]>
2017
10
2
ISSN 2008-1898
516
Vector valued Orlicz-Lorentz sequence spaces and their operator ideals
Vector valued Orlicz-Lorentz sequence spaces and their operator ideals
en
en
In the present paper we introduce and study vector valued Orlicz-Lorentz sequence spaces \(l_{p,q,M,u,\Delta,A}(X)\) on Banach
space \(X\) with the help of a Musilak-Orlicz function \(M\) and for different positive indices p and q. We also study their cross
and topological duals. Finally, we introduce the operator ideals with the help of the corresponding scalar sequence spaces and
s-numbers.
338
353
S. A.
Mohiuddine
Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science
King Abdulaziz University
Saudi Arabia
mohiuddine@gmail.com
K.
Raj
School of Mathematics
Shri Mata Vaishno Devi University
India
kuldipraj68@gmail.com
Lorentz sequence spaces
s-numbers of operators
Musielak-Orlicz function
difference sequence spaces
operator ideals.
Article.1.pdf
[
[1]
L. R. Acharya, Linear operators and approximation quantities, Dissertation, I.I.T. Kanpur, India (2008)
##[2]
A. Alotaibi, K. Raj, S. A. Mohiuddine, Some generalized difference sequence spaces defined by a sequence of moduli in n-normed spaces, J. Funct. Spaces, 2015 (2015 ), 1-8
##[3]
N. De Grande-De Kimpe, Generalized sequence spaces, Bull. Soc. Math. Belg., 23 (1971), 123-166
##[4]
M. Et, R. Çolak, On some generalized difference sequence spaces, Soochow J. Math., 21 (1995), 377-386
##[5]
P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz- Lorentz sequence spaces, Math. Nachr., 281 (2008), 181-198
##[6]
A. Grothendieck, Sur une notion de produit tensoriel topologique d’espaces vectoriels topologiques, et une classe remarquable d’espaces vectoriels liée à cette notion, (French) C. R. Acad. Sci. Paris, 233 (1951), 1556-1558
##[7]
M. Gupta, A. Bhar, Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals, Math. Slovaca, 64 (2014), 1475-1496
##[8]
P. K. Kamthan, M. Gupta , Sequence spaces and series, Lecture Notes in Pure and Applied Mathematics , Marcel Dekker, Inc.,, New York (1981)
##[9]
M. Kato, On Lorentz spaces \(l_{p,q}\{E\}\), Hiroshima Math. J., 6 (1976), 73-93
##[10]
H. Kızmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169-176
##[11]
J. Lindenstrauss, L. Tzafriri, Classical Banach spaces,I , Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin-New York (1977)
##[12]
S. A. Mohiuddine, K. Raj, A. Alotaibi, Generalized spaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spaces, J. Inequal. Appl., 2014 (2014), 1-16
##[13]
J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Springer-Verlag, Berlin (1983)
##[14]
J. Patterson, Generalized sequence spaces and matrix transformations, PhD diss., Dissertation, I.I.T. Kanpur, India (1980)
##[15]
A. Pietsch, s-numbers of operators in Banach spaces, Studia Math., 51 (1974), 201-223
##[16]
A. Pietsch, Eigenvalues and s-numbers, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1987)
##[17]
K. Raj, S. Pandoh, Some vector-valued statistical convergent sequence spaces, Malaya J. Mat., 3 (2015), 161-167
##[18]
K. Raj, S. K. Sharma, Some vector-valued sequence spaces defined by a Musielak-Orlicz function, Rev. Roumaine Math. Pures Appl., 57 (2012), 383-399
##[19]
W. Rudin , Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York- Dsseldorf-Johannesburg (1976)
##[20]
E. Savaş, M. Mursaleen, Matrix transformations in some sequence spaces, Istanbul Üniv. Fen Fak. Mat. Derg., 52 (1993), 1-5
##[21]
Y. Yılmaz, M. K. Özdemir, I. Solak, M. Candan, Operators on some vector valued Orlicz sequence spaces, F. Ü . Fen ve Mühendislik Bilimleri Dergisi, 17 (2005), 59-71
]
Levitin-Polyak well-posedness for lexicographic vector equilibrium problems
Levitin-Polyak well-posedness for lexicographic vector equilibrium problems
en
en
We introduce the notions of Levitin-Poljak (LP) well-posedness and LP well-posedness in the generalized sense for the
lexicographic vector equilibrium problems. Then, we establish some sufficient conditions for lexicographic vector equilibrium
problems to be LP well-posedness at the reference point. Numerous examples are provided to explain that all the assumptions
we impose are very relaxed and cannot be dropped. The results in this paper unify, generalize and extend some known results
in the literature.
354
367
Rabian
Wangkeeree
Department of Mathematics, Faculty of Science
Research Center for Academic Excellence in Mathematics
Naresuan University
Naresuan University
Thailand
Thailand
rabianw@nu.ac.th
Thanatporn
Bantaojai
Department of Mathematics, Faculty of Science
Naresuan University
Thailand
thanatpornmaths@gmail.com
Levitin-polyak well-posedness
lexicographic vector equilibrium problems
metric spaces.
Article.2.pdf
[
[1]
L. Q. Anh, T. Q. Duy, A. Y. Kruger, N. H. Thao, Well-posedness for lexicographic vector equilibrium problems, Constructive nonsmooth analysis and related topics, Springer Optim. Appl., Springer, New York, 87 (2014), 159-174
##[2]
L. Q. Anh, P. Q. Khanh, D. T. M. Van, Well-posedness under relaxed semicontinuity for bilevel equilibrium and optimization problems with equilibrium constraints, J. Optim. Theory Appl., 153 (2012), 42-59
##[3]
L. Q. Anh, P. Q. Khanh, D. T. M Van, J.-C. Yao, Well-posedness for vector quasiequilibria, Taiwanese J. Math., 13 (2009), 713-737
##[4]
J. P. Aubin, H. Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA (1990)
##[5]
J. Banaś, K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1980)
##[6]
M. Bianchi, I. V. Konnov, R. Pini, Lexicographic variational inequalities with applications, Optimization, 56 (2007), 355-367
##[7]
M. Bianchi, I. V. Konnov, R. Pini, Lexicographic and sequential equilibrium problems, J. Global Optim., 46 (2010), 551-560
##[8]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145
##[9]
E. Carlson, Generalized extensive measurement for lexicographic orders, J. Math. Psych., 54 (2010), 345-351
##[10]
L. C. Ceng, N. Hadjisavvas, S. Schaible, J.-C. Yao, Well-posedness for mixed quasivariational-like inequalities, J. Optim. Theory Appl., 139 (2008), 109-225
##[11]
J.-W. Chen, Z.-P. Wan, Y. J. Cho, Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasiequilibrium problems, Math. Methods Oper. Res., 77 (2013), 33-64
##[12]
G. P. Crespi, A. Guerraggio, M. Rocca, Well posedness in vector optimization problems and vector variational inequalities, J. Optim. Theory Appl., 132 (2007), 213-226
##[13]
J. Daneš, On the Istrăţescu’s measure of noncompactness, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 16 (1972), 403-406
##[14]
B. Djafari Rouhani, E. Tarafdar, P. J. Watson, Existence of solutions to some equilibrium problems, J. Optim. Theory Appl., 126 (2005), 97-107
##[15]
V. A. Emelichev, E. E. Gurevsky, K. G. Kuzmin , On stability of some lexicographic integer optimization problem, Control Cybernet., 39 (2010), 811-826
##[16]
Y.-P. Fang, R. Hu, N.-J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89-100
##[17]
Y.-P. Fang, N.-J. Huang, J.-C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems, J. Global Optim., 41 (2008), 117-133
##[18]
F. Flores-Bazán, Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case, SIAM J. Optim., 11 (2001), 675-690
##[19]
E. C. Freuder, R. Heffernan, R. J. Wallace, N. Wilson, Lexicographically-ordered constraint satisfaction problems, Constraints, 15 (2010), 1-28
##[20]
J. Hadamard, Sur le problémes aux dérivées partielles et leur signification physique [On the problems about partial derivatives and their physical signicance] , Bull. Univ. Princeton, 13 (1902), 49-52
##[21]
N. X. Hai, P. Q. Khanh, Existence of solutions to general quasiequilibrium problems and applications, J. Optim. Theory Appl., 133 (2007), 317-327
##[22]
A. D. Ioffe, R. E. Lucchetti, J. P. Revalski, Almost every convex or quadratic programming problem is well posed, Math. Oper. Res., 29 (2004), 369-382
##[23]
K. Kimura, Y.-C. Liou, S.-Y. Wu, J.-C. Yao , Well-posedness for parametric vector equilibrium problems with applications, J. Ind. Manag. Optim., 4 (2008), 313-327
##[24]
I. V. Konnov, M. S. S. Ali, Descent methods for monotone equilibrium problems in Banach spaces, J. Comput. Appl. Math., 188 (2006), 165-179
##[25]
A. S. Konsulova, J. P. Revalski, Constrained convex optimization problemswell-posedness and stability, Numer. Funct. Anal. Optim., 15 (1994), 889-907
##[26]
