186/s13661-015-0493-3 Bound. Value Probl.20152015115X.-N.Lin D.-Q.Jiang X.-Y.LiExistence and uniqueness of solutions for singular (k, n - k) conjugate boundary value problems10.1016/j.camwa.2006.03.019Comput. Math. Appl.200652375382X.-N.Lin D.-Q.Jiang X.-Y.Li Existence and uniqueness of solutions for singular fourth-order boundary value problems10.1016/j.cam.2005.08.016 J. Comput. Appl. Math.2006196155161 L.-S.Liu F.GuoC.-X.WuY.-H.Wu Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces10.1016/j.jmaa.2004.10.069 J. Math. Anal. Appl.2005309638647 L.-S.Liu C.-X.Wu F.GuoExistence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications10.1016/S0898-1221(04)90002-8Comput. Math. Appl.2004471322 L.-S.Liu X.-Q.Zhang J.Jiang Y.-H.WuThe unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problemsJ. Nonlinear Sci. Appl.2016929432958A.Lomtatidze L.Malaguti On a nonlocal boundary value problem for second order nonlinear singular differential equations10.1515/GMJ.2000.133 Georgian Math. J.20007133154 M.-H.Pei S. K.ChangMonotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem10.1016/j.mcm.2010.01.009Math. Comput. Modelling20105112601267 I.PodlubnyFractional differential equationsAn introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA1999 S. G.Samko A. A.Kilbas O. I.Marichev Fractional integrals and derivativesTheory and applications, Edited and with a foreword by S. M. Nikol'skii, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon1993 Y.-P.Sun M.ZhaoPositive solutions for a class of fractional differential equations with integral boundary conditions10.1016/j.aml.2014.03.008Appl. Math. Lett.2014341721Y.-Q.Wang L.-S.Liu Y.-H.Wu Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity10.1016/j.na.2011.06.026Nonlinear Anal.20117464346441 Y.-Q.Wang L.-S.Liu Y.-H.WuPositive solutions for a nonlocal fractional differential equation10.1016/j.na.2011.02.043 Nonlinear Anal.20117435993605J. R. L.WebbUniqueness of the principal eigenvalue in nonlocal boundary value problems10.3934/dcdss.2008.1.177Discrete Contin. Dyn. Syst. Ser. S20081177186J. R. L.Webb Nonlocal conjugate type boundary value problems of higher order10.1016/j.na.2009.01.033Nonlinear Anal.20097119331940J. R. L.WebbExistence of positive solutions for a thermostat model10.1016/j.nonrwa.2011.08.027 Nonlinear Anal. Real World Appl.201213923938 J. R. L.Webb Positive solutions of nonlinear differential equations with Riemann-Stieltjes boundary conditions10.14232/ejqtde.2016.1.86Electron. J. Qual. Theory Differ. Equ.20162016113 P. J. Y.WongTriple positive solutions of conjugate boundary value problems, II10.1016/S0898-1221(00)00178-4 Comput. Math. Appl.200040537557Z.-L.YangPositive solutions for a system of nonlinear Hammerstein integral equations and applications10.1016/j.amc.2012.05.006 Appl. Math. Comput.20122181113811150 B.YangUpper estimate for positive solutions of the (p, n - p) conjugate boundary value problem10.1016/j.jmaa.2012.01.054 J. Math. Anal. Appl.2012390535548C.-J.Yuan X.-D.Wen D.-Q.Jiang Existence and uniqueness of positive solution for nonlinear singular 2mth-order continuous and discrete Lidstone boundary value problems10.1016/S0252-9602(11)60228-2Acta Math. Sci. Ser. B Engl. Ed.201131281291 C.-B.Zhai R.-P.Song Q.-Q.HanThe existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem10.1016/j.camwa.2011.08.003Comput. Math. Appl.20116226392647H.-E.Zhang Iterative solutions for fractional nonlocal boundary value problems involving integral conditions10.1186/s13661-015-0517-zBound. Value Probl.20162016113X.-G.Zhang L.-S.Liu B.Wiwatanapataphee Y.-H.Wu The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition10.1016/j.amc.2014.02.062 Appl. Math. Comput.2014235412422X.-G.Zhang L.-S.Liu Y.-H.WuThe uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium10.1016/j.aml.2014.05.002Appl. Math. Lett.2014372633 X.-Q.Zhang L.-S.Liu Y.-H.WuFixed point theorems for the sum of three classes of mixed monotone operators and applications10.1186/s13663-016-0533-4Fixed Point Theory Appl.20162016122X.-Q.Zhang L.Wang Q.SunExistence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter10.1016/j.amc.2013.10.089Appl. Math. Comput.2014226708718 M.-C.Zhang Y.-M.Yin Z.-L.Wei Existence of positive solution for singular semi-positone (k, n-k) conjugate m-point boundary value problem10.1016/j.camwa.2008.02.016Comput. Math. Appl.20085611461154 Y.-L.Zhao H.-B.Chen L.HuangExistence of positive solutions for nonlinear fractional functional differential equation10.1016/j.camwa.2012.01.081 Comput. Math. Appl.20126434563467Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.02Some common fixed points of multivalued mappings on complex-valued metric spaces with homotopy resultShatanawiWasfi
Department of Mathematics and general courses, Prince Sultan University, Riyadh, Saudi Arabia;Department of Mathematics, Hashemite University Zarqa, Jordan
NoraniMohd Salmi MD
School of mathematical Sciences, Faculty of Science and Technology, University Kebangsaan, Malaysia, 43600 UKM, Selangor, Malaysia
Department of Mathematics, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia
AlsamirHabes
School of mathematical Sciences, Faculty of Science and Technology, University Kebangsaan, Malaysia, 43600 UKM, Selangor, Malaysia
KutbiMarwan Amin
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

The purpose of this article is to generalize common fixed point theorems under contractive condition involving rational expressions on a complete complex-valued metric space. Obtained results in this article extend, generalize, and improve wellknown comparable results in the literature.

47H1054H25Complex-valued metric spacemultivalued mappings$$\alpha^*$$-admissibleclosed ball.
M.Abbas M.Arshad A.Azam Fixed points of asymptotically regular mappings in complex-valued metric spaces10.1515/gmj-2013-0013Georgian Math. J.201320213221 M.Abbas B.Fisher T.NazirWell-Posedness and periodic point property of mappings satisfying a rational inequality in an ordered complex valued metric space Numer. Funct. Anal. Optim.2011243132 M.Arshad J.AhmadOn multivalued contractions in cone metric spaces without normality10.1155/2013/481601 Scientific World J.2013201313 M.Arshad A.Azam P.VetroSome common fixed point results in cone metric spaces10.1155/2009/493965 Fixed Point Theory Appl.20092009111J. H.Asl S.Rezapour N.ShahzadOn fixed points of $$\alpha-\psi$$ -contractive multifunctions10.1186/1687-1812-2012-212Fixed Point Theory Appl.2012201216A.Azam J.Ahmad P.Kumam Common fixed point theorems for multi-valued mappings in complex-valued metric spaces10.1186/1029-242X-2013-578J. Inequal. Appl.20132013112A.Azam B.Fisher M.KhanCommon fixed point theorems in complex valued metric spacesNumer. Funct. Anal. Optim.201132243253S.BanachSur les opérations dans les ensembles abstraits et leur application aux équations intégralesFund. Math.19223133181S.-H.Cho J.-S.BaeFixed point theorems for multivalued maps in cone metric spacesFixed Point Theory Appl.2011201117 C. DiBari P.Vetro $$\phi$$ -pairs and common fixed points in cone metric spaces10.1007/s12215-008-0020-9Rend. Circ. Mat. Palermo200857279285J.HassanzadeaslCommon fixed point theorems for $$\alpha-\psi$$-contractive type mappings Int. J. Anal.2013201317 N.Hussain E.Karapınar P.Salimi F.Akbar$$\alpha$$-admissible mappings and related fixed point theorems10.1186/1029-242X-2013-114J. Inequal. Appl.20132013111N.Hussain E.Karapınar P.Salimi P.VetroFixed point results for $$G^m$$-Meir-Keeler contractive and G-($$\alpha,\psi$$)-Meir- Keeler contractive mappings10.1186/1687-1812-2013-34 Fixed Point Theory Appl.20132013114 E.Karapınar B.SametGeneralized $$\alpha-\psi$$ contractive type mappings and related fixed point theorems with applications10.1155/2012/793486 Abstr. Appl. Anal.20122012117 C.Klin-eam C.SuanoomSome common fixed-point theorems for generalized-contractive-type mappings on complexvalued metric spaces10.1155/2013/604215 Abstr. Appl. Anal.2013201316M. A.Kutbi J.Ahmad A.AzamOn fixed points of $$\alpha-\psi$$-contractive multivalued mappings in cone metric spaces10.1155/2013/313782Abstr. Appl. Anal.2013201316M. A.Kutbi A.Azam J.Ahmad C. DiBariSome common coupled fixed point results for generalized contraction in complex-valued metric spaces10.1155/2013/352927J. Appl. Math.20132013110 B.Mohammadi S.Rezapour N.Shahzad Some results on fixed points of $$\alpha-\psi$$-Ciric generalized multifunctions10.1186/1687-1812-2013-24Fixed Point Theory Appl.20132013110 S. B.NadlerJr.Multi-valued contraction mappings Pacific J. Math.196930475478 F.Rouzkard M.ImdadSome common fixed point theorems on complex valued metric spaces10.1016/j.camwa.2012.02.063 Comput. Math. Appl.20126418661874B.Samet C.Vetro P.VetroFixed point theorems for $$\alpha-\psi$$-contractive type mappings10.1016/j.na.2011.10.014Nonlinear Anal.20127521542165W.Sintunavarat Y. J.Cho P.Kumam Urysohn integral equations approach by common fixed points in complex-valued metric spaces10.1186/1687-1847-2013-49 Adv. Difference Equ.20132013114W.Sintunavarat P.KumamGeneralized common fixed point theorems in complex valued metric spaces and applications10.1186/1029-242X-2012-84 J. Inequal. Appl.20122012112 K.Sitthikul S.Saejung Some fixed point theorems in complex valued metric spaces Fixed Point Theory Appl.20122012111Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.03Generalized $$\mathit{Z}$$-contraction on quasi metric spaces and a fixed point resultŞimşekHakan
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey
YalçinMenşur Tuğba
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey

The simulation function is defined by Khojasteh et al. [F. Khojasteh, S. Shukla, S. Radenović, Filomat, 29 (2015), 1189–1194]. Khojasteh introduced the notion of Z-contraction which is a new type of nonlinear contractions defined by using a specific simulation function. Then, they proved existence and uniqueness of fixed points for Z-contraction mappings. After this work, studies involving simulation functions were performed by various authors [H. H. Alsulami, E. Karapınar, F. Khojasteh, A. F. Roldán-López-de-Hierro, Discrete Dyn. Nat. Soc., 2014 (2014), 10 pages], [M. Olgun, Ö. Biçer, T. Alyildiz, Turkish J. Math., 40 (2016), 832–837]. In this paper, we introduce generalized simulation function on a quasi metric space and we present a fixed point theorem.

47H1054H25Quasi metric spaceleft K-Cauchy sequencesimulation functionsfixed point.
H. H.Alsulami E.Karapınar F.Khojasteh A. F.Roldán-López-de-Hierro A proposal to the study of contractions in quasi-metric spaces10.1155/2014/269286 Discrete Dyn. Nat. Soc.20142014110 I.Altun G.Mınak M.OlgunClassification of completeness of quasi metric space and some new fixed point resultsJ. Nonlinear Funct. Anal.Submitted S.BanachSur les opérations dans les ensembles abstraits et leur application aux équations intégralesFund. Math.19223133181F. E.Browder W. V.PetryshynThe solution by iteration of nonlinear functional equations in Banach spaces10.1090/S0002-9904-1966-11544-6 Bull. Amer. Math. Soc.196672571575V. W.BryantA remark on a fixed-point theorem for iterated mappings10.2307/2313440Amer. Math. Monthly196875399400J.CaristiFixed point theorems for mappings satisfying inwardness conditions10.1090/S0002-9947-1976-0394329-4Trans. Amer. Math. Soc.1976215241251 L. B.ĆirićOn a common fixed point theorem of a GreguštypePubl. Inst. Math. (Beograd) (N.S.)199149174178S.CobzaşCompleteness in quasi-metric spaces and Ekeland Variational Principle10.1016/j.topol.2011.03.003Topology Appl.201115810731084 K.DeimlingMultivalued differential equations De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin1992 M.EdelsteinA theorem on fixed points under isometries10.2307/2313133Amer. Math. Monthly196370298300T. L.HicksFixed point theorems for quasi-metric spacesMath. Japon.198833231236 O.Kada T.Suzuki W.TakahashiNonconvex minimization theorems and fixed point theorems in complete metric spaces Math. Japon.199644381391 R.KannanSome results on fixed points, II10.2307/2316437Amer. Math. Monthly196976405408 F.Khojasteh S.Shukla S.Radenović A new approach to the study of fixed point theory for simulation functions10.2298/FIL1506189KFilomat20152911891194M.Olgun Ö.Biçer T.Alyildiz A new aspect to Picard operators with simulation functions10.3906/mat-1505-26Turkish J. Math.201640832837 I. L.Reilly P. V.Subrahmanyam M. K.Vamanamurthy Cauchy sequences in quasipseudometric spaces10.1007/BF01301400 Monatsh. Math.198293127140B. E.RhoadesA comparison of various definitions of contractive mappings10.1090/S0002-9947-1977-0433430-4 Trans. Amer. Math. Soc.1977226257290A. F. Roldan Lopez deHierro B.Samet$$\varphi$$-admissibility results via extended simulation functions10.1007/s11784-016-0385-xJ. Fixed Point Theory Appl.20162016119J. L.Sieber W. J.PervinCompleteness in quasi-uniform spaces10.1007/BF01370731Math. Ann.19651587981 P. V.SubrahmanyamCompleteness and fixed-points10.1007/BF01472580Monatsh. Math.197580325330 W. A.WilsonOn quasi-metric spaces10.2307/2371174Amer. J. Math.193153675684Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.04Sufficient conditions for ergodic sensitivityWangXiong
Institute for Advanced Study, Shenzhen University, Nanshan District Shenzhen, Guangdong, P. R. China
WuXinxing
School of Sciences, Southwest Petroleum University, Chengdu, Sichuan, 610500, P. R. China
ChenGuanrong
Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China

In this note, some sufficient conditions on the ergodic sensitivity of dynamical systems are obtained, improving the main results in [Q.-L. Huang, Y.-M. Shi, L.-J. Zhang, Appl. Math. Lett., 39 (2015), 31–34] and [R.-S. Li, Y.-M. Shi, Nonlinear Anal., 72 (2010), 2716–2720]. Moreover, it is proved that under these conditions, the second Lyapunov number of a dynamical system is equal to the diameter of its state space.

54H2074H65Sensitivityergodic sensitivityLyapunov number.
C.Abraham G.Biau B.CadreChaotic properties of mappings on a probability space10.1006/jmaa.2001.7754 J. Math. Anal. Appl.2002266420431J.Auslander J. A.YorkeInterval maps, factors of maps, and chaos10.2748/tmj/1178229634 Tôhoku Math. J.198032177188 J.Banks J.Brooks G.Cairns G.Davis P.StaceyOn Devaney’s definition of chaos10.1080/00029890.1992.11995856Amer. Math. Monthly199299332334R. L.DevaneyAn introduction to chaotic dynamical systems Second edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA1989E.Glasner B.WeissSensitive dependence on initial conditions10.1088/0951-7715/6/6/014Nonlinearity1993610671075 R.-B.GuThe large deviations theorem and ergodicity10.1016/j.chaos.2007.01.081 Chaos Solitons Fractals20073413871392L.-F.He X.-H.Yan L.-S.WangWeak-mixing implies sensitive dependence10.1016/j.jmaa.2004.06.066 J. Math. Anal. Appl.2004299300304 W.Huang P.Lu X.-D.Ye Measure-theoretical sensitivity and equicontinuity10.1007/s11856-011-0049-x Israel J. Math.2011183233283Q.-L.Huang Y.-M.Shi L.-J.ZhangSensitivity of non-autonomous discrete dynamical systems10.1016/j.aml.2014.08.007Appl. Math. Lett.2015393134S.Kolyada O.RybakOn the Lyapunov numbers10.4064/cm131-2-4Colloq. Math.2013131209218 R.-S.LiThe large deviations theorem and ergodic sensitivity10.1016/j.cnsns.2012.09.008 Commun. Nonlinear Sci. Numer. Simul.201318819825 R.-S.Li Y.-M.Shi Several sufficient conditions for sensitive dependence on initial conditions10.1016/j.na.2009.11.018Nonlinear Anal.20107227162720T. Y.Li J. A.YorkePeriod three implies chaos10.2307/2318254Amer. Math. Monthly197582985992 T. K. S.MoothathuStronger forms of sensitivity for dynamical systems10.1088/0951-7715/20/9/006Nonlinearity20072021152126 X.-X.WuChaos of transformations induced onto the space of probability measures10.1142/S0218127416502278Internat. J. Bifur. Chaos Appl. Sci. Engrg.201626112 X.-X.WuA remark on topological sequence entropy10.1142/S0218127417501073 Internat. J. Bifur. Chaos Appl. Sci. Engrg.accepted X.-X.Wu G.-R.Chen Sensitivity and transitivity of fuzzified dynamical systems10.1016/j.ins.2017.02.042 Inform. Sci.20173961423X.-X.Wu P.Oprocha G.-R.Chen On various definitions of shadowing with average error in tracing10.1088/0951-7715/29/7/1942Nonlinearity20162919421972X.-X.Wu X.Wang On the iteration properties of large deviations theorem10.1142/S0218127416500541Internat. J. Bifur. Chaos Appl. Sci. Engrg.20162616 X.-X.Wu J.-J.Wang G.-R.ChenF-sensitivity and multi-sensitivity of hyperspatial dynamical systems10.1016/j.jmaa.2015.04.009 J. Math. Anal. Appl.20154291626 X.-X.Wu L.-D.Wang G.-R.ChenWeighted backward shift operators with invariant distributionally scrambled subsets10.1215/20088752-3802705 Ann. Funct. Anal.20178199210J.-D.Yin Z.-L.ZhouWeakly almost periodic points and some chaotic properties of dynamical systems10.1142/S02181274155011511550115Internat. J. Bifur. Chaos Appl. Sci. Engrg.201525110Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.05Stochastic stability analysis for a neutral-type neural networks with Markovian jumping parametersGuoSong
Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu, 223300, P. R. China
DuBo
Department of Mathematics, Huaiyin Normal University, Huaian , Jiangsu, 223300, P. R. China

In this paper, the stability problem is studied for a class of stochastic neutral-type neural networks with Markovian jumping parameters. By using fixed point theorem, the existence and uniqueness of solution for the neural networks system are obtained. Furthermore, based on the Lyapunov-Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions to guarantee the mean square stability of the neural networks. An example is given to show the effectiveness of the proposed stability criterion.

34B15Markovian jumping parameterslinear matrix inequalitymean square stability.
S.Arik Global robust stability analysis of neural networks with discrete time delays10.1016/j.chaos.2005.03.025 Chaos Solitons Fractals20052614071414 S.Boyd L. ElGhaoui E.Feron V.BalakrishnanLinear matrix inequalities in system and control theory SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA1994 H.-W.Chen Z.-M.He J.-L.LiMultiplicity of solutions for impulsive differential equation on the half-line via variational methods10.1186/s13661-016-0524-8Bound. Value Probl.20162016115K.GuAn integral inequality in the stability problem of time-delay systems10.1109/CDC.2000.914233Proceedings of the 39th IEEE Conference on Decision and Control, Sydney Australia2000328052810 Z.-J.Gui W.-G.Ge X.-S.YangPeriodic oscillation for a Hopfield neural networks with neutral delays10.1016/j.physleta.2006.12.013Phys. Lett. A2007364267273M. P.Kennedy L. O.ChuaNeural networks for nonlinear programming10.1109/31.1783 IEEE Trans. Circuits and Systems198835554562Y.-R.Liu Z.-D.Wang X.-H.LiuExponential synchronization of complex networks with Markovian jump and mixed delays10.1016/j.physleta.2008.02.085Phys. Lett. A200837239863998 Y.-R.Liu Z.-D.Wang X.-H.LiuState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delays10.1016/j.physleta.2008.10.045Phys. Lett. A200837271477155 S.-S.Mou H.-J.Gao J.Lam W.-Y.QiangA new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay10.1109/TNN.2007.912593 IEEE Trans. Neural Netw.200819532535 J.Pan X.-Z.Liu W. -C.Xie Exponential stability of a class of complex-valued neural networks with time-varying delays10.1016/j.neucom.2015.02.024Neurocomputing2015164293299J. H.Park C. H.Park O. M.Kwon S. M.LeeA new stability criterion for bidirectional associative memory neural networks of neutral-type10.1016/j.amc.2007.10.032 Appl. Math. Comput.2008199716722 R.Rakkiyappan P.Balasubramaniam J.-D.CaoGlobal exponential stability results for neutral-type impulsive neural networks10.1016/j.nonrwa.2008.10.050 Nonlinear Anal. Real World Appl.201011122130J. E.Slotine W.-P.Li Applied nonlinear controlPrentice-Hall, Englewood Cliffs, New Jersey1991 Z.-D.Wang Y.-R.Liu X.-H.Liu State estimation for jumping recurrent neural networks with discrete and distributed delays10.1016/j.neunet.2008.09.015 Neural Netw.2009224148Z.-H.Xia X.-H.Wang X.-M.Sun Q.WangA secure and dynamic multi-keyword ranked search scheme over encrypted cloud data10.1109/TPDS.2015.2401003 IEEE Trans. Parallel Distrib. Syst.201527340352Y.Xu Z.-M.HeExponential stability of neutral stochastic delay differential equations with Markovian switching10.1016/j.aml.2015.08.019Appl. Math. Lett.2016526473K.-W.Yu C.-H.LienStability criteria for uncertain neutral systems with interval time-varying delays10.1016/j.chaos.2007.01.002 Chaos Solitons Fractals200838650657J.Zhao D. J.Hill T.LiuGlobal bounded synchronization of general dynamical networks with nonidentical nodes10.1109/TAC.2012.2190206 IEEE Trans. Automat. Controll20125726562662Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.06Numerical and exact solutions for time fractional Burgers' equationYokuşAsıf
Department of Actuary, Firat University, Elazig, Turkey
Department of Mathematics, Istanbul Commerce University, Istanbul, Turkey

The main purpose of this paper is to find an exact solution of the traveling wave equation of a nonlinear time fractional Burgers’ equation using the expansion method and the Cole-Hopf transformation. For this purpose, a nonlinear time fractional Burgers’ equation with the initial conditions considered. The finite difference method (FDM for short) which is based on the Caputo formula is used and some fractional differentials are introduced. The Burgers’ equation is linearized by using the Cole- Hopf transformation for a stability of the FDM. It shows that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in $$L_2$$ and $$L_\infty$$. All of obtained results are discussed with an example of the Burgers’ equation including numerical solutions for different situations of the fractional order and the behavior of potentials u is investigated with graphically. All the obtained numerical results in this study are presented in tables. We used the Mathematica software package in performing this numerical study.

65M0635R11Nonlinear time fractional Burgers’ equationan expansion methodfinite difference methodCaputo formulalinear stabilityCole-Hopf transformation.
D. A.Benson S. W.Wheatcraft M. M.MeerschaertApplication of a fractional advection-dispersion equation10.1029/2000WR900031Water Resour. Res.20003614031412 W.Chen L.-J.Ye H.-G.SunFractional diffusion equations by the Kansa method10.1016/j.camwa.2009.08.004 Comput. Math. Appl.20105916141620P. A.Clarkson New similarity solutions for the modified Boussinesq equation10.1088/0305-4470/22/13/029 J. Phys. A19892223552367 S. A.Elwakil S. K.El-labany M. A.Zahran R.Sabry Modified extended tanh-function method for solving nonlinear partial differential equations10.1016/S0375-9601(02)00669-2 Phys. Lett. A2002299179188 E.-G.FanExtended tanh-function method and its applications to nonlinear equations10.1016/S0375-9601(00)00725-8 Phys. Lett. A2000277212218A.GorguisA comparison between Cole-Hopf transformation and the decomposition method for solving Burgers’ equations10.1016/j.amc.2005.02.045 Appl. Math. Comput.2006173126136 S.-M.Guo Y.-B.ZhouThe extended ($$\frac{G'}{G}$$ )-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations10.1016/j.amc.2009.10.008Appl. Math. Comput.201021532143221 J.-H.He X.-H.WuExp-function method for nonlinear wave equations10.1016/j.chaos.2006.03.020 Chaos Solitons Fractals200630700708S. U.Islam A. J.Khattakand I. A.Tirmizi A meshfree method for numerical solution of KdV equation10.1016/j.enganabound.2008.01.003Eng. Anal. Bound. Elem.200832849855 F.Liu P.Zhuang V.Anh I.Turner K.Burrage Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation10.1016/j.amc.2006.08.162Appl. Math. Comput.20071911220 F.Mainardi M.Raberto R.Gorenflo E.Scalas Fractional calculus and continuous-time finance, II, the waiting-time distribution10.1016/S0378-4371(00)00386-1Phys. A2000287468481M. M.Meerschaert C.TadjeranFinite difference approximations for fractional advection-dispersion flow equations10.1016/j.cam.2004.01.033J. Comput. Appl. Math.20041726577 K. S.Miller B.RossAn introduction to the fractional calculus and fractional differential equationsA Wiley-Interscience Publication, John Wiley & Sons, Inc., New York1993 Z. M.Odibat N. T.Shawagfeh Generalized Taylor’s formula10.1016/j.amc.2006.07.102 Appl. Math. Comput.2007186286293K. B.Oldham J.Spanier The fractional calculusTheory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London2006 E. J.Parkes B. R.DuffyAn automated tanh-function method for finding solitary wave solutions to non-linear evolution equations10.1016/0010-4655(96)00104-XComput. Phys. Commun.199698288300I.PodlubnyFractional differential equations An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA1999E.Scalas R.Gorenflo F.MainardiFractional calculus and continuous-time finance10.1016/S0378-4371(00)00255-7Phys. A2000284376384E.SousaFinite difference approximations for a fractional advection diffusion problem10.1016/j.jcp.2009.02.011J. Comput. Phys.200922840384054L.-J.Su W.-Q.Wang Q.-Y.XuFinite difference methods for fractional dispersion equations10.1016/j.amc.2010.04.060Appl. Math. Comput.201021633293334L.-J.Su W.-Q.Wang Z.-X.Yang Finite difference approximations for the fractional advection-diffusion equation10.1016/j.physleta.2009.10.004 Phys. Lett. A200937344054408 M.-L.Wang X.-Z.Li J.-L.ZhangThe ($$\frac{G'}{G}$$ )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics10.1016/j.physleta.2007.07.051 Phys. Lett. A2008372417423A. M.Wazwaz The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd- Bullough equations10.1016/j.chaos.2004.09.122 Chaos Solitons Fractals2005255563 A.YokusSolutions of some nonlinear partial differential equations and comparison of their solutions Ph.D. Thesis, Fırat University, Elazig, Turkey2011 S. B.YusteWeighted average finite difference methods for fractional diffusion equations10.1016/j.jcp.2005.12.006J. Comput. Phys. 2006216264274 G. M.ZaslavskyChaos, fractional kinetics, and anomalous transport10.1016/S0370-1573(02)00331-9 Phys. Rep.2002371461580X.-D.Zheng Y.Chen H.-Q.ZhangGeneralized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation10.1016/S0375-9601(03)00451-1Phys. Lett. A2003311145157 P.Zhuang F.Liu V.Anh I.TurnerNew solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation10.1137/060673114 SIAM J. Numer. Anal.20084610791095Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.07Fuzzy vector metric spaces and some resultsEminoğluŞehla
Department of Mathematics, Faculty of Science, Gazi University, 06500 Teknikokullar, Ankara, Turkey
ÇevikCüneyt
Department of Mathematics, Faculty of Science, Gazi University, 06500 Teknikokullar, Ankara, Turkey

The aim of this paper is to enrich the theory of fuzzy metric spaces through vectors. Additionally we define the concept of fuzzy vector diameter to be able to prove Cantor’s intersection theorem and Baire’s theorem in a different way.

