]> 2017 10 7 ISSN 2008-1898 681
Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations en en 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. 3364 3380 Xinqiu Zhang School of Mathematical Sciences Qufu Normal University China 1257368359@qq.com Lishan Liu School of Mathematical Sciences Department of Mathematics and Statistics Qufu Normal University Curtin University China Australia mathlls@163.com Yonghong Wu Department of Mathematics and Statistics Curtin University Australia Y.Wu@curtin.edu.au Mixed monotone operator fixed point theorem iterative positive solution singular integral equations boundary value problem cone. Article.1.pdf  R. P. Agarwal, On fourth order boundary value problems arising in beam analysis, Differential Integral Equations, 2 (1989), 91-110 ## R. P. Agarwal, Y. M. Chow, Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math., 10 (1984), 203-217 ## A. Cabada, G.-T. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411 ## Z.-W. Cao, D.-Q. Jiang, C.-J. Yuan, D. O’Regan, Existence and uniqueness of solutions for singular integral equation, Positivity, 12 (2008), 725-732 ## Y.-J. Cui, L.-S. Liu, X.-Q. Zhang, Uniqueness and existence of positive solutions for singular differential systems with coupled integral boundary value problems, Abstr. Appl. Anal., 2013 (2013), 1-9 ## D.-J. Guo, Y. J. Cho, J. Zhu, Partial ordering methods in nonlinear problems, Nova Science Publishers, Inc., Hauppauge, NY (2004) ## D.-J. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., , Boston, MA (1988) ## X.-A. Hao, L.-S. Liu, Y.-H. Wu, Q. Sun, Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions, Nonlinear Anal., 73 (2010), 1653-1662 ## H. H. G. Hashem, On successive approximation method for coupled systems of Chandrasekhar quadratic integral equations, J. Egyptian Math. Soc., 23 (2015), 108-112 ## W.-H. Jiang, J.-L. Zhang, Positive solutions for (k, n-k) conjugate boundary value problems in Banach spaces, Nonlinear Anal., 71 (2009), 723-729 ## M. Jleli, B. Samet, Existence of positive solutions to an arbitrary order fractional differential equation via a mixed monotone operator method, Nonlinear Anal. Model. Control, 20 (2015), 367-376 ## A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006) ## K. Q. Lan, Multiple positive solutions of conjugate boundary value problems with singularities, Appl. Math. Comput., 147 (2004), 461-474 ## K. Latrach, M. A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on $$L_1$$ spaces, Nonlinear Anal., 66 (2007), 2325-2333 ## P.-D. Lei, X.-N. Lin, D.-Q. Jiang, Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems, Nonlinear Anal., 69 (2008), 2773-2779 ## F.-Y. Li, Y.-H. Li, Z.-P. Liang, Existence of solutions to nonlinear Hammerstein integral equations and applications, J. Math. Anal. Appl., 323 (2006), 209-227 ## H.-D. Li, L.-S. Liu, Y.-H. Wu, Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl., 2015 (2015), 1-15 ## X.-N. Lin, D.-Q. Jiang, X.-Y. Li, Existence and uniqueness of solutions for singular (k, n - k) conjugate boundary value problems, Comput. Math. Appl., 52 (2006), 375-382 ## X.-N. Lin, D.-Q. Jiang, X.-Y. Li, Existence and uniqueness of solutions for singular fourth-order boundary value problems, J. Comput. Appl. Math., 196 (2006), 155-161 ## L.-S. Liu, F. Guo, C.-X.Wu, Y.-H.Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309 (2005), 638-647 ## L.-S. Liu, C.-X. Wu, F. Guo, Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., 47 (2004), 13-22 ## L.-S. Liu, X.-Q. Zhang, J. Jiang, Y.-H. Wu, The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems, J. Nonlinear Sci. Appl., 9 (2016), 2943-2958 ## A. Lomtatidze, L. Malaguti, On a nonlocal boundary value problem for second order nonlinear singular differential equations, Georgian Math. J., 7 (2000), 133-154 ## M.-H. Pei, S. K. Chang, Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Math. Comput. Modelling, 51 (2010), 1260-1267 ## I. 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, CA (1999) ## S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional 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, Yverdon (1993) ## Y.-P. Sun, M. Zhao, Positive solutions for a class of fractional differential equations with integral boundary conditions, Appl. Math. Lett., 34 (2014), 17-21 ## Y.-Q. Wang, L.-S. Liu, Y.-H. Wu, Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Anal., 74 (2011), 6434-6441 ## Y.-Q. Wang, L.-S. Liu, Y.-H. Wu, Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 74 (2011), 3599-3605 ## J. R. L. Webb, Uniqueness of the principal eigenvalue in nonlocal boundary value problems, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), 177-186 ## J. R. L. Webb, Nonlocal conjugate type boundary value problems of higher order, Nonlinear Anal., 71 (2009), 1933-1940 ## J. R. L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. Real World Appl., 13 (2012), 923-938 ## J. R. L. Webb, Positive solutions of nonlinear differential equations with Riemann-Stieltjes boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-13 ## P. J. Y. Wong, Triple positive solutions of conjugate boundary value problems, II, Comput. Math. Appl., 40 (2000), 537-557 ## Z.-L. Yang, Positive solutions for a system of nonlinear Hammerstein integral equations and applications, Appl. Math. Comput., 218 (2012), 11138-11150 ## B. Yang, Upper estimate for positive solutions of the (p, n - p) conjugate boundary value problem, J. Math. Anal. Appl., 390 (2012), 535-548 ## C.-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 problems, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 281-291 ## C.-B. Zhai, R.-P. Song, Q.-Q. Han, The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem, Comput. Math. Appl., 62 (2011), 2639-2647 ## H.-E. Zhang, Iterative solutions for fractional nonlocal boundary value problems involving integral conditions, Bound. Value Probl., 2016 (2016), 1-13 ## X.-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 condition, Appl. Math. Comput., 235 (2014), 412-422 ## X.-G. Zhang, L.-S. Liu, Y.-H. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 37 (2014), 26-33 ## X.-Q. Zhang, L.-S. Liu, Y.-H. Wu, Fixed point theorems for the sum of three classes of mixed monotone operators and applications, Fixed Point Theory Appl., 2016 (2016), 1-22 ## X.-Q. Zhang, L. Wang, Q. Sun, Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Appl. Math. Comput., 226 (2014), 708-718 ## 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 problem, Comput. Math. Appl., 56 (2008), 1146-1154 ## Y.-L. Zhao, H.-B. Chen, L. Huang, Existence of positive solutions for nonlinear fractional functional differential equation, Comput. Math. Appl., 64 (2012), 3456-3467
Some common fixed points of multivalued mappings on complex-valued metric spaces with homotopy result Some common fixed points of multivalued mappings on complex-valued metric spaces with homotopy result en en 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. 3381 3396 Wasfi Shatanawi Department of Mathematics and general courses Department of Mathematics Prince Sultan University Hashemite University Zarqa Saudi Arabia Jordan wshatanawi@psu.edu.sa;swasfi@hu.edu.jo Mohd Salmi MD Norani School of mathematical Sciences, Faculty of Science and Technology University Kebangsaan Malaysia msn@ukm.my Jamshaid Ahmad Department of Mathematics University of Jeddah Saudi Arabia jamshaid_jasim@yahoo.com Habes Alsamir School of mathematical Sciences, Faculty of Science and Technology University Kebangsaan Malaysia h.alsamer@gmail.com Marwan Amin Kutbi Department of Mathematics King Abdulaziz University Saudi Arabia mkutbi@yahoo.com Complex-valued metric space multivalued mappings $$\alpha^*$$-admissible closed ball. Article.2.pdf  M. Abbas, M. Arshad, A. Azam, Fixed points of asymptotically regular mappings in complex-valued metric spaces, Georgian Math. J., 20 (2013), 213-221 ## M. Abbas, B. Fisher, T. Nazir, Well-Posedness and periodic point property of mappings satisfying a rational inequality in an ordered complex valued metric space, Numer. Funct. Anal. Optim., 243 (2011), 1-32 ## M. Arshad, J. Ahmad, On multivalued contractions in cone metric spaces without normality, Scientific World J., 2013 (2013), 1-3 ## M. Arshad, A. Azam, P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl., 2009 (2009), 1-11 ## J. H. Asl, S. Rezapour, N. Shahzad, On fixed points of $$\alpha-\psi$$ -contractive multifunctions, Fixed Point Theory Appl., 2012 (2012), 1-6 ## A. Azam, J. Ahmad, P. Kumam, Common fixed point theorems for multi-valued mappings in complex-valued metric spaces, J. Inequal. Appl., 2013 (2013), 1-12 ## A. Azam, B. Fisher, M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32 (2011), 243-253 ## S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181 ## S.-H. Cho, J.-S. Bae, Fixed point theorems for multivalued maps in cone metric spaces, Fixed Point Theory Appl., 2011 (2011), 1-7 ## C. Di Bari, P. Vetro, $$\phi$$ -pairs and common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo, 57 (2008), 279-285 ## J. Hassanzadeasl, Common fixed point theorems for $$\alpha-\psi$$-contractive type mappings, Int. J. Anal., 2013 (2013), 1-7 ## N. Hussain, E. Karapınar, P. Salimi, F. Akbar, $$\alpha$$-admissible mappings and related fixed point theorems, J. Inequal. Appl., 2013 (2013), 1-11 ## N. Hussain, E. Karapınar, P. Salimi, P. Vetro, Fixed point results for $$G^m$$-Meir-Keeler contractive and G-($$\alpha,\psi$$)-Meir- Keeler contractive mappings, Fixed Point Theory Appl., 2013 (2013), 1-14 ## E. Karapınar, B. Samet, Generalized $$\alpha-\psi$$ contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal., 2012 (2012), 1-17 ## C. Klin-eam, C. Suanoom, Some common fixed-point theorems for generalized-contractive-type mappings on complexvalued metric spaces, Abstr. Appl. Anal., 2013 (2013), 1-6 ## M. A. Kutbi, J. Ahmad, A. Azam, On fixed points of $$\alpha-\psi$$-contractive multivalued mappings in cone metric spaces, Abstr. Appl. Anal., 2013 (2013), 1-6 ## M. A. Kutbi, A. Azam, J. Ahmad, C. Di Bari, Some common coupled fixed point results for generalized contraction in complex-valued metric spaces, J. Appl. Math., 2013 (2013), 1-10 ## B. Mohammadi, S. Rezapour, N. Shahzad, Some results on fixed points of $$\alpha-\psi$$-Ciric generalized multifunctions, Fixed Point Theory Appl., 2013 (2013), 1-10 ## S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475-478 ## F. Rouzkard, M. Imdad, Some common fixed point theorems on complex valued metric spaces, Comput. Math. Appl., 64 (2012), 1866-1874 ## B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $$\alpha-\psi$$-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165 ## W. Sintunavarat, Y. J. Cho, P. Kumam, Urysohn integral equations approach by common fixed points in complex-valued metric spaces, Adv. Difference Equ., 2013 (2013), 1-14 ## W. Sintunavarat, P. Kumam, Generalized common fixed point theorems in complex valued metric spaces and applications, J. Inequal. Appl., 2012 (2012), 1-12 ## K. Sitthikul, S. Saejung, Some fixed point theorems in complex valued metric spaces, Fixed Point Theory Appl., 2012 (2012), 1-11
Generalized $$\mathit{Z}$$-contraction on quasi metric spaces and a fixed point result Generalized $$\mathit{Z}$$-contraction on quasi metric spaces and a fixed point result en en 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. 3397 3403 Hakan Şimşek Department of Mathematics, Faculty of Science and Arts Kirikkale University Turkey hsimsek@kku.edu.tr;hasimsek@hotmail.com Menşur Tuğba Yalçin Department of Mathematics, Faculty of Science and Arts Kirikkale University Turkey tuugbaa@hotmail.com Quasi metric space left K-Cauchy sequence simulation functions fixed point. Article.3.pdf  H. H. Alsulami, E. Karapınar, F. Khojasteh, A. F. Roldán-López-de-Hierro, A proposal to the study of contractions in quasi-metric spaces, Discrete Dyn. Nat. Soc., 2014 (2014), 1-10 ## I. Altun, G. Mınak, M. Olgun, Classification of completeness of quasi metric space and some new fixed point results, J. Nonlinear Funct. Anal., (Submitted), - ## S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181 ## F. E. Browder, W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72 (1966), 571-575 ## V. W. Bryant, A remark on a fixed-point theorem for iterated mappings, Amer. Math. Monthly, 75 (1968), 399-400 ## J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215 (1976), 241-251 ## L. B. Ćirić, On a common fixed point theorem of a Greguštype, Publ. Inst. Math. (Beograd) (N.S.), 49 (1991), 174-178 ## S. Cobzaş, Completeness in quasi-metric spaces and Ekeland Variational Principle, Topology Appl., 158 (2011), 1073-1084 ## K. Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin (1992) ## M. Edelstein, A theorem on fixed points under isometries, Amer. Math. Monthly, 70 (1963), 298-300 ## T. L. Hicks, Fixed point theorems for quasi-metric spaces, Math. Japon., 33 (1988), 231-236 ## O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996), 381-391 ## R. Kannan, Some results on fixed points, II, Amer. Math. Monthly, 76 (1969), 405-408 ## F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1189-1194 ## M. Olgun, Ö. Biçer, T. Alyildiz, A new aspect to Picard operators with simulation functions, Turkish J. Math., 40 (2016), 832-837 ## I. L. Reilly, P. V. Subrahmanyam, M. K. Vamanamurthy, Cauchy sequences in quasipseudometric spaces, Monatsh. Math., 93 (1982), 127-140 ## B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257-290 ## A. F. Roldan Lopez de Hierro, B. Samet, $$\varphi$$-admissibility results via extended simulation functions, J. Fixed Point Theory Appl., 2016 (2016), 1-19 ## J. L. Sieber, W. J. Pervin, Completeness in quasi-uniform spaces, Math. Ann., 158 (1965), 79-81 ## P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math., 80 (1975), 325-330 ## W. A. Wilson, On quasi-metric spaces, Amer. J. Math., 53 (1931), 675-684
Sufficient conditions for ergodic sensitivity Sufficient conditions for ergodic sensitivity en en 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. 3404 3408 Xiong Wang Institute for Advanced Study Shenzhen University P. R. China wangxiong8686@szu.edu.cn Xinxing Wu School of Sciences Southwest Petroleum University P. R. China wuxinxing5201314@163.com Guanrong Chen Department of Electronic Engineering City University of Hong Kong P. R. China gchen@ee.cityu.edu.hk Sensitivity ergodic sensitivity Lyapunov number. Article.4.pdf  C. Abraham, G. Biau, B. Cadre, Chaotic properties of mappings on a probability space, J. Math. Anal. Appl., 266 (2002), 420-431 ## J. Auslander, J. A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J., 32 (1980), 177-188 ## J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334 ## R. L. Devaney, An introduction to chaotic dynamical systems, Second edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA (1989) ## E. Glasner, B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075 ## R.-B. Gu, The large deviations theorem and ergodicity, Chaos Solitons Fractals, 34 (2007), 1387-1392 ## L.-F. He, X.-H. Yan, L.-S. Wang, Weak-mixing implies sensitive dependence, J. Math. Anal. Appl., 299 (2004), 300-304 ## W. Huang, P. Lu, X.-D. Ye, Measure-theoretical sensitivity and equicontinuity, Israel J. Math., 183 (2011), 233-283 ## Q.-L. Huang, Y.-M. Shi, L.-J. Zhang, Sensitivity of non-autonomous discrete dynamical systems, Appl. Math. Lett., 39 (2015), 31-34 ## S. Kolyada, O. Rybak, On the Lyapunov numbers, Colloq. Math., 131 (2013), 209-218 ## R.-S. Li, The large deviations theorem and ergodic sensitivity, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 819-825 ## R.-S. Li, Y.-M. Shi, Several sufficient conditions for sensitive dependence on initial conditions, Nonlinear Anal., 72 (2010), 2716-2720 ## T. Y. Li, J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992 ## T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126 ## X.-X. Wu, Chaos of transformations induced onto the space of probability measures, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-12 ## X.-X. Wu, A remark on topological sequence entropy, Internat. J. Bifur. Chaos Appl. Sci. Engrg., (accepted), - ## X.-X. Wu, G.-R. Chen, Sensitivity and transitivity of fuzzified dynamical systems, Inform. Sci., 396 (2017), 14-23 ## X.-X. Wu, P. Oprocha, G.-R. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942-1972 ## X.-X. Wu, X. Wang, On the iteration properties of large deviations theorem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-6 ## X.-X. Wu, J.-J. Wang, G.-R. Chen, F-sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26 ## X.-X. Wu, L.-D. Wang, G.-R. Chen, Weighted backward shift operators with invariant distributionally scrambled subsets, Ann. Funct. Anal., 8 (2017), 199-210 ## J.-D. Yin, Z.-L. Zhou, Weakly almost periodic points and some chaotic properties of dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1-10
Stochastic stability analysis for a neutral-type neural networks with Markovian jumping parameters Stochastic stability analysis for a neutral-type neural networks with Markovian jumping parameters en en 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. 3409 3418 Song Guo Department of Mathematics Huaiyin Normal University P. R. China guosong77@hytc.edu.cn Bo Du Department of Mathematics Huaiyin Normal University P. R. China dubo7307@163.com Markovian jumping parameters linear matrix inequality mean square stability. Article.5.pdf  S. Arik, Global robust stability analysis of neural networks with discrete time delays, Chaos Solitons Fractals, 26 (2005), 1407-1414 ## S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1994) ## H.-W. Chen, Z.-M. He, J.-L. Li, Multiplicity of solutions for impulsive differential equation on the half-line via variational methods, Bound. Value Probl., 2016 (2016), 1-15 ## K. Gu, An integral inequality in the stability problem of time-delay systems, Proceedings of the 39th IEEE Conference on Decision and Control, Sydney Australia, 3 (2000), 2805-2810 ## Z.-J. Gui, W.-G. Ge, X.-S. Yang, Periodic oscillation for a Hopfield neural networks with neutral delays, Phys. Lett. A, 364 (2007), 267-273 ## M. P. Kennedy, L. O. Chua, Neural networks for nonlinear programming, IEEE Trans. Circuits and Systems, 35 (1988), 554-562 ## Y.-R. Liu, Z.-D. Wang, X.-H. Liu, Exponential synchronization of complex networks with Markovian jump and mixed delays, Phys. Lett. A, 372 (2008), 3986-3998 ## Y.-R. Liu, Z.-D. Wang, X.-H. Liu, State estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delays, Phys. Lett. A, 372 (2008), 7147-7155 ## S.-S. Mou, H.-J. Gao, J. Lam, W.-Y. Qiang, A new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay, IEEE Trans. Neural Netw., 19 (2008), 532-535 ## J. Pan, X.-Z. Liu, W. -C. Xie, Exponential stability of a class of complex-valued neural networks with time-varying delays, Neurocomputing, 164 (2015), 293-299 ## J. H. Park, C. H. Park, O. M. Kwon, S. M. Lee, A new stability criterion for bidirectional associative memory neural networks of neutral-type, Appl. Math. Comput., 199 (2008), 716-722 ## R. Rakkiyappan, P. Balasubramaniam, J.-D. Cao, Global exponential stability results for neutral-type impulsive neural networks, Nonlinear Anal. Real World Appl., 11 (2010), 122-130 ## J. E. Slotine, W.-P. Li, Applied nonlinear control, Prentice-Hall, Englewood Cliffs, New Jersey (1991) ## Z.-D. Wang, Y.-R. Liu, X.-H. Liu, State estimation for jumping recurrent neural networks with discrete and distributed delays, Neural Netw., 22 (2009), 41-48 ## Z.-H. Xia, X.-H. Wang, X.-M. Sun, Q. Wang, A secure and dynamic multi-keyword ranked search scheme over encrypted cloud data, IEEE Trans. Parallel Distrib. Syst., 27 (2015), 340-352 ## Y. Xu, Z.-M. He, Exponential stability of neutral stochastic delay differential equations with Markovian switching, Appl. Math. Lett., 52 (2016), 64-73 ## K.-W. Yu, C.-H. Lien, Stability criteria for uncertain neutral systems with interval time-varying delays, Chaos Solitons Fractals, 38 (2008), 650-657 ## J. Zhao, D. J. Hill, T. Liu, Global bounded synchronization of general dynamical networks with nonidentical nodes, IEEE Trans. Automat. Controll, 57 (2012), 2656-2662
Numerical and exact solutions for time fractional Burgers' equation Numerical and exact solutions for time fractional Burgers' equation en en 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. 3419 3428 Asıf Yokuş Department of Actuary Firat University Turkey asfyokus@firat.edu.tr Doğan Kaya Department of Mathematics Istanbul Commerce University Turkey dogank@ticaret.edu.tr Nonlinear time fractional Burgers’ equation an expansion method finite difference method Caputo formula linear stability Cole-Hopf transformation. Article.6.pdf  D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403-1412 ## W. Chen, L.-J. Ye, H.-G. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl., 59 (2010), 1614-1620 ## P. A. Clarkson, New similarity solutions for the modified Boussinesq equation, J. Phys. A, 22 (1989), 2355-2367 ## S. A. Elwakil, S. K. El-labany, M. A. Zahran, R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A, 299 (2002), 179-188 ## E.-G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218 ## A. Gorguis, A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers’ equations, Appl. Math. Comput., 173 (2006), 126-136 ## S.-M. Guo, Y.-B. Zhou, The extended ($$\frac{G'}{G}$$ )-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Appl. Math. Comput., 215 (2010), 3214-3221 ## J.-H. He, X.-H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700-708 ## S. U. Islam, A. J. Khattakand, I. A. Tirmizi, A meshfree method for numerical solution of KdV equation, Eng. Anal. Bound. Elem., 32 (2008), 849-855 ## F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191 (2007), 12-20 ## F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuous-time finance, II, the waiting-time distribution, Phys. A, 287 (2000), 468-481 ## M. M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77 ## K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993) ## Z. M. Odibat, N. T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput., 186 (2007), 286-293 ## K. B. Oldham, J. Spanier, The fractional calculus, Theory 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-London (2006) ## E. J. Parkes, B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun., 98 (1996), 288-300 ## I. 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, CA (1999) ## E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Phys. A, 284 (2000), 376-384 ## E. Sousa, Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys., 228 (2009), 4038-4054 ## L.-J. Su, W.-Q. Wang, Q.-Y. Xu, Finite difference methods for fractional dispersion equations, Appl. Math. Comput., 216 (2010), 3329-3334 ## L.-J. Su, W.-Q. Wang, Z.-X. Yang, Finite difference approximations for the fractional advection-diffusion equation, Phys. Lett. A, 373 (2009), 4405-4408 ## M.-L. Wang, X.-Z. Li, J.-L. Zhang, The ($$\frac{G'}{G}$$ )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417-423 ## A. M. Wazwaz, The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd- Bullough equations, Chaos Solitons Fractals, 25 (2005), 55-63 ## A. Yokus, Solutions of some nonlinear partial differential equations and comparison of their solutions, Ph.D. Thesis, Fırat University, Elazig, Turkey (2011) ## S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. , 216 (2006), 264-274 ## G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580 ## X.-D. Zheng, Y. Chen, H.-Q. Zhang, Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Phys. Lett. A, 311 (2003), 145-157 ## P. Zhuang, F. Liu, V. Anh, I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46 (2008), 1079-1095