M. Küçük, M. Soyertem, Y. Küçük, On constructing total orders and solving vector optimization problems with total orders, J. Global Optim., 50 (2011), 235-247
##[27]
C. S. Lalitha, G. Bhatia, Levitin-Polyak well-posedness for parametric quasivariational inequality problem of the Minty type, Positivity, 16 (2012), 527-541
##[28]
E. S. Levitin, B. T. Poljak, Convergence of minimizing sequences in problems on the relative extremum, (Russian) Dokl. Akad. Nauk SSSR, 168 (1966), 997-1000
##[29]
S. J. Li, M. H. Li, Levitin-Polyak well-posedness of vector equilibrium problems, Math. Methods Oper. Res., 69 (2009), 125-140
##[30]
M. B. Lignola, Well-posedness and L-well-posedness for quasivariational inequalities, J. Optim. Theory Appl., 128 (2006), 119-138
##[31]
X. J. Long, N.-J. Huang, K. L. Teo, Levitin-Polyak well-posedness for equilibrium problems with functional constraints, J. Inequal. Appl., 2008 (2008), 1-14
##[32]
P. Loridan, \(\epsilon\)-solutions in vector minimization problems, J. Optim. Theory Appl., 43 (1984), 265-276
##[33]
R. Lucchetti, F. Patrone, A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476
##[34]
M. Margiocco, F. Patrone, L. Pusillo Chicco, A new approach to Tikhonov well-posedness for Nash equilibria, Optimization, 40 (1997), 385-400
##[35]
J. Morgan, V. Scalzo, Pseudocontinuity in optimization and nonzero-sum games, J. Optim. Theory Appl., 120 (2004), 181-197
##[36]
J.-W. Peng, Y. Wang, L.-J. Zhao, Generalized Levitin-Polyak well-posedness of vector equilibrium problems, Fixed Point Theory Appl., 2009 (2009 ), 1-14
##[37]
J.-W. Peng, S.-Y. Wu, Y. Wang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints, J. Global Optim., 52 (2012), 779-795
##[38]
V. Rakočević, Measures of noncompactness and some applications, Filomat, 12 (1998), 87-120
##[39]
J. P. Revalski, Hadamard and strong well-posedness for convex programs, SIAM J. Optim., 7 (1997), 519-526
##[40]
I. Sadeqi, C. G. Alizadeh, Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces, Nonlinear Anal., 74 (2011), 2226-2234
##[41]
A. N. Tikhonov, On the stability of the functional optimization problem, USSR Comput. Math. Math. Phys., 6 (1966), 28-33
##[42]
J. Yu, H. Yang, C. Yu, Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems, Nonlinear Anal., 66 (2007), 777-790
##[43]
T. Zolezzi, Well-posedness criteria in optimization with application to the calculus of variations, Nonlinear Anal., 25 (1995), 437-453
##[44]
T. Zolezzi, Well-posedness and optimization under perturbations, Optimization with data perturbations, II, Ann. Oper. Res., 101 (2001), 351-361
]
Sufficient conditions on existence of solution for nonlinear fractional iterative integral equation
Sufficient conditions on existence of solution for nonlinear fractional iterative integral equation
en
en
In this article, we study nonlinear quadratic iterative integral equations and establish sufficient conditions for the existence
of Volterra solutions for fractional iterative integral equations and solvency in Banach space and \(C_{\ell,\beta}\). In the present work we
use the principle of contraction, Schaefer’s fixed point theorem and the non-expansive operator method as essential tools. In
this study we consider Riemann-Liouville differential operator and prove some related theorems, further provide an example as
an application.
368
376
Faten H. M.
Damag
Department of Mathematics and Institute for Mathematical Research
University Putra Malaysia
Malaysia
faten_212326@hotmail.com
Adem
Kilicman
Department of Mathematics and Institute for Mathematical Research
University Putra Malaysia
Malaysia
akilic@upm.edu.my
Fractional integral equation
existence of solution
Schaefer’s fixed point theorem
non-expansive operator.
Article.3.pdf
[
[1]
A. Atangana, R. T. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Difference Equ., 2016 (2016), 1-13
##[2]
A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454
##[3]
V. Berinde, Iterative approximation of fixed points, Second edition, Lecture Notes in Mathematics, Springer, Berlin (2007)
##[4]
V. Berinde, Existence and approximation of solutions of some first order iterative differential equations, Miskolc Math. Notes, 11 (2010), 13-26
##[5]
S. S. Cheng, J.-G. Si, X.-P.Wang, An existence theorem for iterative functional differential equations, Acta Math. Hungar., 94 (2002), 1-17
##[6]
F. H. Damag, A. Kılıçman, R. A. A. Abdulghafor, Approximate solutions and existence result for some integral equation with modified argument of fractional order, Adv. Difference Equ., (2016), -
##[7]
F. H. Damag, A. Kılıçman, R. W. Ibrahim, Approximate solutions for non-linear iterative fractional differential equations, Innovations Through Mathematical and Statistical Research, Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS2016) , AIP Publishing,1739 ((2016). )
##[8]
F. H. Damag, A. Kılıçman, R. W. Ibrahim, Findings of fractional iterative differential equations involving first order derivative, Int. J. Appl. Comput. Math., (2016), 1-10
##[9]
A. El-Sayed, H. Hashem, Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra, Fract. Calc. Appl. Anal., 16 (2013), 816-826
##[10]
A. Granas, J. Dugundji, Fixed point theory, Springer Science and Business Media, (2013)
##[11]
R.W. Ibrahim, A. Kılıçman, F. H. Damag, Existence and uniqueness for a class of iterative fractional differential equations, Adv. Difference Equ., 2015 (2015 ), 1-13
##[12]
R. W. Ibrahim, A. Kılıçman, F. H. Damag, Extremal solutions by monotone iterative technique for hybrid fractional differential equations, Turkish J. Anal. Number Theory, 4 (2016), 60-66
##[13]
S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65-71
##[14]
M. Lauran , Existence results for some integral equation with modified argument, Gen. Math., 19 (2011), 85-92
##[15]
M. Lauran, Existence results for some nonlinear integral equations, Miskolc Math. Notes, 13 (2012), 67-74
##[16]
M. Lauran, Solution of first iterative differential equations, An. Univ. Craiova Ser. Mat. Inform, 40 (2013), 45-51
##[17]
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)
##[18]
D. O’Regan, Existence results for nonlinear integral equations, J. Math. Anal. Appl., 192 (1995), 705-726
##[19]
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)
##[20]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolskiĭ, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
##[21]
J.-R. Wang, M. Fečkan, Y. Zhou, Fractional order iterative functional differential equations with parameter, Appl. Math. Model., 37 (2013), 6055-6067
##[22]
P.-P. Zhang, X.-B. Gong, Existence of solutions for iterative differential equations, Electron, J. Differential Equations, 2014 (2014 ), 1-10
]
Dynamic reliability evaluation for a multi-state component under stress-strength model
Dynamic reliability evaluation for a multi-state component under stress-strength model
en
en
For many technical systems, stress-strength models are of special importance. Stress-strength models can be described
as an assessment of the reliability of the component in terms of \(X\) and \(Y\) random variables where \(X\) is the random ”stress”
experienced by the component and \(Y\) is the random ”strength” of the component available to overcome the stress. The reliability
of the component is the probability that component is strong enough to overcome the stress applied on it. Traditionally, both
the strength of the component and the applied stress are considered to be both time-independent random variables. But in most
of real life systems, the status of a stress and strength random variables clearly change dynamically with time. Also, in many
important systems, it is very necessary to estimate the reliability of the component without waiting to observe the component
failure. In this paper we study multi-state component where component is subjected to two stresses. In particular, inspired by
the idea of Kullback-Leibler divergence, we aim to propose a new method to compute the dynamic reliability of the component
under stress-strength model. The advantage of the proposed method is that Kullback-Leibler divergence is equal to zero when
the component strength is equal to applied stress. In addition, the formed function can include both stresses when two stresses
exist at the same time. Also, the proposed method provides a simple way and good alternative to compute the reliability of the
component in case of at least one of the stress or strengths quantities depend on time.