54A4054E3506F2046A40Vector metric spacefuzzy vector metric spaceRiesz spacefuzzy diameter.
C. D.Aliprantis K. C.BorderInfinite-dimensional analysis A hitchhiker’s guide, Second edition, Springer-Verlag, Berlin1999C. D.Aliprantis O.BurkinshawPositive operatorsReprint of the 1985 original, Springer, Dordrecht2006C.ÇevikOn continuity of functions between vector metric spaces10.1155/2014/753969 J. Funct. Space2014201416 C.Çevik I.AltunVector metric spaces and some properties10.12775/TMNA.2009.048Topol. Methods Nonlinear Anal.200934375382A.George P.VeeramaniOn some results in fuzzy metric spaces10.1016/0165-0114(94)90162-7Fuzzy Sets and Systems199464395399 A.George P.Veeramani On some results of analysis for fuzzy metric spaces10.1016/S0165-0114(96)00207-2Fuzzy Sets and Systems199790365368I.Kramosil J.MichałekFuzzy metrics and statistical metric spaces Kybernetika (Prague)197511336344 W. A. J.Luxemburg A. C.ZaanenRiesz spaces, Vol. INorth-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York1971 J. R.MunkresTopology: a first coursePrentice-Hall, Inc., Englewood Cliffs, N.J.1975 W.RudinFunctional analysisSecond edition, International Series in Pure and Applied Mathematics, McGraw- Hill, Inc., New York1991 B.Schweizer A.SklarStatistical metric spaces Pacific J. Math.196010313334 B.Schweizer A.SklarProbabilistic metric spaces North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York1983 E. S.ŞuhubiFunctional analysisKluwer Academic Publishers, Dordrecht2003 L. A.ZadehFuzzy sets Information and Control19658338353Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.08On nonexpansive and accretive operators in Banach spacesLiDongfeng
School of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

The purpose of this article is to investigate common solutions of a zero point problem of a accretive operator and a fixed point problem of a nonexpansive mapping via a viscosity approximation method involving a $$\tau$$ -contractive mapping

47H0565J15Accretive operatorapproximation solutionviscosity methodvariational inequality.
I. K.Argyros S.George Iterative regularization methods for nonlinear ill-posed operator equations with m-accretive mappings in Banach spaces10.1016/S0252-9602(15)30056-4Acta Math. Sci. Ser. B Engl. Ed.20153513181324 I. K.Argyros S.George Extending the applicability of a new Newton-like method for nonlinear equations Commun. Optim. Theory2016201619 V.BarbuNonlinear semigroups and differential equations in Banach spacesTranslated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden1976 B. A. BinDehaish A.Latif H. O.Bakodah X.-L.QinA regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces10.1186/s13660-014-0541-z J. Inequal. Appl.20152015114 B. A. BinDehaish X.-L.Qin A.Latif H. O.Bakodah Weak and strong convergence of algorithms for the sum of two accretive operators with applicationsJ. Nonlinear Convex Anal.20151613211336 F. E.BrowderExistence and approximation of solutions of nonlinear variational inequalities10.1073/pnas.56.4.1080Proc. Nat. Acad. Sci. U.S.A.19665610801086R. E.BruckJr.A strongly convergent iterative solution of $$0 \in U(x)$$ for a maximal monotone operator U in Hilbert space10.1016/0022-247X(74)90219-4J. Math. Anal. Appl.197448114126S.-S.Chang H. W. J.Lee C. K.ChanStrong convergence theorems by viscosity approximation methods for accretive mappings and nonexpansive mappingsJ. Appl. Math. Inform.2009275968C. E.Chidume Iterative solutions of nonlinear equations in smooth Banach spaces Nonlinear Anal.19962618231834S. Y.Cho B. A. BinDehaish X.-L.Qin Weak convergence of a splitting algorithm in Hilbert spaces10.11948/2017027J. Appl. Anal. Comput.20177427438S. Y.Cho X.-L.Qin L.Wang Strong convergence of a splitting algorithm for treating monotone operators10.1186/1687-1812-2014-94Fixed Point Theory Appl.20142014115 J. S.Jung Y. J.Cho H.-Y.Zhou Iterative processes with mixed errors for nonlinear equations with perturbed m-accretive operators in Banach spaces10.1016/S0096-3003(01)00239-9 Appl. Math. Comput.2002133389406T.KatoNonlinear semigroups and evolution equations10.2969/jmsj/01940508 J. Math. Soc. Japan196719508520L. S.Liu Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces10.1006/jmaa.1995.1289J. Math. Anal. Appl.1995194114125 A.MoudafiViscosity approximation methods for fixed-points problems10.1006/jmaa.1999.6615J. Math. Anal. Appl.20002414655 X.-L.Qin S. Y.ChoConvergence analysis of a monotone projection algorithm in reflexive Banach spaces10.1016/S0252-9602(17)30016-4 Acta Math. Sci. Ser. B Engl. Ed.201737488502X.-L.Qin S. Y.Cho J. K.KimOn the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings10.1080/02331934.2010.534165Optimization201261805821X.-L.Qin S. Y.Cho L.WangIterative algorithms with errors for zero points of m-accretive operators10.1186/1687-1812-2013-148Fixed Point Theory Appl.20132013117 X.-L.Qin J.-C.Yao Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators10.1186/s13660-016-1163-4 J. Inequal. Appl.2016201619S.Reich On fixed point theorems obtained from existence theorems for differential equations10.1016/0022-247X(76)90232-8 J. Math. Anal. Appl.1976542636R. T.Rockafellar Monotone operators and the proximal point algorithm10.1137/0314056SIAM J. Control Optim.197614877898T.SuzukiStrong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces10.1155/FPTA.2005.103Fixed Point Theory Appl.20052005121 H.-Y.ZhouA characteristic condition for convergence of steepest descent approximation to accretive operator equations10.1016/S0022-247X(02)00122-1J. Math. Anal. Appl.200227116 H.-Y.Zhou Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces10.1016/j.na.2008.08.012 Nonlinear Anal.20097040394046Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.09On common fixed points that belong to the zero set of a certain functionKarapinarErdal
Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey
SametBessem
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
ShahiPriya
St. Andrews College of Arts, Science and Commerce, St. Dominic Road, Bandra (West), Mumbai 400 050, India

We provide sufficient conditions under which the set of common fixed points of two self-mappings $$f, g : X \rightarrow X$$ is nonempty, and every common fixed point of f and g is the zero of a given function $$\varphi:X \rightarrow [0,\infty)$$. Next, we show the usefulness of our obtained result in partial metric fixed point theory.

54H2547H10$$\varphi$$ -admissibilitycommon fixed pointzero setpartial metric.
M.Abbas I.Altun S.RomagueraCommon fixed points of Ćirić-type contractions on partial metric spaces10.5486/PMD.2013.5342Publ. Math. Debrecen201382425438 T.Abdeljawad E.Karapınar K.TaşA generalized contraction principle with control functions on partial metric spaces10.1016/j.camwa.2011.11.035 Comput. Math. Appl.201263716719 D. W.Boyd J. S. W.Wong On nonlinear contractionsProc. Amer. Math. Soc.196920458464 L.Ćirić B.Samet H.Aydi C.VetroCommon fixed points of generalized contractions on partial metric spaces and an application10.1016/j.amc.2011.07.005 Appl. Math. Comput.201121823982406 R.Heckmann Approximation of metric spaces by partial metric spaces10.1023/A:1008684018933Applications of ordered sets in computer science, Braunschweig, (1996), Appl. Categ. Structures199977183E.Karapınar D.O’Regan B.SametOn the existence of fixed points that belong to the zero set of a certain function10.1186/s13663-015-0401-7Fixed Point Theory Appl.20152015114 S. G.MatthewsPartial metric topology10.1111/j.1749-6632.1994.tb44144.x Papers on general topology and applications, Flushing, NY, (1992), Ann. New York Acad. Sci., New York Acad. Sci., New York1994728183197 S.Oltra O.ValeroBanach’s fixed point theorem for partial metric spacesRend. Istit. Mat. Univ. Trieste2004361726S.RomagueraA Kirk type characterization of completeness for partial metric spaces10.1155/2010/493298Fixed Point Theory Appl.2010201016S.RomagueraFixed point theorems for generalized contractions on partial metric spacesTopology Appl.201121823982406 S.Romaguera Matkowski’s type theorems for generalized contractions on (ordered) partial metric spacesAppl. Gen. Topol.201112213220 O.ValeroOn Banach fixed point theorems for partial metric spaces10.4995/agt.2005.1957Appl. Gen. Topol.20056229240Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.10Common fixed points of $$\alpha$$-dominated multivalued mappings on closed balls in a dislocated quasi b-metric spaceAlofiAbdulaziz Saleem Moslem
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
Al-MazrooeiAbdullah Eqal
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia;Department of Mathematics, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia
LeyewBahru Tsegaye
Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa
AbbasMujahid
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia;Department of Mathematics, University of Management and Technology, C-II Johar Town, Lahore, Pakistan

In this paper, we introduce the concept of $$\alpha$$-dominated multivalued mappings and establish the existence of common fixed points of such mappings on a closed ball contained in left/right K-sequentially complete dislocated quasi b-metric spaces. These results improve, generalize, extend, unify, and complement various comparable results in the existing literature. Our results not only extend some primary results to left/right K-sequentially dislocated quasi b-metric spaces but also restrict the contractive conditions on a closed ball only. Some examples are presented to support the results proved herein. Finally as an application, we obtain some common fixed point results for single-valued mappings by an application of the corresponding results for multivalued mappings satisfying the contractive conditions more general than Banach type and Kannan type contractive conditions on closed balls in a left K-sequentially complete dislocated quasi b-metric space endowed with an arbitrary binary relation.

47H0447H0747H10K-sequentially completedislocated quasi b-metric spaces$$\alpha$$-dominated multivalued mappingclosed ballcommon fixed point.
A.Alam M.ImdadRelation-theoretic contraction principle10.1007/s11784-015-0247-y J. Fixed Point Theory Appl.201517693702 M. A.Alghamdi N.Hussain P.SalimiFixed point and coupled fixed point theorems on b-metric-like spaces10.1186/1029-242X-2013-402J. Inequal. Appl.20132013125T. V.An L. Q.Tuyen N. V.DungStone-type theorem on b-metric spaces and applications10.1016/j.topol.2015.02.005Topology Appl.2015185/1865064M.Arshad A.Shoaib I.BegFixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space10.1186/1687-1812-2013-115 Fixed Point Theory Appl.20132013115 A.Azam M.Waseem M.RashidFixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces10.1186/1687-1812-2013-27 Fixed Point Theory Appl.20132013114 I. A.BakhtinThe contraction mapping principle in almost metric space (Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk1989302637S.Banach Sur les opérations dans les ensembles abstraits et leur application aux équations intégralesFund. Math.19223133181M.BoriceanuFixed point theory for multivalued generalized contraction on a set with two b-metricsStud. Univ. Babeş- Bolyai Math.200954314L. B.ĆirićFixed points for generalized multi-valued contractionsMat. Vesnik19729265272S.CzerwikContraction mappings in b-metric spaces Acta Math. Inform. Univ. Ostraviensis19931511S.CzerwikNonlinear set-valued contraction mappings in b-metric spacesAtti Sem. Mat. Fis. Univ. Modena199846263276P.HitzlerGeneralized metrics and topology in logic programming semanticsPh.D. Thesis, School of Mathematics, Applied Mathematics and Statistics, National University Ireland,, University College Cork2001P.Hitzler A. K.SedaDislocated topologiesJ. Electr. Eng.20005137N.Hussain D.Dorić Z.Kadelburg S.RadenovićSuzuki-type fixed point results in metric type spaces10.1186/1687-1812-2012-126Fixed Point Theory Appl.20122012112N.Hussain J. R.Roshan V.Parvaneh M.AbbasCommon fixed point results for weak contractive mappings in ordered b-dislocated metric spaces with applications10.1186/1029-242X-2013-486J. Inequal. Appl.20132013121 N.Hussain PSalimi A.LatifFixed point results for single and set-valued $$\alpha-\eta-\psi-$$contractive mappings10.1186/1687-1812-2013-212Fixed Point Theory Appl.20132013123 R.KannanSome results on fixed points, II10.2307/2316437 Amer. Math. Monthly196976405408C.Klin-eam C.Suanoom Dislocated quasi-b-metric spaces and fixed point theorems for cyclic contractions10.1186/s13663-015-0325-2Fixed Point Theory Appl.20152015112 A.Latif A. A. N.Abdou Multivalued generalized nonlinear contractive maps and fixed points10.1016/j.na.2010.10.017Nonlinear Anal.20117414361444 A.Latif D. T.Luc Variational relation problems: existence of solutions and fixed points of contraction mappings10.1186/1687-1812-2013-315Fixed Point Theory Appl.20132013110A.Latif I.TweddleSome results on coincidence points10.1017/S0004972700032652 Bull. Austral. Math. Soc.199959111117 S.LipschutzSchaum’s outline of theory and problems of set theory and related topics McGraw-Hill, New York1964S. B.NadlerJr. Multi-valued contraction mappingsPacific J. Math.196930475488 J. J.Nieto R.Rodríguez-LópezContractive mapping theorems in partially ordered sets and applications to ordinary differential equationsOrder200522223239J. J.Nieto R.Rodríguez-LópezExistence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations10.1007/s10114-005-0769-0 Acta Math. Sin. (Engl. Ser.)20072322052212M. U.Rahman M.Sarwar Dislocated quasi b-metric space and fixed point theorems Electron. J. Math. Anal. Appl.201641624 A. C. M.Ran M. C. B.Reurings A fixed point theorem in partially ordered sets and some applications to matrix equations10.1090/S0002-9939-03-07220-4Proc. Amer. Math. Soc.200313214351443I. L.Reilly P. V.Subrahmanyam M. K.Vamanamurthy Cauchy sequences in quasipseudometric spaces10.1007/BF01301400Monatsh. Math.198293127140J. R.Roshan N.Hussain S.Sedghi N.ShobkolaeiSuzuki-type fixed point results in b-metric spaces10.1007/s40096-015-0162-9 Math. Sci. (Springer)20159153160 J. R.Roshan V.Parvaneh I.Altun Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations10.1016/j.amc.2013.10.043 Appl. Math. Comput.2014226725737B.Samet M.Turinici Fixed point theorems on a metric space endowed with an arbitrary binary relation and applicationsCommun. Math. Anal.2012138297M. H.Shah N.HussainNonlinear contractions in partially ordered quasi b-metric spaces10.4134/CKMS.2012.27.1.117 Commun. Korean Math. Soc.201227117128A.Shoaib M.Arshadatjana S.Radenović$$\alpha$$-dominated mappings, dislocated metric spaces and fixed point resultsFixed Point Theory Appl. to appeare W. A.WilsonOn quasi-metric spaces10.2307/2371174Amer. J. Math.193153675684F. M.Zeyada G. H.Hassan M. A.AhmedA generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spacesArab. J. Sci. Eng. Sect. A Sci.200631111114 C.-X.Zhu C.-F.Chen X.-Z.ZhangSome results in quasi-b-metric-like spaces10.1186/1029-242X-2014-437 J. Inequal. Appl.2014201418Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.11Naimark-Sacker bifurcation of second order rational difference equation with quadratic termsKulenovicM. R. S.
Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA
MoranjkicS.
Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina
NurkanovicZ.
Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the difference equation $x_{n+1} =\frac{F}{bx_nx_{n-1} + cx^2_{n-1} + f} , n = 0, 1, ... ,$ with non-negative parameters and nonnegative initial conditions $$x_{-1}, x_0$$ such that $$bx_0x_{-1} + cx^2_{-1} + f > 0$$. By using fixed point theorem for monotone maps we find the region of parameters where the unique equilibrium is globally asymptotically stable.

39A1039A2839A30Attractivitybifurcationdifference equationinvariantNaimark-Sacker bifurcationperiodic solution.
A. M.Amleh E.Camouzis G.LadasOn the dynamics of a rational difference equation, IInt. J. Difference Equ.20083135 A. M.Amleh E.Camouzis G.LadasOn the dynamics of a rational difference equation, II Int. J. Difference Equ.20083195225 J. K.Hale H.KocakDynamics and bifurcationsTexts in Applied Mathematics, Springer-Verlag, New York1991 E. A.Janowski M. R. S.KulenovićAttractivity and global stability for linearizable difference equations10.1016/j.camwa.2008.10.064Comput. Math. Appl.20095715921607C. M.Kent H.SedaghatGlobal attractivity in a quadratic-linear rational difference equation with delay10.1080/10236190802040992J. Difference Equ. Appl.200915913925C. M.Kent H.Sedaghat Global attractivity in a rational delay difference equation with quadratic terms10.1080/10236190903049009 J. Difference Equ. Appl.201117457466M. R. S.Kulenović G.Ladas Dynamics of second order rational difference equations With open problems and conjectures, Chapman & Hall/CRC, Boca Raton, FL2001 M. R. S.Kulenović O.Merino Discrete dynamical systems and difference equations with MathematicaChapman & Hall/CRC, Boca Raton, FL2002 M. R. S.Kulenović O.MerinoA global attractivity result for maps with invariant boxes10.3934/dcdsb.2006.6.97Discrete Contin. Dyn. Syst. Ser. B2006697110M. R. S.Kulenović O.MerinoGlobal bifurcation for discrete competitive systems in the plane10.3934/dcdsb.2009.12.133 Discrete Contin. Dyn. Syst. Ser. B200912133149M. R. S.Kulenović O.Merino Invariant manifolds for competitive discrete systems in the plane10.1142/S0218127410027118Internat. J. Bifur. Chaos Appl. Sci. Engrg.20102024712486 M. R. S.Kulenović E.Pilav E.SilićNaimark-Sacker bifurcation of a certain second order quadratic fractional difference equation J. Math. Comput. Sci.2014410251043Y. A.Kuznetsov Elements of applied bifurcation theorySecond edition. Applied Mathematical Sciences, Springer- Verlag, New York1998C.RobinsonStability, symbolic dynamics, and chaos Stud. Adv. Math. Boca Raton, CRC Press, FL1995H.SedaghatGlobal behaviours of rational difference equations of orders two and three with quadratic terms10.1080/10236190802054126J. Difference Equ. Appl.200915215224 S.WigginsIntroduction to applied nonlinear dynamical systems and chaosSecond edition, Texts in Applied Mathematics, Springer-Verlag, New York2003Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.12Norm inequalities of operators and commutators on generalized weighted morrey spacesHuYue
School of Mathematics and Information, Henan Polytechnic University, Jiaozuo 454003, P. R. China
WangYueshan
Department of Mathematics, Jiaozuo University, Jiaozuo 454003, P. R. China

We prove that, if a class of operators, which includes singular integral operator with rough kernel, Bochner-Riesz operator and Marcinkiewicz integral operator, are bounded on weighted Lebesgue spaces and satisfy some local pointwise control, then these operators and associated commutators, formed by a BMO function and these operators, are also bounded on generalized weighted Morrey spaces.

42B3547B3830H35Singular integral with rough kernelBochner-Riesz operatorMarcinkiewicz integralcommutatorweighted Morrey space.
J.Álvarez R. J.Bagby D. S.Kurtz C.Pérez Weighted estimates for commutators of linear operators10.4064/sm-104-2-195-209Studia Math.1993104195209 S.BochnerSummation of multiple Fourier series by spherical means10.2307/1989864Trans. Amer. Math. Soc193640175207Y.Ding D.-S.Fan Y.-B.PanWeighted boundedness for a class of rough Marcinkiewicz integrals10.1512/iumj.1999.48.1696Indiana Univ. Math. J.19994810371055Y.Ding S.-Z.Lu K.Yabuta On commutators of Marcinkiewicz integrals with rough kernel10.1016/S0022-247X(02)00230-5J. Math. Anal. Appl20022756068J.Duoandikoetxea Weighted norm inequalities for homogeneous singular integrals10.1090/S0002-9947-1993-1089418-5 Trans. Amer. Math. Soc.1993336869880 Y.Hu Y.-S.WangMultilinear fractional integral operators on generalized weighted Morrey spaces10.1186/1029-242X-2014-323J. Inequal. Appl.20142014118F.John L.Nirenberg On functions of bounded mean oscillation10.1002/cpa.3160140317Comm. Pure Appl. Math.196114415426 Y.Komori S.ShiraiWeighted Morrey spaces and a singular integral operator10.1002/mana.200610733 Math. Nachr.2009289219231T.Mizuhara Boundedness of some classical operators on generalized Morrey spaces10.1007/978-4-431-68168-7_16Harmonic analysis, Sendai, (1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo1991183189C. B.MorreyJr.On the solutions of quasi-linear elliptic partial differential equations10.2307/1989904Trans. Amer. Math. Soc.193843126166 B.MuckenhouptWeighted norm inequalities for the Hardy maximal functionTrans. Amer. Math. Soc.1972165207226 X. L.Shi Q. Y.Sun Weighted norm inequalities for Bochner-Riesz operators and singular integral operators10.1090/S0002-9939-1992-1136237-1Proc. Amer. Math. Soc.1992116665673E. M.Stein G.Weiss Introduction to Fourier analysis on Euclidean spacesPrinceton Mathematical Series, Princeton University Press, Princeton, N.J.1971 R. L.Wheeden A.ZygmundMeasure and integralAn introduction to real analysis. Pure and Applied Mathematics, Marcel Dekker, Inc., New York-Basel1977Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.13Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spacesCengLu-Chuan
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
WenChing-Feng
Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80702, Taiwan;Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80702, Taiwan

In this paper, we introduce and analyze implicit and explicit iteration methods for solving a variational inequality problem over the set of common fixed points of an infinite family of nonexpansive mappings on a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Strong convergence results are given. Our results improve and extend the corresponding results in the literature.

49J3047H0947J2049M05Nonexpansive mappingfixed pointvariational inequalityglobal convergence.
K.Aoyama H.Iiduka W.Takahashi Weak convergence of an iterative sequence for accretive operators in Banach spaces10.1155/FPTA/2006/35390Fixed Point Theory Appl.20062006113 F. E.Browder W. V.PetryshynConstruction of fixed points of nonlinear mappings in Hilbert space10.1016/0022-247X(67)90085-6J. Math. Anal. Appl.196720197228 N.Buong N. T. H.PhuongStrong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods10.1007/s10957-013-0350-4J. Optim. Theory Appl.2013159399411 N.Buong N. T. QuynhAnhAn implicit iteration method for variational inequalities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces10.1155/2011/276859Fixed Point Theory Appl.20112011110L.-C.Ceng Q. H.Ansari J.-C.YaoMann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces10.1080/01630560802418391Numer. Funct. Anal. Optim.2008299871033 L.-C.Ceng S.-M.Guu J.-C.YaoHybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems10.1186/1687-1812-2012-92 Fixed Point Theory Appl.20122012119 L.-C.Ceng A.Petruşel J.-C.Yao Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of Lipschitz pseudocontractive mappings10.7153/jmi-01-22 J. Math. Inequal.20071243258 L.-C.Ceng J.-C.YaoRelaxed viscosity approximation methods for fixed point problems and variational inequality problems10.1016/j.na.2007.09.019 Nonlinear Anal.20086932993309 Y.Censor A.Gibali S.ReichExtensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space10.1080/02331934.2010.539689Optimization20126111191132 S.-S.ChangSome problems and results in the study of nonlinear analysis10.1016/S0362-546X(97)00388-X Proceedings of the Second World Congress of Nonlinear Analysts, Part 7, Athens, (1996), Nonlinear Anal.19973341974208 I.CioranescuGeometry of Banach spaces, duality mappings and nonlinear problems Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht1990F.Facchinei J.-S.Pang Finite-dimensional variational inequalities and complementarity problems, Vol. ISpringer Series in Operations Research, Springer-Verlag, New York2003 R.Glowinski J.-L.Lions R.TrémolièresNumerical analysis of variational inequalitiesTranslated from the French, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York1981A. N.Iusem B. F.SvaiterA variant of Korpelevich’s method for variational inequalities with a new search strategy10.1080/02331939708844365Optimization199742309321J. S.Jung Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces10.1016/j.jmaa.2004.08.022J. Math. Anal. Appl.2005509520M.Kikkawa W.TakahashiViscosity approximation methods for countable families of nonexpansive mappings in Hilbert spacesRIMS Kokyuroku20061484105113M.Kikkawa W.TakahashiStrong convergence theorems by the viscosity approximation method for a countable family of nonexpansive mappings10.11650/twjm/1500602423Taiwanese J. Math.200812583598N.Kikuchi J. T.OdenContact problems in elasticity: a study of variational inequalities and finite element methodsSIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA1988D.Kinderlehrer G.Stampacchia An introduction to variational inequalities and their applicationsPure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London1980 I. V.KonnovCombined relaxation methods for variational inequalitiesLecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin2001I. V.KonnovEquilibrium models and variational inequalities Mathematics in Science and Engineering, Elsevier B. V., Amsterdam2007G. M.KorpelevičAn extragradient method for finding saddle points and for other problems (Russian) Èkonom. i Mat. Metody197612747756 A.MoudafiViscosity approximation methods for fixed-points problems10.1006/jmaa.1999.6615J. Math. Anal. Appl.20002414655 J. G.O’Hara P.Pillay H.-K.Xu Iterative approaches to convex feasibility problems in Banach spaces10.1016/j.na.2005.07.036 Nonlinear Anal.20066420222042 K.Shimoji W.TakahashiStrong convergence to common fixed points of infinite nonexpansive mappings and applications10.11650/twjm/1500407345Taiwanese J. Math.20015387404 N.Shioji W.TakahashiStrong convergence of approximated sequences for nonexpansive mappings in Banach spaces10.1090/S0002-9939-97-04033-1 Proc. Amer. Math. Soc.199712536413645M. V.Solodov B. F.Svaiter A new projection method for variational inequality problems10.1137/S0363012997317475 SIAM J. Control Optim.199937765776T.SuzukiStrong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals10.1016/j.jmaa.2004.11.017J. Math. Anal. Appl.2005305227239W.TakahashiWeak and strong convergence theorems for families of nonexpansive mappings and their applicationsProceedings of Workshop on Fixed Point Theory, Kazimierz Dolny, (1997), Ann. Univ. Mariae Curie-Skodowska Sect. A199751277292 S.-H.Wang L.-X.Yu B.-H.GuoAn implicit iterative scheme for an infinite countable family of asymptotically nonexpansive mappings in Banach spaces10.1155/2008/350483 Fixed Point Theory Appl. 20082008110 H.-K.XuIterative algorithms for nonlinear operators10.1112/S0024610702003332 J. London Math. Soc.200266240256 I.YamadaThe hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently parallel algorithms in feasibility and optimization and their applicationsHaifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam20018473504Y.-H.Yao R.-D.Chen H.-K.XuSchemes for finding minimum-norm solutions of variational inequalities10.1016/j.na.2009.12.029Nonlinear Anal.20107234473456Y.-H.Yao Y.-C.Liou S. M.KangApproach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method10.1016/j.camwa.2010.03.036 Comput. Math. Appl.20105934723480 Y.-H.Yao M. A.Noor Y.-C.LiouStrong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities10.1155/2012/817436Abstr. Appl. Anal.2012201219 Y.-H.Yao M. A.Noor Y.-C.Liou S. M.Kang Iterative algorithms for general multivalued variational inequalities10.1155/2012/768272Abstr. Appl. Anal.20122012110Y.-H.Yao M.Postolache Y.-C.Liou Z.-S.YaoConstruction algorithms for a class of monotone variational inequalities10.1007/s11590-015-0954-8Optim. Lett.20161015191528H.Zegeye N.Shahzad Y.-H.Yao Minimum-norm solution of variational inequality and fixed point problem in Banach spaces10.1080/02331934.2013.764522Optimization201564453471E.Zeidler Nonlinear functional analysis and its applications, III Variational methods and optimization, Translated from the German by Leo F. Boron, Springer-Verlag, New York1985 L.-C.Zeng J.-C.Yao Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings10.1016/j.na.2005.08.028Nonlinear Anal.20066425072515 H.-Y.Zhou L.Wei Y. J.ChoStrong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces 10.1016/j.amc.2005.02.049Appl. Math. Comput.2006173196212Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.14$$L_p$$-dual geominimal surface areas for the general $$L_p$$-intersection bodiesShenZhonghuan
Department of Mathematics, China Three Gorges University, Yichang, 443002, China
LiYanan
Department of Mathematics, China Three Gorges University, Yichang, 443002, China
WangWeidong
Department of Mathematics, China Three Gorges University, Yichang, 443002, China

For $$0 < p < 1$$, Haberl and Ludwig defined the notions of symmetric and asymmetric $$L_p$$-intersection bodies. Recently, Wang and Li introduced the general $$L_p$$-intersection bodies. In this paper, we give the $$L_p$$-dual geominimal surface area forms for the extremum values and Brunn-Minkowski type inequality of general $$L_p$$-intersection bodies. Further, combining with the $$L_p$$-dual geominimal surface areas, we consider Busemann-Petty type problem for general $$L_p$$-intersection bodies.