Fuzzy vector metric spaces and some results Fuzzy vector metric spaces and some results en en 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. 3429 3436 Şehla Eminoğlu Department of Mathematics, Faculty of Science Gazi University Turkey sehla_eminoglu@hotmail.com Cüneyt Çevik Department of Mathematics, Faculty of Science Gazi University Turkey ccevik@gazi.edu.tr Vector metric space fuzzy vector metric space Riesz space fuzzy diameter. Article.7.pdf  C. D. Aliprantis, K. C. Border, Infinite-dimensional analysis, A hitchhiker’s guide, Second edition, Springer-Verlag, Berlin (1999) ## C. D. Aliprantis, O. Burkinshaw, Positive operators, Reprint of the 1985 original, Springer, Dordrecht (2006) ## C. Çevik, On continuity of functions between vector metric spaces, J. Funct. Space, 2014 (2014), 1-6 ## C. Çevik, I. Altun, Vector metric spaces and some properties, Topol. Methods Nonlinear Anal., 34 (2009), 375-382 ## A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399 ## A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90 (1997), 365-368 ## I. Kramosil, J. Michałek, Fuzzy metrics and statistical metric spaces, Kybernetika (Prague), 11 (1975), 336-344 ## W. A. J. Luxemburg, A. C. Zaanen, Riesz spaces, Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York (1971) ## J. R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1975) ## W. Rudin, Functional analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw- Hill, Inc., New York (1991) ## B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math., 10 (1960), 313-334 ## B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York (1983) ## E. S. Şuhubi, Functional analysis, Kluwer Academic Publishers, Dordrecht (2003) ## L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353
On nonexpansive and accretive operators in Banach spaces On nonexpansive and accretive operators in Banach spaces en en 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 3437 3446 Dongfeng Li School of Information Engineering North China University of Water Resources and Electric Power China sylidf@yeah.net Accretive operator approximation solution viscosity method variational inequality. Article.8.pdf  I. K. Argyros, S. George, Iterative regularization methods for nonlinear ill-posed operator equations with m-accretive mappings in Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 35 (2015), 1318-1324 ## I. K. Argyros, S. George, Extending the applicability of a new Newton-like method for nonlinear equations, Commun. Optim. Theory, 2016 (2016), 1-9 ## V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden (1976) ## B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X.-L. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 1-14 ## B. A. Bin Dehaish, X.-L. Qin, A. Latif, H. O. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321-1336 ## F. E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1080-1086 ## R. E. Bruck, Jr., A strongly convergent iterative solution of $$0 \in U(x)$$ for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl., 48 (1974), 114-126 ## S.-S. Chang, H. W. J. Lee, C. K. Chan, Strong convergence theorems by viscosity approximation methods for accretive mappings and nonexpansive mappings, J. Appl. Math. Inform., 27 (2009), 59-68 ## C. E. Chidume, Iterative solutions of nonlinear equations in smooth Banach spaces, Nonlinear Anal., 26 (1996), 1823-1834 ## S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427-438 ## S. Y. Cho, X.-L. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-15 ## J. S. Jung, Y. J. Cho, H.-Y. Zhou, Iterative processes with mixed errors for nonlinear equations with perturbed m-accretive operators in Banach spaces, Appl. Math. Comput., 133 (2002), 389-406 ## T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508-520 ## L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114-125 ## A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55 ## X.-L. Qin, S. Y. Cho, Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 488-502 ## X.-L. Qin, S. Y. Cho, J. K. Kim, On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings, Optimization, 61 (2012), 805-821 ## X.-L. Qin, S. Y. Cho, L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl., 2013 (2013), 1-17 ## X.-L. Qin, J.-C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 1-9 ## S. Reich, On fixed point theorems obtained from existence theorems for differential equations, J. Math. Anal. Appl., 54 (1976), 26-36 ## R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898 ## T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 1-21 ## H.-Y. Zhou, A characteristic condition for convergence of steepest descent approximation to accretive operator equations, J. Math. Anal. Appl., 271 (2002), 1-6 ## H.-Y. Zhou, Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces, Nonlinear Anal., 70 (2009), 4039-4046
On common fixed points that belong to the zero set of a certain function On common fixed points that belong to the zero set of a certain function en en 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. 3447 3455 Erdal Karapinar Department of Mathematics Atilim University Turkey erdalkarapinar@yahoo.com Bessem Samet Department of Mathematics, College of Science King Saud University Saudi Arabia bsamet@ksu.edu.sa Priya Shahi St. Andrews College of Arts, Science and Commerce India priya.thaparian@gmail.com $$\varphi$$ -admissibility common fixed point zero set partial metric. Article.9.pdf  M. Abbas, I. Altun, S. Romaguera, Common fixed points of Ćirić-type contractions on partial metric spaces, Publ. Math. Debrecen, 82 (2013), 425-438 ## T. Abdeljawad, E. Karapınar, K. Taş, A generalized contraction principle with control functions on partial metric spaces, Comput. Math. Appl., 63 (2012), 716-719 ## D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464 ## L. Ćirić, B. Samet, H. Aydi, C. Vetro, Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput., 218 (2011), 2398-2406 ## R. Heckmann, Approximation of metric spaces by partial metric spaces, Applications of ordered sets in computer science, Braunschweig, (1996), Appl. Categ. Structures, 7 (1999), 71-83 ## E. Karapınar, D. O’Regan, B. Samet, On the existence of fixed points that belong to the zero set of a certain function, Fixed Point Theory Appl., 2015 (2015), 1-14 ## S. G. Matthews, Partial metric topology, Papers on general topology and applications, Flushing, NY, (1992), Ann. New York Acad. Sci., New York Acad. Sci., New York, 728 (1994), 183-197 ## S. Oltra, O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Istit. Mat. Univ. Trieste, 36 (2004), 17-26 ## S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl., 2010 (2010), 1-6 ## S. Romaguera, Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl., 218 (2011), 2398-2406 ## S. Romaguera, Matkowski’s type theorems for generalized contractions on (ordered) partial metric spaces, Appl. Gen. Topol., 12 (2011), 213-220 ## O. Valero, On Banach fixed point theorems for partial metric spaces, Appl. Gen. Topol., 6 (2005), 229-240
Common fixed points of $$\alpha$$-dominated multivalued mappings on closed balls in a dislocated quasi b-metric space Common fixed points of $$\alpha$$-dominated multivalued mappings on closed balls in a dislocated quasi b-metric space en en 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. 3456 3476 Abdulaziz Saleem Moslem Alofi Department of Mathematics King Abdulaziz University Saudi Arabia aalofi1@kau.edu.sa Abdullah Eqal Al-Mazrooei Department of Mathematics Department of Mathematics King Abdulaziz University University of Jeddah Saudi Arabia Saudi Arabia aealmazrooei@uj.edu.sa Bahru Tsegaye Leyew Department of Mathematics and Applied Mathematics University of Pretoria South Africa tsegayebah@gmail.com Mujahid Abbas Department of Mathematics Department of Mathematics King Abdulaziz University University of Management and Technology Saudi Arabia Pakistan mujahid.abbas@up.ac.za K-sequentially complete dislocated quasi b-metric spaces $$\alpha$$-dominated multivalued mapping closed ball common fixed point. Article.10.pdf  A. Alam, M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl., 17 (2015), 693-702 ## M. A. Alghamdi, N. Hussain, P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces, J. Inequal. Appl., 2013 (2013), 1-25 ## T. V. An, L. Q. Tuyen, N. V. Dung, Stone-type theorem on b-metric spaces and applications, Topology Appl., 185/186 (2015), 50-64 ## M. Arshad, A. Shoaib, I. Beg, Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space, Fixed Point Theory Appl., 2013 (2013), 1-15 ## A. Azam, M. Waseem, M. Rashid, Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-14 ## I. A. Bakhtin, The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 30 (1989), 26-37 ## S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181 ## M. Boriceanu, Fixed point theory for multivalued generalized contraction on a set with two b-metrics, Stud. Univ. Babeş- Bolyai Math., 54 (2009), 3-14 ## L. B. Ćirić, Fixed points for generalized multi-valued contractions, Mat. Vesnik, 9 (1972), 265-272 ## S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11 ## S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263-276 ## P. Hitzler, Generalized metrics and topology in logic programming semantics, Ph.D. Thesis, School of Mathematics, Applied Mathematics and Statistics, National University Ireland,, University College Cork (2001) ## P. Hitzler, A. K. Seda, Dislocated topologies, J. Electr. Eng., 51 (2000), 3-7 ## N. Hussain, D. Dorić, Z. Kadelburg, S. Radenović, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012 (2012), 1-12 ## N. Hussain, J. R. Roshan, V. Parvaneh, M. Abbas, Common fixed point results for weak contractive mappings in ordered b-dislocated metric spaces with applications, J. Inequal. Appl., 2013 (2013), 1-21 ## N. Hussain, P Salimi, A. Latif, Fixed point results for single and set-valued $$\alpha-\eta-\psi-$$contractive mappings, Fixed Point Theory Appl., 2013 (2013), 1-23 ## R. Kannan, Some results on fixed points, II, Amer. Math. Monthly, 76 (1969), 405-408 ## C. Klin-eam, C. Suanoom, Dislocated quasi-b-metric spaces and fixed point theorems for cyclic contractions, Fixed Point Theory Appl., 2015 (2015), 1-12 ## A. Latif, A. A. N. Abdou, Multivalued generalized nonlinear contractive maps and fixed points, Nonlinear Anal., 74 (2011), 1436-1444 ## A. Latif, D. T. Luc, Variational relation problems: existence of solutions and fixed points of contraction mappings, Fixed Point Theory Appl., 2013 (2013), 1-10 ## A. Latif, I. Tweddle, Some results on coincidence points, Bull. Austral. Math. Soc., 59 (1999), 111-117 ## S. Lipschutz, Schaum’s outline of theory and problems of set theory and related topics, McGraw-Hill, New York (1964) ## S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475-488 ## J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239 ## J. J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.), 23 (2007), 2205-2212 ## M. U. Rahman, M. Sarwar, Dislocated quasi b-metric space and fixed point theorems, Electron. J. Math. Anal. Appl., 4 (2016), 16-24 ## A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2003), 1435-1443 ## I. L. Reilly, P. V. Subrahmanyam, M. K. Vamanamurthy, Cauchy sequences in quasipseudometric spaces, Monatsh. Math., 93 (1982), 127-140 ## J. R. Roshan, N. Hussain, S. Sedghi, N. Shobkolaei, Suzuki-type fixed point results in b-metric spaces, Math. Sci. (Springer), 9 (2015), 153-160 ## J. R. Roshan, V. Parvaneh, I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput., 226 (2014), 725-737 ## B. Samet, M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal., 13 (2012), 82-97 ## M. H. Shah, N. Hussain, Nonlinear contractions in partially ordered quasi b-metric spaces, Commun. Korean Math. Soc., 27 (2012), 117-128 ## A. Shoaib, M. Arshadatjana, S. Radenović, $$\alpha$$-dominated mappings, dislocated metric spaces and fixed point results, Fixed Point Theory Appl., ( to appeare), - ## W. A. Wilson, On quasi-metric spaces, Amer. J. Math., 53 (1931), 675-684 ## F. M. Zeyada, G. H. Hassan, M. A. Ahmed, A generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spaces, Arab. J. Sci. Eng. Sect. A Sci., 31 (2006), 111-114 ## C.-X. Zhu, C.-F. Chen, X.-Z. Zhang, Some results in quasi-b-metric-like spaces, J. Inequal. Appl., 2014 (2014), 1-8
Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms en en 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. 3477 3489 M. R. S. Kulenovic Department of Mathematics University of Rhode Island USA mkulenovic@mail.uri.edu S. Moranjkic Department of Mathematics University of Tuzla Bosnia and Herzegovina samra.moranjkic@untz.ba Z. Nurkanovic Department of Mathematics University of Tuzla Bosnia and Herzegovina zehra.nurkanovic@untz.ba Attractivity bifurcation difference equation invariant Naimark-Sacker bifurcation periodic solution. Article.11.pdf  A. M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, I, Int. J. Difference Equ., 3 (2008), 1-35 ## A. M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, II, Int. J. Difference Equ., 3 (2008), 195-225 ## J. K. Hale, H. Kocak, Dynamics and bifurcations, Texts in Applied Mathematics, Springer-Verlag, New York (1991) ## E. A. Janowski, M. R. S. Kulenović, Attractivity and global stability for linearizable difference equations, Comput. Math. Appl., 57 (2009), 1592-1607 ## C. M. Kent, H. Sedaghat, Global attractivity in a quadratic-linear rational difference equation with delay, J. Difference Equ. Appl., 15 (2009), 913-925 ## C. M. Kent, H. Sedaghat, Global attractivity in a rational delay difference equation with quadratic terms, J. Difference Equ. Appl., 17 (2011), 457-466 ## M. R. S. Kulenović, G. Ladas, Dynamics of second order rational difference equations, With open problems and conjectures, Chapman & Hall/CRC, Boca Raton, FL (2001) ## M. R. S. Kulenović, O. Merino, Discrete dynamical systems and difference equations with Mathematica, Chapman & Hall/CRC, Boca Raton, FL (2002) ## M. R. S. Kulenović, O. Merino, A global attractivity result for maps with invariant boxes, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 97-110 ## M. R. S. Kulenović, O. Merino, Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149 ## M. R. S. Kulenović, O. Merino, Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2471-2486 ## M. R. S. Kulenović, E. Pilav, E. Silić, Naimark-Sacker bifurcation of a certain second order quadratic fractional difference equation, J. Math. Comput. Sci., 4 (2014), 1025-1043 ## Y. A. Kuznetsov, Elements of applied bifurcation theory, Second edition. Applied Mathematical Sciences, Springer- Verlag, New York (1998) ## C. Robinson, Stability, symbolic dynamics, and chaos, Stud. Adv. Math. Boca Raton, CRC Press, FL (1995) ## H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Difference Equ. Appl., 15 (2009), 215-224 ## S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Second edition, Texts in Applied Mathematics, Springer-Verlag, New York (2003)
Norm inequalities of operators and commutators on generalized weighted morrey spaces Norm inequalities of operators and commutators on generalized weighted morrey spaces en en 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. 3490 3501 Yue Hu School of Mathematics and Information Henan Polytechnic University P. R. China huu3y6@163.com Yueshan Wang Department of Mathematics Jiaozuo University P. R. China wangys1962@163.com Singular integral with rough kernel Bochner-Riesz operator Marcinkiewicz integral commutator weighted Morrey space. Article.12.pdf  J. Álvarez, R. J. Bagby, D. S. Kurtz, C. Pérez, Weighted estimates for commutators of linear operators, Studia Math., 104 (1993), 195-209 ## S. Bochner, Summation of multiple Fourier series by spherical means, Trans. Amer. Math. Soc, 40 (1936), 175-207 ## Y. Ding, D.-S. Fan, Y.-B. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J., 48 (1999), 1037-1055 ## Y. Ding, S.-Z. Lu, K. Yabuta, On commutators of Marcinkiewicz integrals with rough kernel, J. Math. Anal. Appl, 275 (2002), 60-68 ## J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc., 336 (1993), 869-880 ## Y. Hu, Y.-S. Wang, Multilinear fractional integral operators on generalized weighted Morrey spaces, J. Inequal. Appl., 2014 (2014), 1-18 ## F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426 ## Y. Komori, S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr., 289 (2009), 219-231 ## T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic analysis, Sendai, (1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, (1991), 183-189 ## C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166 ## B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226 ## X. L. Shi, Q. Y. Sun, Weighted norm inequalities for Bochner-Riesz operators and singular integral operators, Proc. Amer. Math. Soc., 116 (1992), 665-673 ## E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, Princeton University Press, Princeton, N.J. (1971) ## R. L. Wheeden, A. Zygmund, Measure and integral, An introduction to real analysis. Pure and Applied Mathematics, Marcel Dekker, Inc., New York-Basel (1977)
Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces en en 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. 3502 3518 Lu-Chuan Ceng Department of Mathematics Shanghai Normal University China zenglc@hotmail.com Ching-Feng Wen Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization Department of Medical Research Kaohsiung Medical University Kaohsiung Medical University Hospital Taiwan Taiwan cfwen@kmu.edu.tw Nonexpansive mapping fixed point variational inequality global convergence. Article.13.pdf  K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl., 2006 (2006), 1-13 ## F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228 ## N. Buong, N. T. H. Phuong, Strong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods, J. Optim. Theory Appl., 159 (2013), 399-411 ## N. Buong, N. T. Quynh Anh, An implicit iteration method for variational inequalities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), 1-10 ## L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numer. Funct. Anal. Optim., 29 (2008), 987-1033 ## L.-C. Ceng, S.-M. Guu, J.-C. Yao, Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems, Fixed Point Theory Appl., 2012 (2012), 1-19 ## 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 mappings, J. Math. Inequal., 1 (2007), 243-258 ## L.-C. Ceng, J.-C. Yao, Relaxed viscosity approximation methods for fixed point problems and variational inequality problems, Nonlinear Anal., 69 (2008), 3299-3309 ## Y. Censor, A. Gibali, S. Reich, Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132 ## S.-S. Chang, Some problems and results in the study of nonlinear analysis, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7, Athens, (1996), Nonlinear Anal., 33 (1997), 4197-4208 ## I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1990) ## F. Facchinei, J.-S. Pang, Finite-dimensional variational inequalities and complementarity problems, Vol. I, Springer Series in Operations Research, Springer-Verlag, New York (2003) ## R. Glowinski, J.-L. Lions, R. Trémolières, Numerical analysis of variational inequalities, Translated from the French, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York (1981) ## A. N. Iusem, B. F. Svaiter, A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321 ## J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., (2005), 509-520 ## M. Kikkawa, W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Hilbert spaces, RIMS Kokyuroku, 1484 (2006), 105-113 ## M. Kikkawa, W. Takahashi, Strong convergence theorems by the viscosity approximation method for a countable family of nonexpansive mappings, Taiwanese J. Math., 12 (2008), 583-598 ## N. Kikuchi, J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988) ## D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1980) ## I. V. Konnov, Combined relaxation methods for variational inequalities, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin (2001) ## I. V. Konnov, Equilibrium models and variational inequalities, Mathematics in Science and Engineering, Elsevier B. V., Amsterdam (2007) ## G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747-756 ## A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55 ## J. G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Anal., 64 (2006), 2022-2042 ## K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001), 387-404 ## N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 3641-3645 ## M. V. Solodov, B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776 ## T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239 ## W. Takahashi, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Proceedings of Workshop on Fixed Point Theory, Kazimierz Dolny, (1997), Ann. Univ. Mariae Curie-Skodowska Sect. A, 51 (1997), 277-292 ## S.-H. Wang, L.-X. Yu, B.-H. Guo, An implicit iterative scheme for an infinite countable family of asymptotically nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. , 2008 (2008), 1-10 ## H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256 ## I. Yamada, The 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 applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam, 8 (2001), 473-504 ## Y.-H. Yao, R.-D. Chen, H.-K. Xu, Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447-3456 ## Y.-H. Yao, Y.-C. Liou, S. M. Kang, Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method, Comput. Math. Appl., 59 (2010), 3472-3480 ## Y.-H. Yao, M. A. Noor, Y.-C. Liou, Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities, Abstr. Appl. Anal., 2012 (2012), 1-9 ## Y.-H. Yao, M. A. Noor, Y.-C. Liou, S. M. Kang, Iterative algorithms for general multivalued variational inequalities, Abstr. Appl. Anal., 2012 (2012), 1-10 ## Y.-H. Yao, M. Postolache, Y.-C. Liou, Z.-S. Yao, Construction algorithms for a class of monotone variational inequalities, Optim. Lett., 10 (2016), 1519-1528 ## H. Zegeye, N. Shahzad, Y.-H. Yao, Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453-471 ## E. Zeidler, Nonlinear functional analysis and its applications, III, Variational methods and optimization, Translated from the German by Leo F. Boron, Springer-Verlag, New York (1985) ## L.-C. Zeng, J.-C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal., 64 (2006), 2507-2515 ## H.-Y. Zhou, L. Wei, Y. J. Cho, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces , Appl. Math. Comput., 173 (2006), 196-212