377
385
Sinan
Çalık
Department of Statistics, Faculty of Science
Fırat University
Turkey
scalik@firat.edu.tr
Kullback-Leibler divergence
dynamic reliability
stress-strength model
multi-state component
gamma distribution.
Article.4.pdf
[
[1]
C. Bauckhage, Computing the Kullback-Leibler divergence between two Generalized Gamma distributions, ArXiv, 2014 (2014 ), 1-7
##[2]
R. D. Brunelle, K. C. Kapur , Review and classification of reliability measures for multistate and continuum models, IIE Trans., 31 (1999), 1171-1180
##[3]
S. Chandra, D. B. Owen, On estimating the reliability of a component subject to several different stresses (strengths), Naval Res. Logist. Quart., 22 (1975), 31-39
##[4]
E. Chiodo, D. Fabiani, G. Mazzanti, Bayes inference for reliability of HV insulation systems in the presence of switching voltage surges using a Weibull stress-strength model, IEEE Power Tech. Conf. Proc., Bologna, 3 (2003), -
##[5]
E. Chiodo, G. Mazzanti, Bayesian reliability estimation based on a Weibull stress-strength model for aged power system components subjected to voltage surges, IEEE Trans. Dielectr. Electr. Insul., 13 (2006), 146-159
##[6]
R. Dahlhaus, On the Kullback-Leibler information divergence of locally stationary processes, Stochastic Process. Appl., 62 (1996), 139-168
##[7]
M. N. Do, Fast approximation of Kullback-Leibler distance for dependence trees and hidden Markov models, IEEE Signal Process. Lett., 10 (2003), 115-118
##[8]
N. Ebrahimi, Multistate reliability models, Naval Res. Logist. Quart., 31 (1984), 671-680
##[9]
S. Eryılmaz , Mean residual and mean past lifetime of multi-state systems with identical components, IEEE Trans. Rel., 59 (2010), 644-649
##[10]
S. Eryılmaz, On stress-strength reliability with a time-dependent strength, J. Qual. Reliab. Eng., 2013 (2013 ), 1-6
##[11]
S. Eryılmaz, F. İşçioğlu, Reliability evaluation for a multi-state system under stress-strength setup, Comm. Statist. Theory Methods, 40 (2011), 547-558
##[12]
G. Gökdere, M. Gürcan, Erlang Strength Model for Exponential Effects, Open Phys., 13 (2015), 395-399
##[13]
G. Gökdere, M. Gürcan, Laplace-Stieltjes transform of the system mean lifetime via geometric process model, Open Math., 14 (2016), 384-392
##[14]
G. Gökdere, M. Gürcan, New Reliability Score for Component Strength Using Kullback-Leibler Divergence, Eksploatacja i Niezawodnosc.-Maintenance and Reliability, 18 (2016), 367-372
##[15]
I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Translated from the Russian, Sixth edition, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, Academic Press, Inc., San Diego, CA (2000)
##[16]
L. Guo, M.-M. Zhang, A Time-varying repairable system with repairman vacation and warning device, J. Nonlinear Sci. Appl., 9 (2016), 316-331
##[17]
J. C. Hudson, K. C. Kapur, Reliability analysis for multistate systems with multistate components, IIE Trans., 15 (1983), 127-135
##[18]
F. K. Hwang, Y.-C. Yao, Multistate consecutively-connected systems, IEEE Trans. Rel., 38 (1989), 472-474
##[19]
A. Kossow, W. Preuss, Reliability of linear consecutively-connected systems with multistate components, IEEE Trans. Rel., 44 (1995), 518-522
##[20]
S. Kotz, Y. Lumelskii, M. Pensky, The stress-strength model and its generalizations: theory and applications, World Scientific Publishing Co. Inc., Singapore (2003)
##[21]
S. Kullback, R. A. Leibler, On information and sufficiency, Ann. Math. Statistics, 22 (1951), 79-86
##[22]
W. Kuo, M. J. Zuo, Optimal reliability modeling: principles and applications, John Wiley & Sons, New York, New York (2003)
##[23]
Y. K. Lee, B. U. Park, Estimation of Kullback-Leibler divergence by local likelihood, Ann. Inst. Statist. Math., 58 (2006), 327-340
##[24]
A. Lisnionski, G. Levitin, Multi-state system reliability: assessment, optimization and applications, Series on Guality, Reliability and Engineering Statistics, World Scientific Publishing Co. Inc., Singapore (2003)
##[25]
Z. Rached, F. Alajaji, L. L. Campbell, The Kullback-Leibler divergence rate between Markov sources, IEEE Trans. Inf. Theory, 50 (2004), 917-921
##[26]
G. Yari, A. Mirhabibi, A. Saghafi, Estimation of the Weibull parameters by Kullback-Leibler divergence of survival functions, Appl. Math. Inf. Sci., 7 (2013), 187-192
##[27]
X. Zhang, L. Guo, A new kind of repairable system with repairman vacations, J. Nonlinear Sci. Appl., 8 (2015), 324-333
]
Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces
Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces
en
en
The purpose of this paper is to introduce the implicit midpoint rule of nonexpansive mappings in CAT(0) spaces. The strong
convergence of this method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown
that the limit of the sequence generated by the implicit midpoint rule solves an additional variational inequality. Applications
to nonlinear Volterra integral equations and nonlinear variational inclusion problem are included. The results presented in the
paper extend and improve some recent results announced in the current literature.
386
394
Liang-cai
Zhao
College of Mathematics
Yibin University
China
Shih-sen
Chang
Center for General Education
China Medical University
Taiwan
changss2013@163.com
Lin
Wang
College of Statistics and Mathematics
Yunnan University of Finance and Economics
China
Gang
Wang
College of Statistics and Mathematics
Yunnan University of Finance and Economics
China
Viscosity
implicit midpoint rule
nonexpansive mapping
projection
variational inequality
CAT(0) space.