52A2052A4052A39General $$L_p$$-intersection body$$L_p$$-dual geominimal surface areaextremum valueBrunn-Minkowski inequalityBusemann-Petty problem.
Y.-B.Feng W.-D.WangGeneral $$L_p$$-harmonic Blaschke bodies10.1007/s12044-013-0158-zProc. Indian Acad. Sci. Math. Sci.2014124109119 Y.-B.Feng W.-D.Wang $$L_p$$-dual mixed geominimal surface area10.1017/S0017089513000244 Glasg. Math. J.201456229239 R. J.GardnerGeometric tomographySecond edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, New York2006C.Haberl$$L_p$$ intersection bodies10.1016/j.aim.2007.11.013 Adv. Math.200821725992624 C.Haberl M.Ludwig A characterization of $$L_p$$ intersection bodies10.1155/IMRN/2006/10548Int. Math. Res. Not.20062006129C.Haberl F. E.SchusterAsymmetric affine $$L_p$$ Sobolev inequalities10.1016/j.jfa.2009.04.009J. Funct. Anal.2009257641658 C.Haberl F. E.SchusterGeneral $$L_p$$ affine isoperimetric inequalities10.1016/j.aam.2016.06.007J. Differential Geom.200983126 C.Haberl F. E.Schuster J.XiaoAn asymmetric affine Pólya -Szegö principle10.1007/s00208-011-0640-9 Math. Ann.2012352517542G. H.Hardy J. E.Littlewood G. Pólya InequalitiesReprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge1988Y.-N.LiW.-D.Wang The $$L_p$$-dual mixed geominimal surface area for multiple star bodies10.1186/1029-242X-2014-456 J. Inequal. Appl.20142014110Z.-F.Li W.-D.Wang General $$L_p$$-mixed chord integrals of star bodies10.1186/s13660-016-1006-3 J. Inequal. Appl.20162016112 M.LudwigMinkowski valuations10.1090/S0002-9947-04-03666-9Trans. Amer. Math. Soc.200535741914213M.LudwigIntersection bodies and valuations Amer. J. Math.200612814091428 E.LutwakIntersection bodies and dual mixed volumes10.1016/0001-8708(88)90077-1 Adv. in Math.198871232261 E.LutwakThe Brunn-Minkowski-Firey theory, I10.4310/jdg/1214454097 Mixed volumes and the Minkowski problem, J. Differential Geom.199338131150E.LutwakThe Brunn-Minkowski-Firey theory, II10.1006/aima.1996.0022Affine and geominimal surface areas, Adv. Math.1996118244294L.ParapatitsSL(n)-contravariant $$L_p$$-Minkowski valuations10.1090/S0002-9947-2013-05750-9Trans. Amer. Math. Soc.201436611951211L.ParapatitsSL(n)-covariant $$L_p$$-Minkowski valuations10.1112/jlms/jdt068 J. Lond. Math. Soc.201489397414Y.-N.Pei W.-D.WangA type of Busemann-Petty problems for general $$L_p$$-intersection bodies10.1007/s11859-015-1121-xWuhan Univ. J. Nat. Sci.201520471475Y.-N.Pei W.-D.WangShephard type problems for general $$L_p$$-centroid bodies10.1186/s13660-015-0812-3 J. Inequal. Appl.20152015113C. M.PettyGeominimal surface area10.1007/BF00181363Geometriae Dedicata197437797F. E.Schuster T.WannererGL(n) contravariant Minkowski valuations10.1090/S0002-9947-2011-05364-X Trans. Amer. Math. Soc.2012364815826F. E.Schuster M.Weberndorfer Volume inequalities for asymmetric Wulff shapes10.4310/jdg/1352297808 J. Differential Geom.201292263283 X. Y.Wan W.-D.Wang$$L_p$$-dual geominimal surface area(Chinese) J. Wuhan Univ. Natur. Sci. Ed.201359515518W.-D.Wang Y.-B.FengA general $$L_p$$-version of Petty’s affine projection inequality10.11650/tjm.17.2013.2122 Taiwanese J. Math.201317517528 W.-D.Wang Y.-N.LiBusemann-Petty problems for general $$L_p$$-intersection bodies10.1007/s10114-015-4273-xActa Math. Sin. (Engl. Ser.)201531777786 W.-D.Wang Y.-N.Li General $$L_p$$-intersection bodies10.11650/tjm.19.2015.3493Taiwanese J. Math.20151912471259W.-D.Wang T.-Y.MaAsymmetric $$L_p$$-difference bodies10.1090/S0002-9939-2014-11919-8Proc. Amer. Math. Soc.201414225172527J.-Y.Wang W.-D.WangGeneral $$L_p$$-dual Blaschke bodies and the applications10.1186/s13660-015-0756-7J. Inequal. Appl.20152015111 W.-D.Wang J.-Y.WangExtremum of geometric functionals involving general $$L_p$$-projection bodies10.1186/s13660-016-1076-2 J. Inequal. Appl.20162016116T.WannererGL(n) equivariant Minkowski valuations10.1512/iumj.2011.60.4425Indiana Univ. Math. J.20116016551672 M.Weberndorfer Shadow systems of asymmetric $$L_p$$ zonotopes10.1016/j.aim.2013.02.022 Adv. Math.2013240613635W.Weidong Q.Chen $$L_p$$-dual geominimal surface area10.1186/1029-242X-2011-6J. Inequal. Appl.20112011110W.Weidong W.XiaoyanShephard type problems for general $$L_p$$-projection bodies10.11650/twjm/1500406794Taiwanese J. Math.20121617491762 L.Yan W.-D.WangGeneral $$L_p$$-mixed-brightness integrals10.1186/s13660-015-0708-2J. Inequal. Appl.20152015111D.-P.Ye$$L_p$$ geominimal surface areas and their inequalities10.1093/imrn/rnu009 Int. Math. Res. Not. IMRN2015201524652498 D.-P.Ye B.-C.Zhu J.-Z.ZhouThe mixed $$L_p$$ geominimal surface areas for multiple convex bodies10.1512/iumj.2015.64.5623Indiana Univ. Math. J.20156415131552 F.Yibin W.Weidong L.FenghongSome inequalities on general $$L_p$$-centroid bodies10.7153/mia-18-02 Math. Inequal. Appl.2015183949B.-C.Zhu N.Li J.-Z.ZhouIsoperimetric inequalities for $$L_p$$ geominimal surface area10.1017/S0017089511000292Glasg. Math. J.201153717726B.-C.Zhu J.-Z.Zhou W.-X.XuAffine isoperimetric inequalities for $$L_p$$ geominimal surface area10.1007/978-4-431-55215-4_15Real and complex submanifolds, Springer Proc. Math. Stat., Springer, Tokyo2014106167176 B.-C.Zhu J.-Z.Zhou W.-X.Xu $$L_p$$ mixed geominimal surface area10.1016/j.jmaa.2014.09.035J. Math. Anal. Appl.201542212471263Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.15The viscosity approximation forward-backward splitting method for the implicit midpoint rule of quasi inclusion problems in Banach spacesYangLi
School of Science, South West University of Science and Technology, Mianyang, Sichuan 621010, China
ZhaoFuhai
School of Science, South West University of Science and Technology, Mianyang, Sichuan 621010, China

The purpose of this paper is to introduce a viscosity approximation forward-backward splitting method for the implicit midpoint rule of an accretive operators and m-accretive operators in Banach spaces. The strong convergence of this viscosity method is proved under certain assumptions imposed on the sequence of parameters. The results presented in the paper extend and improve some recent results announced in the current literature. Moreover, some applications to the minimization optimization problem and the linear inverse problem are presented.

47H0947J25Viscosity approximationBanach spacesplitting methodforward-backward algorithmthe implicit midpoint rule.
M. A.Alghamdi N.Shahzad H.-K.XuThe implicit midpoint rule for nonexpansive mappings10.1186/1687-1812-2014-96Fixed Point Theory Appl.2014201419H.Attouch Viscosity solutions of minimization problems10.1137/S1052623493259616 SIAM J. Optim.19966769806W.Auzinger R.FrankAsymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case10.1007/BF01396649 Numer. Math.198956469499 G.Bader P.DeuflhardA semi-implicit mid-point rule for stiff systems of ordinary differential equations10.1007/BF01418331 Numer. Math.198341373398J. B.Baillon G.Haddad Quelques proprits des oprateurs angle-borns et n-cycliquement monotones(French) Israel J. Math.197726137150D. P.Bertsekas J. N.TsitsiklisParallel and distributed computation: numerical methods Englewood Cliffs: Prentice Hall, NJ1989H.Brézis P.-L.LionsProduits infinis de résolvantes10.1007/BF02761171(French) Israel J. Math.197829329345 C.ByrneA unified treatment of some iterative algorithms in signal processing and image reconstruction10.1088/0266-5611/20/1/006Inverse Problems200420103120 G. H. G.Chen R. T.Rockafellar Convergence rates in forward-backward splitting10.1137/S1052623495290179 SIAM J. Optim.19977421444 C.ChidumeGeometric properties of Banach spaces and nonlinear iterations Lecture Notes in Mathematics, Springer- Verlag London, Ltd., London2009S. Y.Cho B. A. BinDehaish X.-L.QinWeak convergence of a splitting algorithm in Hilbert spaces10.11948/2017027J. Appl. Anal. Comput.20177427438 P.CholamjiakA generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces10.1007/s11075-015-0030-6Numer. Algorithms20152015915932I.CioranescuGeometry of Banach spaces, duality mappings and nonlinear problemsMathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht1990 P. L.CombettesIterative construction of the resolvent of a sum of maximal monotone operatorsJ. Convex Anal.200916727748 P. L.Combettes V.R.Wajs Signal recovery by proximal forward-backward splitting10.1137/050626090Multiscale Model. Simul.2005411681200 P.Deuhard Recent progress in extrapolation methods for ordinary differential equations10.1137/1027140SIAM Rev.198527505535J.DouglasJr. H. H.RachfordJr. On the numerical solution of heat conduction problems in two and three space variables10.2307/1993056Trans. Amer. Math. Soc.195682421439 J. C.DunnConvexity, monotonicity, and gradient processes in Hilbert space10.1016/0022-247X(76)90152-9 J. Math. Anal. Appl.197653145158O.GülerOn the convergence of the proximal point algorithm for convex minimization10.1137/0329022SIAM J. Control Optim.199129403419 S.-N.He C.-P.Yang Solving the variational inequality problem defined on intersection of finite level sets10.1155/2013/942315Abstr. Appl. Anal.2013201318 P.-L.Lions B.MercierSplitting algorithms for the sum of two nonlinear operators10.1137/0716071SIAM J. Numer. Anal.197916964979G.López V.Martín-Márquez F.-H.Wang H.-K.XuForward-backward splitting methods for accretive operators in Banach spaces10.1155/2012/109236Abstr. Appl. Anal.20122012125P. E.MaingéApproximation methods for common fixed points of nonexpansive mappings in Hilbert spaces10.1016/j.jmaa.2005.12.066J. Math. Anal. Appl.2007325469479 B.MartinetRégularisation d’inéquations variationnelles par approximations successives(French) Rev. Franaise Informat. Recherche Opérationnelle19704154158D. S.MitrinovićAnalytic inequalitiesIn cooperation with P. M. Vasić, Die Grundlehren der mathematischen Wissenschaften, Band 165 Springer-Verlag, New York-Berlin1970A.Moudafi Viscosity approximation methods for fixed-points problems10.1006/jmaa.1999.6615J. Math. Anal. Appl.20002414655G. B.Passty Ergodic convergence to a zero of the sum of monotone operators in Hilbert space10.1016/0022-247X(79)90234-8J. Math. Anal. Appl.197972383390 D. H.Peaceman H. H.RachfordJr.The numerical solution of parabolic and elliptic differential equations10.1137/0103003J. Soc. Indust. Appl. Math.195532841X.-L.Qin S. Y.Cho L.WangA regularization method for treating zero points of the sum of two monotone operators10.1186/1687-1812-2014-75Fixed Point Theory Appl.20142014110X.-L.Qin J.-C.YaoWeak convergence of a Mann-like algorithm for nonexpansive and accretive operators10.1186/s13660-016-1163-4 J. Inequal. Appl.2016201619 S.ReichStrong convergence theorems for resolvents of accretive operators in Banach spaces10.1016/0022-247X(80)90323-6J. Math. Anal. Appl.198075287292R. T.RockafellarOn the maximal monotonicity of subdifferential mappings Pacific J. Math.197033209216 R. T.RockafellarMonotone operators and the proximal point algorithm10.1137/0314056SIAM J. Control. Optim.197614877898 C.SchneiderAnalysis of the linearly implicit mid-point rule for differential-algebraic equationsElectron. Trans. Numer. Anal.19931110S.SomaliImplicit midpoint rule to the nonlinear degenerate boundary value problems10.1080/00207160211930 Int. J. Comput. Math.200279327332W.Takahashi N.-C.Wong J.-C.YaoTwo generalized strong convergence theorems of Halpern’s type in Hilbert spaces and applications10.11650/twjm/1500406684Taiwanese J. Math.20121611511172 P.TsengA modified forward-backward splitting method for maximal monotone mappings10.1137/S0363012998338806 SIAM J. Control Optim.200038431446 M. vanVeldhuizenAsymptotic expansions of the global error for the implicit midpoint rule (stiff case)10.1007/BF02240190Computing198433185192F.-H.Wang H.-H.Cui On the contraction-proximal point algorithms with multi-parameters10.1007/s10898-011-9772-4J. Global Optim.201254485491 H.-K.Xu Inequalities in Banach spaces with applications10.1016/0362-546X(91)90200-KNonlinear Anal.19911611271138H.-K.XuViscosity approximation methods for nonexpansive mappings10.1016/j.jmaa.2004.04.059 J. Math. Anal. Appl.2004298279291 H.-T.Zegeye N.ShahzadStrong convergence theorems for a common zero for a finite family of m-accretive mappings10.1016/j.na.2006.01.012Nonlinear Anal.20076611611169Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.16Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditionsZhaoDaliang
School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, P. R. China
LiuYansheng
School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, P. R. China

This paper investigates the existence of at least two positive solutions for the following high-order fractional semipositone boundary value problem (SBVP, for short) with coupled integral boundary value conditions: $\begin{cases} D^\alpha_0+u(t)+\lambda f(t,u(t),v(t))=0,\quad t\in (0,1),\\ D^\alpha_0+v(t)+\lambda g(t,u(t),v(t))=0,\quad t\in (0,1),\\ u^{(j)}(0)= v^{(j)}(0)=0,\quad j=0,1,2,...,n-2,\\ D^{\alpha-1}_{0^+}u(1)=\lambda_1\int^{\eta_1}_0 v(t)dt,\\ D^{\alpha-1}_{0^+}v(1)=\lambda_2\int^{\eta_2}_0 u(t)dt, \end{cases}$ where $$n - 1 < \alpha\leq n, n \geq 3, 0 < \eta_1,\eta_2\leq 1, \lambda,\lambda_1,\lambda_2$$ are parameters and satisfy $$\lambda_1\lambda_2(\eta_1\eta_2)^\alpha<\Gamma^2(\alpha+1), D^\alpha_{0^+}$$ is the standard Riemann-Liouville derivative, and f, g are continuous and semipositone. By using the nonlinear alternative of Leray-Schauder type, Krasnoselskii’s fixed point theorems, and the theory of fixed point index on cone, we establish some existence results of multiple positive solutions to the considered fractional SBVP. As applications, two examples are presented to illustrate our main results.

34A0834B1534B18Fractional differential equationssemipositone boundary value problemcoupled integral boundary value conditionsfixed point index.
R. P.Agarwal N.Hussain M. A.TaoudiFixed point theorems in ordered Banach spaces and applications to nonlinear integral equations10.1155/2012/245872Abstr. Appl. Anal.20122012115 R. P.Agarwal M.Meehan D.O’Regan Fixed point theory and applicationsCambridge Tracts in Mathematics, Cambridge University Press, Cambridge2001R. P.Agarwal D.O’Regan A note on existence of nonnegative solutions to singular semi-positone problems10.1016/S0362-546X(98)00181-3 Nonlinear Anal.199936615622 N. A.Asif R. A.KhanPositive solutions to singular system with four-point coupled boundary conditions10.1016%2Fj.jmaa.2011.08.039J. Math. Anal. Appl.2012386848861 A.Cabada G.-T.Wang Positive solutions of nonlinear fractional differential equations with integral boundary value conditions10.1016/j.jmaa.2011.11.065J. Math. Anal. Appl.2012389403411 M.-Q.Feng X.-M.Zhang W.-G.Ge New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions10.1155/2011/720702 Bound. Value Probl.20112011120C. S.Goodrich Existence of a positive solution to systems of differential equations of fractional order10.1016/j.camwa.2011.02.039 Comput. Math. Appl.20116212511268D. J.Guo V.LakshmikanthamNonlinear problems in abstract conesNotes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA1988N.Hussain M. A.Taoudi Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations10.1186/1687-1812-2013-196 Fixed Point Theory Appl.20132013116M.Jia X.-P.LiuMultiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions10.1016/j.amc.2014.01.073Appl. Math. Comput.2014232313323J.-Q.Jiang L.-S.Liu Y.-H.Wu Positive solutions to singular fractional differential system with coupled boundary conditions10.1016/j.cnsns.2013.04.009 Commun. Nonlinear Sci. Numer. Simul.20131830613074A. A.Kilbas H. M.Srivastava J. J.Trujillo Theory and applications of fractional differential equationsNorth-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam2006 V.Lakshmikantham A. S.VatsalaBasic theory of fractional differential equations10.1016/j.na.2007.08.042Nonlinear Anal.20086926772682 K. Q.Lan W.Lin Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations10.1112/jlms/jdq090 J. Lond. Math. Soc.201183449469A.Leung A semilinear reaction-diffusion prey-predator system with nonlinear coupled boundary conditions: equilibrium and stability10.1512/iumj.1982.31.31020 Indiana Univ. Math. J.198231223241 Y.Li Y.-Q.Chen I.PodlubnyStability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability10.1016/j.camwa.2009.08.019Comput. Math. Appl.20105918101821S.-H.Liang J.-H.ZhangPositive solutions for boundary value problems of nonlinear fractional differential equation10.1016/j.na.2009.04.045Nonlinear Anal.20097155455550 Y.-S.Liu Twin solutions to singular semipositone problems10.1016%2FS0022-247X(03)00478-5J. Math. Anal. Appl.2003286248260Y.-S.Liu B.-Q.Yan Multiple solutions of singular boundary value problems for differential systems10.1016/S0022-247X(03)00568-7J. Math. Anal. Appl.2003287540556 Y.Liu W.-Q.Zhang X.-P.LiuA sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann-Liouville derivative10.1016/j.aml.2012.03.018 Appl. Math. Lett.20122519861992J.-X.Mao Z.-Q.Zhao N.-W.Xu On existence and uniqueness of positive solutions for integral boundary value problems Electron. J. Qual. Theory Differ. Equ.2010201018 K. S.Miller B.RossAn introduction to the fractional calculus and fractional differential equationsA Wiley-Interscience Publication, John Wiley & Sons, Inc., New York1993 S. K.Ntouyas G.-T.Wang L.-H.ZhangPositive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments10.7494/OpMath.2011.31.3.433Opuscula Math.201131433442A. S.Perelson D. E.Kirschner R. DeBoerDynamics of HIV infection of $$CD4^+T$$ cells10.1016/0025-5564(93)90043-AMath. Biosci.199311481125I.PodlubnyFractional differential equationsAn introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA1999 S. G.Samko A. A.Kilbas O. I.MarichevFractional integrals and derivatives Theory and applications, Edited and with a foreword by S. M. Nikol'skii, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon1993M.Stojanović R.GorenfloNonlinear two-term time fractional diffusion-wave problem10.1016/j.nonrwa.2009.12.012 Nonlinear Anal. Real World Appl.20101135123523S. W.Vong Positive solutions of singular fractional differential equations with integral boundary conditions10.1016/j.mcm.2012.06.024Math. Comput. Modelling20135710531059J.-F.Xu Z.-L.Wei W.DongUniqueness of positive solutions for a class of fractional boundary value problems10.1016/j.aml.2011.09.065 Appl. Math. Lett.201225590593W.-G.Yang Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions10.1016/j.camwa.2011.11.021Comput. Math. Appl.201263288297X.-J.YangFractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problemsTherm. Sci.2016326326X.-J.Yang H. M.Srivastava J. A. TenreiroMachadoA new fractional derivative without singular kernel: application to the modelling of the steady heat flow10.2298/TSCI151224222Y Therm. Sci.201620753756 X.-J.Yang J. A. TenreiroMachado C.Cattani F.GaoOn a fractal LC-electric circuit modeled by local fractional calculus10.1016/j.cnsns.2016.11.017 Commun. Nonlinear Sci. Numer. Simul.201747200206 C.-J.YuanMultiple positive solutions for (n - 1, 1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations10.14232/ejqtde.2010.1.36Electron. J. Qual. Theory Differ. Equ.20102010112 C.-J.YuanTwo positive solutions for (n - 1, 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations10.1016/j.cnsns.2011.06.008Commun. Nonlinear Sci. Numer. Simul.201217930942X.-U.Zhang Y.-F.HanExistence and uniqueness of positive solutions for higher order nonlocal fractional differential equations10.1016/j.aml.2011.09.058Appl. Math. Lett.201225555560Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.17Some fixed point theorems for contractive mappings of integral typeLiuZeqing
Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, People’s Republic of China
WangYuqing
Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, People’s Republic of China
KangShin Min
Department of Mathematics and the RINS, Gyeongsang National University, Jinju 52828, Korea;Center for General Education, , China Medical University, Taichung 40402, Taiwan
KwunYoung Chel
Department of Mathematics, Dong-A University, Busan 49315, Korea

Four fixed point theorems for mappings satisfying contractive conditions of integral type in complete metric spaces are proved. The results presented in this paper extend and improve a few results existing in literature. Two examples involving the contractive mappings of integral type are constructed.

54H25Contractive mappings of integral typefixed pointcomplete metric space.
M.Abbas M. A.KhanCommon fixed point theorem of two mappings satisfying a generalized weak contractive condition10.1155/2009/131068Int. J. Math. Math. Sci.2009200919H. H.Alsulami E.Karapınar D.O’Regan P.Shahi Fixed points of generalized contractive mappings of integral type Fixed Point Theory Appl., 24 pages. 2014A.BranciariA fixed point theorem for mappings satisfying a general contractive condition of integral type10.1155/S0161171202007524 Int. J. Math. Math. Sci.200229531536 D.Dey A.Ganguly M.SahaFixed point theorems for mappings under general contractive condition of integral typeBull. Math. Anal. Appl.201132734P. N.Dutta B. S.Choudhury A generalisation of contraction principle in metric spaces10.1155/2008/406368Fixed Point Theory Appl.2008200818R.George R.Rajagopalan Common fixed point results for $$\psi-\phi$$ contractions in rectangular metric spacesBull. Math. Anal. Appl.201354452 V.Gupta N.ManiA common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral typeMath. Theory Modeling2011116V.Gupta N.ManiCommon fixed point for two self-maps satisfying a generalized $$^\psi\int_\phi$$ weakly contractive condition of integral type Int. J. Nonlinear Sci.2013166471V.Gupta N.Mani A. K.TripathiA fixed point theorem satisfying a generalized weak contractive condition of integral type Int. J. Math. Anal. (Ruse)2012618831889 V. R.HosseiniCommon fixed point for generalized ($$\phi,\psi$$)-weak contractions contractions mappings condition of integral type Int. J. Math. Anal.2010415351543E.Karapınar P.Shahi K.TasGeneralized $$\alpha-\psi$$-contractive type mappings of integral type and related fixed point theorems10.1186/1029-242X-2014-160J. Inequal. Appl.20142014118 M. A.Kutbi M.Imdad S.Chauhan W.SintunavaratSome integral type fixed point theorems for non-self-mappings satisfying generalized ($$\psi,\phi$$)-weak contractive conditions in symmetric spaces10.1155/2014/519038Abstr. Appl. Anal.20142014111 Z.-Q.Liu J.-L.Li S. M.KangFixed point theorems of contractive mappings of integral type10.1186/1687-1812-2013-300Fixed Point Theory Appl.20132013117 Z.-Q.Liu X.Li S. M.Kang S. Y.ChoFixed point theorems for mappings satisfying contractive conditions of integral type and applications10.1186/1687-1812-2011-64 Fixed Point Theory Appl.20112011118 Z.-Q.Liu H.Wu J. S.Ume S. M.KangSome fixed point theorems for mappings satisfying contractive conditions of integral type10.1186/1687-1812-2014-69 Fixed Point Theory Appl.20142014114N. V.Luong N. X.Thuan A fixed point theorem for $$\psi_{\int \phi}$$-weakly contractive mapping in metric spaces Int. J. Math. Appl.20104233242C.Mongkolkeha P.KumamFixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces10.1155/2011/705943 Int. J. Math. Math. Sci.20112011112B. E.RhoadesSome theorems on weakly contractive maps10.1016/S0362-546X(01)00388-1Proceedings of the Third World Congress of Nonlinear Analysts, Part 4, Catania, (2000), Nonlinear Anal.20014726832693 B. E.RhoadesTwo fixed-point theorems for mappings satisfying a general contractive condition of integral type10.1155/S0161171203208024 Int. J. Math. Math. Sci.2003200340074013Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.18Classification of functions with trivial solutions under $$t$$-equivalenceLiYanqing
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, P. R. China;School of ocean information engineering, Hainan Tropical Ocean University, Sanya, Hainan 572022, P. R. China
PeiDonghe
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, P. R. China
HuangDejian
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, P. R. China;School of ocean information engineering, Hainan Tropical Ocean University, Sanya, Hainan 572022, P. R. China
GaoRuimei
Department of Science, Changchun University of Science and Technology, Changchun, Jilin 130022, P. R. China

We apply singularity theory to study bifurcation problems with trivial solutions. The approach is based on a new equivalence relation called t-equivalence which preserves the trivial solutions. We obtain a sufficient condition for recognizing such bifurcation problems to be t-equivalent and discuss the properties of the bifurcation problems with trivial solutions. Under the action of t-equivalent group, we classify all bifurcation problems with trivial solutions of codimension three or less.

58C2558K40Singularitybifurcationt-equivalenceclassification.
O.Diekmann A beginner’s guide to adaptive dynamicsMathematical modelling of population dynamics, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw2004634786U.Dieckmann R.LawThe dynamical theory of coevolution: a derivation from stochastic ecological processes10.1007/BF02409751J. Math. Biol.199734579612 S.-P.Gao Y.-C.Li Classification of $$(D_4, S^1)$$-equivariant bifurcation problems up to topological codimension 210.1360/02ys0217Sci. China Ser. A200346862871 S.Geritz J.Metz E.Kisdi G.MeszénaDynamics of adaptation and evolutionary branching10.1103/PhysRevLett.78.2024Phys. Rev. Lett.19977820242027M.Golubitsky M.RobertsA classification of degenerate Hopf bifurcations with O(2) symmetry10.1016/0022-0396(87)90119-7J. Differential Equations198769216264 M.Golubitsky D. G.SchaefferA theory for imperfect bifurcation via singularity theory10.1002/cpa.3160320103Comm. Pure Appl. Math.1979322198M.Golubitsky D. G.SchaefferSingularities and groups in bifurcation theoryVol. I, Applied Mathematical Sciences, Springer-Verlag, New York1985 B. L.KeyfitzClassification of one-state-variable bifurcation problems up to codimension seven10.1080/02681118608806002 Dynam. Stability Systems19861141M.Manoel I.StewartThe classification of bifurcations with hidden symmetries10.1112/S0024611500012156Proc. London Math. Soc.200080198234J.Martinet Singularities of smooth functions and mapsTranslated from the French by Carl P. Simon, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge-New York1982M.PetersClassification of two-parameter bifurcations10.1007/BFb0085437Singularity theory and its applications, Part II, Coventry, (1988/1989), Lecture Notes in Math., Springer, Berlin19911463294300J. M.Smith G. R.PriceThe logic of animal conflict10.1038/246015a0Nature19732461518A.Vutha M.GolubitskyNormal forms and unfoldings of singular strategy functions10.1007/s13235-014-0116-0Dyn. Games Appl.20155180213X.-H.Wang M.GolubitskySingularity theory of fitness functions under dimorphism equivalence10.1007/s00285-015-0958-0 J. Math. Biol.201673526573 D.Waxman S.Gavrilets20 questions on adaptive dynamics10.1111/j.1420-9101.2005.00948.xJ. Evol. Biol.20051811391154Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.19Existence for fractional Dirichlet boundary value problem under barrier strip conditionsSongQilin
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China
DongXiaooyu
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China
BaiZhanbing
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China
ChenBo
College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, P. R. China

In this paper, a fixed-point theorem is used to establish existence results for fractional Dirichlet boundary value problem $D^\alpha x(t)=f(t,x(t),D^{\alpha-1}x(t)),\quad x(0)=A,\quad x(1)=B$ where $$1 < \alpha\leq 2,D^\alpha x(t)$$ is the conformable fractional derivative, and $$f : [0, 1] \times R^2 \rightarrow R$$ is a continuous function. The main condition is sign condition. The method used is based upon the theory of fixed-point index.