$$L_p$$-dual geominimal surface areas for the general $$L_p$$-intersection bodies $$L_p$$-dual geominimal surface areas for the general $$L_p$$-intersection bodies en en 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. 3519 3529 Zhonghuan Shen Department of Mathematics China Three Gorges University China 278906478@qq.com Yanan Li Department of Mathematics China Three Gorges University China 502430218@qq.com Weidong Wang Department of Mathematics China Three Gorges University China wdwxh722@163.com General $$L_p$$-intersection body $$L_p$$-dual geominimal surface area extremum value Brunn-Minkowski inequality Busemann-Petty problem. Article.14.pdf  Y.-B. Feng, W.-D. Wang, General $$L_p$$-harmonic Blaschke bodies, Proc. Indian Acad. Sci. Math. Sci., 124 (2014), 109-119 ## Y.-B. Feng, W.-D. Wang, $$L_p$$-dual mixed geominimal surface area, Glasg. Math. J., 56 (2014), 229-239 ## R. J. Gardner, Geometric tomography, Second edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, New York (2006) ## C. Haberl, $$L_p$$ intersection bodies, Adv. Math., 217 (2008), 2599-2624 ## C. Haberl, M. Ludwig, A characterization of $$L_p$$ intersection bodies, Int. Math. Res. Not., 2006 (2006), 1-29 ## C. Haberl, F. E. Schuster, Asymmetric affine $$L_p$$ Sobolev inequalities, J. Funct. Anal., 257 (2009), 641-658 ## C. Haberl, F. E. Schuster, General $$L_p$$ affine isoperimetric inequalities, J. Differential Geom., 83 (2009), 1-26 ## C. Haberl, F. E. Schuster, J. Xiao, An asymmetric affine Pólya -Szegö principle, Math. Ann., 352 (2012), 517-542 ## G. H. Hardy, J. E. Littlewood, G. Pólya , Inequalities, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge (1988) ## Y.-N. Li, W.-D.Wang, The $$L_p$$-dual mixed geominimal surface area for multiple star bodies, J. Inequal. Appl., 2014 (2014), 1-10 ## Z.-F. Li, W.-D. Wang, General $$L_p$$-mixed chord integrals of star bodies, J. Inequal. Appl., 2016 (2016), 1-12 ## M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc., 357 (2005), 4191-4213 ## M. Ludwig, Intersection bodies and valuations, Amer. J. Math., 128 (2006), 1409-1428 ## E. Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math., 71 (1988), 232-261 ## E. Lutwak, The Brunn-Minkowski-Firey theory, I, Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150 ## E. Lutwak, The Brunn-Minkowski-Firey theory, II, Affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294 ## L. Parapatits, SL(n)-contravariant $$L_p$$-Minkowski valuations, Trans. Amer. Math. Soc., 366 (2014), 1195-1211 ## L. Parapatits, SL(n)-covariant $$L_p$$-Minkowski valuations, J. Lond. Math. Soc., 89 (2014), 397-414 ## Y.-N. Pei, W.-D. Wang, A type of Busemann-Petty problems for general $$L_p$$-intersection bodies, Wuhan Univ. J. Nat. Sci., 20 (2015), 471-475 ## Y.-N. Pei, W.-D. Wang, Shephard type problems for general $$L_p$$-centroid bodies, J. Inequal. Appl., 2015 (2015), 1-13 ## C. M. Petty, Geominimal surface area, Geometriae Dedicata, 3 (1974), 77-97 ## F. E. Schuster, T. Wannerer, GL(n) contravariant Minkowski valuations, Trans. Amer. Math. Soc., 364 (2012), 815-826 ## F. E. Schuster, M. Weberndorfer, Volume inequalities for asymmetric Wulff shapes, J. Differential Geom., 92 (2012), 263-283 ## X. Y. Wan, W.-D. Wang, $$L_p$$-dual geominimal surface area, (Chinese) J. Wuhan Univ. Natur. Sci. Ed., 59 (2013), 515-518 ## W.-D. Wang, Y.-B. Feng, A general $$L_p$$-version of Petty’s affine projection inequality, Taiwanese J. Math., 17 (2013), 517-528 ## W.-D. Wang, Y.-N. Li, Busemann-Petty problems for general $$L_p$$-intersection bodies, Acta Math. Sin. (Engl. Ser.), 31 (2015), 777-786 ## W.-D. Wang, Y.-N. Li, General $$L_p$$-intersection bodies, Taiwanese J. Math., 19 (2015), 1247-1259 ## W.-D. Wang, T.-Y. Ma, Asymmetric $$L_p$$-difference bodies, Proc. Amer. Math. Soc., 142 (2014), 2517-2527 ## J.-Y. Wang, W.-D. Wang, General $$L_p$$-dual Blaschke bodies and the applications, J. Inequal. Appl., 2015 (2015), 1-11 ## W.-D. Wang, J.-Y. Wang, Extremum of geometric functionals involving general $$L_p$$-projection bodies, J. Inequal. Appl., 2016 (2016), 1-16 ## T. Wannerer, GL(n) equivariant Minkowski valuations, Indiana Univ. Math. J., 60 (2011), 1655-1672 ## M. Weberndorfer, Shadow systems of asymmetric $$L_p$$ zonotopes, Adv. Math., 240 (2013), 613-635 ## W. Weidong, Q. Chen, $$L_p$$-dual geominimal surface area, J. Inequal. Appl., 2011 (2011), 1-10 ## W. Weidong, W. Xiaoyan, Shephard type problems for general $$L_p$$-projection bodies, Taiwanese J. Math., 16 (2012), 1749-1762 ## L. Yan, W.-D. Wang, General $$L_p$$-mixed-brightness integrals, J. Inequal. Appl., 2015 (2015), 1-11 ## D.-P. Ye, $$L_p$$ geominimal surface areas and their inequalities, Int. Math. Res. Not. IMRN, 2015 (2015), 2465-2498 ## D.-P. Ye, B.-C. Zhu, J.-Z. Zhou, The mixed $$L_p$$ geominimal surface areas for multiple convex bodies, Indiana Univ. Math. J., 64 (2015), 1513-1552 ## F. Yibin, W. Weidong, L. Fenghong, Some inequalities on general $$L_p$$-centroid bodies, Math. Inequal. Appl., 18 (2015), 39-49 ## B.-C. Zhu, N. Li, J.-Z. Zhou, Isoperimetric inequalities for $$L_p$$ geominimal surface area, Glasg. Math. J., 53 (2011), 717-726 ## B.-C. Zhu, J.-Z. Zhou, W.-X. Xu, Affine isoperimetric inequalities for $$L_p$$ geominimal surface area, Real and complex submanifolds, Springer Proc. Math. Stat., Springer, Tokyo, 106 (2014), 167-176 ## B.-C. Zhu, J.-Z. Zhou, W.-X. Xu, $$L_p$$ mixed geominimal surface area, J. Math. Anal. Appl., 422 (2015), 1247-1263
The viscosity approximation forward-backward splitting method for the implicit midpoint rule of quasi inclusion problems in Banach spaces The viscosity approximation forward-backward splitting method for the implicit midpoint rule of quasi inclusion problems in Banach spaces en en 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. 3530 3543 Li Yang School of Science South West University of Science and Technology China scmyxkdyl@163.com Fuhai Zhao School of Science South West University of Science and Technology China scmyxkdzfh@163.com Viscosity approximation Banach space splitting method forward-backward algorithm the implicit midpoint rule. Article.15.pdf  M. A. Alghamdi, N. Shahzad, H.-K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014), 1-9 ## H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769-806 ## W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math., 56 (1989), 469-499 ## G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373-398 ## J. B. Baillon, G. Haddad, Quelques proprits des oprateurs angle-borns et n-cycliquement monotones, (French) Israel J. Math., 26 (1977), 137-150 ## D. P. Bertsekas, J. N. Tsitsiklis, Parallel and distributed computation: numerical methods, Englewood Cliffs: Prentice Hall, NJ (1989) ## H. Brézis, P.-L. Lions, Produits infinis de résolvantes, (French) Israel J. Math., 29 (1978), 329-345 ## C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120 ## G. H. G. Chen, R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim., 7 (1997), 421-444 ## C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, Springer- Verlag London, Ltd., London (2009) ## S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427-438 ## P. Cholamjiak, A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces, Numer. Algorithms, 2015 (2015), 915-932 ## I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1990) ## P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748 ## P. L. Combettes, V. R.Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200 ## P. Deuhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., 27 (1985), 505-535 ## J. Douglas, Jr., H. H. Rachford, Jr., On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc., 82 (1956), 421-439 ## J. C. Dunn, Convexity, monotonicity, and gradient processes in Hilbert space, J. Math. Anal. Appl., 53 (1976), 145-158 ## O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403-419 ## S.-N. He, C.-P. Yang, Solving the variational inequality problem defined on intersection of finite level sets, Abstr. Appl. Anal., 2013 (2013), 1-8 ## P.-L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979 ## G. López, V. Martín-Márquez, F.-H. Wang, H.-K. Xu, Forward-backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012 (2012), 1-25 ## P. E. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479 ## B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, (French) Rev. Franaise Informat. Recherche Opérationnelle, 4 (1970), 154-158 ## D. S. Mitrinović, Analytic inequalities, In cooperation with P. M. Vasić, Die Grundlehren der mathematischen Wissenschaften, Band 165 Springer-Verlag, New York-Berlin (1970) ## A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55 ## G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383-390 ## D. H. Peaceman, H. H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41 ## X.-L. Qin, S. Y. Cho, L. Wang, A regularization method for treating zero points of the sum of two monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-10 ## X.-L. Qin, J.-C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 1-9 ## S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287-292 ## R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216 ## R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control. Optim., 14 (1976), 877-898 ## C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 (1993), 1-10 ## S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327-332 ## W. Takahashi, N.-C. Wong, J.-C. Yao, Two generalized strong convergence theorems of Halpern’s type in Hilbert spaces and applications, Taiwanese J. Math., 16 (2012), 1151-1172 ## P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446 ## M. van Veldhuizen, Asymptotic expansions of the global error for the implicit midpoint rule (stiff case), Computing, 33 (1984), 185-192 ## F.-H.Wang, H.-H. Cui, On the contraction-proximal point algorithms with multi-parameters, J. Global Optim., 54 (2012), 485-491 ## H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138 ## H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291 ## H.-T. Zegeye, N. Shahzad, Strong convergence theorems for a common zero for a finite family of m-accretive mappings, Nonlinear Anal., 66 (2007), 1161-1169
Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions en en 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. 3544 3565 Daliang Zhao School of Mathematics and Statistics Shandong Normal University P. R. China dlzhao928@sdnu.edu.cn Yansheng Liu School of Mathematics and Statistics Shandong Normal University P. R. China ysliu@sdnu.edu.cn Fractional differential equations semipositone boundary value problem coupled integral boundary value conditions fixed point index. Article.16.pdf  R. P. Agarwal, N. Hussain, M. A. Taoudi, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, Abstr. Appl. Anal., 2012 (2012), 1-15 ## R. P. Agarwal, M. Meehan, D. O’Regan, Fixed point theory and applications, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (2001) ## R. P. Agarwal, D. O’Regan, A note on existence of nonnegative solutions to singular semi-positone problems, Nonlinear Anal., 36 (1999), 615-622 ## N. A. Asif, R. A. Khan, Positive solutions to singular system with four-point coupled boundary conditions, J. Math. Anal. Appl., 386 (2012), 848-861 ## A. Cabada, G.-T. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411 ## M.-Q. Feng, X.-M. Zhang, W.-G. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl., 2011 (2011), 1-20 ## C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl., 62 (2011), 1251-1268 ## D. J. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA, (1988), - ## N. Hussain, M. A. Taoudi, Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations, Fixed Point Theory Appl., 2013 (2013), 1-16 ## M. Jia, X.-P. Liu, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions, Appl. Math. Comput., 232 (2014), 313-323 ## J.-Q. Jiang, L.-S. Liu, Y.-H. Wu, Positive solutions to singular fractional differential system with coupled boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3061-3074 ## A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006) ## V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677-2682 ## K. Q. Lan, W. Lin, Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations, J. Lond. Math. Soc., 83 (2011), 449-469 ## A. Leung, A semilinear reaction-diffusion prey-predator system with nonlinear coupled boundary conditions: equilibrium and stability, Indiana Univ. Math. J., 31 (1982), 223-241 ## Y. Li, Y.-Q. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810-1821 ## S.-H. Liang, J.-H. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Anal., 71 (2009), 5545-5550 ## Y.-S. Liu, Twin solutions to singular semipositone problems, J. Math. Anal. Appl., 286 (2003), 248-260 ## Y.-S. Liu, B.-Q. Yan, Multiple solutions of singular boundary value problems for differential systems, J. Math. Anal. Appl., 287 (2003), 540-556 ## Y. Liu, W.-Q. Zhang, X.-P. Liu, A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann-Liouville derivative, Appl. Math. Lett., 25 (2012), 1986-1992 ## J.-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., 2010 (2010), 1-8 ## K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993) ## S. K. Ntouyas, G.-T. Wang, L.-H. Zhang, Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments, Opuscula Math., 31 (2011), 433-442 ## A. S. Perelson, D. E. Kirschner, R. De Boer, Dynamics of HIV infection of $$CD4^+T$$ cells, Math. Biosci., 114 (1993), 81-125 ## I. 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, CA (1999) ## S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional 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, Yverdon (1993) ## M. Stojanović, R. Gorenflo, Nonlinear two-term time fractional diffusion-wave problem, Nonlinear Anal. Real World Appl., 11 (2010), 3512-3523 ## S. W. Vong, Positive solutions of singular fractional differential equations with integral boundary conditions, Math. Comput. Modelling, 57 (2013), 1053-1059 ## J.-F. Xu, Z.-L. Wei, W. Dong, Uniqueness of positive solutions for a class of fractional boundary value problems, Appl. Math. Lett., 25 (2012), 590-593 ## W.-G. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Comput. Math. Appl., 63 (2012), 288-297 ## X.-J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Therm. Sci., (2016), 326-326 ## X.-J. Yang, H. M. Srivastava, J. A. Tenreiro Machado, A new fractional derivative without singular kernel: application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753-756 ## X.-J. Yang, J. A. Tenreiro Machado, C. Cattani, F. Gao, On a fractal LC-electric circuit modeled by local fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 47 (2017), 200-206 ## C.-J. Yuan, Multiple positive solutions for (n - 1, 1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 1-12 ## C.-J. Yuan, Two positive solutions for (n - 1, 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 930-942 ## X.-U. Zhang, Y.-F. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett., 25 (2012), 555-560
Some fixed point theorems for contractive mappings of integral type Some fixed point theorems for contractive mappings of integral type en en 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. 3566 3580 Zeqing Liu Department of Mathematics Liaoning Normal University People’s Republic of China zeqingliu@163.com Yuqing Wang Department of Mathematics Liaoning Normal University People’s Republic of China yuqingwang93@163.com Shin Min Kang Department of Mathematics and the RINS Center for General Education, Gyeongsang National University China Medical University Korea Taiwan smkang@gnu.ac.kr Young Chel Kwun Department of Mathematics Dong-A University Korea yckwun@dau.ac.kr Contractive mappings of integral type fixed point complete metric space. Article.17.pdf  M. Abbas, M. A. Khan, Common fixed point theorem of two mappings satisfying a generalized weak contractive condition, Int. J. Math. Math. Sci., 2009 (2009), 1-9 ## H. H. Alsulami, E. Karapınar, D. O’Regan, P. Shahi, Fixed points of generalized contractive mappings of integral type, Fixed Point Theory Appl., 24 pages. (2014) ## A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci., 29 (2002), 531-536 ## D. Dey, A. Ganguly, M. Saha, Fixed point theorems for mappings under general contractive condition of integral type, Bull. Math. Anal. Appl., 3 (2011), 27-34 ## P. N. Dutta, B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl., 2008 (2008), 1-8 ## R. George, R. Rajagopalan, Common fixed point results for $$\psi-\phi$$ contractions in rectangular metric spaces, Bull. Math. Anal. Appl., 5 (2013), 44-52 ## V. Gupta, N. Mani, A common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral type, Math. Theory Modeling, 1 (2011), 1-6 ## V. Gupta, N. Mani, Common fixed point for two self-maps satisfying a generalized $$^\psi\int_\phi$$ weakly contractive condition of integral type, Int. J. Nonlinear Sci., 16 (2013), 64-71 ## V. Gupta, N. Mani, A. K. Tripathi, A fixed point theorem satisfying a generalized weak contractive condition of integral type, Int. J. Math. Anal. (Ruse), 6 (2012), 1883-1889 ## V. R. Hosseini, Common fixed point for generalized ($$\phi,\psi$$)-weak contractions contractions mappings condition of integral type, Int. J. Math. Anal., 4 (2010), 1535-1543 ## E. Karapınar, P. Shahi, K. Tas, Generalized $$\alpha-\psi$$-contractive type mappings of integral type and related fixed point theorems, J. Inequal. Appl., 2014 (2014), 1-18 ## M. A. Kutbi, M. Imdad, S. Chauhan, W. Sintunavarat, Some integral type fixed point theorems for non-self-mappings satisfying generalized ($$\psi,\phi$$)-weak contractive conditions in symmetric spaces, Abstr. Appl. Anal., 2014 (2014), 1-11 ## Z.-Q. Liu, J.-L. Li, S. M. Kang, Fixed point theorems of contractive mappings of integral type, Fixed Point Theory Appl., 2013 (2013), 1-17 ## Z.-Q. Liu, X. Li, S. M. Kang, S. Y. Cho, Fixed point theorems for mappings satisfying contractive conditions of integral type and applications, Fixed Point Theory Appl., 2011 (2011), 1-18 ## Z.-Q. Liu, H. Wu, J. S. Ume, S. M. Kang, Some fixed point theorems for mappings satisfying contractive conditions of integral type, Fixed Point Theory Appl., 2014 (2014), 1-14 ## N. V. Luong, N. X. Thuan, A fixed point theorem for $$\psi_{\int \phi}$$-weakly contractive mapping in metric spaces, Int. J. Math. Appl., 4 (2010), 233-242 ## C. Mongkolkeha, P. Kumam, Fixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces, Int. J. Math. Math. Sci., 2011 (2011), 1-12 ## B. E. Rhoades, Some theorems on weakly contractive maps, Proceedings of the Third World Congress of Nonlinear Analysts, Part 4, Catania, (2000), Nonlinear Anal., 47 (2001), 2683-2693 ## B. E. Rhoades, Two fixed-point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci., 2003 (2003), 4007-4013
Classification of functions with trivial solutions under $$t$$-equivalence Classification of functions with trivial solutions under $$t$$-equivalence en en 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. 3581 3591 Yanqing Li School of Mathematics and Statistics School of ocean information engineering Northeast Normal University Hainan Tropical Ocean University P. R. China P. R. China liyq516@nenu.edu.cn Donghe Pei School of Mathematics and Statistics Northeast Normal University P. R. China peidh340@nenu.edu.cn Dejian Huang School of Mathematics and Statistics School of ocean information engineering Northeast Normal University Hainan Tropical Ocean University P. R. China P. R. China hdj1107@qzu.edu.cn Ruimei Gao Department of Science Changchun University of Science and Technology P. R. China gaorm135@nenu.edu.cn Singularity bifurcation t-equivalence classification. Article.18.pdf  O. Diekmann, A beginner’s guide to adaptive dynamics, Mathematical modelling of population dynamics, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 63 (2004), 47-86 ## U. Dieckmann, R. Law, The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol., 34 (1997), 579-612 ## S.-P. Gao, Y.-C. Li, Classification of $$(D_4, S^1)$$-equivariant bifurcation problems up to topological codimension 2, Sci. China Ser. A, 46 (2003), 862-871 ## S. Geritz, J. Metz, E. Kisdi, G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Lett., 78 (1997), 2024-2027 ## M. Golubitsky, M. Roberts, A classification of degenerate Hopf bifurcations with O(2) symmetry, J. Differential Equations, 69 (1987), 216-264 ## M. Golubitsky, D. G. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98 ## M. Golubitsky, D. G. Schaeffer, Singularities and groups in bifurcation theory, Vol. I, Applied Mathematical Sciences, Springer-Verlag, New York (1985) ## B. L. Keyfitz, Classification of one-state-variable bifurcation problems up to codimension seven, Dynam. Stability Systems, 1 (1986), 1-41 ## M. Manoel, I. Stewart, The classification of bifurcations with hidden symmetries, Proc. London Math. Soc., 80 (2000), 198-234 ## J. Martinet, Singularities of smooth functions and maps, Translated from the French by Carl P. Simon, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge-New York (1982) ## M. Peters, Classification of two-parameter bifurcations, Singularity theory and its applications, Part II, Coventry, (1988/1989), Lecture Notes in Math., Springer, Berlin, 1463 (1991), 294-300 ## J. M. Smith, G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18 ## A. Vutha, M. Golubitsky, Normal forms and unfoldings of singular strategy functions, Dyn. Games Appl., 5 (2015), 180-213 ## X.-H. Wang, M. Golubitsky, Singularity theory of fitness functions under dimorphism equivalence, J. Math. Biol., 73 (2016), 526-573 ## D. Waxman, S. Gavrilets, 20 questions on adaptive dynamics, J. Evol. Biol., 18 (2005), 1139-1154