Article.5.pdf
[
[1]
B. Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc., 141 (2013), 1029-1039
##[2]
M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, H.-K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014 ), 1-9
##[3]
H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769-806
##[4]
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
##[5]
G. Bader, P. Deuflhard , A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373-398
##[6]
S. Banach, Metric spaces of nonpositive curvature, Springer, New York (1999)
##[7]
I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195-218
##[8]
M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1999)
##[9]
K. S. Brown, Buildings, Springer-Verlag, New York (1989)
##[10]
S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013 ), 1-14
##[11]
H. Dehghan, J. Rooin, A characterization of metric projection in CAT(0) spaces, In: International Conference on Functional Equation, Geometric Functions and Applications (ICFGA 2012), 10-12th May 2012, pp. 41-43. Payame Noor University, Tabriz (2012)
##[12]
S. Dhompongsa, W. A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35-45
##[13]
S. Dhompongsa, B. Panyanak, On \(\Delta\)-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572-2579
##[14]
W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696
##[15]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55
##[16]
S. Saejung, Halpern’s iteration in CAT(0) spaces, Fixed Point Theory Appl., 2010 (2010 ), 1-13
##[17]
C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 (1993), 1-10
##[18]
S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327-332
##[19]
R.Wangkeeree, P. Preechasilp, Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013 ), 1-15
##[20]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256
##[21]
H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291
##[22]
H.-K. Xu, M. A. Alghamdi, N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015 ), 1-12
##[23]
Y.-H. Yao, N. Shahzad, N.-C. Liou, Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015 ), 1-15
##[24]
S.-S. Zhang, Integral equations, Chongqing press, Chongqing (1984)
##[25]
L.-C. Zhao, S.-S. Chang, C.-F. Wen, Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces, J. Nonlinear Sci. Appl., 9 (2016), 4478-4488
]
Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces
Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces
en
en
In this paper, we combine the subgradient extragradient method with the Halpern method for finding a solution of a
variational inequality involving a monotone Lipschitz mapping in Banach spaces. By using the generalized projection operator
and the Lyapunov functional introduced by Alber, we prove a strong convergence theorem. We also consider the problem of
finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of a relatively
nonexpansive mapping. Our results improve some well-known results in Banach spaces or Hilbert spaces.
395
409
Ying
Liu
College of Mathematics and Information Science
Hebei University
China
ly_cyh2013@163.com
Subgradient extragradient method
Halpern method
generalized projection operator
monotone mapping
variational inequality
relatively nonexpansive mapping.
Article.6.pdf
[
[1]
Y. Alber, S. Guerre-Delabriere, On the projection methods for fixed point problems, Analysis (Munich), 21 (2001), 17-39
##[2]
Ya. I. Al’ber, S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J., 4 (1994), 39-54
##[3]
K. Ball, E. A. Carlen, E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., 115 (1994), 463-482
##[4]
N. Buong, Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces, Appl. Math. Comput., 217 (2010), 322-329
##[5]
L.-C. Ceng, N. Hadjisavvas, N.-C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Global Optim., 46 (2010), 635-646
##[6]
Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26 (2011), 827-845
##[7]
Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335
##[8]
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
##[9]
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961
##[10]
H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings and inverse-strongly-monotone mappings, J. Convex Anal., 11 (2004), 69-79
##[11]
H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350
##[12]
H. Iiduka, W. Takahashi, Weak convergence of a projection algorithm for variational inequalities in a Banach space, J. Math. Anal. Appl., 339 (2008), 668-679
##[13]
G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747-756
##[14]
R. Kraikaew, S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412
##[15]
J.-L. Lions, G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-517
##[16]
Y. Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space, Appl. Math. Mech. (English Ed.), 30 (2009), 925-932
##[17]
P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912
##[18]
S.-Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134 (2005), 257-266
##[19]
N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitzcontinuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241
##[20]
K. Nakajo, Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces, Appl. Math. Comput., 271 (2015), 251-258
##[21]
W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428
##[22]
H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138
##[23]
H.-K. Xu , Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113
]
Multiple periodic solutions for second-order discrete Hamiltonian systems
Multiple periodic solutions for second-order discrete Hamiltonian systems
en
en
By applying critical point theory, the multiplicity of periodic solutions to second-order discrete Hamiltonian systems with
partially periodic potentials was considered. It is noticed that, in this paper, the nonlinear term is growing linearly and main
results extend some present results.
410
418
Da-Bin
Wang
Department of Applied Mathematics
Lanzhou University of Technology
People’s Republic of China
wangdb96@163.com
Man
Guo
Department of Applied Mathematics
Lanzhou University of Technology
People’s Republic of China
guoman615@163.com
Discrete Hamiltonian systems
periodic solutions
the generalized saddle point theorem.
Article.7.pdf
[
[1]
D. Baleanu, F. Jarad, Discrete variational principles for higher-order Lagrangians, Nuovo Cimento Soc. Ital. Fis. B, 120 (2005), 931-938
##[2]
H.-H. Bin, Subharmonics with minimal periods for convex discrete Hamiltonian systems, Abstr. Appl. Anal., 2013 (2013 ), 1-9
##[3]
C.-F. Che, X.-P. Xue, Infinitely many periodic solutions for discrete second order Hamiltonian systems with oscillating potential, Adv. Difference Equ., 2012 (2012 ), 1-9
##[4]
H. Gu, T.-Q. An, Existence of periodic solutions for a class of second-order discrete Hamiltonian systems, J. Difference Equ. Appl., 21 (2015), 197-208
##[5]
W. Guan, K. Yang, Existence of periodic solutions for a class of second order discrete Hamiltonian systems, Adv. Difference Equ., 2016 (2016 ), 1-17
##[6]
Z.-M. Guo, J.-S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430
##[7]
F. Jarad, D. Baleanu, Discrete variational principles for Lagrangians linear in velocities, Rep. Math. Phys., 59 (2007), 33-43
##[8]
J. Q. Liu , A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372-385
##[9]
Y.-H. Long, Applications of Clark duality to periodic solutions with minimal period for discrete Hamiltonian systems, [Applications of Clarke duality to periodic solutions with minimal period for discrete Hamiltonian systems], J. Math. Anal. Appl., 342 (2008), 726-741
##[10]
J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer- Verlag, New York (1989)
##[11]
X.-H. Tang, X.-Y. Zhang, Periodic solutions for second-order discrete Hamiltonian systems, J. Difference Equ. Appl., 17 (2011), 1413-1430
##[12]
G.-C. Wu, D. Baleanu, Chaos synchronization of the discrete frational logistic map, Signal Process., 102 (2014), 96-99
##[13]
G.-C. Wu, D. Baleanu, Z.-G. Zeng, S.-D. Zeng , Lattice fractional diffusion equation in terms of a Riesz-Caputo difference, Phys. A, 438 (2015), 335-339
##[14]
Y.-F. Xue, C.-L. Tang, Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system, Nonlinear Anal., 67 (2007), 2072-2080
##[15]
Y.-F. Xue, C.-L. Tang, Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems, Appl. Math. Comput., 196 (2008), 494-500
##[16]
S.-H. Yan, X.-P. Wu, C.-L. Tang, Multiple periodic solutions for second-order discrete Hamiltonian systems, Appl. Math. Comput., 234 (2014), 142-149
##[17]
Y.-W. Ye, C.-L. Tang, Periodic solutions for second-order discrete Hamiltonian system with a change of sign in potential, Appl. Math. Comput., 219 (2013), 6548-6555
##[18]
Q.-Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163
]
New multipled common fixed point theorems in Menger PMT-spaces
New multipled common fixed point theorems in Menger PMT-spaces
en
en
In this work, we introduce the notion of Menger probabilistic metric type space, on the other hand, we introduce a more
general class of auxiliary functions in contractivity condition, following that, we obtain some multipled common fixed point
theorems for a pair of mappings \(T:\underbrace{X\times X\times...\times X}_{m-times}\rightarrow X\)
and \(A : X \rightarrow X\). As an application, we give out an example to demonstrate
the validity of the obtained results.
419
428
Cuiru
Ji
Department of Mathematics
Nanchang University
P. R. China
Chuanxi
Zhu
Department of Mathematics
Nanchang University
P. R. China
chuanxizhu@126.com
Zhaoqi
Wu
Department of Mathematics
Nanchang University
P. R. China
Multipled common fixed point
Menger PMT-spaces
\(\psi\)-contractive mapping.