34A0834B1835J05Barrier stripsfixed-point indexconformable fractional derivative.
T.AbdeljawadOn conformable fractional calculus10.1016/j.cam.2014.10.016 J. Comput. Appl. Math.20152795766 R.Almeida A. B.Malinowska T.OdzijewiczFractional differential equations with dependence on the Caputo- Katugampola derivative10.1115/1.4034432 J. Comput. Nonlinear Dynam.201611061017061027 A.Alsaedi D.Baleanu S.Etemad S.RezapourOn coupled systems of time-fractional differential problems by using a new fractional derivative10.1155/2016/4626940 J. Funct. Spaces2016201618Z.-B.BaiOn positive solutions of a nonlocal fractional boundary value problem10.1016/j.na.2009.07.033Nonlinear Anal.201072916942Z.-B.Bai On solutions of some fractional m-point boundary value problems at resonance Electron. J. Qual. Theory Differ. Equ.20102010115Z.-B.BaiSolvability for a class of fractional m-point boundary value problem at resonance10.1016/j.camwa.2011.03.003Comput. Math. Appl.20116212921302Z.-B.Bai X.-Y.Dong C.YinExistence results for impulsive nonlinear fractional differential equation with mixed boundary conditions10.1186/s13661-016-0573-zBound. Value Probl.20162016111 Z.-B.Bai H.-S.Positive solutions for boundary value problem of nonlinear fractional differential equationJ. Math. Anal. Appl.2005311495505 Z.-B.Bai Y.-H.ZhangThe existence of solutions for a fractional multi-point boundary value problem10.1016/j.camwa.2010.08.030Comput. Math. Appl.20106023642372Z.-B.Bai Y.-H.Zhang Solvability of fractional three-point boundary value problems with nonlinear growth10.1016/j.amc.2011.06.051Appl. Math. Comput.201121817191725Z.-B.Bai S.Zhang S.-J.Sun C.YinMonotone iterative method for fractional differential equationsElectron. J. Differential Equations2016201618Y.-J.Cui Uniqueness of solution for boundary value problems for fractional differential equations10.1016/j.aml.2015.07.002Appl. Math. Lett.2016514854X.-Y.Dong Z.-B.Bai S.-J.SunPositive solutions for some boundary value problems with conformable fractional differential derivatives10.1186/s13661-016-0735-z(Chinese) Acta Math. Sci. Ser. A Chin. Ed.2017378291X.-Y.Dong Z.-B.Bai W.ZhangPositive solutions for nonlinear eigenvalue problems with conformable fractional differential derivativesJ. Shandong Univ. Sci. Technol. Nat. Sci.2016358590X.-Y.Dong Z.-B.Bai S.-Q.Zhang Positive solutions to boundary value problems of p-Laplacian with fractional derivative10.1186/s13661-016-0735-z Bound. Value Probl.20172017115H. H.Dong B. Y.Guo B. S.YinGeneralized fractional supertrace identity for Hamiltonian structure of NLSMKdV hierarchy with self-consistent sources10.1007/s13324-015-0115-3 Anal. Math. Phys.20166199209T.Feng X.-Z.Meng L.-D.Liu S.-J.Gao Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model10.1186/s13660-016-1265-zJ. Inequal. Appl.20162016129 C.-H.Gao Existence of solutions to p-Laplacian difference equations under barrier strips conditionsElectron. J. Differential Equations2007200716L.-M.He X.-Y.Dong Z.-B.Bai B.Chen Solvability of some two-point fractional boundary value problems under barrier strip conditions10.1155/2017/1465623J. Funct. Spaces2017201716Y.-X.Hua X.-H.YuOn the ground state solution for a critical fractional Laplacian equation10.1016/j.na.2013.04.005Nonlinear Anal.201387116125F.Jiao Y.ZhouExistence results for fractional boundary value problem via critical point theory10.1142/S0218127412500861 Internat. J. Bifur. Chaos Appl. Sci. Engrg.201222117P.KelevedjievExistence of solutions for two-point boundary value problems10.1016/0362-546X(94)90035-3 Nonlinear Anal.199422217224P. S.Kelevedjiev S.Tersian Singular and nonsingular first-order initial value problems10.1016/j.jmaa.2010.01.033J. Math. Anal. Appl.2010366516524 R.Khalil M. AlHorani A.Yousef M.SababhehA new definition of fractional derivative10.1016/j.cam.2014.01.002 J. Comput. Appl. Math.20142646570 X.-P.Liu M.Jia W.-G.GeMultiple solutions of a p-Laplacian model involving a fractional derivative10.1186/1687-1847-2013-126 Adv. Difference Equ.20132013112 X.-P.Liu M.Jia W.-G.GeThe method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator10.1016/j.aml.2016.10.001Appl. Math. Lett.2017655662 X.-P.Liu M.Jia B.-F.Wu Existence and uniqueness of solution for fractional differential equations with integral boundary conditionsElectron. J. Qual. Theory Differ. Equ.20092009110 R.-Y.Ma H.LuoExistence of solutions for a two-point boundary value problem on time scales10.1016/S0096-3003(03)00204-2Appl. Math. Comput.2004150139147 X.-Z.Meng S.-N.Zhao T.Feng T.-H.ZhangDynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis10.1016/j.jmaa.2015.07.056 J. Math. Anal. Appl.2016433227242Z.WangA numerical method for delayed fractional-order differential equations10.1155/2013/256071J. Appl. Math.2013201317G.-T.Wang B.Ahmad L.-H.Zhang J. J.Nieto Comments on the concept of existence of solution for impulsive fractional differential equations10.1016/j.cnsns.2013.04.003Commun. Nonlinear Sci. Numer. Simul.201419401403Z.Wang X.Huang J.-P.ZhouA numerical method for delayed fractional-order differential equations: based on G-L definition10.12785/amis/072L22Appl. Math. Inf. Sci.20137525529 J.-R.Wang Y.Zhou M.FečkanOn recent developments in the theory of boundary value problems for impulsive fractional differential equations10.1016/j.camwa.2011.12.064Comput. Math. Appl.20126430083020S.-S.Yan J.-F.Yang X.-H.YuEquations involving fractional Laplacian operator: compactness and application10.1016/j.jfa.2015.04.012J. Funct. Anal.20152694779X.-J.YangA new fractional derivative without singular kernel: Application to the modelling of the steady heat flowTherm. Sci.201620753756X.-J.Yang Fractional Maxwell fluid with fractional derivative without singular kernel Therm. Sci.2016201871X.-J.YangFractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems10.2298/TSCI161216326YTherm. Sci.2017311611171X.-J.Yang D.Baleanu H. M.SrivastavaLocal fractional integral transforms and their applications Elsevier/Academic Press, Amsterdam2015X.-J.Yang J. A. T.MachadoA new fractional operator of variable order: Application in the description of anomalous diffusion10.1016/j.physa.2017.04.054Phys. A2017481276283X.-H.YuSolutions of fractional Laplacian equations and their Morse indices10.1016/j.jde.2015.09.010J. Differential Equations2016260860871S.-Q.ZhangPositive solutions for boundary-value problems of nonlinear fractional differential equations Electron. J. Differential Equations20062006112 W.Zhang Z.-B.Bai S.-J.SuExtremal solutions for some periodic fractional differential equation10.1186/s13662-016-0869-4Adv. Difference Equ.2016201618T.-Q.Zhang X.-Z.Meng Y.Song T.-H.ZhangA stage-structured predator-prey SI model with disease in the prey and impulsive effects10.3846/13926292.2013.840866 Math. Model. Anal.201318505528 Y.-M.Zou Y.-J.Cui Existence results for a functional boundary value problem of fractional differential equations10.1186/1687-1847-2013-233Adv. Difference Equ.20132013125Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.20Stability of fixed points of set-valued mappings and strategic stability of Nash equilibriaXiangShuwen
School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China
XiaShunyou
School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China;School of Mathematics and Computer Science, Guizhou Education University, Guiyang, 550018, P. R. China
HeJihao
School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China
YangYanlong
School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China
LiuChenwei
School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China

In this paper, we mainly focus on the stability of Nash equilibria to any perturbation of strategy sets. A larger perturbation, strong $$\delta$$-perturbation, will be proposed for set-valued mapping. The class of perturbed games considered in the definition of strong $$\delta$$-perturbation is richer than those considered in many other definitions of stability of Nash equilibria. The strong $$\delta$$-perturbation of the best reply correspondence will be used to define an appropriate stable set for Nash equilibria, called SBR-stable set. As an SBR-stable set is stable to any strong $$\delta$$-perturbation and, various perturbations of strategy sets are not beyond the range of strong $$\delta$$-perturbation, it has the stability that various stable sets possess, such as fully stable set, stable set, quasistable set, and essential set. An SBR-stable set is stable to any perturbation of strategy sets, so it will provide convenience for study in strategic stability, which is even used to study any noncooperative game.

90C2990C3191A10StabilityNash equilibriafixed pointstrong $$\delta$$-perturbationstable set.
O.Carbonell-NicolauOn strategic stability in discontinuous games10.1016/j.econlet.2011.06.007Econom. Lett.2011113120123O.Carbonell-NicolauFurther results on essential Nash equilibria in normal-form games10.1007/s00199-014-0829-8Econom. Theory201559277300M. K.FortJr.Points of continuity of semi-continuous functionsPubl. Math. Debrecen19512100102 S.Govindan R.WilsonEssential equilibria10.1073%2Fpnas.0506796102Proc. Natl. Acad. Sci. USA20051021570615711J.HillasOn the definition of the strategic stability of equilibria10.2307/2938320Econometrica19905813651390J.Hillas M.Jansen J.Potters D.Vermeulen On the relation among some definitions of strategic stability10.1287/moor.26.3.611.10585 Math. Oper. Res.200126611635J.Hillas M.Jansen J.Potters D.Vermeulen Independence of inadmissible strategies and best reply stability: a direct proof10.1007/s001820400168 Special issue on stable equilibria, Internat. J. Game Theory200432371377W.-S.Jia S.-W.Xiang J.-H.He Y.-L.YangExistence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leaderfollower games10.1007/s10898-014-0178-y J. Global Optim.201561397405E.Kalai D.Samet Persistent equilibria in strategic games10.1007/BF01769811 Internat. J. Game Theory198413129144E.Klein A. C.Thompson Theory of correspondences Including applications to mathematical economics, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York1984E.Kohlberg J. F.MertensOn the strategic stability of equilibriaEconometrica19865410031037 D. M.Kreps R.WilsonSequential equilibria10.2307/1912767Econometrica198250863894 A.McLennanFixed points of contractible valued correspondences10.1007/BF01268156Internat. J. Game Theory198918175184J. F.MertensStable equilibria–a reformulation, I10.1287/moor.14.4.575 Definition and basic properties, Math. Oper. Res.198914575625 R. B.MyersonRefinements of the Nash equilibrium concept10.1007/BF01753236Internat. J. Game Theory197877380 A. B.Sadanand V.Sadanand Equilibria in non-cooperative games, IPerturbations based refinements of Nash equilibrium, Bull. Econ. Res.199443197224V.ScalzoEssential equilibria of discontinuous games10.1007/s00199-012-0726-y Econom. Theory2013542744V.ScalzoOn the existence of essential and trembling-hand perfect equilibria in discontinuous games10.1007/s40505-013-0021-5 Econ. Theory Bull.20141112R.Selten Reexamination of the perfectness concept for equilibrium points in extensive games10.1007/BF01766400 Internat. J. Game Theory197542555 E. vanDammeStrategic equilibrium10.1016/S1574-0005(02)03004-7Handbook of Game Theory with Economic Applications2002315211596 A. J.Vermeulen J. A. M.Potters M. J. M.JansenOn quasi-stable sets10.1007/BF01254383 Internat. J. Game Theory1996254349 A. J.Vermeulen J. A. M.Potter M. J. M.Jansen On stable sets of equilibria10.1007/978-1-4757-2640-4_12Game theoretical applications to economics and operations research, Bangalore, (1996), Theory Decis. Lib. Ser. C Game Theory Math. Program. Oper. Res., Kluwer Acad. Publ., Boston, MA199718133148W.-T.Wu J.-H.JiangEssential equilibrium points of n-person non-cooperative gamesSci. Sinica19621113071322 S.-W.Xiang G.-D.Liu Y.-H.ZhouOn the strongly essential components of Nash equilibria of infinite n-person games with quasiconcave payoffs10.1016/j.na.2005.03.047Nonlinear Anal.20056312637 J.Yu Q.LuoOn essential components of the solution set of generalized games10.1006/jmaa.1998.6202J. Math. Anal. Appl.1999230303310J.Yu S.-W.XiangOn essential components of the set of Nash equilibrium points10.1016/S0362-546X(98)00193-XNonlinear Anal.199938259264Y.-H.Zhou J.Yu S.-W.XiangEssential stability in games with infinitely many pure strategies10.1007/s00182-006-0063-0Internat. J. Game Theory200735493503Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.21Inequalities on asymmetric $$L_p$$-harmonic radial bodiesLiZhaofeng
Department of Mathematics, China Three Gorges University, Yichang, 443002, China
WangWeidong
Department of Mathematics, China Three Gorges University, Yichang, 443002, China

Lutwak introduced the $$L_p$$-harmonic radial body of a star body. In this paper, we define the notion of asymmetric $$L_p$$- harmonic radial bodies and study their properties. In particular, we obtain the extremum values of dual quermassintegrals and the volume of the polars of the asymmetric $$L_p$$-harmonic radial bodies, respectively.

52A2052A3952A40Star body$$L_p$$-harmonic radial bodyasymmetric $$L_p$$-harmonic radial bodydual quermassintegralspolar.
Y. D.Chai Y. S.LeeHarmonic radial combinations and dual mixed volumes10.4310/AJM.2001.v5.n3.a5Asian J. Math.20015493498 Y.-B.Feng W.-D.WangSome inequalities for $$L_p$$-dual affine surface area10.7153/mia-17-32 Math. Inequal. Appl.201417431441 W. J.FireyMean cross-section measures of harmonic means of convex bodies10.2140/pjm.1961.11.1263Pacific J. Math.19611112631266 W. J.FireyPolar means of convex bodies and a dual to the Brunn-Minkowski theorem10.4153/CJM-1961-037-0Canad. J. Math.196113444453 R. J.GardnerGeometric tomographySecond edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, New York2006C.Haberl$$L_p$$ intersection bodies10.1016/j.aim.2007.11.013 Adv. Math.200821725992624 C.Haberl M.Ludwig A characterization of $$L_p$$ intersection bodies10.1155/IMRN/2006/10548 Int. Math. Res. Not.20062006129C.Haberl F. E.SchusterAsymmetric affine $$L_p$$ Sobolev inequalities10.1016/j.jfa.2009.04.009J. Funct. Anal.2009257641658C.Haberl F. E.SchusterGeneral $$L_p$$ affine isoperimetric inequalities10.4310/jdg/1253804349 J. Differential Geom.200983126 C.Haberl F. E.Schuster J.XiaoAn asymmetric affine Pólya-Szegö principle10.1007/s00208-011-0640-9 Math. Ann.2012352517542M.Ludwig Minkowski valuations10.1090/S0002-9947-04-03666-9 Trans. Amer. Math. Soc.200535741914213M.LudwigIntersection bodies and valuationsAmer. J. Math.200612814091428M.LudwigValuations in the affine geometry of convex bodies10.1142/9789812774644_0005Integral geometry and convexity, World Sci. Publ., Hackensack, NJ20064965 E.LutwakExtended affine surface area10.1016/0001-8708(91)90049-D Adv. Math.1991853968 E.LutwakThe Brunn-Minkowski-Firey theory, II10.1006/aima.1996.0022Affine and geominimal surface areas, Adv. Math.1996118244294 L.ParapatitsSL(n)-contravariant $$L_p$$-Minkowski valuations10.1090/S0002-9947-2013-05750-9Trans. Amer. Math. Soc.201436611951211L.ParapatitsSL(n)-covariant $$L_p$$-Minkowski valuations10.1112/jlms/jdt068J. Lond. Math. Soc.201489397414R.Schneider Convex bodies: the Brunn-Minkowski theoryEncyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge1993F. E.Schuster T.WannererGL(n) contravariant Minkowski valuations10.1090/S0002-9947-2011-05364-X Trans. Amer. Math. Soc.2012364815826 F. E.Schuster M.Weberndorfer Volume inequalities for asymmetric Wulff shapes10.4310/jdg/1352297808 J. Differential Geom.201292263283 W.-D.Wang Q.Chen$$L_p$$-dual geominimal surface area10.1186/1029-242X-2011-6 J. Inequal. Appl.20112011110 W.-D.Wang Y.-B.FengA general $$L_p$$-version of Petty’s affine projection inequality10.11650/tjm.17.2013.2122Taiwanese J. Math.201317517528 W.-D.Wang G.-S.Leng$$L_p$$-dual mixed quermassintegralsIndian J. Pure Appl. Math.200536177188W.-D.Wang Y.-N.LiBusemann-Petty problems for general $$L_p$$-intersection bodies10.1007/s10114-015-4273-xActa Math. Sin. (Engl. Ser.)201531777786W.-D.Wang Y.-N.LiGeneral $$L_p$$-intersection bodies10.11650/tjm.19.2015.3493 Taiwanese J. Math.20151912471259 W.-D.Wang T.-Y.Ma Asymmetric $$L_p$$-difference bodies10.1090/S0002-9939-2014-11919-8 Proc. Amer. Math. Soc.201414225172527T.WannererGL(n) equivariant Minkowski valuations10.1512/iumj.2011.60.4425Indiana Univ. Math. J.20116016551672 M.Weberndorfer Shadow systems of asymmetric $$L_p$$ zonotopes10.1016/j.aim.2013.02.022Adv. Math.2013240613635 F.Yibin W.Weidong L.FenghongSome inequalities on general $$L_p$$-centroid bodies10.7153/mia-18-02Math. Inequal. Appl.2015183949 B.-C.Zhu N.Li J.-Z.Zhou Isoperimetric inequalities for $$L_p$$ geominimal surface area10.1017/S0017089511000292Glasg. Math. J.201153717726Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.22On solving general split equality variational inclusion problems in Banach spaceZhaoJ.
College of Sciences, Qinzhou University, Qinzhou, Guangxi 535000,, P. R. China
LiangY. S.
Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, and College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, P. R. China

In this paper, we are concerned with a new iterative scheme for general split equality variational inclusion problems in Banach spaces. We also show that the iteration converges strongly to a common solution of the general split equality variational inclusion problems (GSEVIP). The results obtained in this paper extend and improve some well-known results in the literature.

47J2234A60General split equality variational problemsstrong convergenceBanach space.
E.Blum W.Oettli From optimization and variational inequalities to equilibrium problems Math. Student199463123145 C.Byrne Iterative oblique projection onto convex sets and the split feasibility problem10.1088/0266-5611/18/2/310Inverse Problems200218441453Y.Censor T.ElfvingA multiprojection algorithm using Bregman projections in a product space 10.1007/BF02142692Numer. Algorithms19948221239Y.Censor T.Elfving N.Kopf T.BortfeldThe multiple-sets split feasibility problem and its applications for inverse problems10.1088/0266-5611/21/6/017 Inverse Problems20052120712084 Y.Censor A.Motova A.SegalPerturbed projections and subgradient projections for the multiple-sets split feasibility problem10.1016/j.jmaa.2006.05.010J. Math. Anal. Appl.200732712441256S.-S.ChangOn Chidume’s open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces10.1006/jmaa.1997.5661 J. Math. Anal. Appl.199721694111S.-S.Chang H. W. J. Lee C. K.Chan W. B.Zhang A modified halpern-type iteration algorithm for totally quasi-$$\phi$$- asymptotically nonexpansive mappings with applications10.1016/j.amc.2011.12.019 Appl. Math. Comput.201221864896497 S.-S.Chang L.Wang L.-J.Qin Z.-L.MaStrongly convergent iterative methods for split equality variational inclusion problems in Banach spaces10.1016/S0252-9602(16)30096-0 Acta Math. Sci. Ser. B Engl. Ed.2016616411650 S.-S.Chang L.Wang X. R.Wang G.WangGeneral split equality equilibrium problems with application to split optimization problems10.1007/s10957-015-0739-3J. Optim. Theory Appl.2015166377390C.-S.ChuangStrong convergence theorems for the split variational inclusion problem in Hilbert spaces10.1186/1687-1812-2013-350Fixed Point Theory Appl.20132013120 P. L.Combettes S. A.Hirstoaga Equilibrium programming in Hilbert spacesJ. Nonlinear Convex Anal.20056117136 L.Gasiński Z.-H.Liu S.Migórski A.Ochal Z.-J.PengHemivariational inequality approach to evolutionary constrained problems on star-shaped sets10.1007/s10957-014-0587-6J. Optim. Theory Appl.2015164514533 K.Goebel W. A.KirkTopics in metric fixed point theory Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge1990Z.-H.LiuExistence results for quasilinear parabolic hemivariational inequalities10.1016/j.jde.2007.09.001J. Differential Equations200824413951409 Z.-H.Liu X.-W.Li D.MotreanuApproximate controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces10.1137/140994058SIAM J. Control Optim.20155332283244 Z.-H.Liu S.-D.ZengEquilibrium problems with generalized monotone mapping and its applications10.1002/mma.3471Math. Methods Appl. Sci.201639152163Z.-H.Liu S.-D.Zeng D.Motreanu Evolutionary problems driven by variational inequalities10.1016/j.jde.2016.01.012J. Differential Equations201626067876799 P. E.MaingéStrong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization10.1007/s11228-008-0102-zSet-Valued Anal.200816899912A.Moudafi E.Al-Shemas Simultaneous iterative methods for split equality problemTrans. Math. Program. Appl.20131111Z.-J.Peng Z.-H.Liu X.-Y.LiuBoundary hemivariational inequality problems with doubly nonlinear operators10.1007/s00208-012-0884-zMath. Ann.201336513391358 S.Suantai P.Cholamjiak Y. J.Cho W.Cholamjiak On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces10.1186/s13663-016-0509-4Fixed Point Theory Appl.20162016116 H.-K.XuA variable Krasnosel’skiı-Mann algorithm and the multiple-set split feasibility problem10.1088/0266-5611/22/6/007Inverse Problems20062220212034 Q.-Z.YangThe relaxed CQ algorithm solving the split feasibility problem10.1088/0266-5611/20/4/014 Inverse Problems20042012611266 J.-L.Zhao Q.-Z.YangSeveral solution methods for the split feasibility problem10.1088/0266-5611/21/5/017Inverse Problems20052117911799Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.23Existence of traveling wave solutions in $$m$$-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interactionZhouKai
Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China;School of Mathematics and Computer, Chizhou University, Chizhou 247000, P. R. China

This paper deals with the existence of traveling wave solutions for $$m$$-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction. By using Schauder’s fixed point theorem and a cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained will be applied to $$m$$-dimensional delayed lattice dynamical systems with Lotka-Volterra type competitive reaction terms and global interaction.

34A3334K3192D25Traveling wave solutionslattice differential systemsdelayupper and lower solutionsSchauder’s fixed point theorem.
P. W.Bates A.ChamjA discrete convolution model for phase transitions10.1007/s002050050189Arch. Ration. Mech. Anal.1999150281305P. W.Bates P. C.Fife X.-F.Ren X.-F.WangTraveling waves in a convolution model for phase transitions10.1007/s002050050037Arch. Rational Mech. Anal.1997138105136J.Bell C.Cosner Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons10.1090/qam/736501Quart. Appl. Math.198442114J. W.Cahn S.-N.Chow E. S. VanVleckSpatially discrete nonlinear diffusion equations10.1216/rmjm/1181072270 Second Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology, Edmonton, AB, (1992), Rocky Mountain J. Math.19952587118J. W.Cahn J.Mallet-Paret E. S. VanVleckTraveling wave solutions for systems of ODEs on a two-dimensional spatial lattice10.1137/S0036139996312703SIAM J. Appl. Math.199859455493S.-N.ChowLattice dynamical systems10.1007/978-3-540-45204-1_1 Dynamical systems, Lecture Notes in Math., Springer, Berlin200318221102S.-N.Chow J.Mallet-Paret W.-X.Shen Traveling waves in lattice dynamical systems10.1006/jdeq.1998.3478J. Differential Equations1998149248291 J.Fang J.-J.Wei X.-Q.Zhao Spreading speeds and travelling waves for non-monotone time-delayed lattice equations10.1098/rspa.2009.0577Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.201046619191934C.-H.Hsu S.-S.LinExistence and multiplicity of traveling waves in a lattice dynamical system10.1006/jdeq.2000.3770 J. Differential Equations2000164431450J.-H.Huang G.LuTraveling wave solutions to systems of delayed lattice differential equations(Chinese) ; translated from Chinese Ann. Math. Ser. A, 25 (2004), 153–164, Chinese J. Contemp. Math.200425125136J.-H.Huang G.Lu S.-G.RuanTraveling wave solutions in delayed lattice differential equations with partial monotonicity10.1016/j.na.2004.10.020Nonlinear Anal.20056013311350G.Lin W.-T.LiTraveling waves in delayed lattice dynamical systems with competition interactions10.1016/j.nonrwa.2010.01.013Nonlinear Anal. Real World Appl.20101136663679G.Lin W.-T.Li M.-J.Ma Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models10.3934/dcdsb.2010.13.393Discrete Contin. Dyn. Syst. Ser. B201013393414G.LinW.-T.Li S.-X.PanTravelling wavefronts in delayed lattice dynamical systems with global interaction10.1080/10236190902828387J. Difference Equ. Appl.20101614291446 Y.Lin Q.-R.Wang K.Zhou Traveling wave solutions in n-dimensional delayed reaction-diffusion systems with mixed monotonicity10.1016/j.cam.2012.11.007J. Comput. Appl. Math.20132431627 S.-W.Ma P.-X.Weng X.-F.ZouAsymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation10.1016/j.na.2005.10.042 Nonlinear Anal.20066518581890 S.-W.Ma X.-F.Zou Propagation and its failure in a lattice delayed differential equation with global interaction10.1016/j.jde.2004.07.014J. Differential Equations2005212129190J.Mallet-ParetTraveling waves in spatially discrete dynamical systems of diffusive type10.1007/978-3-540-45204-1_4Dynamical systems, Lecture Notes in Math., Springer, Berlin20031822231298 H. F.Weinberger M.Lewis B.-T.LiAnalysis of linear determinacy for spread in cooperative models10.1007/s002850200145 J. Math. Biol.200245183218 P.-X.Weng H.-X.Huang J.-H.Wu Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction10.1093/imamat/68.4.409 IMA J. Appl. Math.200368409439J.-H.Wu X.-F.ZouAsymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations10.1006/jdeq.1996.3232J. Differential Equations1997135315357 J.Xia Z.-X.Yu Y.-C.Dong H.-Y.LiTraveling waves for n-species competitive system with nonlocal dispersals and delays10.1016/j.amc.2016.04.025Appl. Math. Comput.2016287/288201213B.Zinner Existence of traveling wavefront solutions for the discrete Nagumo equation10.1016/0022-0396(92)90142-AJ. Differential Equations199296127B.Zinner G.Harris W.HudsonTraveling wavefronts for the discrete Fisher’s equation10.1006/jdeq.1993.1082 J. Differential Equations19931054662Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.24Picard splitting method and Picard CG method for solving the absolute value equationLvChang-Qing
College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou, 350117, P. R. China;School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, 277160, P. R. China
MaChang-Feng
College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou, 350117, P. R. China

In this paper, we combine matrix splitting iteration algorithms, such as, Jacobi, SSOR or SAOR algorithms with Picard method for solving absolute value equation. Then, we propose Picard CG for solving the absolute value equation. We discuss the convergence of those methods we proposed. At last, some examples are provided to illustrate the efficiency and validity of methods that we present.