Existence for fractional Dirichlet boundary value problem under barrier strip conditions Existence for fractional Dirichlet boundary value problem under barrier strip conditions en en 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. 3592 3598 Qilin Song College of Mathematics and System Science Shandong University of Science and Technology P. R. China 912078531@qq.com Xiaooyu Dong College of Mathematics and System Science Shandong University of Science and Technology P. R. China dxy17854288324@163.com Zhanbing Bai College of Mathematics and System Science Shandong University of Science and Technology P. R. China zhanbingbai@163.com Bo Chen College of Mathematics and Statistics Shenzhen University P. R. China chenbo@szu.edu.cn Barrier strips fixed-point index conformable fractional derivative. Article.19.pdf  T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66 ## R. Almeida, A. B. Malinowska, T. Odzijewicz, Fractional differential equations with dependence on the Caputo- Katugampola derivative, J. Comput. Nonlinear Dynam., 11 (2016), 061017-061027 ## A. Alsaedi, D. Baleanu, S. Etemad, S. Rezapour, On coupled systems of time-fractional differential problems by using a new fractional derivative, J. Funct. Spaces, 2016 (2016), 1-8 ## Z.-B. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72 (2010), 916-942 ## Z.-B. Bai, On solutions of some fractional m-point boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 1-15 ## Z.-B. Bai, Solvability for a class of fractional m-point boundary value problem at resonance, Comput. Math. Appl., 62 (2011), 1292-1302 ## Z.-B. Bai, X.-Y. Dong, C. Yin, Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions, Bound. Value Probl., 2016 (2016), 1-11 ## Z.-B. Bai, H.-S. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505 ## Z.-B. Bai, Y.-H. Zhang, The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl., 60 (2010), 2364-2372 ## Z.-B. Bai, Y.-H. Zhang, Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput., 218 (2011), 1719-1725 ## Z.-B. Bai, S. Zhang, S.-J. Sun, C. Yin, Monotone iterative method for fractional differential equations, Electron. J. Differential Equations, 2016 (2016), 1-8 ## Y.-J. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett., 51 (2016), 48-54 ## X.-Y. Dong, Z.-B. Bai, S.-J. Sun, Positive solutions for some boundary value problems with conformable fractional differential derivatives, (Chinese) Acta Math. Sci. Ser. A Chin. Ed., 37 (2017), 82-91 ## X.-Y. Dong, Z.-B. Bai, W. Zhang, Positive solutions for nonlinear eigenvalue problems with conformable fractional differential derivatives, J. Shandong Univ. Sci. Technol. Nat. Sci., 35 (2016), 85-90 ## X.-Y. Dong, Z.-B. Bai, S.-Q. Zhang, Positive solutions to boundary value problems of p-Laplacian with fractional derivative, Bound. Value Probl., 2017 (2017), 1-15 ## H. H. Dong, B. Y. Guo, B. S. Yin, Generalized fractional supertrace identity for Hamiltonian structure of NLSMKdV hierarchy with self-consistent sources, Anal. Math. Phys., 6 (2016), 199-209 ## T. Feng, X.-Z. Meng, L.-D. Liu, S.-J. Gao, Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model, J. Inequal. Appl., 2016 (2016), 1-29 ## C.-H. Gao, Existence of solutions to p-Laplacian difference equations under barrier strips conditions, Electron. J. Differential Equations, 2007 (2007), 1-6 ## L.-M. He, X.-Y. Dong, Z.-B. Bai, B. Chen, Solvability of some two-point fractional boundary value problems under barrier strip conditions, J. Funct. Spaces, 2017 (2017), 1-6 ## Y.-X. Hua, X.-H. Yu, On the ground state solution for a critical fractional Laplacian equation, Nonlinear Anal., 87 (2013), 116-125 ## F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1-17 ## P. Kelevedjiev, Existence of solutions for two-point boundary value problems, Nonlinear Anal., 22 (1994), 217-224 ## P. S. Kelevedjiev, S. Tersian, Singular and nonsingular first-order initial value problems, J. Math. Anal. Appl., 366 (2010), 516-524 ## R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70 ## X.-P. Liu, M. Jia, W.-G. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Difference Equ., 2013 (2013), 1-12 ## X.-P. Liu, M. Jia, W.-G. Ge, The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator, Appl. Math. Lett., 65 (2017), 56-62 ## X.-P. Liu, M. Jia, B.-F. Wu, Existence and uniqueness of solution for fractional differential equations with integral boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2009 (2009), 1-10 ## R.-Y. Ma, H. Luo, Existence of solutions for a two-point boundary value problem on time scales, Appl. Math. Comput., 150 (2004), 139-147 ## X.-Z. Meng, S.-N. Zhao, T. Feng, T.-H. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227-242 ## Z. Wang, A numerical method for delayed fractional-order differential equations, J. Appl. Math., 2013 (2013), 1-7 ## G.-T.Wang, B. Ahmad, L.-H. Zhang, J. J. Nieto, Comments on the concept of existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 401-403 ## Z. Wang, X. Huang, J.-P. Zhou, A numerical method for delayed fractional-order differential equations: based on G-L definition, Appl. Math. Inf. Sci., 7 (2013), 525-529 ## J.-R. Wang, Y. Zhou, M. Fečkan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl., 64 (2012), 3008-3020 ## S.-S. Yan, J.-F. Yang, X.-H. Yu, Equations involving fractional Laplacian operator: compactness and application, J. Funct. Anal., 269 (2015), 47-79 ## X.-J. Yang, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753-756 ## X.-J. Yang, Fractional Maxwell fluid with fractional derivative without singular kernel, Therm. Sci., 20 (2016), 1-871 ## X.-J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Therm. Sci., 3 (2017), 1161-1171 ## X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2015) ## X.-J. Yang, J. A. T. Machado, A new fractional operator of variable order: Application in the description of anomalous diffusion, Phys. A, 481 (2017), 276-283 ## X.-H. Yu, Solutions of fractional Laplacian equations and their Morse indices, J. Differential Equations, 260 (2016), 860-871 ## S.-Q. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differential Equations, 2006 (2006), 1-12 ## W. Zhang, Z.-B. Bai, S.-J. Su, Extremal solutions for some periodic fractional differential equation, Adv. Difference Equ., 2016 (2016), 1-8 ## T.-Q. Zhang, X.-Z. Meng, Y. Song, T.-H. Zhang, A stage-structured predator-prey SI model with disease in the prey and impulsive effects, Math. Model. Anal., 18 (2013), 505-528 ## Y.-M. Zou, Y.-J. Cui, Existence results for a functional boundary value problem of fractional differential equations, Adv. Difference Equ., 2013 (2013), 1-25
Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria en en 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. 3599 3611 Shuwen Xiang School of Mathematics and Statistics Guizhou University P. R. China shwxiang@vip.163.com Shunyou Xia School of Mathematics and Statistics School of Mathematics and Computer Science Guizhou University Guizhou Education University P. R. China P. R. China xiashunyou@126.com Jihao He School of Mathematics and Statistics Guizhou University P. R. China hejihao78@sohu.com Yanlong Yang School of Mathematics and Statistics Guizhou University P. R. China yylong1980@163.com Chenwei Liu School of Mathematics and Statistics Guizhou University P. R. China liuchenwei15@163.com Stability Nash equilibria fixed point strong $$\delta$$-perturbation stable set. Article.20.pdf  O. Carbonell-Nicolau, On strategic stability in discontinuous games, Econom. Lett., 113 (2011), 120-123 ## O. Carbonell-Nicolau, Further results on essential Nash equilibria in normal-form games, Econom. Theory, 59 (2015), 277-300 ## M. K. Fort, Jr., Points of continuity of semi-continuous functions, Publ. Math. Debrecen, 2 (1951), 100-102 ## S. Govindan, R. Wilson, Essential equilibria, Proc. Natl. Acad. Sci. USA, 102 (2005), 15706-15711 ## J. Hillas, On the definition of the strategic stability of equilibria, Econometrica, 58 (1990), 1365-1390 ## J. Hillas, M. Jansen, J. Potters, D. Vermeulen, On the relation among some definitions of strategic stability, Math. Oper. Res., 26 (2001), 611-635 ## J. Hillas, M. Jansen, J. Potters, D. Vermeulen, Independence of inadmissible strategies and best reply stability: a direct proof, Special issue on stable equilibria, Internat. J. Game Theory, 32 (2004), 371-377 ## W.-S. Jia, S.-W. Xiang, J.-H. He, Y.-L. Yang, Existence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leaderfollower games, J. Global Optim., 61 (2015), 397-405 ## E. Kalai, D. Samet, Persistent equilibria in strategic games, Internat. J. Game Theory, 13 (1984), 129-144 ## E. 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 York (1984) ## E. Kohlberg, J. F. Mertens, On the strategic stability of equilibria, Econometrica, 54 (1986), 1003-1037 ## D. M. Kreps, R. Wilson, Sequential equilibria, Econometrica, 50 (1982), 863-894 ## A. McLennan, Fixed points of contractible valued correspondences, Internat. J. Game Theory, 18 (1989), 175-184 ## J. F. Mertens, Stable equilibria–a reformulation, I, Definition and basic properties, Math. Oper. Res., 14 (1989), 575-625 ## R. B. Myerson, Refinements of the Nash equilibrium concept, Internat. J. Game Theory, 7 (1978), 73-80 ## A. B. Sadanand, V. Sadanand, Equilibria in non-cooperative games, I, Perturbations based refinements of Nash equilibrium, Bull. Econ. Res., 43 (1994), 197-224 ## V. Scalzo, Essential equilibria of discontinuous games, Econom. Theory, 54 (2013), 27-44 ## V. Scalzo, On the existence of essential and trembling-hand perfect equilibria in discontinuous games, Econ. Theory Bull., 1 (2014), 1-12 ## R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Internat. J. Game Theory, 4 (1975), 25-55 ## E. van Damme, Strategic equilibrium, Handbook of Game Theory with Economic Applications, 3 (2002), 1521-1596 ## A. J. Vermeulen, J. A. M. Potters, M. J. M. Jansen, On quasi-stable sets, Internat. J. Game Theory, 25 (1996), 43-49 ## A. J. Vermeulen, J. A. M. Potter, M. J. M. Jansen, On stable sets of equilibria, Game theoretical applications to economics and operations research, Bangalore, (1996), Theory Decis. Lib. Ser. C Game Theory Math. Program. Oper. Res., Kluwer Acad. Publ., Boston, MA, 18 (1997), 133-148 ## W.-T. Wu, J.-H. Jiang, Essential equilibrium points of n-person non-cooperative games, Sci. Sinica, 11 (1962), 1307-1322 ## S.-W. Xiang, G.-D. Liu, Y.-H. Zhou, On the strongly essential components of Nash equilibria of infinite n-person games with quasiconcave payoffs, Nonlinear Anal., 63 (2005), 1-2637 ## J. Yu, Q. Luo, On essential components of the solution set of generalized games, J. Math. Anal. Appl., 230 (1999), 303-310 ## J. Yu, S.-W. Xiang, On essential components of the set of Nash equilibrium points, Nonlinear Anal., 38 (1999), 259-264 ## Y.-H. Zhou, J. Yu, S.-W. Xiang, Essential stability in games with infinitely many pure strategies, Internat. J. Game Theory, 35 (2007), 493-503
Inequalities on asymmetric $$L_p$$-harmonic radial bodies Inequalities on asymmetric $$L_p$$-harmonic radial bodies en en 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. 3612 3618 Zhaofeng Li Department of Mathematics China Three Gorges University China kelly0128@163.com Weidong Wang Department of Mathematics China Three Gorges University China wangwd722@163.com Star body $$L_p$$-harmonic radial body asymmetric $$L_p$$-harmonic radial body dual quermassintegrals polar. Article.21.pdf  Y. D. Chai, Y. S. Lee, Harmonic radial combinations and dual mixed volumes, Asian J. Math., 5 (2001), 493-498 ## Y.-B. Feng, W.-D. Wang, Some inequalities for $$L_p$$-dual affine surface area, Math. Inequal. Appl., 17 (2014), 431-441 ## W. J. Firey, Mean cross-section measures of harmonic means of convex bodies, Pacific J. Math., 11 (1961), 1263-1266 ## W. J. Firey, Polar means of convex bodies and a dual to the Brunn-Minkowski theorem, Canad. J. Math., 13 (1961), 444-453 ## R. J. Gardner, Geometric tomography, Second edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, New York (2006) ## C. Haberl, $$L_p$$ intersection bodies, Adv. Math., 217 (2008), 2599-2624 ## C. Haberl, M. Ludwig, A characterization of $$L_p$$ intersection bodies, Int. Math. Res. Not., 2006 (2006), 1-29 ## C. Haberl, F. E. Schuster, Asymmetric affine $$L_p$$ Sobolev inequalities, J. Funct. Anal., 257 (2009), 641-658 ## C. Haberl, F. E. Schuster, General $$L_p$$ affine isoperimetric inequalities, J. Differential Geom., 83 (2009), 1-26 ## C. Haberl, F. E. Schuster, J. Xiao, An asymmetric affine Pólya-Szegö principle, Math. Ann., 352 (2012), 517-542 ## M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc., 357 (2005), 4191-4213 ## M. Ludwig, Intersection bodies and valuations, Amer. J. Math., 128 (2006), 1409-1428 ## M. Ludwig, Valuations in the affine geometry of convex bodies, Integral geometry and convexity, World Sci. Publ., Hackensack, NJ, (2006), 49-65 ## E. Lutwak, Extended affine surface area, Adv. Math., 85 (1991), 39-68 ## E. Lutwak, The Brunn-Minkowski-Firey theory, II, Affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294 ## L. Parapatits, SL(n)-contravariant $$L_p$$-Minkowski valuations, Trans. Amer. Math. Soc., 366 (2014), 1195-1211 ## L. Parapatits, SL(n)-covariant $$L_p$$-Minkowski valuations, J. Lond. Math. Soc., 89 (2014), 397-414 ## R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (1993) ## F. E. Schuster, T. Wannerer, GL(n) contravariant Minkowski valuations, Trans. Amer. Math. Soc., 364 (2012), 815-826 ## F. E. Schuster, M. Weberndorfer, Volume inequalities for asymmetric Wulff shapes, J. Differential Geom., 92 (2012), 263-283 ## W.-D. Wang, Q. Chen, $$L_p$$-dual geominimal surface area, J. Inequal. Appl., 2011 (2011), 1-10 ## W.-D. Wang, Y.-B. Feng, A general $$L_p$$-version of Petty’s affine projection inequality, Taiwanese J. Math., 17 (2013), 517-528 ## W.-D. Wang, G.-S. Leng, $$L_p$$-dual mixed quermassintegrals, Indian J. Pure Appl. Math., 36 (2005), 177-188 ## W.-D. Wang, Y.-N. Li, Busemann-Petty problems for general $$L_p$$-intersection bodies, Acta Math. Sin. (Engl. Ser.), 31 (2015), 777-786 ## W.-D. Wang, Y.-N. Li, General $$L_p$$-intersection bodies, Taiwanese J. Math., 19 (2015), 1247-1259 ## W.-D. Wang, T.-Y. Ma, Asymmetric $$L_p$$-difference bodies, Proc. Amer. Math. Soc., 142 (2014), 2517-2527 ## T. Wannerer, GL(n) equivariant Minkowski valuations, Indiana Univ. Math. J., 60 (2011), 1655-1672 ## M. Weberndorfer, Shadow systems of asymmetric $$L_p$$ zonotopes, Adv. Math., 240 (2013), 613-635 ## F. Yibin, W. Weidong, L. Fenghong, Some inequalities on general $$L_p$$-centroid bodies, Math. Inequal. Appl., 18 (2015), 39-49 ## B.-C. Zhu, N. Li, J.-Z. Zhou, Isoperimetric inequalities for $$L_p$$ geominimal surface area, Glasg. Math. J., 53 (2011), 717-726
On solving general split equality variational inclusion problems in Banach space On solving general split equality variational inclusion problems in Banach space en en 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. 3619 3629 J. Zhao College of Sciences Qinzhou University P. R. China jingzhao100@126.com Y. S. Liang Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, and College of Sciences, Guangxi University for Nationalities Nanning P. R. China 1245470565@qq.com General split equality variational problems strong convergence Banach space. Article.22.pdf  E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145 ## C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453 ## Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space , Numer. Algorithms, 8 (1994), 221-239 ## Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084 ## Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244-1256 ## S.-S. Chang, On Chidume’s open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl., 216 (1997), 94-111 ## S.-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 applications, Appl. Math. Comput., 218 (2012), 6489-6497 ## S.-S. Chang, L. Wang, L.-J. Qin, Z.-L. Ma, Strongly convergent iterative methods for split equality variational inclusion problems in Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 6 (2016), 1641-1650 ## S.-S. Chang, L. Wang, X. R. Wang, G. Wang, General split equality equilibrium problems with application to split optimization problems, J. Optim. Theory Appl., 166 (2015), 377-390 ## C.-S. Chuang, Strong convergence theorems for the split variational inclusion problem in Hilbert spaces, Fixed Point Theory Appl., 2013 (2013), 1-20 ## P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136 ## L. Gasiński, Z.-H. Liu, S. Migórski, A. Ochal, Z.-J. Peng, Hemivariational inequality approach to evolutionary constrained problems on star-shaped sets, J. Optim. Theory Appl., 164 (2015), 514-533 ## K. Goebel, W. A. Kirk, Topics in metric fixed point theory , Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990) ## Z.-H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations, 244 (2008), 1395-1409 ## Z.-H. Liu, X.-W. Li, D. Motreanu, Approximate controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces, SIAM J. Control Optim., 53 (2015), 3228-3244 ## Z.-H. Liu, S.-D. Zeng, Equilibrium problems with generalized monotone mapping and its applications, Math. Methods Appl. Sci., 39 (2016), 152-163 ## Z.-H. Liu, S.-D. Zeng, D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations, 260 (2016), 6787-6799 ## P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912 ## A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problem, Trans. Math. Program. Appl., 1 (2013), 1-11 ## Z.-J. Peng, Z.-H. Liu, X.-Y. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann., 365 (2013), 1339-1358 ## 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 spaces, Fixed Point Theory Appl., 2016 (2016), 1-16 ## H.-K. Xu, A variable Krasnosel’skiı-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034 ## Q.-Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266 ## J.-L. Zhao, Q.-Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799
Existence of traveling wave solutions in $$m$$-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction Existence of traveling wave solutions in $$m$$-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction en en 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. 3630 3642 Kai Zhou Department of Mathematics School of Mathematics and Computer Shanghai Normal University Chizhou University P. R. China P. R. China zk1984@163.com Traveling wave solutions lattice differential systems delay upper and lower solutions Schauder’s fixed point theorem. Article.23.pdf  P. W. Bates, A. Chamj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305 ## P. W. Bates, P. C. Fife, X.-F. Ren, X.-F. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136 ## J. Bell, C. Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), 1-14 ## J. W. Cahn, S.-N. Chow, E. S. Van Vleck, Spatially discrete nonlinear diffusion equations, Second Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology, Edmonton, AB, (1992), Rocky Mountain J. Math., 25 (1995), 87-118 ## J. W. Cahn, J. Mallet-Paret, E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1998), 455-493 ## S.-N. Chow, Lattice dynamical systems, Dynamical systems, Lecture Notes in Math., Springer, Berlin, 1822 (2003), 1-102 ## S.-N. Chow, J. Mallet-Paret, W.-X. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291 ## J. Fang, J.-J. Wei, X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1919-1934 ## C.-H. Hsu, S.-S. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system, J. Differential Equations, 164 (2000), 431-450 ## J.-H. Huang, G. Lu, Traveling 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., 25 (2004), 125-136 ## J.-H. Huang, G. Lu, S.-G. Ruan, Traveling wave solutions in delayed lattice differential equations with partial monotonicity, Nonlinear Anal., 60 (2005), 1331-1350 ## G. Lin, W.-T. Li, Traveling waves in delayed lattice dynamical systems with competition interactions, Nonlinear Anal. Real World Appl., 11 (2010), 3666-3679 ## G. Lin, W.-T. Li, M.-J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414 ## G. Lin,W.-T. Li, S.-X. Pan, Travelling wavefronts in delayed lattice dynamical systems with global interaction, J. Difference Equ. Appl., 16 (2010), 1429-1446 ## Y. Lin, Q.-R. Wang, K. Zhou, Traveling wave solutions in n-dimensional delayed reaction-diffusion systems with mixed monotonicity, J. Comput. Appl. Math., 243 (2013), 16-27 ## S.-W. Ma, P.-X. Weng, X.-F. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal., 65 (2006), 1858-1890 ## S.-W. Ma, X.-F. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190 ## J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical systems, Lecture Notes in Math., Springer, Berlin, 1822 (2003), 231-298 ## H. F. Weinberger, M. Lewis, B.-T. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218 ## P.-X. Weng, H.-X. Huang, J.-H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439 ## J.-H. Wu, X.-F. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Differential Equations, 135 (1997), 315-357 ## J. Xia, Z.-X. Yu, Y.-C. Dong, H.-Y. Li, Traveling waves for n-species competitive system with nonlocal dispersals and delays, Appl. Math. Comput., 287/288 (2016), 201-213 ## B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27 ## B. Zinner, G. Harris, W. Hudson, Traveling wavefronts for the discrete Fisher’s equation, J. Differential Equations, 105 (1993), 46-62
Picard splitting method and Picard CG method for solving the absolute value equation Picard splitting method and Picard CG method for solving the absolute value equation en en 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. 3643 3654 Chang-Qing Lv College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications School of Mathematics and Statistics Fujian Normal University Zaozhuang University P. R. China P. R. China Chang-Feng Ma College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications Fujian Normal University P. R. China macf@fjnu.edu.cn Absolute value equation Picard algorithm matrix splitting iteration method conjugate gradient method. Article.24.pdf  Z.-Z. Bai, X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59 (2009), 2923-2936 ## R. W. Cottle, G. B. Dantzig, Complementary pivot theory of mathematical programming, Linear Algebra and Appl., 1 (1968), 103-125 ## R.W. Cottle, J.-S. Pang, R. E. Stone, The linear complementarity problem, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA (1992) ## T.-X. Gu, X.-W. Xu, X.-P. Liu, H.-B. An, X.-D. Hang, Iterative methods and preconditioning techniques, (Chinese) Science Press, Beijing (2015) ## S.-L. Hu, Z.-H. Huang, A note on absolute value equations, Optim. Lett., 4 (2010), 417-424 ## S. Ketabchi, H. Moosaei, An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side, Comput. Math. Appl., 64 (2012), 1882-1885 ## S. Ketabchi, H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, J. Optim. Theory Appl., 152 (2012), 1080-1087 ## S. Ketabchi, H. Moosaei, S. Fallahi, Optimal error correction of the absolute value equation using a genetic algorithm, Math. Comput. Model., 57 (2013), 2339-2342 ## O. L. Mangasarian, Linear complementarity problems solvable by a single linear program, Math. Programming, 10 (1976), 263-270 ## O. L. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3-8 ## O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43-53 ## O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101-108 ## O. L. Mangasarian, Primal-dual bilinear programming solution of the absolute value equation, Optim. Lett., 6 (2012), 1527-1533 ## O. L. Mangasarian, Absolute value equation solution via dual complementarity, Optim. Lett., 7 (2013), 625-630 ## O. L. Mangasarian, Absolute value equation solution via linear programming, J. Optim. Theory Appl., 161 (2014), 870-876 ## O. L. Mangasarian, R. R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359-367 ## M. A. Noor, J. Iqbal, K. I. Noor, E. Al-Said, On an iterative method for solving absolute value equations, Optim. Lett., 6 (2012), 1027-1033 ## J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, (1970), - ## O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363-372 ## J. Rohn, A theorem of the alternatives for the equation $$Ax + B|x| = b$$, Linear Multilinear Algebra, 52 (2004), 421-426 ## J. Rohn, An algorithm for solving the absolute value equation, Electron. J. Linear Algebra, 18 (2009), 589-599 ## J. Rohn, On unique solvability of the absolute value equation, Optim. Lett., 3 (2009), 603-606 ## J. Rohn, An algorithm for computing all solutions of an absolute value equation, Optim. Lett., 6 (2012), 851-856 ## J. Rohn, V. Hooshyarbakhsh, R. Farhadsefat, An iterative method for solving absolute value equations and sufficient conditions for unique solvability, Optim. Lett., 8 (2014), 35-44 ## D. K. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), 2191-2202 ## A.-X. Wang, H.-J. Wang, Y.-K. Deng, Interval algorithm for absolute value equations, Cent. Eur. J. Math., 9 (2011), 1171-1184