Article.8.pdf
[
[1]
A. A. N. Abdou, Y. J. Cho, R. Saadati, Distance type and common fixed point theorems in Menger probabilistic metric type spaces, Appl. Math. Comput., 265 (2015), 1145-1154
##[2]
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations Intégrales, Fund. Math., 3 (1922), 133-181
##[3]
S.-S. Chang, Y. J. Cho, S. M. Kang, Nonlinear operator theory in probabilistic metric spaces, Nova Science Publishers, Inc., Huntington, NY (2001)
##[4]
Y. J. Cho, M. Grabiec, V. Radu, On nonsymmetric topological and probabilistic structures, Nova Science Publishers, Inc., New York (2006)
##[5]
B. S. Choudhury, K. Das, A new contraction principle in Menger spaces, Acta Math. Sin. (Engl. Ser.), 24 (2008), 1379-1386
##[6]
B. S. Choudhury, K. Das, A coincidence point result in Menger spaces using a control function , Chaos Solitons Fractals, 42 (2009), 3058-3063
##[7]
P. N. Dutta, B. S. Choudhury, K. Das, Some fixed point results in Menger spaces using a control function, Surv. Math. Appl., 4 (2009), 41-52
##[8]
O. Hadžić, E. Pap, Fixed point theory in probabilistic metric spaces, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (2001)
##[9]
M. S. Khan, M. Swalen, S. Sessa, Fixed points theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), 1-9
##[10]
M. A. Kutbi, D. Gopal, C. Vetro, W. Sintunavarat, Further generalization of fixed point theorems in Menger PM-spaces, Fixed Point Theory Appl., 2015 (2015), 1-10
##[11]
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
##[12]
K. Menger , Statistical metrics, Proc. Nat. Acad. Sci. U. S. A., 28 (1942), 535-537
##[13]
D. Miheţ, Altering distances in probabilistic Menger spaces , Nonlinear Anal., 71 (2009), 2734-2738
##[14]
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
##[15]
B. Schweizer, A. Sklat, Probabilistic metric spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York (1983)
##[16]
V. M. Sehgal, A. T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory, 6 (1972), 72-102
##[17]
C.-X. Zhu, Several nonlinear operator problems in the Menger PN space, Nonlinear Anal., 65 (2006), 1281-1284
##[18]
C.-X. Zhu, Research on some problems for nonlinear operators , Nonlinear Anal., 71 (2009), 4568-4571
##[19]
C.-X. Zhu, Z. Wei, Z.-Q. Wu, W.-Q. Xu, Multidimensional common fixed point theorems under probabilistic \(\phi\)-contractive conditions in multidimensional Menger probabilistic metric spaces, Fixed Point Thenry Appl., 2015 (2015 ), 1-15
]
A sharp generalization on cone b-metric space over Banach algebra
A sharp generalization on cone b-metric space over Banach algebra
en
en
The aim of this paper is to generalize a famous result for Banach-type contractive mapping from \(\rho(k)\in[0,\frac{1}{s})\) to \(\rho(k)\in[0,1)\)
in cone b-metric space over Banach algebra with coefficient \(s\geq 1\), where \(\rho(k)\) is the spectral radius of the generalized Lipschitz
constant \(k\). Moreover, some similar generalizations for the contractive constant \(k\) from \(k\in[0,\frac{1}{s})\) to \(k \in [0, 1)\) in cone b-metric
space and in b-metric space are also obtained. In addition, two examples are given to illustrate that our generalizations are in
fact real generalizations.
429
435
Huaping
Huang
School of Mathematical Sciences
Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education
China
mathhhp@163.com
Stojan
Radenovic
Faculty of Mechanical Engineering
University of Belgrade
Serbia
radens@beotel.net
Guantie
Deng
School of Mathematical Sciences
Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education
China
denggt@bnu.edu.cn
Cone b-metric space over Banach algebra
fixed point
c-sequence
iterative sequence.
Article.9.pdf
[
[1]
I. A. Bakhtin , The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, (1989), 26-37
##[2]
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), -
##[3]
M. Boriceanu, M. Bota, A. Petruşel, Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8 (2010), 367-377
##[4]
S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11
##[5]
L. Gajić, V. Rakočević, Quasi-contractions on a nonnormal cone metric space , Funct. Anal. Appl., 46 (2012), 62-65
##[6]
H.-P. Huang, S. Radenović, Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications, J. Nonlinear Sci. Appl., 8 (2015), 787-799
##[7]
H.-P. Huang, S. Radenović, Some fixed point results of generalized Lipschitz mappings on cone b-metric spaces over Banach algebras, J. Comput. Anal. Appl., 20 (2016), 566-583
##[8]
H.-P. Huang, S.-Y. Xu, Correction: Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory Appl., 2014 (2014 ), 1-5
##[9]
S. Janković, Z. Kadelburg, S. Radenović, On cone metric spaces: a survey, Nonlinear Anal., 74 (2011), 2591-2601
##[10]
M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed point results in metric-type spaces, Fixed Point Theory Appl., 2010 (2015), 1-15
##[11]
Z. Kadelburg, S. Radenović, V. Rakočević, Remarks on ''Quasi-contraction on a cone metric space'', Appl. Math. Lett., 22 (2009), 1674-1679
##[12]
P. K. Mishra, S. Sachdeva, S. K. Banerjee, Some fixed point theorems in b-metric space, Turkish J. Anal. Number Theory, 2 (2014), 19-22
##[13]
W. Sintunavarat, Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations, ev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 110 (2016), 585-600
##[14]
W. Sintunavarat , Nonlinear integral equations with new admissibility types in b-metric spaces, J. Fixed Point Theory Appl., 18 (2016), 397-416
##[15]
O. Yamaod, W. Sintunavarat, Y. J. Cho, Common fixed point theorems for generalized cyclic contraction pairs in b-metric spaces with applications, Fixed Point Theory Appl., 2015 (2015), 1-18
##[16]
O. Yamaod, W. Sintunavarat, Y. J. Cho, Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces, Open Math., 14 (2016), 128-145
]
Dynamics of stochastic hybrid Gilpin-Ayala system with impulsive perturbations
Dynamics of stochastic hybrid Gilpin-Ayala system with impulsive perturbations
en
en
This paper is mainly concerned with the dynamics of the stochastic Gilpin-Ayala model under regime switching with
impulsive perturbations. The goal is to analyze the effects of Markov chain and impulse on the dynamics. Some asymptotic
properties are considered and sufficient criteria for stochastic permanence, extinction, non-persistence in the mean and weak
persistence are obtained. The critical value among the extinction, non-persistence in the mean and weak persistence is explored.
Our results demonstrate that the dynamics of the model have close relations with the impulse and the stationary distribution of
the Markov chain.
436
450
Ruihua
Wu
College of Science
China University of Petroleum (East China)
P. R. China
wu_ruihua@hotmail.com
Gilpin-Ayala model
Markov chain
impulsive perturbations
stochastic permanence
extinction.