65H1047H10Absolute value equationPicard algorithmmatrix splitting iteration methodconjugate gradient method.
Z.-Z.Bai X.YangOn HSS-based iteration methods for weakly nonlinear systems10.1016/j.apnum.2009.06.005 Appl. Numer. Math.20095929232936 R. W.Cottle G. B.Dantzig Complementary pivot theory of mathematical programmingLinear Algebra and Appl.19681103125 R.W.Cottle J.-S.Pang R. E.StoneThe linear complementarity problemComputer Science and Scientific Computing, Academic Press, Inc., Boston, MA1992T.-X.Gu X.-W.Xu X.-P.Liu H.-B.An X.-D.HangIterative methods and preconditioning techniques(Chinese) Science Press, Beijing2015S.-L.Hu Z.-H.HuangA note on absolute value equations10.1007/s11590-009-0169-y Optim. Lett.20104417424S.Ketabchi H.MoosaeiAn efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side10.1016/j.camwa.2012.03.015 Comput. Math. Appl.20126418821885 S.Ketabchi H.Moosaei Minimum norm solution to the absolute value equation in the convex case10.1007/s10957-012-0044-3J. Optim. Theory Appl.201215210801087 S.Ketabchi H.Moosaei S.FallahiOptimal error correction of the absolute value equation using a genetic algorithm10.1016/j.mcm.2011.11.068Math. Comput. Model.20135723392342 O. L.Mangasarian Linear complementarity problems solvable by a single linear program10.1007/BF01580671 Math. Programming197610263270O. L.MangasarianAbsolute value equation solution via concave minimization10.1007/s11590-006-0005-6 Optim. Lett.2007138 O. L.MangasarianAbsolute value programming10.1007/s10589-006-0395-5Comput. Optim. Appl.2007364353O. L.MangasarianA generalized Newton method for absolute value equations10.1007/s11590-008-0094-5Optim. Lett.20093101108O. L.MangasarianPrimal-dual bilinear programming solution of the absolute value equation10.1007/s11590-011-0347-6Optim. Lett.2012615271533 O. L.Mangasarian Absolute value equation solution via dual complementarity10.1007/s11590-012-0469-5Optim. Lett.20137625630 O. L.MangasarianAbsolute value equation solution via linear programming10.1007/s10957-013-0461-y J. Optim. Theory Appl.2014161870876 O. L.Mangasarian R. R.MeyerAbsolute value equations10.1016/j.laa.2006.05.004 Linear Algebra Appl.2006419359367M. A.Noor J.Iqbal K. I.Noor E.Al-SaidOn an iterative method for solving absolute value equations10.1007/s11590-011-0332-0 Optim. Lett.2012610271033 J. M.Ortega W. C.RheinboldtIterative solution of nonlinear equations in several variablesAcademic Press, New York-London1970 O.ProkopyevOn equivalent reformulations for absolute value equations10.1007/s10589-007-9158-1 Comput. Optim. Appl.200944363372J.Rohn A theorem of the alternatives for the equation $$Ax + B|x| = b$$10.1080/0308108042000220686Linear Multilinear Algebra200452421426J.Rohn An algorithm for solving the absolute value equation10.13001/1081-3810.1332Electron. J. Linear Algebra200918589599 J.Rohn On unique solvability of the absolute value equation10.1007/s11590-009-0129-6Optim. Lett.20093603606J.Rohn An algorithm for computing all solutions of an absolute value equation10.1007/s11590-011-0305-3 Optim. Lett.20126851856 J.Rohn V.Hooshyarbakhsh R.FarhadsefatAn iterative method for solving absolute value equations and sufficient conditions for unique solvability10.1007/s11590-012-0560-yOptim. Lett.201483544 D. K.SalkuyehThe Picard-HSS iteration method for absolute value equations10.1007/s11590-014-0727-9Optim. Lett.2014821912202A.-X.Wang H.-J.Wang Y.-K.Deng Interval algorithm for absolute value equations10.2478/s11533-011-0067-2Cent. Eur. J. Math.2011911711184Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.25$$(L,M)$$-fuzzy convex structuresShiFu-Gui
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
XiuZhen-Yu
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610000, China

In this paper, the notion of $$(L,M)$$-fuzzy convex structures is introduced. It is a generalization of L-convex structures and $$M$$-fuzzifying convex structures. In our definition of $$(L,M)$$-fuzzy convex structures, each $$L$$-fuzzy subset can be regarded as an $$L$$-convex set to some degree. The notion of convexity preserving functions is also generalized to lattice-valued case. Moreover, under the framework of $$(L,M)$$-fuzzy convex structures, the concepts of quotient structures, substructures and products are presented and their fundamental properties are discussed. Finally, we create a functor $$\omega$$ from MYCS to LMCS and show that MYCS can be embedded in LMCS as a coreflective subcategory, where MYCS and LMCS denote the category of $$M$$-fuzzifying convex structures and the category of $$(L,M)$$-fuzzy convex structures, respectively.

03E7252A01quotient structuressubstructures$$(L،M)$$-fuzzy convex structure$$(L،M)$$-fuzzy convexity preserving functionproducts.
N.Ajmal K. V.ThomasFuzzy lattices10.1016/0020-0255(94)90124-4Inform. Sci.199479271291M.BergerConvexity Amer. Math. Monthly199097650678 P.DwingerCharacterization of the complete homomorphic images of a completely distributive complete lattice, INederl. Akad. Wetensch. Indag. Math.198285403414 J.-M.Fang P.-W.ChenOne-to-one correspondence between fuzzifying topologies and fuzzy preorders10.1016/j.fss.2007.03.016Fuzzy Sets and Systems200715818141822H.-L.Huang F.-G.Shi L-fuzzy numbers and their properties10.1016/j.ins.2007.10.001Inform. Sci.200817811411151Q.Jin L.-Q. LiOn the embedding of convex spaces in stratified L-convex spaces10.1186/s40064-016-3255-5SpringerPlus20165110 F.JinmingI-fuzzy Alexandrov topologies and specialization orders10.1016/j.fss.2007.05.001Fuzzy Sets and Systems200715823592374T.KubiakOn fuzzy topologies Ph.D. Thesis, Adam Mickiewicz University, Poznan, Poland1985M.LassakOn metric B-convexity for which diameters of any set and its hull are equal Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.197725969975 Y.MaruyamaLattice-valued fuzzy convex geometry RIMS Kokyuroku200916412237C. V.Negoiţă D. A.RalescuApplications of fuzzy sets to systems analysisTranslated from the Romanian, ISR— Interdisciplinary Systems Research, Birkhäuser Verlag, Basel-Stuttgart1975111187 B.Pang F.-G.ShiSubcategories of the category of L-convex spaces10.1016/j.fss.2016.02.014Fuzzy Sets and Systems20173136174B.Pang Y.ZhaoCharacterizations of L-convex spaces10.22111/ijfs.2016.2595Iran. J. Fuzzy Syst.2016135161M. V.RosaA study of fuzzy convexity with special reference to separation propertiesPh.D. Thesis, Cochin University of Science and Technology, Kerala, India1994M. V.RosaOn fuzzy topology fuzzy convexity spaces and fuzzy local convexity10.1016/0165-0114(94)90076-0 Fuzzy Sets and Systems19946297100F.-G.ShiTheory of $$L_\beta$$-nested sets and $$L_\alpha$$-nested sets and its applications(Chinese) Fuzzy Syst. Math.199546572 F.-G.ShiL-fuzzy relations and L-fuzzy subgroups J. Fuzzy Math.20008491499 F.-G.Shi E.-Q.Li The restricted hull operator of M-fuzzifying convex structures10.3233/IFS-151765J. Intell. Fuzzy Syst.201530409421 F.-G.Shi Z.-Y.XiuA new approach to the fuzzification of convex structures10.1155/2014/249183J. Appl. Math.20142014112 V. P.Soltan d-convexity in graphs(Russian) Dokl. Akad. Nauk SSSR1983272535537A. P.ŠostakOn a fuzzy topological structureRend. Circ. Mat. Palermo19851189103 M. L. J. van deVelTheory of convex structuresNorth-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam1993 J. vanMill Supercompactness and Wallman spacesMathematical Centre Tracts, Mathematisch Centrum, Amsterdam1977J. C.Varlet Remarks on distributive latticesBull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.19752311431147G.-J.WangTheory of topological molecular lattices10.1016/0165-0114(92)90301-JFuzzy Sets and Systems199247351376 Z.-Y.Xiu F.-G.ShiM-fuzzifying interval spaces Iran. J. Fuzzy Syst.201714145162M.-S.YingA new approach for fuzzy topology, I10.1016/0165-0114(91)90100-5Fuzzy Sets and Systems199139303321L. A.Zadeh Fuzzy setsInformation and Control19658338353Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.26Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spacesZhangMin
College of Science, China University of Petroleum, Qingdao, Shandong 266580, People’s Republic of China
RuiJie
College of Science, China University of Petroleum, Qingdao, Shandong 266580, People’s Republic of China

In this paper, d-dimensional (dD) quasi-periodically forced nonlinear Schrödinger equation with a general nonlinearity $iu_t - \Delta u +M_\xi u + \varepsilon\phi (t)(u + h(|u| ^2)u) = 0, \quad x\in \mathbb{T}^d,\quad t\in \mathbb{R}$ under periodic boundary conditions is studied, where $$M_\xi$$ is a real Fourier multiplier and $$\varepsilon$$ is a small positive parameter, $$\phi (t)$$ is a real analytic quasi-periodic function in t with frequency vector $$\omega=(\omega_1,\omega_2,...,\omega_m)$$ , and $$h(|u| ^2)$$ is a real analytic function near $$u = 0$$ with $$h(0) = 0$$. It is shown that, under suitable hypothesis on $$\phi (t)$$, there are many quasi-periodic solutions for the above equation via KAM theory.

35Q4135G50Quasi-periodically forcedKAM theorySchrödinger equationquasi-periodic solutions.
D.Bambusi S.GraffiTime quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods10.1007/s002200100426Comm. Math. Phys.2001219465480 M.Berti P.BolleSobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential10.1088/0951-7715/25/9/2579Nonlinearity20122525792613 M.Berti P.BolleQuasi-periodic solutions with Sobolev regularity of NLS on $$\mathbb{T}^d$$ with a multiplicative potential10.4171/JEMS/361 J. Eur. Math. Soc. (JEMS)201315229286J.Bourgain Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE10.1155/S1073792894000516 Internat. Math. Res. Notices19941994121J. BourgainConstruction of periodic solutions of nonlinear wave equations in higher dimension10.1007/BF01902055Geom. Funct. Anal.19955629639 J.Bourgain Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations10.2307/121001Ann. of Math.1998148363439J.BourgainNonlinear Schrödinger equationsHyperbolic equations and frequency interactions, Park City, UT, (1995), IAS/Park City Math. Ser., Amer. Math. Soc., Providence, RI199953157 J.BourgainGreen’s function estimates for lattice Schrödinger operators and applicationsAnnals of Mathematics Studies, Princeton University Press, Princeton, NJ2005 W.Craig C. E.WayneNewton’s method and periodic solutions of nonlinear wave equations10.1002/cpa.3160461102Comm. Pure Appl. Math.19934614091498 H. L.Eliasson S. B.Kuksin On reducibility of Schrödinger equations with quasiperiodic in time potentials10.1007/s00220-008-0683-2Comm. Math. Phys.2009286125135H. L.Eliasson S. B.KuksinKAM for the nonlinear Schrödinger equation10.4007/annals.2010.172.371Ann. of Math.2010172371435J.-S.Geng X.-D.Xu J.-G.You An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation10.1016/j.aim.2011.01.013Adv. Math.201122653615402J.-S.Geng Y.-F.YiQuasi-periodic solutions in a nonlinear Schrödinger equation10.1016/j.jde.2006.07.027J. Differential Equations2007233512542 J.-S.Geng J.-G.YouA KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions10.1016/j.jde.2004.09.013 J. Differential Equations2005209156 J.-S.Geng J.-G.YouA KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces10.1007/s00220-005-1497-0Comm. Math. Phys.2006262343372J.-S.Geng J.-G.YouKAM tori for higher dimensional beam equations with constant potentials10.1088/0951-7715/19/10/007Nonlinearity20061924052423J.-S.Geng J.-G.YouKAM theorem for higher dimensional nonlinear Schrödinger equations10.1007/s10884-013-9296-3 J. Dynam. Differential Equations201325451476S. B.KuksinNearly integrable infinite-dimensional Hamiltonian systemsLecture Notes in Mathematics, Springer- Verlag, Berlin1993S. B.Kuksin J.Pöschel Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation10.2307/2118656Ann. of Math.1996143149179Z.-G.Liang J.-G. YouQuasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity10.1137/S0036141003435011SIAM J. Math. Anal.20053619651990 J.Pöschel A KAM-theorem for some nonlinear partial differential equations Ann. Scuola Norm. Sup. Pisa Cl. Sci.199623119148J.PöschelQuasi-periodic solutions for a nonlinear wave equation10.1007/BF02566420Comment. Math. Helv.199671269296 C.Procesi M.Procesi A KAM algorithm for the resonant non-linear Schrödinger equation10.1016/j.aim.2014.12.004Adv. Math.2015272399470 C. E.WaynePeriodic and quasi-periodic solutions of nonlinear wave equations via KAM theory10.1007/BF02104499Comm. Math. Phys.1990127479528.-X.Xu J.-G.You Persistence of lower-dimensional tori under the first Melnikov’s non-resonance condition10.1016/S0021-7824(01)01221-1 J. Math. Pures Appl.20018010451067 X.-P.YuanQuasi-periodic solutions of completely resonant nonlinear wave equations10.1016/j.jde.2005.12.012J. Differential Equations2006230213274Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.27Determinant and inverse of a Gaussian Fibonacci skew-Hermitian Toeplitz matrixJiangZhaolin
Department of Mathematics, Linyi University, Linyi 276000, P. R. China
SunJixiu
Department of Mathematics, Linyi University, Linyi 276000, P. R. China;School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, P. R. China

In this paper, we consider the determinant and the inverse of the Gaussian Fibonacci skew-Hermitian Toeplitz matrix. We first give the definition of the Gaussian Fibonacci skew-Hermitian Toeplitz matrix. Then we compute the determinant and inverse of the Gaussian Fibonacci skew-Hermitian Toeplitz matrix by constructing the transformation matrices.

15B0515A0915A15Gaussian Fibonacci numberskew-Hermitian Toeplitz matrixdeterminantinverse.
M.Akbulak D.Bozkurt On the norms of Toeplitz matrices involving Fibonacci and Lucas numbersHacet. J. Math. Stat.2008378995D.Bozkurt T.-Y.Tam Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas Numbers10.1016/j.amc.2012.06.039Appl. Math. Comput.2012219544551A.BuckleyOn the solution of certain skew symmetric linear systems10.1137/0714035 SIAM J. Numer. Anal.197714566570R. H.Chan X.-Q.Jin Circulant and skew-circulant preconditioners for skew-Hermitian type Toeplitz systems10.1007/BF01933178BIT199131632646 X.-T.Chen Z.-L.Jiang J.-M.Wang Determinants and inverses of Fibonacci and Lucas skew symmetric Toeplitz matricesBritish J. Math. Comput. Sci.201619121L.DazhengFibonacci-Lucas quasi-cyclic matrices A special tribute to Calvin T. Long, Fibonacci Quart.200240280286U.Grenander G.SzegöToeplitz forms and their applicationsCalifornia Monographs in Mathematical Sciences University of California Press, Berkeley-Los Angeles1958 G.Heining K.RostAlgebraic methods for Toeplitz-like matrices and operators Mathematical Research, Akademie- Verlag, Berlin1984 A. F.HoradamFurther appearence of the Fibonacci sequence Fibonacci Quart.196314142A.Ipek K.ArıOn Hessenberg and pentadiagonal determinants related with Fibonacci and Fibonacci-like numbers10.1016/j.amc.2013.12.071Appl. Math. Comput.2014229433439Z.-L.Jiang Y.-P.Gong Y.Gao Circulant type matrices with the sum and product of Fibonacci and Lucas numbers10.1155/2014/375251Abstr. Appl. Anal.20142014112 Z.-L.Jiang Y.-P.Gong Y.Gao Invertibility and explicit inverses of circulant-type matrices with k-Fibonacci and k-Lucas numbers10.1155/2014/238953 Abstr Appl. Anal.20142014110 X.-Y.Jiang K.-C.HongExact determinants of some special circulant matrices involving four kinds of famous numbers10.1155/2014/273680 Abstr. Appl. Anal.20142014112X.-Y.Jiang K.-C.Hong Explicit inverse matrices of Tribonacci skew circulant type matrices10.1016/j.amc.2015.05.103Appl. Math. Comput.201526893102X.-Y.Jiang K.-C.HongSkew cyclic displacements and inversions of two innovative patterned matrices10.1016/j.amc.2017.03.024Appl. Math. Comput.2017308174184Z.-L.Jiang J.-W.ZhouA note on spectral norms of even-order r-circulant matrices10.1016/j.amc.2014.11.020Appl. Math. Comput.2014250368371 J.Li Z.-L.Jiang F.-L.Lu Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers10.1155/2014/381829Abstr. Appl. Anal.2014201419 L.Liu Z.-L.JiangExplicit form of the inverse matrices of Tribonacci circulant type matrices10.1155/2015/169726Abstr. Appl. Anal.20152015110 M.Merca A note on the determinant of a Toeplitz-Hessenberg matrix10.2478/spma-2013-0003Spec. Matrices201421016 B. N.Mukherjee S. S.MaitiOn some properties of positive definite Toeplitz matrices and their possible applications10.1016/0024-3795(88)90326-6Linear Algebra Appl.1988102211240 L.RodmanPairs of Hermitian and skew-Hermitian quaternionic matrices: canonical forms and their applications10.1016/j.laa.2006.12.017 Linear Algebra Appl.20084299811019 S.-Q.Shen J.-M.Cen Y.Hao On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers10.1016/j.amc.2011.04.072 Appl. Math. Comput.201121797909797 J.-X.Sun Z.-L.Jiang Computing the determinant and inverse of the complex Fibonacci Hermitian Toeplitz matrix10.9734/BJMCS/2016/30398British J. Mathe. Comput. Sci.201619116 Y.-P.Zheng S.-G.Shon Exact determinants and inverses of generalized Lucas skew circulant type matrices10.1016/j.amc.2015.08.021Appl. Math. Comput.2015270105113 Y.-P.Zheng S.-G.Shon J.-Y.KimCyclic displacements and decompositions of inverse matrices for CUPL Toeplitz matrices10.1016/j.jmaa.2017.06.016 J. Math. Anal. Appl.2017455727741J.-W.Zhou Z.-L.JiangThe spectral norms of g-circulant matrices with classical Fibonacci and Lucas numbers entries10.1016/j.amc.2014.02.020 Appl. Math. Comput.2014233582587Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.28Common fixed point theorems in non-Archimedean fuzzy metric-like spaces with applicationsZhaoHaiqing
Department of Mathematics and Physics, Baoding 071003, China
LuYanxia
Department of Mathematics and Physics, Baoding 071003, China
SridaratPhikul
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
SuantaiSuthep
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
ChoYeol Je
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Korea;Center for General Education, China Medical University, Taichung 40402, Taiwan

In this paper, we introduce the new concept called a non-Archimedean fuzzy metric-like space and prove some common fixed point theorems in this space. Our results extend some corresponding ones in the literature. Also, we give some examples to illustrate the main results. Finally, as applications, we consider the existence problem of solutions of integral equations by our main results.

47H0547H0954H25Fuzzy metric spacemetric-like spaceCauchy sequencefixed point theorem.
H.Aydi A.Felhi S.SahmimFixed points of multivalued nonself almost contractions in metric-like spaces10.1007/s40096-015-0156-7 Math. Sci. (Springer)20159103108H.Aydi E.KarapınarFixed point results for generalized $$\alpha-\psi$$-contractions in metric-like spaces and applicationsElectron. J. Differential Equations20152015115 Y. J.Cho M.Grabiec V.RaduOn nonsymmetric topological and probabilistic structuresNova Science Publishers, Inc., New York2006Z.DengFuzzy pseudo-metric spaces10.1016/0022-247X(82)90255-4J. Math. Anal. Appl.1982867495M. A.ErcegMetric spaces in fuzzy set theory10.1016/0022-247X(79)90189-6 J. Math. Anal. Appl.197969205230J.-X. FangOn fixed point theorems in fuzzy metric spaces10.1016/0165-0114(92)90271-5 Fuzzy Sets and Systems199246107113A.George P.VeeramaniOn some results in fuzzy metric spaces10.1016/0165-0114(94)90162-7Fuzzy Sets and Systems199464395399 M. GrabiecFixed points in fuzzy metric spaces10.1016/0165-0114(88)90064-4Fuzzy Sets and Systems198827385389 V.Gregori A.SapenaOn fixed-point theorems in fuzzy metric spaces10.1016/S0165-0114(00)00088-9Fuzzy Sets and Systems2002125245252V. I.IstrăţescuAn introduction to theory of probabilistic metric spaces with applications(Romanian) Ed. , Tehnică Bucureşti1974O.Kaleva S.SeikkalaOn fuzzy metric spaces10.1016/0165-0114(84)90069-1 Fuzzy Sets and Systems198412215229 O.Kramosil J.MichálekFuzzy metrics and statistical metric spacesKybernetika (Prague)197511336344D.MiheţA Banach contraction theorem in fuzzy metric spaces10.1016/S0165-0114(03)00305-1Fuzzy Sets and Systems2004144431439D.Miheţ On fuzzy contractive mappings in fuzzy metric spaces10.1016/j.fss.2006.11.012Fuzzy Sets and Systems2007158915921 D.MiheţA class of contractions in fuzzy metric spaces10.1016/j.fss.2009.09.018 Fuzzy Sets and Systems201016111311137 S. N.Mishra N.Sharma S. L.SinghCommon fixed points of maps on fuzzy metric spaces Internat. J. Math. Math. Sci.199417253258 A. F. Roldán López deHierro M. de laSen Some fixed point theorems in Menger probabilistic metric-like spaces10.1186/s13663-015-0421-3 Fixed Point Theory Appl.20152015116 S.Sharma Common fixed point theorems in fuzzy metric spaces10.1016/S0165-0114(01)00112-9Fuzzy Sets and Systems2002127345352Y.-H.Shen D.Qiu W.ChenFixed point theorems in fuzzy metric spaces10.1016/j.aml.2011.08.002Appl. Math. Lett.201225138141B.Singh M. S.ChauhanCommon fixed points of compatible maps in fuzzy metric spacesFuzzy Sets and Systems2000115471475R.VasukiA common fixed point theorem in a fuzzy metric space10.1016/S0165-0114(96)00342-9Fuzzy Sets and Systems199897395397Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.29Coincidence point results via generalized $$(\psi,\phi)$$-weak contractions in partial ordered $$b$$-metric spaces with applicationSarwarMuhammad
Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan
JamalNoor
Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan
LiYongjin
Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China

In this manuscript, some coincidence point and fixed point results via generalized $$(\psi,\phi)$$-weak contractive condition are established. The presented work explicitly generalize some recent results from the existing literature in the setting of partial order b-metric spaces. An example is provided to show the authenticity of the derived results.

47H1054H25coincidence pointweak compatible mappingincreasing pairs of mapsGeneralized $$(\psi،\phi)$$-weak contractionpartial ordered complete b-metric spaces.
M.Abbas D.Đorić Common fixed point for generalized $$(\psi,\phi)$$-weak contractionsMath. Un. of Nis. Serbia.201010110M.Abbas T.Nazir S.Radenović Common fixed points of four maps in partially ordered metric spaces10.1016/j.aml.2011.03.038 Appl. Math. Lett.20112415201526 A.Aghajani M.Abbas J. R.RoshanCommon fixed point of generalized weak contractive mappings in partially ordered b-metric spaces10.2478/s12175-014-0250-6Math. Slovaca201464941960 R.Allahyari R.Arab A. SholeHaghighiA generalization on weak contractions in partially ordered b-metric spaces and its application to quadratic integral equations10.1186/1029-242X-2014-355 J. Inequal. Appl.20142014115 I.Altun H.Simsek Some fixed point theorems on ordered metric spaces and application10.1155/2010/621469 Fixed Point Theory Appl.20102010117 S.Czerwik Nonlinear set-valued contraction mappings in b-metric spacesAtti Sem. Mat. Fis. Univ. Modena199846263276 D.DorićCommon fixed point for generalized $$(\psi,\phi)$$-weak contractions10.1016/j.aml.2009.08.001Appl. Math. Lett.20092218961900 P. N.Dutta B. S.Choudhury A generalisation of contraction principle in metric spaces10.1155/2008/406368 Fixed Point Theory Appl.20081818 J.Esmaily S. M.Vaezpour B. E.RhoadesCoincidence point theorem for generalized weakly contractions in ordered metric spaces10.1016/j.amc.2012.07.054Appl. Math. Comput.201221915361548 M.Jovanović Z.Kadelburg S.Radenović Common fixed point results in metric-type spaces10.1155/2010/978121Fixed Point Theory Appl.20102010115G.Jungck Compatible mappings and common fixed points10.1155/S0161171286000935Internat. J. Math. Math. Sci.19864771779 G.Jungck Common fixed points for noncontinuous nonself maps on nonmetric spacesFar East J. Math. Sci.19964199215P. P.Murthy K.Tas U. DeviPatelCommon fixed point theorems for generalized $$(\phi,\psi)$$-weak contraction condition in complete metric spaces10.1186/s13660-015-0647-y J. Inequal. Appl.20152015114 H. K.Nashine B.SametFixed point results for mappings satisfying $$(\psi,\phi)$$-weakly contractive condition in partially ordered metric spaces10.1016/j.na.2010.11.024Nonlinear Anal.20117422012209 S.Radenović Z.KadelburgGeneralized weak contractions in partially ordered metric spaces10.1016/j.camwa.2010.07.008 Comput. Math. Appl.20106017761783 A. C. M. Ran M. C. B.ReuringsA fixed point theorem in partially ordered sets and some applications to matrix equations10.1090/S0002-9939-03-07220-4Proc. Amer. Math. Soc.200413214351443 K. P. R.Rao I.Altun K. R. K.Rao N.Srinivasarao A common fixed point theorem for four maps under $$(\psi,\phi)$$ contractive condition of integral type in ordered partial metric spaces10.12785/msl/040107 Math. Sci. Lett.201542531A.Razani V.Parvaneh M.Abbas A common fixed point for generalized $$(\psi,\phi)_{f,g}$$-weak contractions10.1007/s11253-012-0611-7Ukrainian Math. J.20126317561769 J. R.Roshan V.Parvaneh I.Altun Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations10.1016/j.amc.2013.10.043Appl. Math. Comput.2014226725737J. R.Roshan V.Parvaneh S.Radenović M.RajovićSome coincidence point results for generalized $$(\psi,\varphi)$$-weakly contractions in ordered b-metric spaces10.1186/s13663-015-0313-6 Fixed Point Theory Appl.20152015121J. R.Roshan V.Parvaneh S.Sedghi N.ShobkolaeiW.ShatanawiCommon fixed points of almost generalized $$(\psi,\varphi)_s$$- contractive mappings in ordered b-metric spaces10.1186/1687-1812-2013-159 Fixed Point Theory Appl.20132013123 W.Shatanawi B.Samet On $$(\psi,\phi)$$-weakly contractive condition in partially ordered metric spaces10.1016/j.camwa.2011.08.033 Comput. Math. Appl.20116232043214Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.30Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problemsJungJong Soo
Department of Mathematics, Dong-A University, Busan 49315, Korea

In this paper, we introduce two modified hybrid iterative methods (one implicit method and one explicit method) for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality problem for a continuous monotone mapping and the set of fixed points of a continuous pseudocontractive mapping in Hilbert spaces, and show under suitable control conditions that the sequences generated by the proposed iterative methods converge strongly to a common element of three sets, which solves a certain variational inequality. As a direct consequence, we obtain the unique minimum-norm common point of three sets. The results in this paper substantially improve upon, develop and complement the previous well-known results in this area.