$$(L,M)$$-fuzzy convex structures $$(L,M)$$-fuzzy convex structures en en 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. 3655 3669 Fu-Gui Shi School of Mathematics and Statistics Beijing Institute of Technology China fuguishi@bit.edu.cn Zhen-Yu Xiu College of Applied Mathematics Chengdu University of Information Technology China xiuzhenyu112@sohu.com quotient structures substructures $$(L،M)$$-fuzzy convex structure $$(L،M)$$-fuzzy convexity preserving function products. Article.25.pdf  N. Ajmal, K. V. Thomas, Fuzzy lattices, Inform. Sci., 79 (1994), 271-291 ## M. Berger, Convexity, Amer. Math. Monthly, 97 (1990), 650-678 ## P. Dwinger, Characterization of the complete homomorphic images of a completely distributive complete lattice, I, Nederl. Akad. Wetensch. Indag. Math., 85 (1982), 403-414 ## J.-M. Fang, P.-W. Chen, One-to-one correspondence between fuzzifying topologies and fuzzy preorders, Fuzzy Sets and Systems, 158 (2007), 1814-1822 ## H.-L. Huang, F.-G. Shi, L-fuzzy numbers and their properties, Inform. Sci., 178 (2008), 1141-1151 ## Q. Jin, L.-Q. Li , On the embedding of convex spaces in stratified L-convex spaces, SpringerPlus, 5 (2016), 1-10 ## F. Jinming, I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets and Systems, 158 (2007), 2359-2374 ## T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz University, Poznan, Poland (1985) ## M. Lassak, On metric B-convexity for which diameters of any set and its hull are equal, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25 (1977), 969-975 ## Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 1641 (2009), 22-37 ## C. V. Negoiţă, D. A. Ralescu, Applications of fuzzy sets to systems analysis, Translated from the Romanian, ISR— Interdisciplinary Systems Research, Birkhäuser Verlag, Basel-Stuttgart, 11 (1975), 1-187 ## B. Pang, F.-G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets and Systems, 313 (2017), 61-74 ## B. Pang, Y. Zhao, Characterizations of L-convex spaces, Iran. J. Fuzzy Syst., 13 (2016), 51-61 ## M. V. Rosa, A study of fuzzy convexity with special reference to separation properties, Ph.D. Thesis, Cochin University of Science and Technology, Kerala, India (1994) ## M. V. Rosa, On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets and Systems, 62 (1994), 97-100 ## F.-G. Shi, Theory of $$L_\beta$$-nested sets and $$L_\alpha$$-nested sets and its applications, (Chinese) Fuzzy Syst. Math., 4 (1995), 65-72 ## F.-G. Shi, L-fuzzy relations and L-fuzzy subgroups, J. Fuzzy Math., 8 (2000), 491-499 ## F.-G. Shi, E.-Q. Li, The restricted hull operator of M-fuzzifying convex structures, J. Intell. Fuzzy Syst., 30 (2015), 409-421 ## F.-G. Shi, Z.-Y. Xiu, A new approach to the fuzzification of convex structures, J. Appl. Math., 2014 (2014), 1-12 ## V. P. Soltan, d-convexity in graphs, (Russian) Dokl. Akad. Nauk SSSR, 272 (1983), 535-537 ## A. P. Šostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo, 11 (1985), 89-103 ## M. L. J. van de Vel, Theory of convex structures, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam (1993) ## J. van Mill, Supercompactness and Wallman spaces, Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam (1977) ## J. C. Varlet, Remarks on distributive lattices, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 23 (1975), 1143-1147 ## G.-J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992), 351-376 ## Z.-Y. Xiu, F.-G. Shi, M-fuzzifying interval spaces, Iran. J. Fuzzy Syst., 14 (2017), 145-162 ## M.-S. Ying, A new approach for fuzzy topology, I, Fuzzy Sets and Systems, 39 (1991), 303-321 ## L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353
Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces en en 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. 3670 3693 Min Zhang College of Science China University of Petroleum People’s Republic of China zhangminmath@163.com Jie Rui College of Science China University of Petroleum People’s Republic of China rjhygl@163.com Quasi-periodically forced KAM theory Schrödinger equation quasi-periodic solutions. Article.26.pdf  D. Bambusi, S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480 ## M. Berti, P. Bolle, Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity, 25 (2012), 2579-2613 ## M. Berti, P. Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on $$\mathbb{T}^d$$ with a multiplicative potential, J. Eur. Math. Soc. (JEMS), 15 (2013), 229-286 ## J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices, 1994 (1994), 1-21 ## J. Bourgain , Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639 ## J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439 ## J. Bourgain, Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions, Park City, UT, (1995), IAS/Park City Math. Ser., Amer. Math. Soc., Providence, RI, 5 (1999), 3-157 ## J. Bourgain, Green’s function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ (2005) ## W. Craig, C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498 ## H. L. Eliasson, S. B. Kuksin, On reducibility of Schrödinger equations with quasiperiodic in time potentials, Comm. Math. Phys., 286 (2009), 125-135 ## H. L. Eliasson, S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435 ## J.-S. Geng, X.-D. Xu, J.-G. You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402 ## J.-S. Geng, Y.-F. Yi, Quasi-periodic solutions in a nonlinear Schrödinger equation, J. Differential Equations, 233 (2007), 512-542 ## J.-S. Geng, J.-G. You, A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions, J. Differential Equations, 209 (2005), 1-56 ## J.-S. Geng, J.-G. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343-372 ## J.-S. Geng, J.-G. You, KAM tori for higher dimensional beam equations with constant potentials, Nonlinearity, 19 (2006), 2405-2423 ## J.-S. Geng, J.-G. You, KAM theorem for higher dimensional nonlinear Schrödinger equations, J. Dynam. Differential Equations, 25 (2013), 451-476 ## S. B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, Springer- Verlag, Berlin (1993) ## S. B. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179 ## Z.-G. Liang, J.-G. You , Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity, SIAM J. Math. Anal., 36 (2005), 1965-1990 ## J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148 ## J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296 ## C. Procesi, M. Procesi, A KAM algorithm for the resonant non-linear Schrödinger equation, Adv. Math., 272 (2015), 399-470 ## C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528 ## .-X. Xu, J.-G. You, Persistence of lower-dimensional tori under the first Melnikov’s non-resonance condition, J. Math. Pures Appl., 80 (2001), 1045-1067 ## X.-P. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274
Determinant and inverse of a Gaussian Fibonacci skew-Hermitian Toeplitz matrix Determinant and inverse of a Gaussian Fibonacci skew-Hermitian Toeplitz matrix en en 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. 3694 3707 Zhaolin Jiang Department of Mathematics Linyi University P. R. China jzh1208@sina.com Jixiu Sun Department of Mathematics School of Mathematics and Statistics Linyi University Shandong Normal University P. R. China P. R. China sunjixiu1314@163.com Gaussian Fibonacci number skew-Hermitian Toeplitz matrix determinant inverse. Article.27.pdf  M. Akbulak, D. Bozkurt, On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers, Hacet. J. Math. Stat., 37 (2008), 89-95 ## D. Bozkurt, T.-Y. Tam, Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas Numbers, Appl. Math. Comput., 219 (2012), 544-551 ## A. Buckley, On the solution of certain skew symmetric linear systems, SIAM J. Numer. Anal., 14 (1977), 566-570 ## R. H. Chan, X.-Q. Jin, Circulant and skew-circulant preconditioners for skew-Hermitian type Toeplitz systems, BIT, 31 (1991), 632-646 ## X.-T. Chen, Z.-L. Jiang, J.-M. Wang, Determinants and inverses of Fibonacci and Lucas skew symmetric Toeplitz matrices, British J. Math. Comput. Sci., 19 (2016), 1-21 ## L. Dazheng, Fibonacci-Lucas quasi-cyclic matrices, A special tribute to Calvin T. Long, Fibonacci Quart., 40 (2002), 280-286 ## U. Grenander, G. Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences University of California Press, Berkeley-Los Angeles (1958) ## G. Heining, K. Rost, Algebraic methods for Toeplitz-like matrices and operators, Mathematical Research, Akademie- Verlag, Berlin (1984) ## A. F. Horadam, Further appearence of the Fibonacci sequence, Fibonacci Quart., 1 (1963), 41-42 ## A. Ipek, K. Arı, On Hessenberg and pentadiagonal determinants related with Fibonacci and Fibonacci-like numbers, Appl. Math. Comput., 229 (2014), 433-439 ## Z.-L. Jiang, Y.-P. Gong, Y. Gao, Circulant type matrices with the sum and product of Fibonacci and Lucas numbers, Abstr. Appl. Anal., 2014 (2014), 1-12 ## Z.-L. Jiang, Y.-P. Gong, Y. Gao, Invertibility and explicit inverses of circulant-type matrices with k-Fibonacci and k-Lucas numbers, Abstr Appl. Anal., 2014 (2014), 1-10 ## X.-Y. Jiang, K.-C. Hong, Exact determinants of some special circulant matrices involving four kinds of famous numbers, Abstr. Appl. Anal., 2014 (2014), 1-12 ## X.-Y. Jiang, K.-C. Hong, Explicit inverse matrices of Tribonacci skew circulant type matrices, Appl. Math. Comput., 268 (2015), 93-102 ## X.-Y. Jiang, K.-C. Hong, Skew cyclic displacements and inversions of two innovative patterned matrices, Appl. Math. Comput., 308 (2017), 174-184 ## Z.-L. Jiang, J.-W. Zhou, A note on spectral norms of even-order r-circulant matrices, Appl. Math. Comput., 250 (2014), 368-371 ## J. Li, Z.-L. Jiang, F.-L. Lu, Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers, Abstr. Appl. Anal., 2014 (2014), 1-9 ## L. Liu, Z.-L. Jiang, Explicit form of the inverse matrices of Tribonacci circulant type matrices, Abstr. Appl. Anal., 2015 (2015), 1-10 ## M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices, 2 (2014), 10-16 ## B. N. Mukherjee, S. S. Maiti, On some properties of positive definite Toeplitz matrices and their possible applications, Linear Algebra Appl., 102 (1988), 211-240 ## L. Rodman, Pairs of Hermitian and skew-Hermitian quaternionic matrices: canonical forms and their applications, Linear Algebra Appl., 429 (2008), 981-1019 ## S.-Q. Shen, J.-M. Cen, Y. Hao, On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput., 217 (2011), 9790-9797 ## J.-X. Sun, Z.-L. Jiang, Computing the determinant and inverse of the complex Fibonacci Hermitian Toeplitz matrix, British J. Mathe. Comput. Sci., 19 (2016), 1-16 ## Y.-P. Zheng, S.-G. Shon, Exact determinants and inverses of generalized Lucas skew circulant type matrices, Appl. Math. Comput., 270 (2015), 105-113 ## Y.-P. Zheng, S.-G. Shon, J.-Y. Kim, Cyclic displacements and decompositions of inverse matrices for CUPL Toeplitz matrices, J. Math. Anal. Appl., 455 (2017), 727-741 ## J.-W. Zhou, Z.-L. Jiang, The spectral norms of g-circulant matrices with classical Fibonacci and Lucas numbers entries, Appl. Math. Comput., 233 (2014), 582-587
Common fixed point theorems in non-Archimedean fuzzy metric-like spaces with applications Common fixed point theorems in non-Archimedean fuzzy metric-like spaces with applications en en 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. 3708 3718 Haiqing Zhao Department of Mathematics and Physics China haiqingzhaoncepu@163.com Yanxia Lu Department of Mathematics and Physics China yanxialuncepu@163.com Phikul Sridarat Department of Mathematics, Faculty of Science Chiang Mai University Thailand phikul_s@cmu.ac.th Suthep Suantai Department of Mathematics, Faculty of Science Chiang Mai University Thailand suthep.s@cmu.ac.th Yeol Je Cho Department of Mathematics Education and RINS Center for General Education Gyeongsang National University China Medical University Korea Taiwan yjcho@gnu.ac.kr Fuzzy metric space metric-like space Cauchy sequence fixed point theorem. Article.28.pdf  H. Aydi, A. Felhi, S. Sahmim, Fixed points of multivalued nonself almost contractions in metric-like spaces, Math. Sci. (Springer), 9 (2015), 103-108 ## H. Aydi, E. Karapınar, Fixed point results for generalized $$\alpha-\psi$$-contractions in metric-like spaces and applications, Electron. J. Differential Equations, 2015 (2015), 1-15 ## Y. J. Cho, M. Grabiec, V. Radu, On nonsymmetric topological and probabilistic structures, Nova Science Publishers, Inc., New York (2006) ## Z. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl., 86 (1982), 74-95 ## M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69 (1979), 205-230 ## J.-X. Fang , On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 46 (1992), 107-113 ## A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399 ## M. Grabiec , Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27 (1988), 385-389 ## V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 125 (2002), 245-252 ## V. I. Istrăţescu, An introduction to theory of probabilistic metric spaces with applications, (Romanian) Ed. , Tehnică Bucureşti (1974) ## O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984), 215-229 ## O. Kramosil, J. Michálek, Fuzzy metrics and statistical metric spaces, Kybernetika (Prague), 11 (1975), 336-344 ## D. Miheţ, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems, 144 (2004), 431-439 ## D. Miheţ, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems, 158 (2007), 915-921 ## D. Miheţ, A class of contractions in fuzzy metric spaces, Fuzzy Sets and Systems, 161 (2010), 1131-1137 ## S. N. Mishra, N. Sharma, S. L. Singh, Common fixed points of maps on fuzzy metric spaces, Internat. J. Math. Math. Sci., 17 (1994), 253-258 ## A. F. Roldán López de Hierro, M. de la Sen, Some fixed point theorems in Menger probabilistic metric-like spaces, Fixed Point Theory Appl., 2015 (2015), 1-16 ## S. Sharma, Common fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 127 (2002), 345-352 ## Y.-H. Shen, D. Qiu, W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Lett., 25 (2012), 138-141 ## B. Singh, M. S. Chauhan, Common fixed points of compatible maps in fuzzy metric spaces, Fuzzy Sets and Systems, 115 (2000), 471-475 ## R. Vasuki, A common fixed point theorem in a fuzzy metric space, Fuzzy Sets and Systems, 97 (1998), 395-397
Coincidence point results via generalized $$(\psi,\phi)$$-weak contractions in partial ordered $$b$$-metric spaces with application Coincidence point results via generalized $$(\psi,\phi)$$-weak contractions in partial ordered $$b$$-metric spaces with application en en 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. 3719 3731 Muhammad Sarwar Department of Mathematics University of Malakand Pakistan sarwarswati@gmail.com Noor Jamal Department of Mathematics University of Malakand Pakistan noorjamalmphil791@gmail.com Yongjin Li Department of Mathematics Sun Yat-sen University P. R. China stslyj@mail.sysu.edu.cn coincidence point weak compatible mapping increasing pairs of maps Generalized $$(\psi،\phi)$$-weak contraction partial ordered complete b-metric spaces. Article.29.pdf  M. Abbas, D. Đorić, Common fixed point for generalized $$(\psi,\phi)$$-weak contractions, Math. Un. of Nis. Serbia., 10 (2010), 1-10 ## M. Abbas, T. Nazir, S. Radenović, Common fixed points of four maps in partially ordered metric spaces, Appl. Math. Lett., 24 (2011), 1520-1526 ## A. Aghajani, M. Abbas, J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64 (2014), 941-960 ## R. Allahyari, R. Arab, A. Shole Haghighi, A generalization on weak contractions in partially ordered b-metric spaces and its application to quadratic integral equations, J. Inequal. Appl., 2014 (2014), 1-15 ## I. Altun, H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl., 2010 (2010), 1-17 ## S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263-276 ## D. Dorić, Common fixed point for generalized $$(\psi,\phi)$$-weak contractions, Appl. Math. Lett., 22 (2009), 1896-1900 ## P. N. Dutta, B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl., 18 (2008), 1-8 ## J. Esmaily, S. M. Vaezpour, B. E. Rhoades, Coincidence point theorem for generalized weakly contractions in ordered metric spaces, Appl. Math. Comput., 219 (2012), 1536-1548 ## M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed point results in metric-type spaces, Fixed Point Theory Appl., 2010 (2010), 1-15 ## G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 4 (1986), 771-779 ## G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci., 4 (1996), 199-215 ## P. P. Murthy, K. Tas, U. Devi Patel, Common fixed point theorems for generalized $$(\phi,\psi)$$-weak contraction condition in complete metric spaces, J. Inequal. Appl., 2015 (2015), 1-14 ## H. K. Nashine, B. Samet, Fixed point results for mappings satisfying $$(\psi,\phi)$$-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 2201-2209 ## S. Radenović, Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Comput. Math. Appl., 60 (2010), 1776-1783 ## A. C. M. Ran , M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435-1443 ## 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 spaces, Math. Sci. Lett., 4 (2015), 25-31 ## A. Razani, V. Parvaneh, M. Abbas, A common fixed point for generalized $$(\psi,\phi)_{f,g}$$-weak contractions, Ukrainian Math. J., 63 (2012), 1756-1769 ## J. R. Roshan, V. Parvaneh, I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput., 226 (2014), 725-737 ## J. R. Roshan, V. Parvaneh, S. Radenović, M. Rajović, Some coincidence point results for generalized $$(\psi,\varphi)$$-weakly contractions in ordered b-metric spaces, Fixed Point Theory Appl., 2015 (2015), 1-21 ## J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei,W. Shatanawi, Common fixed points of almost generalized $$(\psi,\varphi)_s$$- contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-23 ## W. Shatanawi, B. Samet, On $$(\psi,\phi)$$-weakly contractive condition in partially ordered metric spaces, Comput. Math. Appl., 62 (2011), 3204-3214
Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems en en 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. 3732 3754 Jong Soo Jung Department of Mathematics Dong-A University Korea jungjs@dau.ac.kr Hybrid iterative method generalized mixed equilibrium problem continuous monotone mapping continuous pseudocontractive mapping variational inequality fixed point $$\rho$$-Lipschitzian and $$\eta$$-strongly monotone mapping metric projection. Article.30.pdf  R. P. Agarwal, D. O’Regan, D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Topological Fixed Point Theory and Its Applications, Springer, New York (2009) ## E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145 ## L.-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 mappings, J. Comput. Appl. Math., 223 (2009), 967-974 ## L.-C. Ceng, S.-M. Guu, J.-C. Yao, Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems, Fixed Point Theory Appl., 2012 (2012), 1-19 ## L.-C. Ceng, J.-C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214 (2008), 186-201 ## S.-S. Chang, H. W. Joseph Lee, C. K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal., 70 (2009), 3307-3319 ## S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427-438 ## V. Colao, G. Marino, H.-K. Xu, An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl., 344 (2008), 340-352 ## P. I. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136 ## S. D. Flåm, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming, 78 (1997), 29-41 ## H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350 ## C. Jaiboon, P. Kumam, A general iterative method for addressing mixed equilibrium problems and optimization problems, Nonlinear Anal., 73 (2010), 1180-1202 ## J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, Appl. Math. Comput., 213 (2009), 498-505 ## J. S. Jung, Some results on a general iterative method for k-strictly pseudo-contractive mappings, Fixed Point Theory Appl., 2011 (2011), 1-11 ## J. S. Jung, Iterative methods for mixed equilibrium problems and strictly pseudocontractive mappings, Fixed point Theory Appl., 2012 (2012), 1-19 ## J. S. Jung, Weak convergence theorems for strictly pseudocontractive mappings and generalized mixed equilibrium problems, J. Appl. Math., 2012 (2012), 1-18 ## J. S. Jung, Weak convergence theorems for generalized mixed equilibrium problems, monotone mappings and pseudocontractive mappings, J. Korean Math. Soc., 52 (2015), 1179-1194 ## P. Katchang, T. Jitpeera, P. Kumam, Strong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method, Nonlinear Anal. Hybrid Syst., 4 (2010), 838-852 ## S.-T. Lv, Some results on a two-step iterative algorithm in Hilbert spaces, J. Nonlinear Funct. Anal., 2015 (2015), 1-10 ## G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 315-321 ## A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal., 9 (2008), 37-43 ## J.-W. Peng, J.-C. Yao, A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese J. Math., 12 (2008), 1401-1432 ## S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 336 (2007), 445-468 ## S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput., 197 (2008), 548-558 ## Y.-F. Su, M.-J. Shang, X.-L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal., 69 (2008), 2709-2719 ## T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239 ## A. Tada, W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl., 133 (2007), 359-370 ## S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506-515 ## S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal., 69 (2008), 1025-1033 ## W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428 ## J.-F. Tang, S.-S. Chang, J. Dong, Split equality fixed point problem for two quasi-asymptotically pseudocontractive mappings, J. Nonlinear Funct. Anal., 2017 (2017), 1-15 ## S. Wang, C.-S. Hu, G.-Q. Chai, Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems, Appl. Math. Comput., 215 (2010), 3891-3898 ## U. Witthayarat, A. A. N. Abdou, Y. J. Cho, Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), 1-14 ## H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659-678 ## I. Yamada, The 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 applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam, 8 (2001), 473-504 ## Y.-H. Yao, Y. J. Cho, R.-D. Chen, An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems, Nonlinear Anal., 71 (2009), 3363-3373 ## Y.-H. Yao, Y.-C. Liou, J.-C. Yao, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory Appl., 2007 (2007), 1-12 ## Y.-H. Yao, M. A. Noor, Y.-C. Liou, On iterative methods for equilibrium problems, Nonlinear Anal., 70 (2009), 497-509 ## Y.-H. Yao, M. A. Noor, S. Zainab, Y.-C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl., 354 (2009), 319-329 ## Y.-H. Yao, Z.-S. Yao, A. A. N. Abdou, Y. J. Cho, Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis, Fixed Point Theory Appl., 2015 (2015), 1-13 ## H. Zegeye, An iterative approximation method for a common fixed point of two pseudocontractive mappings, ISRN Math. Anal., 2011 (2011), 1-14 ## S.-S. Zhang, Generalized mixed equilibrium problem in Banach spaces, Appl. Math. Mech. (English Ed.), 30 (2009), 1105-1112