Article.10.pdf
[
[1]
S. Ahmad, I. M. Stamova , Asymptotic stability of competitive systems with delays and impulsive perturbations, J. Math. Anal. Appl., 334 (2007), 686-700
##[2]
W. J. Anderson, Continuous-time Markov chains, An applications-oriented approach, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York (1991)
##[3]
D. Baınov, P. Simeonov, Impulsive differential equations: periodic solutions and applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1993)
##[4]
F.-D. Chen , Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays, Nonlinear Anal., 7 (2006), 1205-1222
##[5]
N. H. Du, R. Kon, K. Sato, Y. Takeuchi, Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422
##[6]
M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpin-Ayala competition model, Comput. Math. Appl., 40 (2000), 1141-1151
##[7]
T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370
##[8]
T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419
##[9]
M.-X. He, F.-D. Chen, Dynamic behaviors of the impulsive periodic multi-species predator-prey system, Comput. Math. Appl., 57 (2009), 248-265
##[10]
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546
##[11]
J. Hou, Z.-D. Teng, S.-J. Gao, Permanence and global stability for nonautonomous N-species Lotka-Valterra competitive system with impulses, Nonlinear Anal. Real World Appl., 11 (2010), 1882-1896
##[12]
D.-Q. Jiang, N.-Z. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172
##[13]
D.-Q. Jiang, N.-Z. Shi, X.-Y. Li , Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597
##[14]
V. Lakshmikantham, D. D. Baınov, P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ (1989)
##[15]
X.-Y. Li, D.-Q. Jiang, X.-R. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448
##[16]
C.-X. Li, J.-T. Sun, Stability analysis of nonlinear stochastic differential delay systems under impulsive control , Phys. Lett. A, 374 (2010), 1154-1158
##[17]
C.-X. Li, J.-T. Sun, R.-Y. Sun, Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects, J. Franklin Inst., 347 (2010), 1186-1198
##[18]
C.-X. Li, J.-P. Shi, J.-T. Sun, Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks, Nonlinear Anal., 74 (2011), 3099-3111
##[19]
B.-S. Lian, S.-G. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339 (2008), 419-428
##[20]
M. Liu, C.-Z. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597-625
##[21]
M. Liu, C.-Z. Bai, Dynamics of a stochastic one-prey two-predator model with Lévy jumps, Appl. Math. Comput., 284 (2016), 308-321
##[22]
M. Liu, K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Modelling, 54 (2011), 2139-2154
##[23]
M. Liu, K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching II, Math. Comput. Modelling, 55 (2012), 405-418
##[24]
M. Liu, K. Wang, Dynamics and simulations of a logistic model with impulsive perturbations in a random environment, Math. Comput. Simulation, 92 (2013), 53-75
##[25]
M. Liu, K. Wang, Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations, Math. Comput. Modelling, 57 (2013), 909-925
##[26]
M. Liu, K. Wang, On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl., 63 (2012), 871-886
##[27]
M. Liu, K. Wang, Dynamics of a two-prey one-predator system in random environments, J. Nonlinear Sci., 23 (2013), 751-775
##[28]
M. Liu, K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457
##[29]
Q. Luo, X.-R. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84
##[30]
X.-R. Mao, G. G. Yin, C.-G. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43 (2007), 264-273
##[31]
X.-R. Mao, C.-G. Yuan, Stochastic differential equations with Markovian switching, Imperial College Press, London (2006)
##[32]
R. Sakthivel, J. Luo, Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett., 79 (2009), 1219-1223
##[33]
M. Slatkin, The dynamics of a population in a Markovian environment, Ecol, 59 (1978), 249-256
##[34]
Y. Takeuchi, N. H. Du, N. T. Hieu, K. Sato , Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957
##[35]
R.-H. Wu, X.-L. Zou, K. Wang, Asymptotic properties of a stochastic Lotka-Volterra cooperative system with impulsive perturbations, Nonlinear Dynam., 77 (2014), 807-817
##[36]
J.-R. Yan, Stability for impulsive delay differential equations, Nonlinear Anal., 63 (2005), 66-80
]
From fuzzy metric spaces to modular metric spaces: a fixed point approach
From fuzzy metric spaces to modular metric spaces: a fixed point approach
en
en
We propose an intuitive theorem which uses some concepts of auxiliary functions for establishing existence and uniqueness
of the fixed point of a self-mapping. First we work in the setting of fuzzy metric spaces in the sense of George and Veeramani,
then we deduce some consequences in modular metric spaces. Finally, a sample homotopy result is derived making use of the
main theorem.
451
464
Fairouz
Tchier
Mathematics Department, College of Science (Malaz)
King Saud University
King Saudi Arabia
ftchier@ksu.edu.sa
Calogero
Vetro
Department of Mathematics and Computer Science
University of Palermo
Italy
calogero.vetro@unipa.it
Francesca
Vetro
Department of Energy, Information Engineering and Mathematical Models (DEIM)
University of Palermo
Italy
francesca.vetro@unipa.it
Fixed point
fuzzy metric space
modular metric space.
Article.11.pdf
[
[1]
A. A. N. Abdou, M. A. Khamsi, Fixed point results of pointwise contractions in modular metric spaces, Fixed Point Theory Appl., 2013 (2013 ), 1-11
##[2]
R. P. Agarwal, M. Meehan, D. O’Regan , Fixed point theory and applications, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (2002)
##[3]
H. Argoubi, B. Samet, C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl., 8 (2015), 1082-1094
##[4]
B. Azadifar, G. Sadeghi, R. Saadati, C.-K. Park, Integral type contractions in modular metric spaces, J. Inequal. Appl., 2013 (2013), 1-14
##[5]
V. V. Chistyakov , Modular metric spaces, I, Basic concepts, Nonlinear Anal., 72 (2010), 1-14
##[6]
V. V. Chistyakov, Modular metric spaces,II, Application to superposition operators, Nonlinear Anal., 72 (2010), 15-30
##[7]
C. Di Bari, C. Vetro, Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space, J. Fuzzy Math., 13 (2005), 973-982
##[8]
A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399
##[9]
V. Gregori, A. López-Crevillén, S. Morillas, A. Sapena, On convergence in fuzzy metric spaces, Topology Appl., 156 (2009), 3002-3006
##[10]
V. Gregori, S. Morillas, A. Sapena, On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems, 161 (2010), 2193-2205
##[11]
V. Gregori, S. Romaguera, On completion of fuzzy metric spaces, Theme: Fuzzy intervals, Fuzzy Sets and Systems, 130 (2002), 399-404
##[12]
N. Hussain, P. Salimi , Implicit contractive mappings in modular metric and fuzzy metric spaces, Scientific World J., 2014 (2014), 1-12
##[13]
F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1189-1194
##[14]
D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems, 158 (2007), 915-921
##[15]
L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338-353
]
Tripled random coincidence point and common fixed point results of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras
Tripled random coincidence point and common fixed point results of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras
en
en
In this paper, based on the concept of cone b-metric space over Banach algebra, which was introduced by Huang and
Radenovic [H.-P. Huang, S. Radenović, J. Nonlinear Sci. Appl., 8 (2015), 787–799], we obtain some tripled common random
fixed point and tripled random fixed point theorems with several generalized Lipschitz constants in such spaces. We consider
the obtained assertions without the assumption of normality of cones. The presented results generalize some coupled common
fixed point theorems in the existing literature.
465
482
Binghua
Jiang
School of Mathematics and Statistics
Hubei Normal University
China
jbh510@163.com
Zelin
Cai
School of Mathematics and Statistics
Hubei Normal University
China
woshicaizelin@163.com
Jinyang
Chen
School of Mathematics and Statistics
Hubei Normal University
China
984121640@qq.com
Huaping
Huang
School of Mathematical Sciences
Beijing Normal University
China
mathhhp@163.com
Tripled random fixed point
tripled random coincidence point
cone b-metric space over Banach algebra
generalized Lipschitz constant
tripled common random fixed point.