49J3049J4047H0947H1047J2047J2547J0549M05Hybrid iterative methodgeneralized mixed equilibrium problemcontinuous monotone mappingcontinuous pseudocontractive mappingvariational inequalityfixed point$$\rho$$-Lipschitzian and $$\eta$$-strongly monotone mappingmetric projection.
R. P.Agarwal D.O’Regan D. R.SahuFixed point theory for Lipschitzian-type mappings with applicationsTopological Fixed Point Theory and Its Applications, Springer, New York2009 E.Blum W.Oettli From optimization and variational inequalities to equilibrium problems Math. Student199463123145L.-C.Ceng S.Al-Homidan Q. H.Ansari J.-C.Yao An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings10.1016/j.cam.2008.03.032J. Comput. Appl. Math.2009223967974 L.-C.Ceng S.-M.Guu J.-C.YaoHybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems10.1186/1687-1812-2012-92Fixed Point Theory Appl.20122012119L.-C.Ceng J.-C.Yao A hybrid iterative scheme for mixed equilibrium problems and fixed point problems10.1016/j.cam.2007.02.022 J. Comput. Appl. Math.2008214186201S.-S.Chang H. W. JosephLee C. K.Chan A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization10.1016/j.na.2008.04.035Nonlinear Anal.20097033073319 S. Y.Cho B. A. BinDehaish X.-L.Qin Weak convergence of a splitting algorithm in Hilbert spaces10.11948/2017027J. Appl. Anal. Comput.20177427438V.Colao G.Marino H.-K.XuAn iterative method for finding common solutions of equilibrium and fixed point problems10.1016/j.jmaa.2008.02.041J. Math. Anal. Appl.2008344340352P. I.Combettes S. A.HirstoagaEquilibrium programming in Hilbert spacesJ. Nonlinear Convex Anal.20056117136S. D.Flåm A. S.AntipinEquilibrium programming using proximal-like algorithms10.1007/BF02614504 Math. Programming1997782941 H.Iiduka W.TakahashiStrong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings10.1016/j.na.2003.07.023 Nonlinear Anal.200561341350 C.Jaiboon P.Kumam A general iterative method for addressing mixed equilibrium problems and optimization problems10.1016/j.na.2010.04.041Nonlinear Anal.20107311801202 J. S.JungStrong convergence of composite iterative methods for equilibrium problems and fixed point problems10.1016/j.amc.2009.03.048 Appl. Math. Comput.2009213498505J. S.JungSome results on a general iterative method for k-strictly pseudo-contractive mappings10.1186/1687-1812-2011-24Fixed Point Theory Appl.20112011111 J. S.Jung Iterative methods for mixed equilibrium problems and strictly pseudocontractive mappings10.1186/1687-1812-2012-184 Fixed point Theory Appl.20122012119 J. S.JungWeak convergence theorems for strictly pseudocontractive mappings and generalized mixed equilibrium problems10.1155/2012/384108J. Appl. Math.20122012118 J. S.JungWeak convergence theorems for generalized mixed equilibrium problems, monotone mappings and pseudocontractive mappings10.4134/JKMS.2015.52.6.1179J. Korean Math. Soc.20155211791194 P.Katchang T.Jitpeera P.KumamStrong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method10.1016/j.nahs.2010.07.001Nonlinear Anal. Hybrid Syst.20104838852 S.-T.LvSome results on a two-step iterative algorithm in Hilbert spaces J. Nonlinear Funct. Anal.20152015110 G. J.MintyOn the generalization of a direct method of the calculus of variations10.1090/S0002-9904-1967-11732-4Bull. Amer. Math. Soc.196773315321 A.Moudafi Weak convergence theorems for nonexpansive mappings and equilibrium problems J. Nonlinear Convex Anal.200893743 J.-W.Peng J.-C.YaoA new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems10.11650/twjm/1500405033Taiwanese J. Math.20081214011432 S.Plubtieng R.PunpaengA general iterative method for equilibrium problems and fixed point problems in Hilbert spaces10.1016/j.jmaa.2007.02.044 J. Math. Anal. Appl.2007336445468 S.Plubtieng R.PunpaengA new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings10.1016/j.amc.2007.07.075Appl. Math. Comput.2008197548558 Y.-F.Su M.-J.Shang X.-L.Qin An iterative method of solution for equilibrium and optimization problems10.1016/j.na.2007.08.045 Nonlinear Anal.20086927092719 T.SuzukiStrong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals10.1016/j.jmaa.2004.11.017 J. Math. Anal. Appl.2005305227239A.Tada W.Takahashi Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem10.1007/s10957-007-9187-zJ. Optim. Theory Appl.2007133359370 S.Takahashi W.Takahashi Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces10.1016/j.jmaa.2006.08.036J. Math. Anal. Appl.2007331506515 S.Takahashi W.TakahashiStrong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space10.1016/j.na.2008.02.042Nonlinear Anal.20086910251033W.Takahashi M.ToyodaWeak convergence theorems for nonexpansive mappings and monotone mappings10.1023/A:1025407607560 J. Optim. Theory Appl.2003118417428J.-F.Tang S.-S.Chang J.DongSplit equality fixed point problem for two quasi-asymptotically pseudocontractive mappings J. Nonlinear Funct. Anal.20172017115 S.Wang C.-S.Hu G.-Q.ChaiStrong convergence of a new composite iterative method for equilibrium problems and fixed point problems10.1016/j.amc.2009.11.036 Appl. Math. Comput.201021538913898U.Witthayarat A. A. N.Abdou Y. J.ChoShrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces10.1186/s13663-015-0448-5Fixed Point Theory Appl.20152015114 H. K.XuAn iterative approach to quadratic optimization10.1023/A:1023073621589J. Optim. Theory Appl.2003116659678 I.YamadaThe hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings10.1016/S1570-579X(01)80028-8 Inherently parallel algorithms in feasibility and optimization and their applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam20018473504 Y.-H.Yao Y. J.Cho R.-D.ChenAn iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems10.1016/j.na.2009.01.236 Nonlinear Anal.20097133633373Y.-H.Yao Y.-C.Liou J.-C.YaoConvergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings10.1155/2007/64363 Fixed Point Theory Appl.20072007112 Y.-H.Yao M. A.Noor Y.-C.LiouOn iterative methods for equilibrium problems10.1016/j.na.2007.12.021 Nonlinear Anal.200970497509 Y.-H.Yao M. A.Noor S.Zainab Y.-C.LiouMixed equilibrium problems and optimization problems10.1016/j.jmaa.2008.12.055J. Math. Anal. Appl.2009354319329 Y.-H.Yao Z.-S.Yao A. A. N.Abdou Y. J.ChoSelf-adaptive algorithms for proximal split feasibility problems and strong convergence analysis10.1186/s13663-015-0462-7Fixed Point Theory Appl.20152015113 H.ZegeyeAn iterative approximation method for a common fixed point of two pseudocontractive mappings10.5402/2011/621901 ISRN Math. Anal.20112011114S.-S.Zhang Generalized mixed equilibrium problem in Banach spaces10.1007/s10483-009-0904-6 Appl. Math. Mech. (English Ed.)20093011051112Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.31Analysis of an SVEIR model with age-dependence vaccination, latency and relapseWangJinliang
School of Mathematical Science, Heilongjiang University, Harbin 150080, China
DongXiu
School of Mathematical Science, Heilongjiang University, Harbin 150080, China
SunHongquan
School of Mathematical Science, Heilongjiang University, Harbin 150080, China

In this paper, we propose an epidemic model with age-dependence vaccination, latency and relapse. We derive the positivity and boundedness of solutions and find the basic reproduction number. Asymptotic smoothness, the existence of global compact attractor and uniform persistence of the model are investigated. By constructing Lyapunov functionals, we establish global stability of the equilibria in a threshold type.

34D3092D30Vaccination agelatency agerelapse ageglobal stabilityLyapunov function.
C. J.Browne S. S.PilyuginGlobal analysis of age-structured within-host virus model 10.3934/dcdsb.2013.18.1999Discrete Contin. Dyn. Syst. Ser. B20131819992017Y.-M.Chen S.-F.Zou J.-Y.YangGlobal analysis of an SIR epidemic model with infection age and saturated incidence10.1016/j.nonrwa.2015.11.001Nonlinear Anal. Real World Appl.2016301631 R. D.Demasse A.Ducrot An age-structured within-host model for multistrain malaria infections10.1137/120890351SIAM J. Appl. Math.201373572593X.-C.Duan S.-L.Yuan X.-Z.LiGlobal stability of an SVIR model with age of vaccination10.1016/j.amc.2013.10.073Appl. Math. Comput.2014226528540 W. J.Edmunds G. F.Medley D. J.Nokes A. J.Hall H. C.Whittle The influence of age on the development of the hepatitis B carrier state10.1098/rspb.1993.0102Proc. R. Soc. Lond. B Biol. Sci.1993253197201 D.Ganem A. M.PrinceHepatitis B virus infection—natural history and clinical consequences10.1056/NEJMra031087N. Engl. J. Med.200435011181129 J. K.Hale Functional differential equationsApplied Mathematical Sciences, Springer-Verlag New York, New York- Heidelberg1971J. K.HaleAsymptotic behavior of dissipative systems Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI1988J. K.Hale P.WaltmanPersistence in infinite-dimensional systems10.1137/0520025 SIAM J. Math. Anal.198920388395 M. IannelliMathematical theory of age-structured population dynamicsAppl. Math. Monogr. C.N.R., Giardini Editori e Stampatori in Pisa1995L.-L.Liu J.-L.Wang X.-N.LiuGlobal stability of an SEIR epidemic model with age-dependent latency and relapse10.1016/j.nonrwa.2015.01.001Nonlinear Anal. Real World Appl.2015241835P.MagalCompact attractors for time-periodic age-structured population modelsElectron. J. Differential Equations20012001135 P.Magal C. C.McCluskeyTwo-group infection age model including an application to nosocomial infection10.1137/120882056SIAM J. Appl. Math.20137310581095 P.Magal C. C.McCluskey G. F.Webb Lyapunov functional and global asymptotic stability for an infection-age model10.1080/00036810903208122 Appl. Anal.20108911091140P.Magal X.-Q.Zhao Global attractors and steady states for uniformly persistent dynamical systems10.1137/S0036141003439173SIAM J. Math. Anal.200537251275 C. C.McCluskeyGlobal stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes10.3934/mbe.2012.9.819Math. Biosci. Eng.20129819841 L.-L.Rong Z.-L.Feng A. S.Perelson Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy10.1137/060663945SIAM J. Appl. Math.200767731756 H. R.ThiemeSemiflows generated by Lipschitz perturbations of non-densely defined operators Differential Integral Equations1990310351066 H. R.Thieme Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators10.1016/j.jde.2011.01.007J. Differential Equations201125037723801P. van denDriessche L.Wang X.-F.ZouModeling diseases with latency and relapse10.3934/mbe.2007.4.205Math. Biosci. Eng.20074205219P. van denDriessche X.-F.Zou Modeling relapse in infectious diseases10.1016/j.mbs.2006.09.017Math. Biosci.200720789103J. A.Walker Dynamical systems and evolution equations Theory and applications, Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York-London1980 J.-L.Wang R.Zhang T.KuniyaGlobal dynamics for a class of age-infection HIV models with nonlinear infection rate10.1016/j.jmaa.2015.06.040J. Math. Anal. Appl.2015432289313 J.-L.Wang R.Zhang T.KuniyaThe stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes10.1080/17513758.2015.1006696 J. Biol. Dyn.2015973101J.-L.Wang R.Zhang T.KuniyaA note on dynamics of an age-of-infection cholera model10.3934/mbe.2016.13.227 Math. Biosci. Eng.201613227247 J.-L.Wang R.Zhang T.KuniyaThe dynamics of an SVIR epidemiological model with infection age10.1093/imamat/hxv039IMA J. Appl. Math.201681321343G. F.WebbTheory of nonlinear age-dependent population dynamicsMonographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York1985J.-X.Yang Z.-P.Qiu X.-Z.Li Global stability of an age-structured cholera model10.3934/mbe.2014.11.641Math. Biosci. Eng.201411641665Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.32Two type quasi-contractions on quasi metric spaces and some fixed point resultsŞimşekHakan
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey
AltunIshak
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey;College of Science, King Saud University, Riyadh, Saudi Arabia

In this paper, we introduce new concepts of quasi-contractions of type (A) and of type (B) in a quasi metric space and we present the differences between of them. Then we present some fixed point results. In the light of the theorems it is shown that, although the Hausdorffness condition of quasi metric space is needed for the mapping of quasi contraction of type (A), it is not necessary to guarantee the existence of fixed point for the mapping of quasi contraction of type (B).

54H2547H10Quasi metric spaceleft K-Cauchy sequenceleft K-completenessfixed pointquasi contraction.
E.Alemany S.Romaguera On right K-sequentially complete quasi-metric spaces10.1023/A:1006559507624 Acta Math. Hungar.199775267278I.Altun M.Olgun G.MınakClassification of completeness of quasi metric space and some new fixed point resultsNonlinear Funct. Anal. Appl.201722371384 L.Ćirić Fixed Point Theory Contraction Mapping Principle Faculty of Mechanical Enginearing, University of Belgrade, Beograd2003S.CobzaşFunctional analysis in asymmetric normed spacesBirkhuser-Springer, Basel2013H.Dağ G.Mınak I.AltunSome fixed point results for multivalued F-contractions on quasi metric spaces10.1007/s13398-016-0285-3Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM2017111177187 H.Dağ S.Romaguera P. TiradoThe Banach contraction principle in quasi-metric spaces revisitedProceeding of the Workshop on Applied Topological Structures WATS’1520152531 J. C.KellyBitopological spaces10.1112/plms/s3-13.1.71 Proc. London Math. Soc.1963137189H.-P. A.KünziNonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology10.1007/978-94-017-0470-0_3 Handbook of the history of general topology (Springer)20013853968 I. L.Reilly P. V.Subrahmanyam M. K.VamanamurthyCauchy sequences in quasi-pseudo-metric spaces10.1007/BF01301400 Monatsh. Math.198293127140S.Romaguera A. GutiérrezA note on Cauchy sequences in quasi-pseudometric spacesGlasnik Mat.198621191200 S.Romaguera Left K-completeness in quasi-metric spaces10.1002/mana.19921570103 Math. Nachr.19921571523 M.Sarwar M. U.Rahman G.AliSome fixed point results in dislocated quasi metric (dq-metric) spaces10.1186/1029-242X-2014-278 J. Inequal. Appl.20142014111Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.33Spatio-temporal chaos in duopoly gamesLiRisong
School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China
ZhaoYu
School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China
LuTianxiu
Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China;Artificial Intelligence Key Laboratory of Sichuan Province, Zigong, Sichuan, 643000, P. R. China
JiangRu
School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China
WangHongqing
School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China
LiangHaihua
School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China

Suppose that $$G$$ and $$H$$ are two given closed subintervals of $$R$$, and that $$q : G \rightarrow H$$ and $$p : H \rightarrow G$$ are continuous maps. Let $$\Gamma (s, t) = (p(t), q(s))$$ be a Cournot map over the space $$G \times H$$. In this paper, we study spatio-temporal chaos of such a Cournot map. In particular, it is shown that if $$p$$ and $$q$$ are onto maps, then the following are equivalent: 1) $$\Gamma$$ is spatio-temporally chaotic; 2) $$\Gamma^2\mid_{\Lambda_1}$$ is spatio-temporally chaotic; 3) $$\Gamma^2\mid_{\Lambda_2}$$ is spatio-temporally chaotic; 4) $$\Gamma\mid_{\Lambda_1\cup\Lambda_2}$$ is spatio-temporally chaotic. Moreover, it is proved that if $$p$$ and $$q$$ are onto maps, then $$p \circ q$$ is spatio-temporally chaotic if and only if so is $$q \circ p$$. Also, we give two examples which show that for the above results, it is necessary to assume that $$p$$ and $$q$$ are onto maps.

37D4554H2037B4026A1828D20Spatio-temporal chaosLi-Yorke sensitivityduopoly game.
E.Akin S.KolyadaLi-Yorke sensitivity10.1088/0951-7715/16/4/313Nonlinearity20031614211433G. I.Bischi C.Mammana L.Gardini Multistability and cyclic attractors in duopoly games10.1016/S0960-0779(98)00130-1Chaos Solitons Fractals200011543564 J. S.CánovasChaos in duopoly games Nonlinear Stud.2000797104 J. S.Cánovas M. RuízMarín Chaos on MPE-sets of duopoly games10.1016/S0960-0779(03)00156-5Dedicated to our teacher, mentor and friend, Nobel laureate, Ilya Prigogine, Chaos Solitons Fractals200419179183 J. S. CánovasPeña G. SolerLópez M. RuizMarín Distributional chaos of Cournot maps10.1515/ans-2001-0203Adv. Nonlinear Stud.200117987R. A.Dana L.MontrucchioDynamic complexity in duopoly games10.1016/0022-0531(86)90006-2J. Econom. Theory1986444056H.KatoEverywhere chaotic homeomorphisms on manifolds and k-dimensional Menger manifolds10.1016/0166-8641(96)00008-9Topology Appl.199672117 M.KopelSimple and complex adjustment dynamics in Cournot duopoly models10.1016/S0960-0779(96)00070-7 Complex dynamics in economic and social systems, Umea, (1995), Chaos Solitons Fractals1996720312048R.-S.LiA note on distributional chaos of periodically adsorbing systems (Chinese) J. Systems Sci. Math. Sci.201232237243 R.-S.Li A note on stronger forms of sensitivity for dynamical systems10.1016/j.chaos.2012.02.003Chaos Solitons Fractals201245753758 R.-S.Li H.-Q.Wang Y.Zhao Kato’s chaos in duopoly games10.1016/j.chaos.2016.01.006 Chaos Solitons Fractals2016846972 T. Y.Li J. A.YorkePeriod three implies chaos10.2307/2318254Amer. Math. Monthly197282985992T.-X.Lu P.-Y.ZhuFurther discussion on chaos in duopoly games10.1016/j.chaos.2013.03.012Chaos Solitons Fractals2013524548 P.Oprocha P.WilczyńskiShift spaces and distributional chaos10.1016/j.chaos.2005.09.069 Chaos Solitons Fractals200731347355 R.PikulaOn some notions of chaos in dimension zero10.4064%2Fcm107-2-1Colloq. Math.2007107167177 T.PuuChaos in duopoly pricing Chaos Solitons Fractals19911573581T.Puu I.SushkoOligopoly and complex dynamicsSpringer, New York2002D.RandExotic phenomena in games and duopoly models10.1016/0304-4068(78)90022-8J. Math. Econom.19785173184D.Ruelle F.TakensOn the nature of turbulence10.1007/BF01646553Comm. Math. Phys.197120167192 B.Schweizer J.SmítalMeasures of chaos and a spectral decomposition of dynamical systems on the interval10.2307/2154504 Trans. Amer. Math. Soc.1994344737754 J.Smítal M.Štefánková Distributional chaos for triangular maps10.1016/j.chaos.2003.12.105Chaos Solitons Fractals20042111251128 L.-D.Wang G.-F.Huang S.-M.HuanDistributional chaos in a sequence10.1016/j.na.2006.09.005 Nonlinear Anal.20076721312136 X.-X.WuChaos of transformations induced onto the space of probability measures10.1142/S0218127416502278Internat. J. Bifur. Chaos Appl. Sci. Engrg.201626112 X.-X.Wu A remark on topological sequence entropy10.1142/S0218127417501073 Internat. J. Bifur. Chaos Appl. Sci. Engrg.20172717X.-X.Wu G.-R.ChenSensitivity and transitivity of fuzzified dynamical systems10.1016/j.ins.2017.02.042Inform. Sci.20173961423X.-X.Wu P.Oprocha G.-R.ChenOn various definitions of shadowing with average error in tracing10.1088/0951-7715/29/7/1942Nonlinearity20162919421972X.-X.Wu J.-J.WangA remark on accessibility10.1016/j.chaos.2016.05.015Chaos Solitons Fractals201691115117X.-X.Wu X.WangOn the iteration properties of large deviations theorem10.1142/S0218127416500541 Internat. J. Bifur. Chaos Appl. Sci. Engrg.20162616X.-X.Wu J.-J.Wang G.-R.Chen F-sensitivity and multi-sensitivity of hyperspatial dynamical systems10.1016/j.jmaa.2015.04.009J. Math. Anal. Appl.20154291626X.-X.Wu X.Wang G.-R.ChenOn the large deviations of weaker types10.1142/S0218127417501279 Internat. J. Bifur. Chaos Appl. Sci. Engrg.201727112X.-X.Wu L.-D.Wang G.-R.ChenWeighted backward shift operators with invariant distributionally scrambled subsets10.1215/20088752-3802705Ann. Funct. Anal.20178199210X.-X.Wu L.-D.Wang J.-H.LiangThe Chain Properties and Average Shadowing Property of Iterated Function Systems10.1007/s12346-016-0220-1 Qual. Theory Dyn. Syst.2016201619 X.-X.Wu L.-D.Wang J.-H.Liang The chain properties and Li-Yorke sensitivity of zadehs extension on the space of upper semi-continuous fuzzy sets Iran. J. Fuzzy Syst.AcceptedJournal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.34Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spacesWangLiben
Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China;Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China
ZhangXingyong
Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China
FangHui
Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China;Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system $\begin{cases} -div(\phi_1(|\nabla u|)\nabla u)= F_u(x,u,v),\,\,\,\,\, \texttt{in} \Omega,\\ -div(\phi_2(|\nabla v|)\nabla v)= F_v(x,u,v),\,\,\,\,\, \texttt{in} \Omega,\\ u=v=0,\,\,\,\,\, \texttt{on} \partial \Omega. \end{cases}$ where $$\Omega$$ is a bounded domain in $$R^N(N \geq 2)$$ with smooth boundary $$\partial\Omega$$ , functions $$\phi_i(t)t (i = 1, 2)$$ are increasing homeomorphisms from $$R^+$$ onto $$R^+$$. When $$F$$ satisfies some $$(\phi_1,\phi_2)$$-superlinear and subcritical growth conditions at infinity, by using the mountain pass theorem we obtain that system has a nontrivial solution, and when $$F$$ satisfies an additional symmetric condition, by using the symmetric mountain pass theorem, we obtain that system has infinitely many solutions. Some of our results extend and improve those corresponding results in Carvalho et al. [M. L. M. Carvalho, J. V. A. Goncalves, E. D. da Silva, J. Math. Anal. Appl., 426 (2015), 466–483].

35J2035J5035J55Orlicz-Sobolev spacesmountain pass theoremsymmetric mountain theorem.
R. A.Adams J. F.FournierSobolev spacesSecond edition, Pure and Applied Mathematics (Amsterdam), Elsevier/ Academic Press, Amsterdam2003K.Adriouch A. ElHamidiThe Nehari manifold for systems of nonlinear elliptic equations10.1016/j.na.2005.06.003 Nonlinear Anal.20066421492167 G. A.Afrouzi S.HeidarkhaniExistence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $$(p_1, . . . , p_n)$$-Laplacian10.1016/j.na.2007.11.038Nonlinear Anal.200970135143C. O.Alves G. M.Figueiredo J. A.SantosStrauss and Lions type results for a class of Orlicz-Sobolev spaces and applications10.12775/TMNA.2014.055 Topol. Methods Nonlinear Anal.201444435456G.AnelloOn the multiplicity of critical points for parameterized functionals on reflexive Banach spaces10.4064%2Fsm213-1-4Stud. Math.20122134960 P.Bartolo V.Benci D.FortunatoAbstract critical point theorems and applications to some nonlinear problems with ”strong” resonance at infinity10.1016/0362-546X(83)90115-3 Nonlinear Anal.198379811012L.Boccardo D. Guedes deFigueiredoSome remarks on a system of quasilinear elliptic equations10.1007/s00030-002-8130-0NoDEA Nonlinear Differential Equations Appl.20029309323 G.Bonanno G. MolicaBisci D.O’ReganInfinitely many weak solutions for a class of quasilinear elliptic systems10.1016/j.mcm.2010.02.004Math. Comput. Modelling201052152160 G.Bonanno G. MolicaBisci V. D. RădulescuQuasilinear elliptic non-homogeneous Dirichlet problems through Orlicz- Sobolev spaces10.1016/j.na.2011.12.016Nonlinear Anal.20127544414456Y.Bozhkov E. MitidieriExistence of multiple solutions for quasilinear systems via fibering method10.1016/S0022-0396(02)00112-2 J. Differential Equations2003190239267 F.Cammaroto L.Vilasi Multiple solutions for a nonhomogeneous Dirichlet problem in Orlicz-Sobolev spaces10.1016/j.amc.2012.05.039Appl. Math. Comput.20122181151811527 M. L. M.Carvalho J. V. A.Goncalves E. D. daSilva On quasilinear elliptic problems without the Ambrosetti-Rabinowitz condition10.1016/j.jmaa.2015.01.023 J. Math. Anal. Appl.2015426466483 N. T.Chung H. Q.ToanOn a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz-Sobolev spaces10.1016/j.amc.2013.02.011 Appl. Math. Comput.201321978207829P.Clément M.García-Huidobro R.Manásevich K.Schmitt Mountain pass type solutions for quasilinear elliptic equations10.1007/s005260050002Calc. Var. Partial Differential Equations2000113362P. DeNápoli M. C.MarianiMountain pass solutions to equations of p-Laplacian type10.1016/S0362-546X(03)00105-6 Nonlinear Anal.20035412051219 A. ElKhalil M.Ouanan A.TouzaniExistence and regularity of positive solutions for an elliptic systemProceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., Southwest Texas State Univ., San Marcos, TX20029171182 F.Fang Z. TanExistence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting10.1016%2Fj.jmaa.2011.11.078J. Math. Anal. Appl.2012389420428N.Fukagai M.Ito K.Narukawa Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $$R^N$$10.1619/fesi.49.235Funkcial. Ekvac.200649235267N.Fukagai M.Ito K.NarukawaQuasilinear elliptic equations with slowly growing principal part and critical Orlicz- Sobolev nonlinear term10.1017/S0308210507000765Proc. Roy. Soc. Edinburgh Sect. A200913973106N.Fukagai K.Narukawa On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems10.1007/s10231-006-0018-xAnn. Mat. Pura Appl.2007186539564 M.García-Huidobro V. K.Le R.Manásevich K.SchmittOn principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting10.1007/s000300050073 NoDEA Nonlinear Differential Equations Appl.19996207225 J. P.Gossez Orlicz-Sobolev spaces and nonlinear elliptic boundary value problemsNonlinear analysis, function spaces and applications, Proc. Spring School, Horni Bradlo, (1978), Teubner, Leipzig19795994 J.Huentutripay R.ManásevichNonlinear eigenvalues for a quasilinear elliptic system in Orlicz-Sobolev spaces10.1007/s10884-006-9049-7J. Dynam. Differential Equations200618901929 V. K.LeSome existence results and properties of solutions in quasilinear variational inequalities with general growths10.1007/s12591-009-0025-7Differ. Equ. Dyn. Syst.200917343364 G. M.Lieberman Boundary regularity for solutions of degenerate elliptic equations10.1016/0362-546X(88)90053-3Nonlinear Anal.19881212031219G. M.LiebermanThe natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations10.1080/03605309108820761Comm. Partial Differential Equations199116311361J.-J.Liu X.-Y.ShiExistence of three solutions for a class of quasilinear elliptic systems involving the (p(x), q(x))-Laplacian10.1016/j.na.2008.10.094Nonlinear Anal.200971550557M.Mihăilescu D.Repovš Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces10.1016/j.amc.2011.01.050 Appl. Math. Comput.201121766246632P.Pucci J.SerrinThe strong maximum principle revisited10.1016/j.jde.2003.05.001J. Differential Equations2004196166 P. H.RabinowitzMinimax methods in critical point theory with applications to differential equationsCBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI1986 M. M.Rao Z. D.RenApplications of Orlicz spaces Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York2002B.Ricceri A further refinement of a three critical points theorem10.1016/j.na.2011.07.064 Nonlinear Anal.20117474467454 J. A.SantosMultiplicity of solutions for quasilinear equations involving critical Orlicz-Sobolev nonlinear termsElectron. J. Differential Equations20132013113 L.-B.Wang X.-Y.Zhang H.FangMultiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces10.11650/tjm/7887Taiwanese J. Math.2017132 T.-F.WuThe Nehari manifold for a semilinear elliptic system involving sign-changing weight functions10.1016/j.na.2007.01.004Nonlinear Anal.20086817331745 F.-L.Xia G.-X.WangExistence of solution for a class of elliptic systems J. Hunan Agric. Univ. Nat. Sci.200733362366 J. F.ZhaoStructure theory of Banach spaces(Chinese) Wuhan Univ. Press, Wuhan1991Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.35On multi-valued weak quasi-contractions in b-metric spacesHussainNawab
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
MitrovićZoran D.
Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina

We introduce some generalizations of the contractions for multi-valued mappings and establish some fixed point theorems for multi-valued mappings in b-metric spaces. Our results generalize and extend several known results in b-metric and metric spaces. Some examples are included which illustrate the cases when the new results can be applied while the old ones cannot.