Analysis of an SVEIR model with age-dependence vaccination, latency and relapse Analysis of an SVEIR model with age-dependence vaccination, latency and relapse en en 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. 3755 3776 Jinliang Wang School of Mathematical Science Heilongjiang University China jinliangwang@hlju.edu.cn Xiu Dong School of Mathematical Science Heilongjiang University China ngxiaoxiu@163.com Hongquan Sun School of Mathematical Science Heilongjiang University China sunhongquan@hlju.edu.cn Vaccination age latency age relapse age global stability Lyapunov function. Article.31.pdf  C. J. Browne, S. S. Pilyugin, Global analysis of age-structured within-host virus model , Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017 ## Y.-M. Chen, S.-F. Zou, J.-Y. Yang, Global analysis of an SIR epidemic model with infection age and saturated incidence, Nonlinear Anal. Real World Appl., 30 (2016), 16-31 ## R. D. Demasse, A. Ducrot, An age-structured within-host model for multistrain malaria infections, SIAM J. Appl. Math., 73 (2013), 572-593 ## X.-C. Duan, S.-L. Yuan, X.-Z. Li, Global stability of an SVIR model with age of vaccination, Appl. Math. Comput., 226 (2014), 528-540 ## 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 state, Proc. R. Soc. Lond. B Biol. Sci., 253 (1993), 197-201 ## D. Ganem, A. M. Prince, Hepatitis B virus infection—natural history and clinical consequences, N. Engl. J. Med., 350 (2004), 1118-1129 ## J. K. Hale, Functional differential equations, Applied Mathematical Sciences, Springer-Verlag New York, New York- Heidelberg (1971) ## J. K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI (1988) ## J. K. Hale, P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395 ## M. Iannelli , Mathematical theory of age-structured population dynamics, Appl. Math. Monogr. C.N.R., Giardini Editori e Stampatori in Pisa (1995) ## L.-L. Liu, J.-L. Wang, X.-N. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. Real World Appl., 24 (2015), 18-35 ## P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 1-35 ## P. Magal, C. C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095 ## P. Magal, C. C. McCluskey, G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140 ## P. Magal, X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275 ## C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841 ## L.-L. Rong, Z.-L. Feng, A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. Appl. Math., 67 (2007), 731-756 ## H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066 ## H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801 ## P. van den Driessche, L. Wang, X.-F. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219 ## P. van den Driessche, X.-F. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103 ## J. A. Walker, Dynamical systems and evolution equations, Theory and applications, Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York-London (1980) ## J.-L. Wang, R. Zhang, T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, J. Math. Anal. Appl., 432 (2015), 289-313 ## J.-L. Wang, R. Zhang, T. Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes, J. Biol. Dyn., 9 (2015), 73-101 ## J.-L. Wang, R. Zhang, T. Kuniya, A note on dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 13 (2016), 227-247 ## J.-L. Wang, R. Zhang, T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, IMA J. Appl. Math., 81 (2016), 321-343 ## G. F. Webb, Theory of nonlinear age-dependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1985) ## J.-X. Yang, Z.-P. Qiu, X.-Z. Li, Global stability of an age-structured cholera model, Math. Biosci. Eng., 11 (2014), 641-665
Two type quasi-contractions on quasi metric spaces and some fixed point results Two type quasi-contractions on quasi metric spaces and some fixed point results en en 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). 3777 3783 Hakan Şimşek Department of Mathematics, Faculty of Science and Arts Kirikkale University Turkey hsimsek@kku.edu.tr Ishak Altun Department of Mathematics, Faculty of Science and Arts College of Science Kirikkale University King Saud University Turkey Saudi Arabia ishakaltun@yahoo.com Quasi metric space left K-Cauchy sequence left K-completeness fixed point quasi contraction. Article.32.pdf  E. Alemany, S. Romaguera, On right K-sequentially complete quasi-metric spaces, Acta Math. Hungar., 75 (1997), 267-278 ## I. Altun, M. Olgun, G. Mınak, Classification of completeness of quasi metric space and some new fixed point results, Nonlinear Funct. Anal. Appl., 22 (2017), 371-384 ## L. Ćirić, Fixed Point Theory Contraction Mapping Principle, Faculty of Mechanical Enginearing, University of Belgrade, Beograd (2003) ## S. Cobzaş, Functional analysis in asymmetric normed spaces, Birkhuser-Springer, Basel (2013) ## H. Dağ, G. Mınak, I. Altun, Some fixed point results for multivalued F-contractions on quasi metric spaces, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 111 (2017), 177-187 ## H. Dağ, S. Romaguera, P. Tirado , The Banach contraction principle in quasi-metric spaces revisited, Proceeding of the Workshop on Applied Topological Structures WATS’15, (2015), 25-31 ## J. C. Kelly, Bitopological spaces, Proc. London Math. Soc., 13 (1963), 71-89 ## H.-P. A. Künzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, Handbook of the history of general topology (Springer), 3 (2001), 853-968 ## I. L. Reilly, P. V. Subrahmanyam, M. K. Vamanamurthy, Cauchy sequences in quasi-pseudo-metric spaces, Monatsh. Math., 93 (1982), 127-140 ## S. Romaguera, A. Gutiérrez , A note on Cauchy sequences in quasi-pseudometric spaces, Glasnik Mat., 21 (1986), 191-200 ## S. Romaguera, Left K-completeness in quasi-metric spaces, Math. Nachr., 157 (1992), 15-23 ## M. Sarwar, M. U. Rahman, G. Ali, Some fixed point results in dislocated quasi metric (dq-metric) spaces, J. Inequal. Appl., 2014 (2014), 1-11
Spatio-temporal chaos in duopoly games Spatio-temporal chaos in duopoly games en en 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. 3784 3791 Risong Li School of Mathematic and Computer Science Guangdong Ocean University P. R. China gdoulrs@163.com Yu Zhao School of Mathematic and Computer Science Guangdong Ocean University P. R. China datom@189.cn Tianxiu Lu Department of Mathematics Artificial Intelligence Key Laboratory of Sichuan Province Sichuan University of Science and Engineering P. R. China P. R. China lubeeltx@163.com Ru Jiang School of Mathematic and Computer Science Guangdong Ocean University P. R. China jiru1995@163.com Hongqing Wang School of Mathematic and Computer Science Guangdong Ocean University P. R. China wanghq3333@126.com Haihua Liang School of Mathematic and Computer Science Guangdong Ocean University P. R. China lhhlucy@126.com Spatio-temporal chaos Li-Yorke sensitivity duopoly game. Article.33.pdf  E. Akin, S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433 ## G. I. Bischi, C. Mammana, L. Gardini, Multistability and cyclic attractors in duopoly games, Chaos Solitons Fractals, 11 (2000), 543-564 ## J. S. Cánovas, Chaos in duopoly games, Nonlinear Stud., 7 (2000), 97-104 ## J. S. Cánovas, M. Ruíz Marín, Chaos on MPE-sets of duopoly games, Dedicated to our teacher, mentor and friend, Nobel laureate, Ilya Prigogine, Chaos Solitons Fractals, 19 (2004), 179-183 ## J. S. Cánovas Peña, G. Soler López, M. Ruiz Marín, Distributional chaos of Cournot maps, Adv. Nonlinear Stud., 1 (2001), 79-87 ## R. A. Dana, L. Montrucchio, Dynamic complexity in duopoly games, J. Econom. Theory, 44 (1986), 40-56 ## H. Kato, Everywhere chaotic homeomorphisms on manifolds and k-dimensional Menger manifolds, Topology Appl., 72 (1996), 1-17 ## M. Kopel, Simple and complex adjustment dynamics in Cournot duopoly models, Complex dynamics in economic and social systems, Umea, (1995), Chaos Solitons Fractals, 7 (1996), 2031-2048 ## R.-S. Li, A note on distributional chaos of periodically adsorbing systems, (Chinese) J. Systems Sci. Math. Sci., 32 (2012), 237-243 ## R.-S. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45 (2012), 753-758 ## R.-S. Li, H.-Q. Wang, Y. Zhao, Kato’s chaos in duopoly games, Chaos Solitons Fractals, 84 (2016), 69-72 ## T. Y. Li, J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1972), 985-992 ## T.-X. Lu, P.-Y. Zhu, Further discussion on chaos in duopoly games, Chaos Solitons Fractals, 52 (2013), 45-48 ## P. Oprocha, P. Wilczyński, Shift spaces and distributional chaos, Chaos Solitons Fractals, 31 (2007), 347-355 ## R. Pikula, On some notions of chaos in dimension zero, Colloq. Math., 107 (2007), 167-177 ## T. Puu, Chaos in duopoly pricing, Chaos Solitons Fractals, 1 (1991), 573-581 ## T. Puu, I. Sushko, Oligopoly and complex dynamics, Springer, New York (2002) ## D. Rand, Exotic phenomena in games and duopoly models, J. Math. Econom., 5 (1978), 173-184 ## D. Ruelle, F. Takens, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192 ## B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754 ## J. Smítal, M. Štefánková, Distributional chaos for triangular maps, Chaos Solitons Fractals, 21 (2004), 1125-1128 ## L.-D. Wang, G.-F. Huang, S.-M. Huan, Distributional chaos in a sequence, Nonlinear Anal., 67 (2007), 2131-2136 ## X.-X. Wu, Chaos of transformations induced onto the space of probability measures, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-12 ## X.-X. Wu, A remark on topological sequence entropy, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1-7 ## X.-X. Wu, G.-R. Chen, Sensitivity and transitivity of fuzzified dynamical systems, Inform. Sci., 396 (2017), 14-23 ## X.-X. Wu, P. Oprocha, G.-R. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942-1972 ## X.-X. Wu, J.-J. Wang, A remark on accessibility, Chaos Solitons Fractals, 91 (2016), 115-117 ## X.-X. Wu, X. Wang, On the iteration properties of large deviations theorem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-6 ## X.-X. Wu, J.-J. Wang, G.-R. Chen, F-sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26 ## X.-X. Wu, X. Wang, G.-R. Chen, On the large deviations of weaker types, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1-12 ## X.-X. Wu, L.-D. Wang, G.-R. Chen, Weighted backward shift operators with invariant distributionally scrambled subsets, Ann. Funct. Anal., 8 (2017), 199-210 ## X.-X. Wu, L.-D. Wang, J.-H. Liang, The Chain Properties and Average Shadowing Property of Iterated Function Systems, Qual. Theory Dyn. Syst., 2016 (2016), 1-9 ## 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., (Accepted), -
Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces en en 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]. 3792 3814 Liben Wang Faculty of Civil Engineering and Mechanics Department of Mathematics, Faculty of Science Kunming University of Science and Technology Kunming University of Science and Technology P. R. China P. R. China Xingyong Zhang Department of Mathematics, Faculty of Science Kunming University of Science and Technology P. R. China zhangxingyong1@163.com Hui Fang Faculty of Civil Engineering and Mechanics Department of Mathematics, Faculty of Science Kunming University of Science and Technology Kunming University of Science and Technology P. R. China P. R. China Orlicz-Sobolev spaces mountain pass theorem symmetric mountain theorem. Article.34.pdf  R. A. Adams, J. F. Fournier, Sobolev spaces, Second edition, Pure and Applied Mathematics (Amsterdam), Elsevier/ Academic Press, Amsterdam (2003) ## K. Adriouch, A. El Hamidi, The Nehari manifold for systems of nonlinear elliptic equations, Nonlinear Anal., 64 (2006), 2149-2167 ## G. A. Afrouzi, S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $$(p_1, . . . , p_n)$$-Laplacian, Nonlinear Anal., 70 (2009), 135-143 ## C. O. Alves, G. M. Figueiredo, J. A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435-456 ## G. Anello, On the multiplicity of critical points for parameterized functionals on reflexive Banach spaces, Stud. Math., 213 (2012), 49-60 ## P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with ”strong” resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012 ## L. Boccardo, D. Guedes de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 309-323 ## G. Bonanno, G. Molica Bisci, D. O’Regan, Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Modelling, 52 (2010), 152-160 ## G. Bonanno, G. Molica Bisci, V. D. Rădulescu , Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz- Sobolev spaces, Nonlinear Anal., 75 (2012), 4441-4456 ## Y. Bozhkov, E. Mitidieri , Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239-267 ## F. Cammaroto, L. Vilasi, Multiple solutions for a nonhomogeneous Dirichlet problem in Orlicz-Sobolev spaces, Appl. Math. Comput., 218 (2012), 11518-11527 ## M. L. M. Carvalho, J. V. A. Goncalves, E. D. da Silva, On quasilinear elliptic problems without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 426 (2015), 466-483 ## N. T. Chung, H. Q. Toan, On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz-Sobolev spaces, Appl. Math. Comput., 219 (2013), 7820-7829 ## P. Clément, M. García-Huidobro, R. Manásevich, K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33-62 ## P. De Nápoli, M. C. Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal., 54 (2003), 1205-1219 ## A. El Khalil, M. Ouanan, A. Touzani, Existence and regularity of positive solutions for an elliptic system, Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., Southwest Texas State Univ., San Marcos, TX, 9 (2002), 171-182 ## F. Fang, Z. Tan , Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., 389 (2012), 420-428 ## N. Fukagai, M. Ito, K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $$R^N$$, Funkcial. Ekvac., 49 (2006), 235-267 ## N. Fukagai, M. Ito, K. Narukawa, Quasilinear elliptic equations with slowly growing principal part and critical Orlicz- Sobolev nonlinear term, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 73-106 ## N. Fukagai, K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl., 186 (2007), 539-564 ## M. García-Huidobro, V. K. Le, R. Manásevich, K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 207-225 ## J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, Nonlinear analysis, function spaces and applications, Proc. Spring School, Horni Bradlo, (1978), Teubner, Leipzig, (1979), 59-94 ## J. Huentutripay, R. Manásevich, Nonlinear eigenvalues for a quasilinear elliptic system in Orlicz-Sobolev spaces, J. Dynam. Differential Equations, 18 (2006), 901-929 ## V. K. Le, Some existence results and properties of solutions in quasilinear variational inequalities with general growths, Differ. Equ. Dyn. Syst., 17 (2009), 343-364 ## G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219 ## G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361 ## J.-J. Liu, X.-Y. Shi, Existence of three solutions for a class of quasilinear elliptic systems involving the (p(x), q(x))-Laplacian, Nonlinear Anal., 71 (2009), 550-557 ## M. Mihăilescu, D. Repovš, Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces, Appl. Math. Comput., 217 (2011), 6624-6632 ## P. Pucci, J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66 ## P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986) ## M. M. Rao, Z. D. Ren, Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (2002) ## B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal., 74 (2011), 7446-7454 ## J. A. Santos, Multiplicity of solutions for quasilinear equations involving critical Orlicz-Sobolev nonlinear terms, Electron. J. Differential Equations, 2013 (2013), 1-13 ## L.-B. Wang, X.-Y. Zhang, H. Fang, Multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, Taiwanese J. Math., (2017), 1-32 ## T.-F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal., 68 (2008), 1733-1745 ## F.-L. Xia, G.-X. Wang, Existence of solution for a class of elliptic systems, J. Hunan Agric. Univ. Nat. Sci., 33 (2007), 362-366 ## J. F. Zhao, Structure theory of Banach spaces, (Chinese) Wuhan Univ. Press, Wuhan (1991)
On multi-valued weak quasi-contractions in b-metric spaces On multi-valued weak quasi-contractions in b-metric spaces en en 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. 3815 3823 Nawab Hussain Department of Mathematics King Abdulaziz University Saudi Arabia nhusain@kau.edu.sa Zoran D. Mitrović Faculty of Electrical Engineering University of Banja Luka Bosnia and Herzegovina zoran.mitrovic@etf.unibl.org Fixed points b-metric space set-valued mapping. Article.35.pdf  M. Abbas, N. Hussain, B. E. Rhoades, Coincidence point theorems for multivalued f-weak contraction mappings and applications, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 105 (2011), 261-272 ## A. Amini-Harandi, Fixed point theory for set-valued quasi-contraction maps in metric spaces, Appl. Math. Lett., 24 (2011), 1791-1794 ## H. Aydi, M. F. Bota, E. Karapınar, S. Mitrović, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl., 2012 (2012), 1-8 ## I. A. Bakhtin, The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 30 (1989), 26-37 ## M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl., 326 (2007), 772-782 ## S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727-730 ## L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math. (Beograd) (N.S.), 12 (1971), 19-26 ## L. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267-273 ## M. Cosentino, P. Salimi, P. Vetro, Fixed point results on metric-type spaces, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1237-1253 ## S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11 ## S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263-276 ## N. Hussain, A. Amini-Harandi, J. Y. Cho, Approximate endpoints for set-valued contractions in metric spaces, Fixed Point Theory Appl., 2010 (2010), 1-13 ## N. Hussain, D. Dorić, Z. Kadelburg, S. Radenović, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012 (2012), 1-12 ## 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 applications, Fixed Point Theory Appl., 2013 (2013), 1-18 ## N. Hussain, R. Saadati, R. P. Agrawal, On the topology and wt-distance on metric type spaces, Fixed Point Theory Appl., 2014 (2014), 1-14 ## N. Hussain, M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl., 62 (2011), 1677-1684 ## R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76 ## M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl., 2010 (2010), 1-7 ## M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73 (2010), 3123-3129 ## R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19 (2017), 2153-2163 ## S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475-488 ## S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121-124 ## J. R. Roshan, V. Parvaneh, I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput., 226 (2014), 725-737 ## J. R. Roshan, V. Parvaneh, S. Radenović, M Rajović, Some coincidence point results for generalized ($$\psi,\varphi$$)-weakly contractions in ordered b-metric spaces, Fixed Point Theory Appl., 2015 (2015), 1-21 ## J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei,W. Shatanawi, Common fixed points of almost generalized $$(\psi,\varphi)_s$$- contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-23 ## S. Shukla, S. Radenović, C. Vetro, Set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces, Int. J. Math. Math. Sci., 2014 (2014), 1-9
$$L^{2}(\mathbb{R}^{n})$$ estimate of the solution to the Navier-Stokes equations with linearly growth initial data $$L^{2}(\mathbb{R}^{n})$$ estimate of the solution to the Navier-Stokes equations with linearly growth initial data en en 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. 3824 3833 Minghua Yang School of Information Technology Jiangxi University of Finance and Economics P. R. China ymh20062007@163.com Navier-Stokes equations linearly growing data Ornstein-Uhlenbeck operators Article.36.pdf  G.-P. Gao, C. Cattani, X.-J. Yang, About local fractional three-dimensional compressible Navier-Stokes equations in cantortype cylindrical co-ordinate system, Therm. Sci., 20 (2016), 1-847 ## Y. Giga, O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, Kluwer Acad. Publ., Dordrecht, 1,2 (2003), 549-562 ## M. Hieber, O. Sawada, The Navier-Stokes equations in $$R^n$$ with linearly growing initial data, Arch. Ration. Mech. Anal., 175 (2005), 269-285 ## T. Kato , Strong $$L^p$$-solutions of the Navier-Stokes equation in $$R^m$$, with applications to weak solutions, Math. Z., 187 (1984), 471-480 ## G. Metafune, D. Pallara, E. Priola, Spectrum of Ornstein-Uhlenbeck operators in $$L^p$$ spaces with respect to invariant measures, J. Funct. Anal., 196 (2002), 40-60 ## G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an $$L^p$$-space with invariant measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 471-485 ## T. Miyakawa, M. E. Schonbek, On 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., 126 (2001), 443-455 ## O. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the whole space, Bol. Soc. Parana. Mat., 22 (2004), 75-96 ## O. Sawada, Y. Taniuchi, On the Boussinesq flow with nondecaying initial data, Funkcial. Ekvac., 47 (2004), 225-250 ## K. Wang, S.-Y. Liu, Analytical study of time-fractional Navier-Stokes equation by using transform methods, Adv. Difference Equ., 2016 (2016), 1-12 ## X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016) ## X.-J. Yang, Z.-Z. Zhang, H. M. Srivastava, Some new applications for heat and fluid flows via fractional derivatives without singular kernel, Therm. Sci., 20 (2016), 1-833