Article.12.pdf
[
[1]
C. T. Aage, J. N. Salunke, Some fixed point theorems for expansion onto mappings on cone metric spaces, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1101-1106
##[2]
M. Abbas, M. Ali Khan, S. Radenović, Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. Math. Comput., 217 (2010), 195-202
##[3]
A. Aliouche, C. Simpson, Fixed points and lines in 2-metric spaces , Adv. Math., 229 (2012), 668-690
##[4]
A. Alotaibi, S. M. Alsulami, Coupled coincidence points for monotone operators in partially ordered metric spaces, Fixed Point Theory Appl., 2011 (2011), 1-13
##[5]
I. Altun, B. Damjanović, D. Djorić, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett., 23 (2010), 310-316
##[6]
M. Asadi, B. E. Rhoades, H. Soleimani, Some notes on the paper ''The equivalence of cone metric spaces and metric spaces'', Fixed Point Theory Appl., 2012 (2012 ), 1-4
##[7]
H. Aydi, M. Abbas, W. Sintunavarat, P. Kumam, Tripled fixed point of W-compatible mappings in abstract metric spaces, Fixed Point Theory Appl., 2012 (2012 ), 1-20
##[8]
I. A. Bakhtin , The contraction mapping principle in almost metric space, Functional analysis Gos. Ped. Inst. Unianowsk, 30 (1989), 26-37
##[9]
S.-B. Chen, W. Li, S.-P. Tian, Z.-Y. Mao, On optimization problems in quasi-metric spaces, in: Proc. 5th International Conf. on Machine Learning and Cybernetics, Dalian, (2006), 13-16
##[10]
C.-F. Chen, C.-X. Zhu, Fixed point theorems for n times reasonable expansive mapping, Fixed Point Theory Appl., 2008 (2008), 1-6
##[11]
S. Chouhan, C. Malviya, A fixed point theorem for expansive type mappings in cone metric spaces, Int. Math. Forum, 6 (2011), 891-897
##[12]
L. Ćirić, V. Lakshmikantham, Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces, Stoch. Anal. Appl., 27 (2009), 1246-1259
##[13]
W.-S. Du, E. Karapınar, A note on cone b-metric and its related results: generalizations or equivalence?, Fixed Point Theory Appl., 2013 (2013), 1-7
##[14]
Z. M. Fadail, A. G. B. Ahmad, Coupled coincidence point and common coupled fixed point results in cone b-metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-14
##[15]
Y. Han, S.-Y. Xu, Some new theorems of expanding mappings without continuity in cone metric spaces, Fixed point Theory Appl., 2013 (2013 ), 1-9
##[16]
C. J. Himmelberg, Measurable relations, Fund. Math., 87 (1975), 53-72
##[17]
H.-P. Huang, S. Radenović, Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications, J. Nonlinear Sci. Appl., 8 (2015), 787-799
##[18]
L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476
##[19]
N. Hussain, A. Latif, N. Shafqat, Weak contractive inequalities and compatible mixed monotone random operators in ordered metric spaces, J. Inequal. Appl., 2012 (2012 ), 1-20
##[20]
N. Hussian, M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl., 62 (2011), 1677-1684
##[21]
M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed point results in metric-type spaces, Fixed point Theory Appl., 2010 (2010), 1-15
##[22]
Z. Kadelburg, S. Radenović, Fixed point results in C-algebra-valued metric spaces are direct consequences of their standard metric counterparts, Fixed Point Theory Appl., 2016 (2016 ), 1-6
##[23]
Z. Kadelburg, S. Radenović, V. Rakočević , A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., 24 (2011), 370-374
##[24]
E. Karapınar, Generalizations of Caristi Kirk’s theorem on partial metric spaces, Fixed Point Theory Appl., 2011 (2011), 1-7
##[25]
E. Karapınar, N. Van Luong, N. X. Thuan, Coupled coincidence points for mixed monotone operators in partially ordered metric spaces, Arab. J. Math. (Springer), 1 (2012), 329-339
##[26]
T.-C. Lin, Random approximations and random fixed point theorems for non-self-maps, Proc. Amer. Math. Soc., 103 (1988), 1129-1135
##[27]
H. Liu, S.-Y. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013 (2013 ), 1-10
##[28]
S. Radenović, Common fixed points under contractive conditions in cone metric spaces, Comput. Math. Appl., 58 (2009), 1273-1278
##[29]
S. Radenović, Remarks on some coupled coincidence point results in partially ordered metric spaces, Arab J. Math. Sci., 20 (2014), 29-39
##[30]
S. Rezapour, R. Hamlbarani, Some notes on the paper: ''Cone metric spaces and fixed point theorems of contractive mappings'', J. Math. Anal. Appl., 332 (2007), 1468–1476, by L.-G. Huang, X. Zhang, J. Math. Anal. Appl., 345 (2008), 719-724
##[31]
W. Rudin, Functional analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw- Hill, Inc., New York (1991)
##[32]
S. Saadati, S. M. Vaezpour, P. Vetro, B. E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Math. Comput. Modelling, 52 (2010), 797-801
##[33]
I. Sahni, M. Telci, Fixed points of contractive mappings on complete cone metric spaces, Hacet. J. Math. Stat., 38 (2009), 59-67
##[34]
B. Samet, C. Vetro, Coupled fixed point, F-invariant set and fixed point of N-order, Ann. Funct. Anal., 1 (2010), 46-56
##[35]
W. Shatanawi, F. Awawdeh, Some fixed and coincidence point theorems for expansive maps in cone metric spaces, Fixed point Theory Appl., 2012 (2012 ), 1-10
##[36]
W. Shatanawi, Z. Mustafa, On coupled random fixed point results in partially ordered metric spaces, Mat. Vesnik, 64 (2012), 139-146
##[37]
Y.-H, Shen, D. Qiu, W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Lett., 25 (2012), 138-141
##[38]
L. Shi, S.-Y. Xu, Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces, Fixed Point Theory Appl., 2013 (2013 ), 1-11
##[39]
S.-Y. Xu, S. Radenović, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed point Theory Appl., 2014 (2014 ), 1-12
]
A projected fixed point algorithm with Meir-Keeler contraction for pseudocontractive mappings
A projected fixed point algorithm with Meir-Keeler contraction for pseudocontractive mappings
en
en
In this paper, we introduce a projected algorithm with Meir-Keeler contraction for finding the fixed points of the pseudocontractive
mappings. We prove that the presented algorithm converges strongly to the fixed point of the pseudocontractive
mapping in Hilbert spaces.
483
491
Yonghong
Yao
Department of Mathematics
Tianjin Polytechnic University
China
yaoyonghong@aliyun.com
Naseer
Shahzad
Department of Mathematics
King Abdulaziz University
Saudi Arabia
nshahzad@kau.edu.sa
Yeong-Cheng
Liou
Department of Information Management
Center for General Education
Cheng Shiu University
Kaohsiung Medical University
Taiwan
Taiwan
simplex_liou@hotmail.com
Li-Jun
Zhu
School of Mathematics and Information Science
Beifang University of Nationalities
China
zhulijun1995@sohu.com
Projected algorithm
pseudocontractive mapping
fixed point.
Article.13.pdf
[
[1]
L.-C. Ceng, A. Petrusel, J.-C. Yao, Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings, Appl. Math. Comput., 209 (2009), 162-176
##[2]
C. E. Chidume, M. Abbas, B. Ali , Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings, Appl. Math. Comput., 194 (2007), 1-6
##[3]
C. E. Chidume, S. A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc., 129 (2001), 2359-2363
##[4]
Y. J. Cho, S. M. Kang, X.-L. Qin, Strong convergence of an implicit iterative process for an infinite family of strict pseudocontractions, Bull. Korean Math. Soc., 47 (2010), 1259-1268
##[5]
L. Ćirić, A. Rafiq, N. Cakić, J. S. Ume, Implicit Mann fixed point iterations for pseudo-contractive mappings, Appl. Math. Lett., 22 (2009), 581-584
##[6]
W.-P. Guo, M. S. Choi, Y. J. Cho, Convergence theorems for continuous pseudocontractive mappings in Banach spaces, J. Inequal. Appl., 2014 (2014 ), 1-10
##[7]
S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150
##[8]
A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326-329
##[9]
C. Moore, B. V. C. Nnoli, Strong convergence of averaged approximants for Lipschitz pseudocontractive maps, J. Math. Anal. Appl., 260 (2001), 269-278
##[10]
U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585
##[11]
X.-L. Qin, Y. J. Cho, H.-Y. Zhou, Strong convergence theorems of fixed point for quasi-pseudo-contractions by hybrid projection algorithms, Fixed Point Theory, 11 (2010), 347-354
##[12]
N. Shahzad, N. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal., 71 (2009), 838-844
##[13]
T. Suzuki, Moudafi’s viscosity approximations with Meir-Keeler contractions, J. Math. Anal. Appl., 325 (2007), 342-352
##[14]
M. Tsukada , Convergence of best approximations in a smooth Banach space, J. Approx. Theory, 40 (1984), 301-309
##[15]
A. Udomene , Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces, Nonlinear Anal., 67 (2007), 2403-2414
##[16]
Y.-H. Yao, Y.-C. Liou, G. Marino, A hybrid algorithm for pseudo-contractive mappings, Nonlinear Anal., 71 (2009), 4997-5002
##[17]
Y.-H. Yao, Y.-C. Liou, J.-C. Yao, Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction, Fixed Point Theory Appl., 2015 (2015 ), 1-19
##[18]
Y.-H. Yao, M. A. Noor, S. Zainab, Y.-C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl., 345 (2009), 319-329
##[19]
Y.-H. Yao, M. Postolache, S. M. Kang, Strong convergence of approximated iterations for asymptotically pseudocontractive mappings, Fixed Point Theory Appl., 2014 (2014 ), 1-13
##[20]
Y.-H. Yao, M. Postolache, Y.-C. Liou, Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings, Fixed Point Theory Appl., 2013 (2013 ), 1-8
##[21]
Y.-H. Yao, M. Postolache, Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013 ), 1-12
##[22]
Y.-H. Yao, N. Shahzad, New methods with perturbations for non-expansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011 ), 1-9
##[23]
Y.-H. Yao, N, Shahzad, Implicit and explicit methods for finding fixed points of strictly pseudo-contractive mappings in Banach spaces, J. Nonlinear Convex Anal., 13 (2012), 183-194
##[24]
H. Zegeye, E. U. Ofoedu, N. Shahzad, Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings, Appl. Math. Comput., 216 (2010), 3439-3449
##[25]
H. Zegeye, N. Shahzad, Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal., 70 (2009), 2707-2716
##[26]
H. Zegeye, N. Shahzad, Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces, Optim. Lett., 5 (2011), 691-704
##[27]
H. Zegeye, N. Shahzad, An algorithm for a common fixed point of a family of pseudocontractive mappings, Fixed Point Theory Appl., 2013 (2013 ), 1-14
##[28]
H. Zegeye, N. Shahzad, M. A. Alghamdi , Convergence of Ishikawa’s iteration method for pseudocontractive mappings, Nonlinear Anal., 74 (2011), 7304-7311
##[29]
H. Zegeye, N. Shahzad, T. Mekonen, Viscosity approximation methods for pseudocontractive mappings in Banach spaces, Appl. Math. Comput., 185 (2007), 538-546
##[30]
H.-Y. Zhou, Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 343 (2008), 546-556
##[31]
H.-Y. Zhou, Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces, Nonlinear Anal., 70 (2009), 4039-4046
]
Mild solutions to nonlocal impulsive differential inclusions governed by a noncompact semigroup
Mild solutions to nonlocal impulsive differential inclusions governed by a noncompact semigroup
en
en
In this paper, we study the existence of mild solutions to impulsive differential inclusions with nonlocal conditions in general
Banach spaces when operator semigroup is not compact. By using measure of noncompactness and multivalued analysis, we
give some sufficient conditions on the existence results where the impulsive items and the nonlocal items are compact and
Lipschitz continuous, respectively. An example concerning with the partial differential equation is also presented.