47H10Fixed pointsb-metric spaceset-valued mapping.
M.Abbas N.Hussain B. E.Rhoades Coincidence point theorems for multivalued f-weak contraction mappings and applications10.1007/s13398-011-0036-4 Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM2011105261272A.Amini-HarandiFixed point theory for set-valued quasi-contraction maps in metric spaces10.1016/j.aml.2011.04.033Appl. Math. Lett.20112417911794H.Aydi M. F.Bota E.Karapınar S.Mitrović A fixed point theorem for set-valued quasi-contractions in b-metric spaces10.1186/1687-1812-2012-88Fixed Point Theory Appl.2012201218 I. A.BakhtinThe contraction mapping principle in almost metric space(Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk1989302637M.Berinde V.BerindeOn a general class of multi-valued weakly Picard mappings10.1016/j.jmaa.2006.03.016 J. Math. Anal. Appl.2007326772782S. K.ChatterjeaFixed-point theoremsC. R. Acad. Bulgare Sci.197225727730L. B.ĆirićGeneralized contractions and fixed-point theorems Publ. Inst. Math. (Beograd) (N.S.)1971121926 L. B.Ćirić A generalization of Banach’s contraction principle10.1090/S0002-9939-1974-0356011-2 Proc. Amer. Math. Soc.197445267273 M.Cosentino P.Salimi P.VetroFixed point results on metric-type spaces10.1016/S0252-9602(14)60082-5Acta Math. Sci. Ser. B Engl. Ed.20143412371253S.CzerwikContraction mappings in b-metric spacesActa Math. Inform. Univ. Ostraviensis19931511S.Czerwik Nonlinear set-valued contraction mappings in b-metric spacesAtti Sem. Mat. Fis. Univ. Modena199846263276 N.Hussain A.Amini-Harandi J. Y.ChoApproximate endpoints for set-valued contractions in metric spaces10.1155/2010/614867Fixed Point Theory Appl.20102010113N.Hussain D.Dorić Z.Kadelburg S.RadenovićSuzuki-type fixed point results in metric type spaces10.1186/1687-1812-2012-126 Fixed Point Theory Appl.20122012112 N.Hussain V.Parvaneh J. R.Roshan Z.Kadelburg Fixed points of cyclic weakly $$( \psi,\varphi, L,A, B)$$-contractive mappings in ordered b-metric spaces with applications10.1186/1687-1812-2013-256Fixed Point Theory Appl.20132013118 N.Hussain R.Saadati R. P.Agrawal On the topology and wt-distance on metric type spaces10.1186/1687-1812-2014-88 Fixed Point Theory Appl.20142014114N.Hussain M. H.Shah KKM mappings in cone b-metric spaces10.1016/j.camwa.2011.06.004 Comput. Math. Appl.20116216771684 R.KannanSome results on fixed points Bull. Calcutta Math. Soc.1968607176M. A.KhamsiRemarks on cone metric spaces and fixed point theorems of contractive mappings10.1155/2010/315398 Fixed Point Theory Appl.2010201017 M. A.Khamsi N.HussainKKM mappings in metric type spaces10.1016/j.na.2010.06.084Nonlinear Anal.20107331233129 R.Miculescu A.Mihail New fixed point theorems for set-valued contractions in b-metric spaces10.1007/s11784-016-0400-2J. Fixed Point Theory Appl.20171921532163 S. B.NadlerJr.Multi-valued contraction mappings Pacific J. Math.196930475488 S.Reich Some remarks concerning contraction mappings10.4153/CMB-1971-024-9Canad. Math. Bull.197114121124 J. R.Roshan V.Parvaneh I.AltunSome coincidence point results in ordered b-metric spaces and applications in a system of integral equations10.1016/j.amc.2013.10.043Appl. Math. Comput.2014226725737J. R.Roshan V.Parvaneh S.Radenović MRajović Some coincidence point results for generalized ($$\psi,\varphi$$)-weakly contractions in ordered b-metric spaces10.1186/s13663-015-0313-6Fixed Point Theory Appl.20152015121J. R.Roshan V.Parvaneh S.Sedghi N.ShobkolaeiW.ShatanawiCommon fixed points of almost generalized $$(\psi,\varphi)_s$$- contractive mappings in ordered b-metric spaces10.1186/1687-1812-2013-159 Fixed Point Theory Appl.20132013123S.Shukla S.Radenović C.Vetro Set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces10.1155/2014/652925 Int. J. Math. Math. Sci.2014201419Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.36$$L^{2}(\mathbb{R}^{n})$$ estimate of the solution to the Navier-Stokes equations with linearly growth initial dataYangMinghua
School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330032, P. R. China

In this article, we consider the incompressible Navier-Stokes equations with linearly growing initial data $$U_0 := u_0(x)-Mx$$. Here $$M$$ is an $$n \times n$$ matrix, $$trM = 0, M^2$$ is symmetric and $$u_0 \in L^{2}(\mathbb{R}^{n}) \cap L^{n}(\mathbb{R}^{n})$$. Under these conditions, we consider $$v(t) := u(t) - e^{-tA}u_0$$, where $$u(x) := U(x) - Mx$$ and $$U(x)$$ is the mild solution of the incompressible Navier-Stokes equations with linearly growing initial data. We shall show that $$D^\beta v(t)$$ on the $$L^{2}(\mathbb{R}^{n})$$ norm like $$t^{\frac{-|\beta|-1}{2}-\frac{n}{4}}$$ for all $$|\beta|\geq 0$$. Navier-Stokes equations, linearly growing data, Ornstein-Uhlenbeck operators, $$L^{2}(\mathbb{R}^{n})$$ estimates.

35Q3035B4076A1535B65Navier-Stokes equationslinearly growing dataOrnstein-Uhlenbeck operators
G.-P.Gao C.Cattani X.-J.YangAbout local fractional three-dimensional compressible Navier-Stokes equations in cantortype cylindrical co-ordinate systemTherm. Sci.2016201847Y.Giga O.SawadaOn regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problemNonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, Kluwer Acad. Publ., Dordrecht20031,2549562 M.Hieber O.Sawada The Navier-Stokes equations in $$R^n$$ with linearly growing initial data10.1007/s00205-004-0347-0Arch. Ration. Mech. Anal.2005175269285 T. KatoStrong $$L^p$$-solutions of the Navier-Stokes equation in $$R^m$$, with applications to weak solutions10.1007/BF01174182 Math. Z.1984187471480 G.Metafune D.Pallara E.PriolaSpectrum of Ornstein-Uhlenbeck operators in $$L^p$$ spaces with respect to invariant measures10.1006/jfan.2002.3978J. Funct. Anal.20021964060G.Metafune J.Prüss A.Rhandi R.SchnaubeltThe domain of the Ornstein-Uhlenbeck operator on an $$L^p$$-space with invariant measureAnn. Sc. Norm. Super. Pisa Cl. Sci.20021471485 T.Miyakawa M. E.SchonbekOn optimal decay rates for weak solutions to the Navier-Stokes equations in $$R^n$$Proceedings of Partial Differential Equations and Applications, Olomouc, (1999), Math. Bohem.2001126443455 O.SawadaThe Navier-Stokes flow with linearly growing initial velocity in the whole space10.5269/bspm.v22i2.7484Bol. Soc. Parana. Mat.2004227596 O.Sawada Y.TaniuchiOn the Boussinesq flow with nondecaying initial data10.1619/fesi.47.225Funkcial. Ekvac.200447225250K.Wang S.-Y.Liu Analytical study of time-fractional Navier-Stokes equation by using transform methods10.1186/s13662-016-0783-9 Adv. Difference Equ.20162016112 X.-J.Yang D.Baleanu H. M.SrivastavaLocal fractional integral transforms and their applications Elsevier/Academic Press, Amsterdam2016 X.-J.Yang Z.-Z.Zhang H. M.SrivastavaSome new applications for heat and fluid flows via fractional derivatives without singular kernel10.2298/TSCI16S3833YTherm. Sci.2016201833Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.37The iterative methods with higher order convergence for solving a system of nonlinear equationsChenZhongyuan
Research Center for Science Technology and Society, Fuzhou University of International Studies and Trade, Fuzhou 350202, P. R. China
QiuXiaofang
Research Center for Science Technology and Society, Fuzhou University of International Studies and Trade, Fuzhou 350202, P. R. China
LinSongbin
Research Center for Science Technology and Society, Fuzhou University of International Studies and Trade, Fuzhou 350202, P. R. China
ChenBaoguo
Research Center for Science Technology and Society, Fuzhou University of International Studies and Trade, Fuzhou 350202, P. R. China

In this paper, two variants of iterative methods with higher order convergence are developed in order to solve a system of nonlinear equations. It is proved that these two new methods have cubic convergence. Some numerical examples are given to show the efficiency and the performance of the new iterative methods, which confirm the good theoretical properties of the approach.

65H10System of nonlinear equationsiterative methodshigher convergence ratenumerical examples.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China;Department of Mathematics, Dalian Minzu University, Dalian, 116600, People’s Republic of China
ZhaoYingcui
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China
ChuZhenyan
Department of Mathematics, Dalian Minzu University, Dalian, 116600, People’s Republic of China

In this paper, we define and study Li-Yorke chaos and distributional chaos along a sequence for finitely generated semigroup actions. Let X be a compact space with metric d and G be a semigroup generated by $$f_1, f_2, ..., f_m$$ which are finitely many continuous mappings from X to itself. Then we show if (X,G) is transitive and there exists a common fixed point for all the above mappings, then (X,G) is chaotic in the sense of Li-Yorke. And we give a sufficient condition for (X,G) to be uniformly distributionally chaotic along a sequence and chaotic in the strong sense of Li-Yorke. At the end of this paper, an example on the one-sided symbolic dynamical system for (X,G) to be chaotic in the strong sense of Li-Yorke and uniformly distributionally chaotic along a sequence is given.

54H2022B9937B05Li-Yorke chaosdistributional chaos along a sequencefinitely generated semigroup actions.
G.Cairns A.Kolganova A.Nielsen Topological transitivity and mixing notions for group actions10.1216/rmjm/1181068757Rocky Mountain J. Math.200737371397E.Kontorovich M. MegrelishviliA note on sensitivity of semigroup actions10.1007/s00233-007-9033-5Semigroup Forum200876133141 F.PoloSensitive dependence on initial conditions and chaotic group actions10.1090/S0002-9939-10-10286-XProc. Amer. Math. Soc.201013828152826O. V.RybakLi-Yorke sensitivity for semigroup actions10.1007/s11253-013-0811-9 Ukrainian Math. J.201365752759L.-D.Wang Y.-N.Li Y.-L.Gao H.LiuDistributional chaos of time-varying discrete dynamical systems10.4064/ap107-1-3Ann. Polon. Math.20131074957L.-D.Wang G.-F.Liao S.-M.HuanDistributional chaos in a sequence10.1016/j.na.2006.09.005Nonlinear Anal.20076721312136X.-X.Wu Chaos of transformations induced onto the space of probability measures10.1142/S0218127416502278 Internat. J. Bifur. Chaos Appl. Sci. Engrg.201626112X.-X.WuA remark on topological sequence entropy10.1142/S0218127417501073 Internat. J. Bifur. Chaos Appl. Sci. Engrg.20172717X.-X.Wu G.-R.ChenSensitivity and transitivity of fuzzified dynamical systems10.1016/j.ins.2017.02.042 Inform. Sci.20173961423X.-X.Wu P.Oprocha G.-R.ChenOn various definitions of shadowing with average error in tracing10.1088/0951-7715/29/7/1942Nonlinearity20162919421972 X.-X.Wu X.WangOn the iteration properties of large deviations theorem10.1142/S0218127416500541 Internat. J. Bifur. Chaos Appl. Sci. Engrg.20162616X.-X.Wu J.-J.Wang G.-R.ChenF-sensitivity and multi-sensitivity of hyperspatial dynamical systems10.1016/j.jmaa.2015.04.009J. Math. Anal. Appl.20154291626 X.-X.Wu X.Wang G.-R.ChenOn the large deviations of weaker types10.1142/S0218127417501279 Internat. J. Bifur. Chaos Appl. Sci. Engrg.201727112 X.-X.Wu L.-D.Wang G.-R.Chen Weighted backward shift operators with invariant distributionally scrambled subsets10.1215/20088752-3802705 Ann. Funct. Anal.20178199210X.-X.Wu L.-D.Wang J.-H.LiangThe chain properties and average shadowing property of iterated function systems10.1007/s12346-016-0220-1Qual. Theory Dyn. Syst.2016201619 X.-X.Wu L.-D.Wang J.-H.LiangThe chain properties and Li-Yorke sensitivity of zadehs extension on the space of upper semi-continuous fuzzy setsIran. J. Fuzzy Syst.AcceptedJ.-C.XiongChaos in a topologically transitive system10.1360/04ys0120Sci. China Ser. A200548929939Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.39Fourier series of higher-order ordered Bell functionsKimTaekyun
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China;Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea
KimDae San
Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea
JangGwan-Woo
Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea
KwonJongkyum
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea

In this paper, we consider higher-order ordered Bell functions and derive their Fourier series expansions. Moreover, we express those functions in terms of Bernoulli functions.

11B8342A16Fourier serieshigher-order ordered Bell functionshigher-order ordered Bell polynomials.
M.Abramowitz I. A.StegunHandbook of mathematical functions with formulas, graphs, and mathematical tables National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.1964 A.CayleyOn the analytical forms called trees10.1080/14786445908642782 Philos. Mag.18599374378 L.ComtetAdvanced combinatorics The art of finite and infinite expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht1974 D. V.Dolgy D. S.Kim T.Kim T.MansourSums of finite products of ordered Bell functionspreprint I. J.GoodThe number of orderings of n candidates when ties are permittedFibonacci Quart.1975131118 G.-W.Jang D. S.Kim T.Kim T.MansourFourier series of functions related to Bernoulli polynomials10.23001/ascm2017.27.1.49 Adv. Stud. Contemp. Math.2017274962 T.Kim Euler numbers and polynomials associated with zeta functions10.1155/2008/581582Abstr. Appl. Anal.20082008111 T.Kim D. S.KimOn $$\lambda$$-Bell polynomials associated with umbral calculus10.1134/S1061920817010058Russ. J. Math. Phys.2017246978 D. S.Kim T.Kim Fourier series of higher-order Euler functions and their applications Bull. Korean Math. Soc.to appearT.Kim D. S.Kim Some formulas of ordered Bell numbers and polynomials arising from umbral calculus10.17777/pjms2017.20.4.659preprintT.Kim D. S.Kim S.-H.Rim D.-V.DolgyFourier series of higher-order Bernoulli functions and their applications10.1186/s13660-016-1282-yJ. Inequal. Appl.2017201717A.Knopfmacher M. E.MaysA survey of factorization counting functions10.1142/S1793042105000315 Int. J. Number Theory20051563581 J. E.Marsden Elementary classical analysisWith the assistance of Michael Buchner, Amy Erickson, Adam Hausknecht, Dennis Heifetz, Janet Macrae and William Wilson, and with contributions by Paul Chernoff, István Fáry and Robert Gulliver, W. H. Freeman and Co., San Francisco1974 M.Mor A. S.FraenkelCayley permutations 10.1016/0012-365X(84)90136-5Discrete Math.198448101112 A.SklarOn the factorization of squarefree integers10.2307/2032169Proc. Amer. Math. Soc.19523701705 D. G.Zill W. R.Wright M. R.Cullen Advanced engineering mathematicsFourth edition, Jones and Bartlett Publishers, Mississauga, Ontario2011Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.40A regularization algorithm for a splitting feasibility problem in Hilbert spacesLatifAbdul
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah-21589, Saudi Arabia
QinXiaolong
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Sichuan, China

In this article, we investigate a split feasibility problem via a regularization iterative algorithm. Strong convergence theorems of solutions for the split feasibility are established in the framework of Hilbert spaces. We also apply our main results to the split equality problem.

47H0547H0990C33Metric projectionmonotone operatornonexpansive mappingsplit feasibility problemvariational inequality.
I. K.Argyros S.George S. M.ErappaExpanding the applicability of the generalized Newton method for generalized equationsCommun. Optim. Theory20172017112B. A. BinDehaish A.Latif H. OBakodah X.-L.QinA regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces10.1186/s13660-014-0541-zJ. Inequal. Appl.20152015114B. A. BinDehaish X.-L.Qin A.Latif H.Bakodah Weak and strong convergence of algorithms for the sum of two accretive operators with applicationsJ. Nonlinear Convex Anal.20151613211336C.ByrneA unified treatment of some iterative algorithms in signal processing and image reconstruction10.1088/0266-5611/20/1/006 Inverse Problems200420103120 Y.Censor T.Bortfeld B.Martin A.TrofimovA unified approach for inversion problems in intensity-modulated radiation therapy10.1088/0031-9155/51/10/001 Phys. Med. Biol.20065123532365 Y.Censor T.Elfving A multiprojection algorithm using Bregman projections in a product space10.1007/BF02142692 Numer. Algorithms19948221239 Y.Censor T.Elfving N.Kopf T.Bortfeld The multiple-sets split feasibility problem and its applications for inverse problems10.1088/0266-5611/21/6/017Inverse Problems20052120712084 J. W.Chen E.Kobis M. A.Kobis J.-C.Yao Optimality conditions for solutions of constrained inverse vector variational inequalities by means of nonlinear scalarization J. Nonlinear Var. Anal.20171145158S. Y.Cho B. A. BinDehaish X.-L.QinWeak convergence of a splitting algorithm in Hilbert spaces10.11948/2017027J. Appl. Anal. Comput.20177427438S. Y.Cho X.-L.Qin L.Wang Strong convergence of a splitting algorithm for treating monotone operators10.1186/1687-1812-2014-94Fixed Point Theory Appl.20142014115N.-N.Fang Y.-P.Gong Viscosity iterative methods for split variational inclusion problems and fixed point problems of a nonexpansive mapping Commun. Optim. Theory20162016115L. S.Liu Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces10.1006/jmaa.1995.1289 J. Math. Anal. Appl.1995194114125A.Moudafi E.Al-Shemas Simultaneous iterative methods for split equality problemTrans. Math. Program. Appl.20131111 X.-L.Qin S. Y.ChoConvergence analysis of a monotone projection algorithm in reflexive Banach spaces10.1016/S0252-9602(17)30016-4Acta Math. Sci. Ser. B Engl. Ed.201737488502 X.-L.Qin S. Y.Cho L.WangA regularization method for treating zero points of the sum of two monotone operators10.1186/1687-1812-2014-75Fixed Point Theory Appl.20142014110 X.-L.Qin J.-C.YaoWeak convergence of a Mann-like algorithm for nonexpansive and accretive operators10.1186/s13660-016-1163-4J. Inequal. Appl.2016201619D. R.Sahu J. C.YaoA generalized hybrid steepest descent method and applications J. Nonlinear Var. Anal.20171111126J.-F.Tang S.-S.Chang J.Dong Split equality fixed point problem for two quasi-asymptotically pseudocontractive mappings J. Nonlinear Funct. Anal.20172017115 H.-K.Xu Inequalities in Banach spaces with applications10.1016/0362-546X(91)90200-KNonlinear Anal.19911611271138 H.-Y.Zhou P.-Y.Wang Adaptively relaxed algorithms for solving the split feasibility problem with a new step size10.1186/1029-242X-2014-448J. Inequal. Appl.20142014122Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.41Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equationsWangChao
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China;Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., TX 78363-8202, Kingsville, USA
AgarwalRavi P.
Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., TX 78363-8202, Kingsville, USA
O'ReganDonal
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

In this paper, based on the concept and properties of almost-complete closedness time scales (ACCTS), we investigate the existence of weighted pseudo double-almost periodic mild solutions for non-autonomous impulsive evolution equations. We also consider the exponential stability of weighted pseudo double-almost periodic solutions. Finally, we conclude our paper by providing several illustrative applications to different types of dynamic equations and mathematical models. These applications justify the practical usefulness of the established theoretical results.

34N0543A6034A3734C27Time scalesweighted pseudo double-almost periodic solutionimpulsive evolution equations.
R. P.Agarwal M.BohnerBasic calculus on time scales and some of its applications10.1007/BF03322019Results Math.199935322R. P.Agarwal M.Bohner D.O’Regan A.PetersonDynamic equations on time scales: a survey10.1016/S0377-0427(01)00432-0Dynamic equations on time scales, J. Comput. Appl. Math.2002141126R. P.Agarwal D.O’ReganSome comments and notes on almost periodic functions and changing-periodic time scales Electr. J. Math. Anal. Appl.20186125136M. U.Akhmet M.TuranThe differential equations on time scales through impulsive differential equations10.1016/j.na.2005.12.042Nonlinear Anal.20066520432060S.BochnerBeiträge zur Theorie der fastperiodischen Funktionen10.1007/BF01209156 (German) Math. Ann.192796119147 M.Bohner G. S.GuseinovDouble integral calculus of variations on time scales10.1016/j.camwa.2006.10.032Comput. Math. Appl.2007544557M.Bohner A.PetersonDynamic equations on time scalesAn introduction with applications, Birkhäuser Boston, Inc., Boston, MA2001T.Caraballo D.Cheban Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition, I10.1016/j.jde.2008.04.001 J. Differential Equations2009246108128 T.Caraballo D.Cheban Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition, II10.1016/j.jde.2008.07.025 J. Differential Equations200924611641186C.Cuevas E.Hernández M. RabeloThe existence of solutions for impulsive neutral functional differential equations10.1016/j.camwa.2009.04.008 Comput. Math. Appl.200958744757C.Cuevas A.Sepúlveda H.Soto Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations10.1016/j.amc.2011.06.054Appl. Math. Comput.201121817351745 T.Diagana Weighted pseudo almost periodic functions and applications10.1016/j.crma.2006.10.008 C. R. Math. Acad. Sci. Paris2006343643646 T.DiaganaExistence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equationsNonlinear Anal.201174600615A. M.FinkAlmost periodic differential equationsLecture Notes in Mathematics, Springer-Verlag, Berlin-New York1974J.Gao Q.-R.Wang L.-W.ZhangExistence and stability of almost-periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales10.1016/j.amc.2014.03.051Appl. Math. Comput.2014237639649 E.Hernández M.Rabello H. R.HenríquezExistence of solutions for impulsive partial neutral functional differential equations10.1016/j.jmaa.2006.09.043 J. Math. Anal. Appl.200733111351158S.Hilger Ein Maßkettenkalkäul mit anwendung auf zentrumsmannigfaltigkeitenPh.D thesis, Universität Wäurzburg1988S.HilgerAnalysis on measure chains–a unified approach to continuous and discrete calculus10.1007/BF03323153 Results Math.1990181856 B.JacksonPartial dynamic equations on time scales10.1016/j.cam.2005.02.011 J. Comput. Appl. Math.2006186391415 E. R.Kaufmann Y. N.RaffoulPeriodic solutions for a neutral nonlinear dynamical equation on a time scale10.1016%2Fj.jmaa.2006.01.063 J. Math. Anal. Appl.2006319315325 J.Liang J.Zhang T.-J.XiaoComposition of pseudo almost automorphic and asymptotically almost automorphic functions10.1016/j.jmaa.2007.09.065 J. Math. Anal. Appl.200834014931499 C.PötzscheExponential dichotomies for dynamic equations on measure chains10.1016/S0362-546X(01)00230-9Proceedings of the Third World Congress of Nonlinear Analysts, Part 2, Catania, (2000), Nonlinear Anal.200147873884A. M.Samoĭlenko N. A.Perestyuk Impulsive differential equations With a preface by Yu. A. Mitropolskiĭand a supplement by S. I. Trofimchuk, Translated from the Russian by Y. Chapovsky, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, World Scientific Publishing Co., Inc., River Edge, NJ1995 G. T.StamovAlmost periodic solutions of impulsive differential equationsLecture Notes in Mathematics, Springer, Heidelberg2012 I.StamovaStability analysis of impulsive functional differential equationsDe Gruyter Expositions in Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin2009C. C.Tisdell A.ZaidiBasic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling10.1016/j.na.2007.03.043Nonlinear Anal.20086835043524 C.WangAlmost periodic solutions of impulsive BAM neural networks with variable delays on time scales10.1016/j.cnsns.2013.12.038Commun. Nonlinear Sci. Numer. Simul.20141928282842C.Wang R. P.AgarwalA further study of almost periodic time scales with some notes and applications10.1155/2014/267384Abstr. Appl. Anal.20142014111C.Wang R. P.AgarwalRelatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations10.1186/s13662-015-0650-0 Adv. Difference Equ.2015201519C.Wang R. P.Agarwal A classification of time scales and analysis of the general delays on time scales with applications10.1002/mma.3590Math. Methods Appl. Sci.20163915681590C.Wang R. P.AgarwalAlmost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales 10.1016/j.cnsns.2015.12.003Commun. Nonlinear Sci. Numer. Simul.201636238251 C.Wang R. P.Agarwal D.O’ReganCompactness criteria and new impulsive functional dynamic equations on time scales10.1186/s13662-016-0921-4Adv. Difference Equ.20162016141C.Wang R. P.Agarwal D.O’Regan Periodicity, almost periodicity for time scales and related functions10.1515/msds-2016-0003Nonauton. Dyn. Syst.201632441C.Wang R. P.Agarwal D.O’ReganPiecewise double-almost periodic functions on almost-complete closedness time scales with generalizationssubmittedC. Y.Zhang Pseudo-almost-periodic solutions of some differential equations10.1006/jmaa.1994.1005J. Math. Anal. Appl.19941816276C. Y.ZhangPseudo almost periodic solutions of some differential equations, II10.1006/jmaa.1995.1189J. Math. Anal. Appl.1995192543561Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.42Strong convergence of some iterative algorithms for a general system of variational inequalitiesJungJong Soo
Department of Mathematics, Dong-A University, Busan 49315, Korea

In this paper, we introduce two iterative algorithms (one implicit algorithm and one explicit algorithm) for finding a common element of the solution set of a general system of variational inequalities for continuous monotone mappings and the fixed point set of a continuous pseudocontractive mapping in a Hilbert space. First, this system of variational inequalities is proven to be equivalent to a fixed point problem of nonexpansive mapping. Then we establish strong convergence of the sequence generated by the proposed iterative algorithms to a common element of the solution set and the fixed point set, which is the unique solution of a certain variational inequality.

47J2047H0547H0947H1049J4049M05Composite iterative algorithmgeneral system of variational inequatlitescontinuous monotone mappingcontinuous peudocontractive mapping$$\rho$$-Lipschitzian$$\eta$$-strongly monotone mappingvariational inequalitystrongly positive bounded linear operatorfixed points.
A. S. M.Alofi A.Latif A. E.Al-Marzooei J. C.Yao Composite viscosity iterative methods for general systems of variational inequalities and fixed point problem in Hilbert spacesJ. Nonlinear Convex Anal.201617669682L.-C.Ceng C.-Y.Wang J.-C.Yao Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities10.1007/s00186-007-0207-4 Math. Methods Oper. Res.200867375390J.-M.Chen L.-J.Zhang T.-G.FanViscosity approximation methods for nonexpansive mappings and monotone mappings10.1016/j.jmaa.2006.12.088J. Math. Anal. Appl.200733414501461 K.Goebel W. A.Kirk Topics in metric fixed point theoryCambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge1990H.Iiduka W.TakahashiStrong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings10.1016/j.na.2003.07.023Nonlinear Anal.200561341350 J. S.JungA general composite iterative method for strictly pseudocontractive mappings in Hilbert spaces10.1186/1687-1812-2014-173Fixed Point Theory Appl.20142014121 J. S.JungA composite extragradient-like algorithm for inverse-strongly monotone mappings and strictly pseudocontractive mappingsLinear Nonlinear Anal.20151271285 G. M.KorpelevičAn extragradient method for finding saddle points and for other problems (Russian) Èkonom. i Mat. Metody197612747756P.-L.Lions G.StampacchiaVariational inequalities10.1002/cpa.3160200302Comm. Pure Appl. Math.196720493519 F.-S.Liu M. Z.NashedRegularization of nonlinear ill-posed variational inequalities and convergence rates10.1023/A:1008643727926Set-Valued Anal.19986313344G.Marino H.-K.XuA general iterative method for nonexpansive mappings in Hilbert spaces10.1016/j.jmaa.2005.05.028J. Math. Anal. Appl.20063184352 G. J.Minty On the generalization of a direct method of the calculus of variations10.1090/S0002-9904-1967-11732-4 Bull. Amer. Math. Soc.196773315321W.Takahashi M.ToyodaWeak convergence theorems for nonexpansive mappings and monotone mappings10.1023/A:1025407607560J. Optim. Theory Appl.2003118417428Y.TangStrong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings10.1186/1687-1812-2013-273Fixed Point Theory Appl.20132013111H.-K.Xu Iterative algorithms for nonlinear operators10.1112/S0024610702003332J. London Math. Soc.200266240256I.YamadaThe hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings10.1016/S1570-579X(01)80028-8 Inherently parallel algorithms in feasibility and optimization and their applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam20018473504H.ZegeyeAn iterative approximation method for a common fixed point of two pseudocontractive mappings10.5402/2011/621901 ISRN Math. Anal.20112011114H.Zegeye N.Shahzad Strong convergence of an iterative method for pseudo-contractive and monotone mappings10.1007/s10898-011-9755-5 J. Global Optim.201254173184Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.43Controllability of fractional impulsive neutral stochastic functional differential equations via Kuratowski measure of noncompactnessHuJunhao
School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China
YangJiashun
School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China
YuanChenggui
Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK

In this paper, the controllability problem for a class of fractional impulsive neutral stochastic functional differential equations is considered in infinite dimensional space. By using Kuratowski measure of noncompactness and Mönch fixed point theorem, the sufficient conditions of controllability of the equations are obtained under the assumption that the semigroup generated by the linear part of the equations is not compact. At the end, an example is provided to illustrate the proposed result.