Chaos for finitely generated semigroup actions Chaos for finitely generated semigroup actions en en 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. 3843 3850 Lidong Wang School of Mathematical Sciences Department of Mathematics Dalian University of Technology Dalian Minzu University People’s Republic of China People’s Republic of China wld0707@126.com Yingcui Zhao School of Mathematical Sciences Dalian University of Technology People’s Republic of China zycchaos@126.com Zhenyan Chu Department of Mathematics Dalian Minzu University People’s Republic of China chuzhenyan8@163.com Li-Yorke chaos distributional chaos along a sequence finitely generated semigroup actions. Article.38.pdf  G. Cairns, A. Kolganova, A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain J. Math., 37 (2007), 371-397 ## E. Kontorovich, M. Megrelishvili , A note on sensitivity of semigroup actions, Semigroup Forum, 76 (2008), 133-141 ## F. Polo, Sensitive dependence on initial conditions and chaotic group actions, Proc. Amer. Math. Soc., 138 (2010), 2815-2826 ## O. V. Rybak, Li-Yorke sensitivity for semigroup actions, Ukrainian Math. J., 65 (2013), 752-759 ## L.-D. Wang, Y.-N. Li, Y.-L. Gao, H. Liu, Distributional chaos of time-varying discrete dynamical systems, Ann. Polon. Math., 107 (2013), 49-57 ## L.-D. Wang, G.-F. Liao, S.-M. Huan, Distributional chaos in a sequence, Nonlinear Anal., 67 (2007), 2131-2136 ## X.-X. Wu, Chaos of transformations induced onto the space of probability measures, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-12 ## X.-X. Wu, A remark on topological sequence entropy, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1-7 ## X.-X. Wu, G.-R. Chen, Sensitivity and transitivity of fuzzified dynamical systems, Inform. Sci., 396 (2017), 14-23 ## X.-X. Wu, P. Oprocha, G.-R. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942-1972 ## X.-X. Wu, X. Wang, On the iteration properties of large deviations theorem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-6 ## X.-X. Wu, J.-J. Wang, G.-R. Chen, F-sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26 ## X.-X. Wu, X. Wang, G.-R. Chen, On the large deviations of weaker types, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1-12 ## X.-X. Wu, L.-D. Wang, G.-R. Chen, Weighted backward shift operators with invariant distributionally scrambled subsets, Ann. Funct. Anal., 8 (2017), 199-210 ## X.-X. Wu, L.-D. Wang, J.-H. Liang, The chain properties and average shadowing property of iterated function systems, Qual. Theory Dyn. Syst., 2016 (2016), 1-9 ## 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., (Accepted), - ## J.-C. Xiong, Chaos in a topologically transitive system, Sci. China Ser. A, 48 (2005), 929-939
Fourier series of higher-order ordered Bell functions Fourier series of higher-order ordered Bell functions en en 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. 3851 3855 Taekyun Kim Department of Mathematics, College of Science Department of Mathematics Tianjin Polytechnic University Kwangwoon University China Republic of Korea tkkim@kw.ac.kr Dae San Kim Department of Mathematics Sogang University Republic of Korea dskim@sogang.ac.kr Gwan-Woo Jang Department of Mathematics Kwangwoon University Republic of Korea jgw5687@naver.com Jongkyum Kwon Department of Mathematics Education and RINS Gyeongsang National University Republic of Korea mathkjk26@gnu.ac.kr Fourier series higher-order ordered Bell functions higher-order ordered Bell polynomials. Article.39.pdf  M. Abramowitz, I. A. Stegun, Handbook 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. Cayley, On the analytical forms called trees, Philos. Mag., 9 (1859), 374-378 ## L. Comtet, Advanced combinatorics , The art of finite and infinite expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht (1974) ## D. V. Dolgy, D. S. Kim, T. Kim, T. Mansour, Sums of finite products of ordered Bell functions, , (preprint), - ## I. J. Good, The number of orderings of n candidates when ties are permitted, Fibonacci Quart., 13 (1975), 11-18 ## G.-W. Jang, D. S. Kim, T. Kim, T. Mansour, Fourier series of functions related to Bernoulli polynomials, Adv. Stud. Contemp. Math., 27 (2017), 49-62 ## T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal., 2008 (2008), 1-11 ## T. Kim, D. S. Kim, On $$\lambda$$-Bell polynomials associated with umbral calculus, Russ. J. Math. Phys., 24 (2017), 69-78 ## D. S. Kim, T. Kim, Fourier series of higher-order Euler functions and their applications, Bull. Korean Math. Soc., (to appear), - ## T. Kim, D. S. Kim, Some formulas of ordered Bell numbers and polynomials arising from umbral calculus, , (preprint), - ## T. Kim, D. S. Kim, S.-H. Rim, D.-V. Dolgy, Fourier series of higher-order Bernoulli functions and their applications, J. Inequal. Appl., 2017 (2017), 1-7 ## A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, Int. J. Number Theory, 1 (2005), 563-581 ## J. E. Marsden, Elementary classical analysis, With 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 Francisco (1974) ## M. Mor, A. S. Fraenkel, Cayley permutations , Discrete Math., 48 (1984), 101-112 ## A. Sklar, On the factorization of squarefree integers, Proc. Amer. Math. Soc., 3 (1952), 701-705 ## D. G. Zill, W. R. Wright, M. R. Cullen, Advanced engineering mathematics, Fourth edition, Jones and Bartlett Publishers, Mississauga, Ontario (2011)
A regularization algorithm for a splitting feasibility problem in Hilbert spaces A regularization algorithm for a splitting feasibility problem in Hilbert spaces en en 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. 3856 3862 Abdul Latif Department of Mathematics King Abdulaziz University Saudi Arabia alatif@kau.edu.sa Xiaolong Qin Institute of Fundamental and Frontier Sciences University of Electronic Science and Technology of China China qxlxajh@163.com Metric projection monotone operator nonexpansive mapping split feasibility problem variational inequality. Article.40.pdf  I. K. Argyros, S. George, S. M. Erappa, Expanding the applicability of the generalized Newton method for generalized equations, Commun. Optim. Theory, 2017 (2017), 1-12 ## B. A. Bin Dehaish, A. Latif, H. O Bakodah, X.-L. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 1-14 ## B. A. Bin Dehaish, X.-L. Qin, A. Latif, H. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321-1336 ## C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120 ## Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365 ## Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239 ## Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084 ## 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., 1 (2017), 145-158 ## S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427-438 ## S. Y. Cho, X.-L. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-15 ## N.-N. Fang, Y.-P. Gong, Viscosity iterative methods for split variational inclusion problems and fixed point problems of a nonexpansive mapping, Commun. Optim. Theory, 2016 (2016), 1-15 ## L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114-125 ## A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problem, Trans. Math. Program. Appl., 1 (2013), 1-11 ## X.-L. Qin, S. Y. Cho, Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 488-502 ## X.-L. Qin, S. Y. Cho, L. Wang, A regularization method for treating zero points of the sum of two monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-10 ## X.-L. Qin, J.-C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 1-9 ## D. R. Sahu, J. C. Yao, A generalized hybrid steepest descent method and applications, J. Nonlinear Var. Anal., 1 (2017), 111-126 ## J.-F. Tang, S.-S. Chang, J. Dong, Split equality fixed point problem for two quasi-asymptotically pseudocontractive mappings, J. Nonlinear Funct. Anal., 2017 (2017), 1-15 ## H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138 ## H.-Y. Zhou, P.-Y. Wang, Adaptively relaxed algorithms for solving the split feasibility problem with a new step size, J. Inequal. Appl., 2014 (2014), 1-22
Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations en en 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. 3863 3886 Chao Wang Department of Mathematics Department of Mathematics Yunnan University Texas A&M University-Kingsville People’s Republic of China USA chaowang@ynu.edu.cn Ravi P. Agarwal Department of Mathematics Texas A&M University-Kingsville USA Agarwal@tamuk.edu Donal O'Regan School of Mathematics, Statistics and Applied Mathematics National University of Ireland Ireland donal.oregan@nuigalway.ie Time scales weighted pseudo double-almost periodic solution impulsive evolution equations. Article.41.pdf  R. P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22 ## R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: a survey, Dynamic equations on time scales, J. Comput. Appl. Math., 141 (2002), 1-26 ## R. P. Agarwal, D. O’Regan, Some comments and notes on almost periodic functions and changing-periodic time scales, Electr. J. Math. Anal. Appl., 6 (2018), 125-136 ## M. U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Anal., 65 (2006), 2043-2060 ## S. Bochner, Beiträge zur Theorie der fastperiodischen Funktionen, (German) Math. Ann., 96 (1927), 119-147 ## M. Bohner, G. S. Guseinov, Double integral calculus of variations on time scales, Comput. Math. Appl., 54 (2007), 45-57 ## M. Bohner, A. Peterson, Dynamic equations on time scales, An introduction with applications, Birkhäuser Boston, Inc., Boston, MA (2001) ## T. Caraballo, D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition, I, J. Differential Equations, 246 (2009), 108-128 ## T. Caraballo, D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition, II, J. Differential Equations, 246 (2009), 1164-1186 ## C. Cuevas, E. Hernández, M. Rabelo , The existence of solutions for impulsive neutral functional differential equations, Comput. Math. Appl., 58 (2009), 744-757 ## C. Cuevas, A. Sepúlveda, H. Soto, Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations, Appl. Math. Comput., 218 (2011), 1735-1745 ## T. Diagana, Weighted pseudo almost periodic functions and applications, C. R. Math. Acad. Sci. Paris, 343 (2006), 643-646 ## T. Diagana, Existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations, Nonlinear Anal., 74 (2011), 600-615 ## A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York (1974) ## J. Gao, Q.-R. Wang, L.-W. Zhang, Existence and stability of almost-periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales, Appl. Math. Comput., 237 (2014), 639-649 ## E. Hernández, M. Rabello, H. R. Henríquez, Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl., 331 (2007), 1135-1158 ## S. Hilger, Ein Maßkettenkalkäul mit anwendung auf zentrumsmannigfaltigkeiten, Ph.D thesis, Universität Wäurzburg (1988) ## S. Hilger, Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56 ## B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math., 186 (2006), 391-415 ## E. R. Kaufmann, Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl., 319 (2006), 315-325 ## J. Liang, J. Zhang, T.-J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl., 340 (2008), 1493-1499 ## C. Pötzsche, Exponential dichotomies for dynamic equations on measure chains, Proceedings of the Third World Congress of Nonlinear Analysts, Part 2, Catania, (2000), Nonlinear Anal., 47 (2001), 873-884 ## A. 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, NJ (1995) ## G. T. Stamov, Almost periodic solutions of impulsive differential equations, Lecture Notes in Mathematics, Springer, Heidelberg (2012) ## I. Stamova, Stability analysis of impulsive functional differential equations, De Gruyter Expositions in Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin (2009) ## C. C. Tisdell, A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal., 68 (2008), 3504-3524 ## C. Wang, Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2828-2842 ## C. Wang, R. P. Agarwal, A further study of almost periodic time scales with some notes and applications, Abstr. Appl. Anal., 2014 (2014), 1-11 ## C. Wang, R. P. Agarwal, Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations, Adv. Difference Equ., 2015 (2015), 1-9 ## C. Wang, R. P. Agarwal, A classification of time scales and analysis of the general delays on time scales with applications, Math. Methods Appl. Sci., 39 (2016), 1568-1590 ## C. Wang, R. P. Agarwal, Almost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales , Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 238-251 ## C. Wang, R. P. Agarwal, D. O’Regan, Compactness criteria and new impulsive functional dynamic equations on time scales, Adv. Difference Equ., 2016 (2016), 1-41 ## C. Wang, R. P. Agarwal, D. O’Regan, Periodicity, almost periodicity for time scales and related functions, Nonauton. Dyn. Syst., 3 (2016), 24-41 ## C. Wang, R. P. Agarwal, D. O’Regan, Piecewise double-almost periodic functions on almost-complete closedness time scales with generalizations, , (submitted), - ## C. Y. Zhang, Pseudo-almost-periodic solutions of some differential equations, J. Math. Anal. Appl., 181 (1994), 62-76 ## C. Y. Zhang, Pseudo almost periodic solutions of some differential equations, II, J. Math. Anal. Appl., 192 (1995), 543-561
Strong convergence of some iterative algorithms for a general system of variational inequalities Strong convergence of some iterative algorithms for a general system of variational inequalities en en 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. 3887 3902 Jong Soo Jung Department of Mathematics Dong-A University Korea jungjs@dau.ac.kr Composite iterative algorithm general system of variational inequatlites continuous monotone mapping continuous peudocontractive mapping $$\rho$$-Lipschitzian $$\eta$$-strongly monotone mapping variational inequality strongly positive bounded linear operator fixed points. Article.42.pdf  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 spaces, J. Nonlinear Convex Anal., 17 (2016), 669-682 ## L.-C. Ceng, C.-Y. Wang, J.-C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res., 67 (2008), 375-390 ## J.-M. Chen, L.-J. Zhang, T.-G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl., 334 (2007), 1450-1461 ## K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990) ## H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350 ## J. S. Jung, A general composite iterative method for strictly pseudocontractive mappings in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 1-21 ## J. S. Jung, A composite extragradient-like algorithm for inverse-strongly monotone mappings and strictly pseudocontractive mappings, Linear Nonlinear Anal., 1 (2015), 271-285 ## G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747-756 ## P.-L. Lions, G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519 ## F.-S. Liu, M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344 ## G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52 ## G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 315-321 ## W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428 ## Y. Tang, Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings, Fixed Point Theory Appl., 2013 (2013), 1-11 ## H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256 ## I. Yamada, The 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 applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam, 8 (2001), 473-504 ## H. Zegeye, An iterative approximation method for a common fixed point of two pseudocontractive mappings, ISRN Math. Anal., 2011 (2011), 1-14 ## H. Zegeye, N. Shahzad, Strong convergence of an iterative method for pseudo-contractive and monotone mappings, J. Global Optim., 54 (2012), 173-184
Controllability of fractional impulsive neutral stochastic functional differential equations via Kuratowski measure of noncompactness Controllability of fractional impulsive neutral stochastic functional differential equations via Kuratowski measure of noncompactness en en 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. 3903 3915 Junhao Hu School of Mathematics and Statistics South-Central University for Nationalities China junhaohu74@163.com Jiashun Yang School of Mathematics and Statistics South-Central University for Nationalities China Chenggui Yuan Department of Mathematics Swansea University UK C.Yuan@swansea.ac.uk Controllability fractional differential equations impulsive stochastic differential equations Kuratowski measure of noncompactness Mönch fixed point theorem. Article.43.pdf  R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033 ## P. Balasubramaniam, J. P. Dauer, Controllability of semilinear stochastic delay evolution equations in Hilbert spaces, Int. J. Math. Math. Sci., 31 (2002), 157-166 ## P. Balasubramaniam, J. Y. Park, P. Muthukumar, Approximate controllability of neutral stochastic functional differential systems with infinite delay, Stoch. Anal. Appl., 28 (2010), 389-400 ## H.-B. Bao, J.-D. Cao, Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 215 (2009), 1732-1743 ## S. Das, D. Pandey, N. Sukavanam, Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1509-1523 ## A. Dehici, N. Redjel, Measure of noncompactness and application to stochastic differential equations , Adv. Difference Equ., 2016 (2016), 1-17 ## S. Duan, J.-H. Hu, Y. Li, Exact controllability of nonlinear stochastic impulsive evolution differential inclusions with infinite delay in Hilbert spaces, Int. J. Nonlinear Sci. Numer. Simul., 12 (2011), 23-33 ## T. Guendouzi, Existence and controllability of fractional-order impulsive stochastic system with infinite delay, Discuss. Math. Differ. Incl. Control Optim., 33 (2013), 65-87 ## D. J. Guo, Impulsive integral equations in Banach spaces and applications, J. Appl. Math. Stochastic Anal., 5 (1992), 111-122 ## H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., (1983), 1351-1371 ## L.-R. Huang, F.-Q. Deng, Razumikhin-type theorems on stability of neutral stochastic functional differential equations, IEEE Trans. Automat. Control, 53 (2008), 1718-1723 ## Y.-G. Kao, Q.-X. Zhu, W.-H. Qi, Exponential stability and instability of impulsive stochastic functional differential equations with Markovian switching, Appl. Math. Comput., 271 (2015), 795-804 ## A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006) ## V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ (1989) ## K.-X. Li, J.-G. Peng, Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941-959 ## X.-R. Mao, Razumikhin-type theorems on exponential stability of stochastic functional-differential equations, Stochastic Process. Appl., 65 (1996), 233-250 ## X.-R. Mao, Stochastic differential equations and their applications, Second edition, Woodhead Publishing Limited, Cambridge (2007) ## K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993) ## H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980), 985-999 ## A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York (1983) ## I. 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, CA (1999) ## M. D. Quinn, N. Carmichael, An approach to nonlinear control problems using fixed-point methods, degree theory and pseudo-inverses, Numer. Funct. Anal. Optim., 7 (1984/85), 197-219 ## R. Sakthivel, R. Ganesh, S. Suganya, Approximate controllability of fractional neutral stochastic system with infinite delay, Rep. Math. Phys., 70 (2012), 291-311 ## R. Sakthivel, N. I. Mahmudov, J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340 ## R. Sakthivel, S. Suganya, S. M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660-668 ## R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optimization, 15 (1977), 407-411 ## R. Triggiani, ddendum: ”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., 18 (1980), 98-99 ## Q. Wang, X.-Z. Liu, Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method, Appl. Math. Lett., 20 (2007), 839-845 ## T. Yang, Impulsive systems and control: theory and applications, Nova Science Publishers, Inc., New York (2001) ## R.-P. Ye, Existence of solutions for impulsive partial neutral functional differential equation with infinite delay, Nonlinear Anal., 73 (2010), 155-162 ## Y.-C. Zang, J.-P. Li, Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Bound. Value Probl., 2013 (2013), 1-13 ## Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077
New exact solution of generalized biological population model New exact solution of generalized biological population model en en 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. 3916 3929 Omer Acan Art and Science Faculty, Department of Mathematics Siirt University Turkey omeracan@yahoo.com Maysaa Mohamed Al Qurashi Faculty of Art and Science, Department of Mathematics King Saud University Saudi Arabia maysaa@ksu.edu.sam Dumitru Baleanu Faculty of Art and Science, Department of Mathematics Institute of Space Sciences Cankaya University Turkey Romania dumitru@cankaya.edu.tr Numerical solution biological populations model reduced differential transform method conformable derivative partial differential equations. Article.44.pdf  T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66 ## O. Acan, The existence and uniqueness of periodic solutions for a kind of forced rayleigh equation, Gazi Univ. J. Sci., 29 (2016), 645-650 ## O. Acan, D. Baleanu, A new numerical technique for solving fractional partial differential equations, ArXiv, 2017 (2017), 1-12 ## O. Acan, D. Baleanu, M. M. A. Qurashi, M. G. Sakar, Analytical approximate solutions of (n + 1)-dimensional fractal heat-like and wave-like equations, Entropy, 19 (2017), 1-14 ## O. Acan, O. Firat, Y. Keskin, The use of conformable variational iteration method, conformable reduced differential transform method and conformable homotopy analaysis method for solving different types of nonlinear partial differential equations, Proceedings of the 3rd International Conference on Recent Advances in Pure and Applied Mathematics, Bodrum, Turkey (2016) ## O. Acan, O. Firat, Y. Keskin, G. Oturanc, Solution of conformable fractional partial differential equations by reduced differential transform method, Selcuk J. Appl. Math., (2016), - ## O. Acan, O. Firat, Y. Keskin, G. Oturanc, Conformable variational iteration method, New Trends Math. Sci., 5 (2017), 172-178 ## O. Acan, Y. Keskin, Approximate solution of Kuramoto-Sivashinsky equation using reduced differential transform method, Proc. Int. Conf. Numer. Anal. Appl. Math., AIP Publishing, 1648 (2015), 1-470003 ## O. Acan, Y. Keskin, Reduced differential transform method for (2+1) dimensional type of the Zakharov-Kuznetsov ZK(n,n) equations, Proc. Int. Conf. Numer. Anal. Appl. Math., AIP Publishing, 1648 (2015), 1-370015 ## O. Acan, Y. Keskin, A comparative study of numerical methods for solving (n + 1) dimensional and third-order partial differential equations, J. Comput. Theor. Nanosci., 13 (2016), 8800-8807 ## O. Acan, Y. Keskin, A new technique of Laplace Pade reduced differential transform method for (1 + 3) dimensional wave equations, New Trends Math. Sci., 5 (2017), 164-171 ## D. R. Anderson, D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137 ## A. A. M. Arafa, S. Z. Rida, H. Mohamed, Homotopy analysis method for solving biological population model, Commun. Theor. Phys. (Beijing), 56 (2011), 797-800 ## A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898 ## D. Baleanu, A. H. Bhrawy, R. A. Van Gorder, New trends on fractional and functional differential equations, Abstr. Appl. Anal., 2015 (2015), 1-2 ## D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus, Models and numerical methods, Second edition, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017) ## D. Baleanu, K. Sayevand, Performance evaluation of matched asymptotic expansions for fractional differential equations with multi-order, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 59 (2016), 3-12 ## D. Baleanu, J. A. Tenreiro Machado, C. Cattani, M. C. Baleanu, X.-J. Yang, Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators, Abstr. Appl. Anal., 2014 (2014), 1-6 ## N. Baykus, M. Sezer, Solution of high-order linear Fredholm integro-differential equations with piecewise intervals, Numer. Methods Partial Differential Equations, 27 (2011), 1327-1339 ## W. S. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math., 290 (2015), 150-158 ## M. Ekici, M. Mirzazadeh, M. Eslami, Q. Zhou, S. P. Moshokoa, A. Biswas, M. Belic, Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives, Optik, 127 (2016), 10659-10669 ## A. M. A. El-Sayed, S. Z. Rida, A. A. M. Arafa, Exact solutions of fractional-order biological population model, Commun. Theor. Phys. (Beijing), 52 (2009), 992-996 ## A. Gökdoğan, E. Ünal, E. Çelik, Existence and uniqueness theorems for sequential linear conformable fractional differential equations, Miskolc Math. Notes, 17 (2016), 267-279 ## A. K. Golmankhaneh, X.-J. Yang, D. Baleanu, Einstein field equations within local fractional calculus, Rom. J. Phys., 60 (2015), 22-31 ## Z.-H. Guo, O. Acan, S. Kumar, Sumudu transform series expansion method for solving the local fractional Laplace equation in fractal thermal problems, Therm. Sci., 20 (2016), 1-739 ## W. S. C. Gurney, R. M. Nisbet, The regulation of inhomogeneous populations, J. Theor. Biol., 52 (1975), 441-457 ## M. E. Gurtin, R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49 ## Z. Hacioglu, N. Baykus Savasaneril, H. Kose, Solution of Dirichlet problem for a square region in terms of elliptic functions, New Trends Math. Sci., 3 (2016), 98-103 ## H. Karayer, D. Demirhan, F. Büyükklç, Conformable fractional Nikiforov-Uvarov method, Commun. Theor. Phys. (Beijing), 66 (2016), 12-18 ## R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70 ## A. Kurt, Y. C¸ enesiz, O. Tasbozan, On the solution of Burgers equation with the new fractional derivative, Open Phys., 13 (2015), 355-360 ## Y.-Q. Liu, Z.-L. Li, Y.-Y. Zhang, Homotopy perturbation method to fractional biological population equation, Fract. Differ. Calc., 1 (2011), 117-124 ## Y.-G. Lu, Hölder estimates of solutions of biological population equations, Appl. Math. Lett., 13 (2000), 123-126 ## M. Ma, D. Baleanu, Y. S. Gasimov, X.-J. Yang, New results for multidimensional diffusion equations in fractal dimensional space, Rom. J. Phys., 61 (2016), 784-794 ## K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993) ## A. Okubo, Diffusion and ecological problems: mathematical models, An extended version of the Japanese edition, Ecology and diffusion, Translated by G. N. Parker. Biomathematics, Springer-Verlag, Berlin-New York (1980) ## K. B. Oldham, J. Spanier, The fractional calculus, Theory 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-London (1974) ## I. 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, CA (1999) ## P. Roul, Application of homotopy perturbation method to biological population model, Appl. Appl. Math., 5 (2010), 1369-1379 ## S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional 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, Yverdon (1993) ## S. Sarwar, M. A. Zahid, S. Iqbal, Mathematical study of fractional-order biological population model using optimal homotopy asymptotic method, Int. J. Biomath., 9 (2016), 1-17 ## N. B. Savasaneril, H. Delibas, Analytic solution for two-dimensional heat equation for an ellipse region, New Trends Math. Sci., 70 (2016), 65-70 ## F. Shakeri, M. Dehghan, Numerical solution of a biological population model using He’s variational iteration method, Comput. Math. Appl., 54 (2007), 1197-1209 ## B. K. Singh, A novel approach for numeric study of 2D biological population model, Cogent Math., 3 (2016), 1-15 ## J. Singh, D. Kumar, A. Kılıçman, Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations, Abstr. Appl. Anal., 2014 (2014), 1-12 ## V. K. Srivastava, S. Kumar, M. K. Awasthi, B. K. Singh, Two-dimensional time fractional-order biological population model and its analytical solution, Egyptian J. Basic Appl. Sci., 1 (2014), 71-76 ## E. Ünal, A. Gökdoğan, Solution of conformable fractional ordinary differential equations via differential transform method, Optik, 128 (2017), 264-273 ## X.-J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Therm. Sci., 21 (2017), 1161-1171 ## X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett., 47 (2015), 54-60 ## X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016) ## X.-J. Yang, F. Gao, H. M. Srivastava, New rheological models within local fractional derivative, Rom. Rep. Phys., 69 (2017), 1-12 ## X.-J. Yang, H. M. Srivastava, C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67 (2015), 752-761 ## X.-J. Yang, J. A. Tenreiro Machado, A new fractional operator of variable order: application in the description of anomalous diffusion, Phys. A, 481 (2017), 276-283 ## X.-J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26 (2016), 1-5
Some properties of g-p-frames in complex Banach spaces Some properties of g-p-frames in complex Banach spaces en en 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. 3930 3938 Xiao Tan School of Mathematics and Information Science North Minzu University China 13649507784@163.com Yongdong Huang School of Mathematics and Information Science North Minzu University China nxhyd74@126.com g-p-frame g-q-Riesz basis dual frames analysis operator. Article.45.pdf  M. R. Abdollahpour, M. H. Faroughi, A. Rahimi, pg-frames in Banach spaces, Methods Funct. Anal. Topology, 13 (2007), 201-210 ## M. R. Abdollahpour, A. Najati, P. Gavruta, Multipliers of pg-Bessel sequences in Banach spaces, ArXiv, 2015 (2015), 1-14 ## A. Aldroubi, Q. Sun, W.-S. Tang, p-frames and shift invariant subspaces of $$L^p$$, J. Fourier Anal. Appl., 7 (2001), 1-22 ## C. D. Aliprantis, K. C. Border, Infinite-dimensional analysis, A hitchhiker’s guide, Second edition, Springer-Verlag, Berlin (1999) ## O. Christensen, D. T. Stoeva, p-frames in separable Banach spaces, Frames. Adv. Comput. Math., 18 (2003), 117-126 ## J. B. Conway, A course in functional analysis, Graduate Texts in Mathematics, Springer-Verlag, New York (1985) ## R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341-366 ## A. Khosravi, K. Musazadeh, Fusion frames and g-frames, J. Math. Anal. Appl., 342 (2008), 1068-1083 ## X.-C. Xiao, X.-M. Zeng, Some properties of g-frames in Hilbert $$C^*$$-modules, J. Math. Anal. Appl., 363 (2010), 399-408 ## X.-C. Xiao, Y.-C. Zhu, X.-M. Zeng, Generalized p-frame in separable complex Banach spaces, Int. J. Wavelets Multiresolut. Inf. Process., 8 (2010), 133-148
Common fixed point results for probabilistic $$\varphi$$-contractions in generalized probabilistic metric spaces Common fixed point results for probabilistic $$\varphi$$-contractions in generalized probabilistic metric spaces en en 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. 3939 3962 Jingfeng Tian College of Science and Technology North China Electric Power University P. R. China tianjf@ncepu.edu.cn Ximei Hu China Mobile Group Hebei Co., Ltd. P. R. China huxm_bd@163.com Coupled fixed point fixed point metric space probabilistic $$\varphi$$-contractions gauge function. Article.46.pdf  R. P. Agarwal, E. Karapınar, Remarks on some coupled fixed point theorems in G-metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-33 ## V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 4889-4897 ## L. Ćirić, R. P. Agarwal, B. Samet, Mixed monotone-generalized contractions in partially ordered probabilistic metric spaces, Fixed Point Theory Appl., 2011 (2011), 1-13 ## J.-X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Anal., 71 (2009), 1833-1843 ## T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393 ## D. J. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal., 11 (1987), 623-632 ## O. Hadžić, A fixed point theorem in Menger spaces, Publ. Inst. Math. (Beograd) (N.S.), 20 (1979), 107-112 ## O. Hadžić, Fixed point theorems for multivalued mappings in probabilistic metric spaces, Fuzzy Sets and Systems, 88 (1997), 219-226 ## J. Jachymski, On probabilistic $$\phi$$-contractions on Menger spaces, Nonlinear Anal., 73 (2010), 2199-2203 ## M. Jleli, E. Karapınar, B. Samet, Further generalizations of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 1-9 ## M. Jleli, E. Karapınar, B. Samet, On cyclic ($$\psi,\phi$$)-contractions in Kaleva-Seikkala’s type fuzzy metric spaces, J. Intell. Fuzzy Systems, 27 (2014), 2045-2053 ## E. Karapınar, N. V. Luong, Quadruple fixed point theorems for nonlinear contractions, Comput. Math. Appl., 64 (2012), 1839-1848 ## V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341-4349 ## T. Luo, C.-X. Zhu, Z.-Q. Wu, Tripled common fixed point theorems under probabilistic $$\phi$$-contractive conditions in generalized Menger probabilistic metric spaces, Fixed Point Theory Appl., 2014 (2014), 1-17 ## K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U. S. A., 28 (1942), 535-537 ## Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289-297 ## V. I. Opoĭcev, Heterogeneous and combined-concave operators, (Russian) Sibirsk. Mat. Z ., 16 (1975), 781-792 ## B. Samet, On the approximation of fixed points for a new class of generalized Berinde mappings, Carpathian J. Math., 32 (2016), 363-374 ## B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York (1983) ## S. Sedghi, I. Altun, N. Shobe, Coupled fixed point theorems for contractions in fuzzy metric spaces, Nonlinear Anal., 72 (2010), 1298-1304 ## ] J. Wu, Some fixed-point theorems for mixed monotone operators in partially ordered probabilistic metric spaces, Fixed Point Theory Appl., 2014 (2014), 1-12 ## J.-Z. Xiao, X.-H. Zhu, Y.-F. Cao, Common coupled fixed point results for probabilistic $$\phi$$-contractions in Menger spaces, Nonlinear Anal., 74 (2011), 4589-4600 ## C.-L. Zhou, S.-H. Wang, L. Ćirić, S. M. Alsulami, Generalized probabilistic metric spaces and fixed point theorems, Fixed Point Theory Appl., 2014 (2014), 1-15
Common fixed point for mappings satisfying new contractive condition and applications to integral equations Common fixed point for mappings satisfying new contractive condition and applications to integral equations en en 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. 3988 3999 Feng Gu Institute of Applied Mathematics and Department of Mathematics Hangzhou Normal University China mathgufeng@163.com Hongqing Ye Hangzhou Wenchang High School China weili5412783137@qq.com Common fixed point generalized metric space integral equation. Article.48.pdf  M. Abbas, T. Nazir, S. Radenović, Some periodic point results in generalized metric spaces, Appl. Math. Comput., 217 (2010), 4094-4099 ## M. Abbas, B. E. Rhoades, Common fixed point results for noncommuting mappings without continuity in generalized metric spaces, Appl. Math. Comput., 215 (2009), 262-269 ## H. Aydi, W. Shatanawi, C. Vetro, On generalized weakly G-contraction mapping in G-metric spaces, Comput. Math. Appl., 62 (2011), 4222-4229 ## F. Gu, W. Gao, W. Tian, Fixed point theorem and the iterative convergence of nonlinear operator, Harbin Science and Technology Press, Harbin, China (2002) ## F. Gu, Z.-Z. Yang, Some new common fixed point results for three pairs of mappings in generalized metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-21 ## F. Gu, Y. Yin, Common fixed point for three pairs of self-maps satisfying common (E.A) property in generalized metric spaces, Abstr. Appl. Anal., 2013 (2013), 1-11 ## F. Gu, D. Zhang, The common fixed point theorems for six self-mappings with twice power type $$\Phi$$-contraction condition, Thai J. Math., 10 (2012), 587-603 ## M. Jleli, B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl, 2012 (2012), 1-7 ## E. Karapınar, R. P. Agarwal, Further fixed point results on G-metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-14 ## Z. Mustafa, H. Aydi, E. Karapınar, On common fixed points in G-metric spaces using (E.A) property, Comput. Math. Appl., 64 (2012), 1944-1956 ## Z. Mustafa, H. Obiedat, H. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl., 2008 (2008), 1-12 ## Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289-297 ## B. Samet, C. Vetro, F. Vetro, Remarks on G-metric spaces, Int. J. Anal., 2013 (2013), 1-6 ## W. Shatanawi, Fixed point theory for contractive mappings satisfying $$\Phi$$-maps in G-metric spaces, Fixed Point Theory Appl., 2010 (2010), 1-9 ## 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 spaces, Fixed Point Theory Appl., 2012 (2012), 1-9 ## H.-Q. Ye, F. Gu, A new common fixed point theorem for a class of four power type contraction mappings, J. Hangzhou Univ. Natur. Sci. Ed., 10 (2011), 520-523 ## H.-Q. Ye, F. Gu, Common fixed point theorems for a class of twice power type contraction maps in G-metric spaces, Abstr. Appl. Anal., 2012 (2012), 1-19
Quasi associated continued fractions and Hankel determinants of Dixon elliptic functions via Sumudu transform Quasi associated continued fractions and Hankel determinants of Dixon elliptic functions via Sumudu transform en en 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. 4000 4014 Adem Kilicman Department of Mathematics, Faculty of Science Universiti Putra Malaysia Malaysia akilic@upm.edu.my Rathinavel Silambarasan M. Tech IT-Networking, Department of Information Technology, School of Information Technology and Engineering VIT University India silambu_vel@yahoo.co.in Omer Altun Department of Mathematics and Institute for Mathematical research Universiti Putra Malaysia Malaysia omeraltun11@yahoo.com Dixon elliptic functions quasi associated continued fractions Hankel determinants Sumudu transform three term recurrence. Article.49.pdf  O. S. Adams, Elliptic functions applied to conformal world maps, US Government Printing Office, Washington (1925) ## W. A. Al-Salam, L. Carlitz, Some determinants of Bernoulli, Euler and related numbers, Portugal. Math., 18 (1959), 91-99 ## R. Bacher, P. Flajolet, Pseudo-factorials, elliptic functions, and continued fractions, Ramanujan J., 21 (2009), 71-97 ## F. B. M. Belgacem, E. H. Al-Shemas, R. Silambarasan, Sumudu computation of the transient magnetic field in a lossy medium, Appl. Math. Inf. Sci., 11 (2017), 209-217 ## F. B. M. Belgacem, R. Silambarasan, A distinctive Sumudu treatment of trigonometric functions, J. Comput. Appl. Math., 312 (2017), 74-81 ## F. B. M. Belgacem, R. Silambarasan, Further distinctive investigations of the Sumudu transform, AIP Conf. Proc., 1798 (2017), 020025-1 ## F. B. M. Belgacem, R. Silambarasan, Sumudu transform of Dumont bimodular Jacobi elliptic functions for arbitrary powers, AIP Conf. Proc., 1798 (2017), 020026-1 ## L. Carlitz, Some orthogonal polynomials related to elliptic functions, Duke Math. J., 27 (1960), 443-459 ## E. V. F. Conrad, Some continued fraction expansions of Laplace transforms of elliptic functions, Thesis (Ph.D.)–The Ohio State University, ProQuest LLC, Ann Arbor, MI, (2002), 1-87 ## E. V. F. Conrad, P. Flajolet, The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion, Sém. Lothar. Combin., 54 (2005/07), 1-44 ## A. C. Dixon, On the doubly periodic functions arising out of the curve $$x^3 + y^3 - 3\alpha xy = 1$$, Quart. J. Pure Appl. Math., 24 (1890), 167-233 ## A. C. Dixon, The elementary properties of the elliptic functions, with examples, Macmillan, London and New York (1894) ## D. Dumont, Le parametrage de la courbe d’equation $$x^3 + y^3 = 1$$, Une introduction elementaire aux fonctions elliptiques, preprint (1988) ## H. Eltayeb, A. Kılıçman, On double sumudu transform and double Laplace transform, Malays. J. Math. Sci., 4 (2010), 17-30 ## H. Eltayeb, A. Kılıçman, R. P. Agarwal, On integral transforms and matrix functions, Abstr. Appl. Anal., 2011 (2011), 1-15 ## M. E. H. Ismail, D. R. Masson, Some continued fractions related to elliptic functions, Continued fractions: from analytic number theory to constructive approximation, Columbia, MO, (1998), Contemp. Math., Amer. Math. Soc., Providence, RI, 236 (1999), 149-166 ## 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. (1980) ## A. Kılıçman, H. Eltayeb, On a new integral transform and differential equations, Math. Probl. Eng., 2010 (2010), 1-13 ## A. Kılıçman, H. Eltayeb, Some remarks on the Sumudu and Laplace transforms and applications to differential equations, ISRN Appl. Math., 2012 (2012), 1-13 ## A. Kılıçman, H. Eltayeb, K. A. M. Atan, A note on the comparison between Laplace and Sumudu transforms, Bull. Iranian Math. Soc., 37 (2011), 131-141 ## A. Kılıçman, V. G. Gupta, B. Shrama, On the solution of fractional Maxwell equations by Sumudu transform, J. Math. Res., 2 (2010), 147-151 ## J. C. Langer, D. A. Singer, The trefoil, Milan J. Math., 82 (2014), 161-182 ## D. F. Lawden, Elliptic functions and applications, Applied Mathematical Sciences, Springer-Verlag, New York (1989) ## L. Lorentzen, H. Waadeland, Continued fractions with applications, Studies in Computational Mathematics, North- Holland Publishing Co., Amsterdam (1992) ## A. M. Mahdy, A. S. Mohamed, A. A. Mtawa, Implementation of the homotopy perturbation Sumudu transform method for solving Klein-Gordon equation, Appl. Math., 6 (2015), 617-628 ## S. C. Milne , Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, Reprinted from Ramanujan J.,/ 6 (2002), 7–149 , With a preface by George E. Andrews, Developments in Mathematics, Kluwer Academic Publishers, Boston, MA (2002) ## T. Muir, A treatise on the theory of determinants, Revised and enlarged by William H. Metzler, Dover Publications, Inc., New York (1960) ## M. A. Ramadan, M. S. Al-Luhaibi, Application of sumudu decomposition method for solving linear and nonlinear Klein- Gordon equations, Int. J. Soft Comput. Eng., 3 (2014), 138-141 ## J. Singh, D. Kumar, A. Kilicman, Application of homotopy perturbation Sumudu transform method for solving heat and wave-like equations, Malays. J. Math. Sci., 7 (2013), 79-95 ## H. S. Wall, Note on the expansion of a power series into a continued fraction, Bull. Amer. Math. Soc., 51 (1945), 97-105 ## H. S. Wall, Analytic theory of continued fractions, D. Van Nostrand Company, Inc., New York, N. Y. (1948) ## X.-J. Yang, A new integral transform with an application in heat-transfer problem, Therm. Sci., 20 (2016), 1-677 ## X.-J. Yang, A new integral transform operator for solving the heat-diffusion problem, Appl. Math. Lett., 64 (2017), 193-197
Some fixed point results via measure of noncompactness Some fixed point results via measure of noncompactness en en 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. 4015 4024 Chi-Ming Chen Institute for Computational and Modeling Science National Tsing Hua University Taiwan chenchiming@mx.nthu.edu.tw Erdal Karapinar Department of Mathematics Nonlinear Analysis and Applied Mathematics (NAAM) Research Group Atılım University King Abdulaziz University Turkey Saudi Arabia erdalkarapinar@yahoo.com Measure of noncompactness Meir-Keeler-type set contraction fixed points. Article.50.pdf  R. Agarwal, M. Meehan, D. O’Regan, Fixed point theory and applications, Cambridge University Press, United kingdom (2001) ## A. Aghajani, M. Mursaleen, A. S. Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta Math. Sci., 35 (2015), 552-566 ## R. R. Akhmerov, M. I. Kamenski, A. S. Potapov, A. E. Rodkina, B. N. Sadovski, Measures of Noncompactness and Condensing Operators, Translated from the 1986 Russian original by A. Iacob. Oper. Theory Adv. Appl., 55 (1992), 1-244 ## J. Banaś, Measures of noncompactness in the space of continuous tempered functions, Demonstratio Math., 14 (1981), 127-133 ## J. Banaś, Measures of noncompactness in the study of solutions of nonlinear differential and integral equations, Cent. Eur. J. Math., 10 (2012), 2003-2011 ## J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, New York (1980) ## C.-M. Chen, T.-H. Chang, Fixed Point Theorems for a Weaker Meir-Keeler Type $$\psi$$-Set Contraction in Metric Spaces, Fixed Point Theory Appl., 2009 (2009), 1-8 ## G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova, 24 (1955), 84-92 ## W.-S. Du, On coincidence point and fixed point theorems for nonlinear multivalued maps, Topol. Appl, 159 (2012), 49-56 ## K. Kuratowski, Sur les espaces complets, Fund. Math., 15 (1930), 301-309 ## B. de Malafosse, E. Malkowsky, V. Rakocevic, Measure of noncompactness of operators and matrices on the spaces c and c0, Int. J. Math. Math. Sci., 2006 (2006), 1-5 ## A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326-329 ## M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73 (2010), 2541-2557 ## J. M. A. Toledano, T. D. Benavides, G. L. Azedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhuser Verlag, Basel (1997)
Some Hermite-Hadamard type inequalities for harmonically extended $$s$$-convex functions Some Hermite-Hadamard type inequalities for harmonically extended $$s$$-convex functions en en In this paper, we establish some inequalities of Hermite-Hadamard type for functions whose derivatives absolute values are harmonically extended s-convex functions. 4025 4033 Chun-Long Li College of Mathematics Inner Mongolia University for Nationalities China lichunlong70@163.com Shan-He Wu Department of Mathematics Longyan University China shanhewu@163.com Harmonically extended s-convex function Hermite-Hadamard type inequalities integral inequalities. Article.51.pdf  W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen, (German) Publ. Inst. Math. (Beograd) (N.S.), 23 (1978), 13-20 ## F.-X. Chen, S.-H. Wu, Some Hermite-Hadamard type inequalities for harmonically s-convex functions, Scientific World J., 2014 (2014), 1-7 ## S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91-95 ## H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100-111 ## S. Hussain, M. I. Bhatti, M. Iqbal, Hadamard-type inequalities for s-convex functions, I, Punjab Univ. J. Math. (Lahore), 41 (2009), 51-60 ## İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942 ## İ. İşcan, S.-H. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237-244 ## U. S. Kirmaci, M. Klaričić Bakula, M. E. Özdemir, J. Pečarić, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 193 (2007), 26-35 ## M. A. Noor, K. I. Noor, M. U. Awan, S. Costache, Some integral inequalities for harmonically h-convex functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77 (2015), 5-16 ## C. E. M. Pearce, J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formulæ, Appl. Math. Lett., 13 (2000), 51-55 ## S.-H. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, Rocky Mountain J. Math., 39 (2009), 1741-1749 ## S.-H. Wu, B. Sroysang, J.-S. Xie, Y.-M. Chu, Parametrized inequality of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex, SpringerPlus, 4 (2015), 1-9 ## B.-Y. Xi, R.-F. Bai, F. Qi, Hermite-Hadamard type inequalities for the m- and $$(\alpha,m)$$-geometrically convex functions, Aequationes Math., 84 (2012), 261-269 ## B.-Y. Xi, F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex functions and applications, Hacet. J. Math. Stat., 42 (2013), 243-257 ## B.-Y. Xi, F. Qi, Hermite-Hadamard type inequalities for geometrically r-convex functions, Studia Sci. Math. Hungar., 51 (2014), 530-546 ## B.-Y. Xi, F. Qi, Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means, J. Nonlinear Convex Anal., 16 (2015), 873-890 ## B.-Y. Xi, T.-Y. Zhang, F. Qi, Some inequalities of Hermite–Hadamard type for m-harmonic-arithmetically convex functions, ScienceAsia, 41 (2015), 357-361