26A3365C3093B05Controllabilityfractional differential equationsimpulsive stochastic differential equationsKuratowski measure of noncompactnessMönch fixed point theorem.
R. P.Agarwal M.Benchohra S.HamaniA survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions10.1007/s10440-008-9356-6Acta Appl. Math.20101099731033P.Balasubramaniam J. P.Dauer Controllability of semilinear stochastic delay evolution equations in Hilbert spaces10.1155/S0161171202111318 Int. J. Math. Math. Sci.200231157166 P.Balasubramaniam J. Y.Park P.MuthukumarApproximate controllability of neutral stochastic functional differential systems with infinite delay10.1080/07362990802405695Stoch. Anal. Appl.201028389400 H.-B.Bao J.-D.CaoExistence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay10.1016/j.amc.2009.07.025Appl. Math. Comput.200921517321743 S.Das D.Pandey N.Sukavanam Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay10.1016/S0252-9602(16)30086-8 Acta Math. Sci. Ser. B Engl. Ed.20163615091523 A.Dehici N.RedjelMeasure of noncompactness and application to stochastic differential equations 10.1186/s13662-016-0748-zAdv. Difference Equ.20162016117S.Duan J.-H.Hu Y.LiExact controllability of nonlinear stochastic impulsive evolution differential inclusions with infinite delay in Hilbert spaces10.1515/ijnsns.2011.023Int. J. Nonlinear Sci. Numer. Simul.2011122333T.GuendouziExistence and controllability of fractional-order impulsive stochastic system with infinite delay10.7151/dmdico.1144Discuss. Math. Differ. Incl. Control Optim.2013336587D. J.Guo Impulsive integral equations in Banach spaces and applications10.1155/S104895339200008XJ. Appl. Math. Stochastic Anal.19925111122H. P.Heinz On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions10.1016/0362-546X(83)90006-8Nonlinear Anal.198313511371 L.-R.Huang F.-Q.Deng Razumikhin-type theorems on stability of neutral stochastic functional differential equations10.1109/TAC.2008.929383 IEEE Trans. Automat. Control20085317181723 Y.-G.Kao Q.-X.Zhu W.-H.QiExponential stability and instability of impulsive stochastic functional differential equations with Markovian switching10.1016/j.amc.2015.09.063Appl. Math. Comput.2015271795804A. A.Kilbas H. M.Srivastava J. J.TrujilloTheory and applications of fractional differential equations North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam2006V.Lakshmikantham D. D.Bainov P. S.SimeonovTheory of impulsive differential equationsSeries in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ1989 K.-X.Li J.-G.Peng Controllability of fractional neutral stochastic functional differential systems10.1007/s00033-013-0369-2 Z. Angew. Math. Phys.201465941959 X.-R.MaoRazumikhin-type theorems on exponential stability of stochastic functional-differential equations10.1016/S0304-4149(96)00109-3Stochastic Process. Appl.199665233250 X.-R.Mao Stochastic differential equations and their applicationsSecond edition, Woodhead Publishing Limited, Cambridge2007 K. S.Miller B.Ross An introduction to the fractional calculus and fractional differential equationsA Wiley-Interscience Publication, John Wiley & Sons, Inc., New York1993H.MönchBoundary value problems for nonlinear ordinary differential equations of second order in Banach spaces10.1016/0362-546X(80)90010-3 Nonlinear Anal.19804985999A.PazySemigroups of linear operators and applications to partial differential equationsApplied Mathematical Sciences, Springer-Verlag, New York1983I.Podlubny Fractional differential equations An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA1999M. D.Quinn N.Carmichael An approach to nonlinear control problems using fixed-point methods, degree theory and pseudo-inverses10.1080/01630568508816189 Numer. Funct. Anal. Optim.1984/857197219 R.Sakthivel R.Ganesh S.Suganya Approximate controllability of fractional neutral stochastic system with infinite delay10.1016/S0034-4877(12)60047-0Rep. Math. Phys.201270291311 R.Sakthivel N. I.Mahmudov J. J.NietoControllability for a class of fractional-order neutral evolution control systems10.1016/j.amc.2012.03.093Appl. Math. Comput.20122181033410340R.Sakthivel S.Suganya S. M.AnthoniApproximate controllability of fractional stochastic evolution equations10.1016/j.camwa.2011.11.024Comput. Math. Appl.201263660668R.TriggianiA note on the lack of exact controllability for mild solutions in Banach spaces10.1137/0315028 SIAM J. Control Optimization197715407411R.Triggianiddendum: ”A note on the lack of exact controllability for mild solutions in Banach spaces” [SIAM J. Control Optim., 15 (1977), 407–411], SIAM J. Control Optim.1980189899 Q.Wang X.-Z.LiuImpulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method10.1016/j.aml.2006.08.016Appl. Math. Lett.200720839845 T.Yang Impulsive systems and control: theory and applications Nova Science Publishers, Inc., New York2001 R.-P.YeExistence of solutions for impulsive partial neutral functional differential equation with infinite delay10.1016/j.na.2010.03.008Nonlinear Anal.201073155162Y.-C.Zang J.-P.LiApproximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions10.1186/1687-2770-2013-193 Bound. Value Probl.20132013113 Y.Zhou F.Jiao Existence of mild solutions for fractional neutral evolution equations10.1016/j.camwa.2009.06.026Comput. Math. Appl.20105910631077Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.44New exact solution of generalized biological population modelAcanOmer
Art and Science Faculty, Department of Mathematics, Siirt University, Siirt, Turkey
QurashiMaysaa Mohamed Al
Faculty of Art and Science, Department of Mathematics, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia
BaleanuDumitru
Faculty of Art and Science, Department of Mathematics, Cankaya University, Ankara, Turkey;Institute of Space Sciences, Magurele-Bucharest, Romania

In this study, a mathematical model of the generalized biological population model (GBPM) gets a new exact solution with a conformable derivative operator (CDO). The new exact solution of this model will be obtained by a new approximate analytic technique named three dimensional conformable reduced differential transform method (TCRDTM). By using this technique, it is possible to find new exact solution as well as closed analytical approximate solution of a partial differential equations (PDEs). Three numerical applications of GBPM are given to check the accuracy, effectiveness, and convergence of the TCRDTM. In these applications, obtained new exact solutions in conformable sense are compared with the exact solutions in Caputo sense in literature. The comparisons are illustrated in 3D graphics. The results show that when $$\alpha\rightarrow 1$$, the exact solutions in conformable and Caputo sense converge to each other. In other cases, exact solutions different from each other are obtained.

35R1174H15Numerical solutionbiological populations modelreduced differential transform methodconformable derivativepartial differential equations.
School of Mathematics and Information Science, North Minzu University, Yin Chuan, 750021, China
HuangYongdong
School of Mathematics and Information Science, North Minzu University, Yin Chuan, 750021, China

In this paper, we introduce the concept of dual frame of g-p-frame, and give the sufficient condition for a g-p-frame to have dual frames. Using operator theory and methods of functional analysis, we get some new properties of g-p-frame. In addition, we also characterize g-p-frame and g-q-Riesz bases by using analysis operator of g-p-Bessel sequence.

42C1542C40g-p-frameg-q-Riesz basisdual framesanalysis operator.
M. R.Abdollahpour M. H.Faroughi A.Rahimipg-frames in Banach spacesMethods Funct. Anal. Topology200713201210 M. R.Abdollahpour A.Najati P.GavrutaMultipliers of pg-Bessel sequences in Banach spacesArXiv20152015114A.Aldroubi Q.Sun W.-S.Tangp-frames and shift invariant subspaces of $$L^p$$10.1007/s00041-001-0001-2 J. Fourier Anal. Appl.20017122C. D.Aliprantis K. C.BorderInfinite-dimensional analysis A hitchhiker’s guide, Second edition, Springer-Verlag, Berlin1999O.Christensen D. T.Stoeva p-frames in separable Banach spaces10.1023/A:1021364413257Frames. Adv. Comput. Math.200318117126J. B.Conway A course in functional analysisGraduate Texts in Mathematics, Springer-Verlag, New York1985R. J.Duffin A. C.SchaefferA class of nonharmonic Fourier series10.2307/1990760Trans. Amer. Math. Soc.195272341366 A.Khosravi K.MusazadehFusion frames and g-frames10.1016/j.jmaa.2008.01.002J. Math. Anal. Appl.200834210681083X.-C.Xiao X.-M.ZengSome properties of g-frames in Hilbert $$C^*$$-modules10.1016/j.jmaa.2009.08.043J. Math. Anal. Appl.2010363399408X.-C.Xiao Y.-C.Zhu X.-M.ZengGeneralized p-frame in separable complex Banach spaces10.1142/S0219691310003419Int. J. Wavelets Multiresolut. Inf. Process.20108133148Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.46Common fixed point results for probabilistic $$\varphi$$-contractions in generalized probabilistic metric spacesTianJingfeng
College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, P. R. China
HuXimei
China Mobile Group Hebei Co., Ltd., Baoding, Hebei Province, 071000, P. R. China

In this paper, we present some new fixed point and common fixed point (common coupled fixed point, common tripled fixed point, and common quadruple fixed point) theorems of probabilistic contractions with a gauge function $$\varphi$$ in generalized probabilistic metric spaces proposed by Zhou et al. [C.-L. Zhou, S.-H. Wang, L. Ćirić, S. M. Alsulami, Fixed Point Theory Appl., 2014 (2014), 15 pages]. Our results extend some existing results. Moreover, an example is given to illustrate our main results.

47H1054E70Coupled fixed pointfixed pointmetric spaceprobabilistic $$\varphi$$-contractionsgauge function.
R. P.Agarwal E.KarapınarRemarks on some coupled fixed point theorems in G-metric spaces10.1186/1687-1812-2013-2 Fixed Point Theory Appl.20132013133 V.Berinde M.BorcutTripled fixed point theorems for contractive type mappings in partially ordered metric spaces10.1016/j.na.2011.03.032Nonlinear Anal.20117448894897L.Ćirić R. P.Agarwal B.SametMixed monotone-generalized contractions in partially ordered probabilistic metric spaces10.1186/1687-1812-2011-56Fixed Point Theory Appl.20112011113 J.-X.Fang Common fixed point theorems of compatible and weakly compatible maps in Menger spaces10.1016/j.na.2009.01.018 Nonlinear Anal.20097118331843 T. GnanaBhaskar V.LakshmikanthamFixed point theorems in partially ordered metric spaces and applications10.1016/j.na.2005.10.017Nonlinear Anal.20066513791393D. J.Guo V.Lakshmikantham Coupled fixed points of nonlinear operators with applications10.1016/0362-546X(87)90077-0Nonlinear Anal.198711623632O.HadžićA fixed point theorem in Menger spacesPubl. Inst. Math. (Beograd) (N.S.)197920107112O.HadžićFixed point theorems for multivalued mappings in probabilistic metric spaces10.1016/S0165-0114(96)00072-3Fuzzy Sets and Systems199788219226J.JachymskiOn probabilistic $$\phi$$-contractions on Menger spaces10.1016/j.na.2010.05.046Nonlinear Anal.20107321992203M.Jleli E.Karapınar B.SametFurther generalizations of the Banach contraction principle10.1186/1029-242X-2014-439 J. Inequal. Appl.2014201419 M.Jleli E.Karapınar B.SametOn cyclic ($$\psi,\phi$$)-contractions in Kaleva-Seikkala’s type fuzzy metric spaces10.3233/IFS-141170J. Intell. Fuzzy Systems20142720452053 E.Karapınar N. V.Luong Quadruple fixed point theorems for nonlinear contractions10.1016/j.camwa.2012.02.061 Comput. Math. Appl.20126418391848 V.Lakshmikantham L.ĆirićCoupled fixed point theorems for nonlinear contractions in partially ordered metric spacesNonlinear Anal.20097043414349T.Luo C.-X.Zhu Z.-Q.WuTripled common fixed point theorems under probabilistic $$\phi$$-contractive conditions in generalized Menger probabilistic metric spaces10.1186/1687-1812-2014-158Fixed Point Theory Appl.20142014117K.MengerStatistical metrics10.1073/pnas.28.12.535Proc. Nat. Acad. Sci. U. S. A.194228535537 Z.Mustafa B.SimsA new approach to generalized metric spaces J. Nonlinear Convex Anal.20067289297V. I.Opoĭcev Heterogeneous and combined-concave operators10.1007/BF00967133(Russian) Sibirsk. Mat. Z .197516781792 B.Samet On the approximation of fixed points for a new class of generalized Berinde mappings Carpathian J. Math.201632363374B.Schweizer A.SklarProbabilistic metric spacesNorth-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York1983S.Sedghi I.Altun N.ShobeCoupled fixed point theorems for contractions in fuzzy metric spaces10.1016/j.na.2009.08.018Nonlinear Anal.20107212981304] J.WuSome fixed-point theorems for mixed monotone operators in partially ordered probabilistic metric spaces10.1186/1687-1812-2014-49Fixed Point Theory Appl.20142014112J.-Z.Xiao X.-H.Zhu Y.-F.CaoCommon coupled fixed point results for probabilistic $$\phi$$-contractions in Menger spaces10.1016/j.na.2011.04.030Nonlinear Anal.20117445894600C.-L.Zhou S.-H.Wang L.Ćirić S. M.AlsulamiGeneralized probabilistic metric spaces and fixed point theorems10.1186/1687-1812-2014-91Fixed Point Theory Appl.20142014115Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.47Hybrid implicit steepest-descent methods for triple hierarchical variational inequalities with hierarchical variational inequality constraintsCengLu-Chuan
Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
LiouYeong-Cheng
Department of Healthcare Administration and Medical Informatics, and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan;Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan
WenChing-Feng
Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
LoChing-Hua
Department of Management, Yango University, Fujian 350015, China

In this paper, we introduce and analyze a hybrid implicit steepest-descent algorithm for solving the triple hierarchical variational inequality problem with the hierarchical variational inequality constraint for finitely many nonexpansive mappings in a real Hilbert space. The proposed algorithm is based on Korpelevich’s extragradient method, hybrid steepest-descent method, Mann’s implicit iteration method, and Halpern’s iteration method. Under mild conditions, the strong convergence of the iteration sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.

49J3047H0947J2049M05Hybrid implicit steepest-descent algorithmtriple hierarchical variational inequalityMann’s implicit iteration methodnonexpansive mappinginverse-strong monotonicityglobal convergence.
Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China
YeHongqing
Hangzhou Wenchang High School, Hangzhou, Zhejiang 311121, China

In this paper, we prove some common fixed point theorems for three self-mappings satisfying various new contractive conditions in complete G-metric spaces. We also discuss that these mappings are G-continuous on such a common fixed point. And a non-trivial example is provided to support our new result in the framework of nonsymmetric G-metric spaces. At the end of the results, we give an existence theorem for common solution of three integral equations. The results obtained in this paper differ from the recent relative results in literature.

47H1054H25Common fixed pointgeneralized metric spaceintegral equation.
M.Abbas T.Nazir S.RadenovićSome periodic point results in generalized metric spaces10.1016/j.amc.2010.10.026Appl. Math. Comput.201021740944099M.Abbas B. E.RhoadesCommon fixed point results for noncommuting mappings without continuity in generalized metric spaces10.1016/j.amc.2009.04.085Appl. Math. Comput.2009215262269H.Aydi W.Shatanawi C.VetroOn generalized weakly G-contraction mapping in G-metric spaces10.1016/j.camwa.2011.10.007Comput. Math. Appl.20116242224229F.Gu W.Gao W.TianFixed point theorem and the iterative convergence of nonlinear operatorHarbin Science and Technology Press, Harbin, China2002F.Gu Z.-Z.YangSome new common fixed point results for three pairs of mappings in generalized metric spaces10.1186/1687-1812-2013-174 Fixed Point Theory Appl.20132013121F.Gu Y.YinCommon fixed point for three pairs of self-maps satisfying common (E.A) property in generalized metric spaces10.1155/2013/808092Abstr. Appl. Anal.20132013111F.Gu D.Zhang The common fixed point theorems for six self-mappings with twice power type $$\Phi$$-contraction conditionThai J. Math.201210587603M.Jleli B.SametRemarks on G-metric spaces and fixed point theorems10.1186/1687-1812-2012-210Fixed Point Theory Appl2012201217 E.Karapınar R. P.AgarwalFurther fixed point results on G-metric spaces10.1186/1687-1812-2013-154 Fixed Point Theory Appl.20132013114 Z.Mustafa H.Aydi E.KarapınarOn common fixed points in G-metric spaces using (E.A) property10.1016/j.camwa.2012.03.051 Comput. Math. Appl.20126419441956 Z.Mustafa H.Obiedat H.AwawdehSome fixed point theorem for mapping on complete G-metric spaces10.1155/2008/189870 Fixed Point Theory Appl.20082008112Z.Mustafa B.SimsA new approach to generalized metric spacesJ. Nonlinear Convex Anal.20067289297B.Samet C.Vetro F.VetroRemarks on G-metric spaces Int. J. Anal.2013201316W.ShatanawiFixed point theory for contractive mappings satisfying $$\Phi$$-maps in G-metric spacesFixed Point Theory Appl.2010201019 N.Tahat H.Aydi E.Karapınar W.Shatanawi Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces10.1186/1687-1812-2012-48 Fixed Point Theory Appl.2012201219 H.-Q.Ye F.Gu A new common fixed point theorem for a class of four power type contraction mappings10.3969/j.issn.1674-232X.2011.06.009 J. Hangzhou Univ. Natur. Sci. Ed.201110520523H.-Q.Ye F.GuCommon fixed point theorems for a class of twice power type contraction maps in G-metric spaces10.1155/2012/736214 Abstr. Appl. Anal.20122012119Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.49Quasi associated continued fractions and Hankel determinants of Dixon elliptic functions via Sumudu transformKilicmanAdem
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
SilambarasanRathinavel
M. Tech IT-Networking, Department of Information Technology, School of Information Technology and Engineering, VIT University, Vellore, Tamilnadu, India
AltunOmer
Department of Mathematics and Institute for Mathematical research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

In this work, Sumudu transform of Dixon elliptic functions for higher arbitrary powers $$sm^N(x, \alpha);N \geq 1, sm^N(x, \alpha)cm(x, \alpha); N \geq 0$$ and $$sm^N(x, \alpha)cm^2(x, \alpha);N \geq 0$$ by considering modulus $$\alpha \neq 0$$ is obtained as three term recurrences and hence expanded as product of quasi associated continued fractions where the coefficients are functions of $$\alpha$$. Secondly the coefficients of quasi associated continued fractions are used for Hankel determinants calculations by connecting the formal power series (Maclaurin series) and associated continued fractions.

11A5511C2033E0544A10Dixon elliptic functionsquasi associated continued fractionsHankel determinantsSumudu transformthree term recurrence.
O. S.Adams Elliptic functions applied to conformal world maps US Government Printing Office, Washington1925W. A.Al-Salam L.Carlitz Some determinants of Bernoulli Euler and related numbers, Portugal. Math.1959189199R.Bacher P.FlajoletPseudo-factorials, elliptic functions, and continued fractions10.1007/s11139-009-9186-9Ramanujan J.2009217197 F. B. M.Belgacem E. H.Al-Shemas R.SilambarasanSumudu computation of the transient magnetic field in a lossy medium10.18576/amis/110126 Appl. Math. Inf. Sci.201711209217F. B. M.Belgacem R.Silambarasan A distinctive Sumudu treatment of trigonometric functions10.1016/j.cam.2015.12.036J. Comput. Appl. Math.20173127481 F. B. M.Belgacem R.SilambarasanFurther distinctive investigations of the Sumudu transform10.1063/1.4972617 AIP Conf. Proc.201717980200251F. B. M.Belgacem R.SilambarasanSumudu transform of Dumont bimodular Jacobi elliptic functions for arbitrary powers10.1063/1.4972618AIP Conf. Proc.201717980200261L.CarlitzSome orthogonal polynomials related to elliptic functions10.1215/S0012-7094-60-02742-3 Duke Math. J.196027443459 E. V. F.Conrad Some continued fraction expansions of Laplace transforms of elliptic functionsThesis (Ph.D.)–The Ohio State University, ProQuest LLC, Ann Arbor, MI2002187 E. V. F.Conrad P.FlajoletThe Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursionSém. Lothar. Combin.2005/0754144 A. C.DixonOn the doubly periodic functions arising out of the curve $$x^3 + y^3 - 3\alpha xy = 1$$Quart. J. Pure Appl. Math.189024167233A. C.DixonThe elementary properties of the elliptic functions, with examplesMacmillan, London and New York1894 D.DumontLe parametrage de la courbe d’equation $$x^3 + y^3 = 1$$Une introduction elementaire aux fonctions elliptiques, preprint1988 H.Eltayeb A.Kılıçman On double sumudu transform and double Laplace transform Malays. J. Math. Sci.201041730H.Eltayeb A.Kılıçman R. P.AgarwalOn integral transforms and matrix functions10.1155/2011/207930Abstr. Appl. Anal.20112011115M. E. H.Ismail D. R.MassonSome continued fractions related to elliptic functions10.1090/conm/236/03495Continued fractions: from analytic number theory to constructive approximation, Columbia, MO, (1998), Contemp. Math., Amer. Math. Soc., Providence, RI1999236149166 W. B.Jones W. J.Thron Continued fractions Analytic theory and applications, With a foreword by Felix E. Browder, With an introduction by Peter Henrici, Encyclopedia of Mathematics and its Applications, Addison- Wesley Publishing Co., Reading, Mass.1980A.Kılıçman H.EltayebOn a new integral transform and differential equations10.1155/2010/463579 Math. Probl. Eng.20102010113 A.Kılıçman H.EltayebSome remarks on the Sumudu and Laplace transforms and applications to differential equations10.5402/2012/591517 ISRN Appl. Math.20122012113 A.Kılıçman H.Eltayeb K. A. M.AtanA note on the comparison between Laplace and Sumudu transformsBull. Iranian Math. Soc.201137131141 A.Kılıçman V. G.Gupta B.ShramaOn the solution of fractional Maxwell equations by Sumudu transformJ. Math. Res.20102147151J. C.Langer D. A.Singer The trefoilMilan J. Math.201482161182 D. F.Lawden Elliptic functions and applications Applied Mathematical Sciences, Springer-Verlag, New York1989L.Lorentzen H.Waadeland Continued fractions with applications Studies in Computational Mathematics, North- Holland Publishing Co., Amsterdam1992 A. M.Mahdy A. S.Mohamed A. A.MtawaImplementation of the homotopy perturbation Sumudu transform method for solving Klein-Gordon equation10.4236/am.2015.63056 Appl. Math.20156617628S. C. MilneInfinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functionsReprinted from Ramanujan J.,/ 6 (2002), 7–149 , With a preface by George E. Andrews, Developments in Mathematics, Kluwer Academic Publishers, Boston, MA2002 T.MuirA treatise on the theory of determinants Revised and enlarged by William H. Metzler, Dover Publications, Inc., New York1960 M. A.Ramadan M. S.Al-LuhaibiApplication of sumudu decomposition method for solving linear and nonlinear Klein- Gordon equationsInt. J. Soft Comput. Eng.20143138141J.Singh D.Kumar A.KilicmanApplication of homotopy perturbation Sumudu transform method for solving heat and wave-like equationsMalays. J. Math. Sci.201377995 H. S.WallNote on the expansion of a power series into a continued fraction10.1090/S0002-9904-1945-08280-9Bull. Amer. Math. Soc.19455197105H. S.Wall Analytic theory of continued fractions D. Van Nostrand Company, Inc., New York, N. Y.1948X.-J.YangA new integral transform with an application in heat-transfer problemTherm. Sci.2016201677X.-J.YangA new integral transform operator for solving the heat-diffusion problem10.1016/j.aml.2016.09.011Appl. Math. Lett.201764193197Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.50Some fixed point results via measure of noncompactnessChenChi-Ming
Institute for Computational and Modeling Science, National Tsing Hua University, Taiwan
KarapinarErdal
Department of Mathematics, Atılım University, 06586 Incek, Ankara, Turkey;Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, King Abdulaziz University, 21589, Jeddah, Saudi Arabia

In this paper, by using the measure of noncompactness and Meir-Keeler type mappings, we prove some new fixed point theorems for some certain mappings, namely, the weaker $$\varphi$$-Meir-Keeler type contractions, asymptotic weaker $$\varphi$$-Meir-Keeler type contractions, asymptotic sequence $$\{\phi_i\}$$-Meir-Keeler type contraction, $$\xi$$-generalized comparison type contraction, and R- functional type $$\psi$$-contractions. Our results improve and hence cover the well-known Darbo’s fixed point theorem, and several related recent fixed point results.

54H2547H10Measure of noncompactnessMeir-Keeler-type set contractionfixed points.
R.Agarwal M.Meehan D.O’ReganFixed point theory and applicationsCambridge University Press, United kingdom2001A.Aghajani M.Mursaleen A. S.HaghighiFixed point theorems for Meir-Keeler condensing operators via measure of noncompactness10.1016/S0252-9602(15)30003-5 Acta Math. Sci.201535552566 R. R.Akhmerov M. I.Kamenski A. S.Potapov A. E.Rodkina B. N.SadovskiMeasures of Noncompactness and Condensing Operators10.1007/978-3-0348-5727-7 Translated from the 1986 Russian original by A. Iacob. Oper. Theory Adv. Appl.1992551244 J.Banaś Measures of noncompactness in the space of continuous tempered functionsDemonstratio Math.198114127133 J.Banaś Measures of noncompactness in the study of solutions of nonlinear differential and integral equations10.2478/s11533-012-0120-9Cent. Eur. J. Math.20121020032011J.Banaś K.GoebelMeasures of Noncompactness in Banach SpacesLecture Notes in Pure and Applied Mathematics, New York1980 C.-M.Chen T.-H.Chang Fixed Point Theorems for a Weaker Meir-Keeler Type $$\psi$$-Set Contraction in Metric Spaces10.1155/2009/129124 Fixed Point Theory Appl.2009200918G.Darbo Punti uniti in trasformazioni a codominio non compattoRend. Sem. Mat. Univ. Padova1955248492W.-S.DuOn coincidence point and fixed point theorems for nonlinear multivalued maps10.1016/j.topol.2011.07.021Topol. Appl20121594956K.KuratowskiSur les espaces completsFund. Math.193015301309B. deMalafosse E.Malkowsky V.Rakocevic Measure of noncompactness of operators and matrices on the spaces c and c010.1155/IJMMS/2006/46930 Int. J. Math. Math. Sci.2006200615A.Meir E.KeelerA theorem on contraction mappingsJ. Math. Anal. Appl.196928326329M.Mursaleen A. K.NomanCompactness by the Hausdorff measure of noncompactness10.1016/j.na.2010.06.030 Nonlinear Anal.20107325412557 J. M. A.Toledano T. D.Benavides G. L.Azedo Measures of Noncompactness in Metric Fixed Point TheoryBirkhuser Verlag, Basel1997Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.51Some Hermite-Hadamard type inequalities for harmonically extended $$s$$-convex functionsLiChun-Long
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China
WuShan-He
Department of Mathematics, Longyan University, Longyan, Fujian 364012, China

In this paper, we establish some inequalities of Hermite-Hadamard type for functions whose derivatives absolute values are harmonically extended s-convex functions.

26A5126D1541A55Harmonically extended s-convex functionHermite-Hadamard type inequalitiesintegral inequalities.
W. W.BrecknerStetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen(German) Publ. Inst. Math. (Beograd) (N.S.)1978231320 F.-X.Chen S.-H.Wu Some Hermite-Hadamard type inequalities for harmonically s-convex functions10.1155/2014/279158Scientific World J.2014201417S. S.Dragomir R. P.AgarwalTwo inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula10.1016/S0893-9659(98)00086-XAppl. Math. Lett.1998119195H.Hudzik L.MaligrandaSome remarks on s-convex functions10.1007/BF01837981 Aequationes Math.199448100111S.Hussain M. I.Bhatti M.IqbalHadamard-type inequalities for s-convex functions, IPunjab Univ. J. Math. (Lahore)2009415160İ.İşcanHermite-Hadamard type inequalities for harmonically convex functions Hacet. J. Math. Stat.201443935942 İ.İşcan S.-H.WuHermite-Hadamard type inequalities for harmonically convex functions via fractional integrals10.1016/j.amc.2014.04.020 Appl. Math. Comput.2014238237244 U. S.Kirmaci M. KlaričićBakula M. E.Özdemir J.Pečarić Hadamard-type inequalities for s-convex functions10.1016/j.amc.2007.03.030 Appl. Math. Comput.20071932635M. A.Noor K. I.Noor M. U.Awan S.CostacheSome integral inequalities for harmonically h-convex functionsPolitehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys.201577516 C. E. M.Pearce J.PečarićInequalities for differentiable mappings with application to special means and quadrature formulæ10.1016/S0893-9659(99)00164-0 Appl. Math. Lett.2000135155S.-H.Wu On the weighted generalization of the Hermite-Hadamard inequality and its applications10.1216/RMJ-2009-39-5-1741 Rocky Mountain J. Math.20093917411749 S.-H.Wu B.Sroysang J.-S.Xie Y.-M.ChuParametrized inequality of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex10.1186/s40064-015-1633-zSpringerPlus2015419 B.-Y.Xi R.-F.Bai F.QiHermite-Hadamard type inequalities for the m- and $$(\alpha,m)$$-geometrically convex functions10.1007/s00010-011-0114-xAequationes Math.201284261269 B.-Y.Xi F.QiSome Hermite-Hadamard type inequalities for differentiable convex functions and applicationsHacet. J. Math. Stat.201342243257 B.-Y.Xi F.QiHermite-Hadamard type inequalities for geometrically r-convex functions10.1556/SScMath.51.2014.4.1294Studia Sci. Math. Hungar.201451530546 B.-Y.Xi F.Qi Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means J. Nonlinear Convex Anal.201516873890 B.-Y.Xi T.-Y.Zhang F.Qi Some inequalities of Hermite–Hadamard type for m-harmonic-arithmetically convex functions10.2306/scienceasia1513-1874.2015.41.357ScienceAsia201541357361Journal of Nonlinear Sciences and Applications(JNSA) 2008-1898 2008-1901Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.010.07.52Existence result for a class of coupled fractional differential systems with integral boundary value conditionsQiTingting
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, P. R. China
LiuYansheng
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, P. R. China
ZouYumei
Department of Mathematics, Shandong University of Science and Technology, Qingdao 266590, P. R. China