492
503
Shaochun
Ji
Faculty of Mathematics and Physics
Huaiyin Institute of Technology
P. R. China
jiscmath@163.com
Differential inclusions
impulsive conditions
fixed point theorems
measure of noncompactness
nonlocal conditions.
Article.14.pdf
[
[1]
N. Abada, M. Benchohra, H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations, 246 (2009), 3834-3863
##[2]
R. P. Agarwal, M. Meehan, D. O’Regan, Fixed point theory and applications, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (2001)
##[3]
S. Aizicovici, V. Staicu, Multivalued evolution equations with nonlocal initial conditions in Banach spaces, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 361-376
##[4]
J. Banaś, K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1980)
##[5]
V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste Romania, Bucharest; Noordhoff International Publishing, Leiden (1976)
##[6]
M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York (2006)
##[7]
M. Benchohra, J. J. Nieto, A. Ouahab, Impulsive differential inclusions involving evolution operators in separable Banach spaces, Ukrainian Math. J., 64 (2012), 991-1018
##[8]
I. Benedetti, N. V. Loi, L. Malaguti, Nonlocal problems for differential inclusions in Hilbert spaces, Set-Valued Var. Anal., 22 (2014), 639-656
##[9]
L. Byszewski, V. Lakshmikantham , Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 40 (1991), 11-19
##[10]
T. Cardinali, P. Rubbioni, Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains, Nonlinear Anal., 69 (2008), 73-84
##[11]
J.-F. Couchouron, M. Kamenskii, An abstract topological point of view and a general averaging principle in the theory of differential inclusions, Nonlinear Anal., 42 (2000), 1101-1129
##[12]
K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin (1985)
##[13]
Z.-B. Fan, G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709-1727
##[14]
E. Hernández, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649
##[15]
S.-C. Ji, G. Li, Existence results for impulsive differential inclusions with nonlocal conditions, Comput. Math. Appl., 62 (2011), 1908-1915
##[16]
S.-C. Ji, G, Li, A unified approach to nonlocal impulsive differential equations with the measure of noncompactness, Adv. Difference Equ., 2012 (2012), 1-14
##[17]
M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin (2001)
##[18]
V. Lakshmikantham, D. D. Baınov, P. S. Simeonov, Theory of impulsive differential equations , Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ (1989)
##[19]
J. Liang, J. H. Liu, T.-J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Modelling, 49 (2009), 798-804
##[20]
S. K. Ntouyas, P. C. Tsamatos , Global existence for semilinear evolution equations with nonlocal condition, J. Math. Anal. Appl., 210 (1997), 679-687
##[21]
A. Pazy , Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York (1983)
##[22]
H. R. Thieme, ''Integrated semigroups'' and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447
##[23]
J.-R. Wang, A. G. Ibrahim, M. Fečkan, Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comput., 257 (2015), 103-118
##[24]
X.-M. Xue, Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlinear Anal., 70 (2009), 2593-2601
]
Hyers-Ulam-Rassias stability of non-linear delay differential equations
Hyers-Ulam-Rassias stability of non-linear delay differential equations
en
en
In this paper, we prove the Hyers-Ulam-Rassias stability and Hyers-Ulam stability of delay differential equation of the form
\[y^{(n)}=F(t,\{y^{(i)}(t)\}^{n-1}_{i=0},\{y^{(i)}(t-\lambda)\}^{n-1}_{i=0}),\]
with Lipschitz condition by using fixed point approach. The results of the paper generalize most of the results concerning the
stability of delay differential equations in the existing literature.
504
510
Akbar
Zada
Department of Mathematics
University of Peshawar
Pakistan
zadababo@yahoo.com
Shah
Faisal
Department of Mathematics
University of Peshawar
Pakistan
shahfaisal8763@gmail.com
Yongjin
Li
Department of Mathematics
Sun Yat-Sen University
P. R. China
stslyj@mail.sysu.edu.cn
Fixed point theorem
Hyers-Ulam stability
Hyers-Ulam-Rassias stability
non-linear delay differential equations.
Article.15.pdf
[
[1]
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66
##[2]
G.-Y. Choi, S.-M. Jung, Invariance of Hyers-Ulam stability of linear differential equations and its applications, Adv. Difference Equ., 2015 (2015 ), 1-14
##[3]
J. B. Diaz, B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305-309
##[4]
J.-H. Huang, Q. H. Alqifiary, Y.-J. Li, Superstability of differential equations with boundary conditions, Electron. J. Differential Equations, 2014 (2014 ), 1-8
##[5]
J.-H. Huang, S.-M. Jung, Y.-J. Li, On Hyers-Ulam stability of nonlinear differential equations, Bull. Korean Math. Soc., 52 (2015), 685-697
##[6]
J.-H. Huang, Y.-J. Li, Hyers-Ulam stability of linear functional differential equations, J. Math. Anal. Appl., 426 (2015), 1192-1200
##[7]
J.-H. Huang, Y.-J. Li, Hyers-Ulam stability of delay differential equations of first order, Math. Nachr., 289 (2016), 60-66
##[8]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 222-224
##[9]
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135-1140
##[10]
S.-M. Jung, A fixed point approach to the stability of differential equations \(\acute{y} = F(x, y)\), Bull. Malays. Math. Sci. Soc., 33 (2010), 47-56
##[11]
S.-M. Jung, J. Brzdęk, Hyers-Ulam stability of the delay equation \(\acute{y}(t) = \lambda y(t-\tau)\), Abstr. Appl. Anal., 2010 (2010 ), 1-10
##[12]
S.-M. Jung, J. Roh, The linear differential equations with complex constant coefficients and Schrödinger equations, Appl. Math. Lett., 66 (2016), 1-6
##[13]
Y. Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA (1993)
##[14]
Y.-J. Li, Y. Shen, Hyers-Ulam stability of nonhomogeneous linear differential equations of second order, Int. J. Math. Math. Sci., 2009 (2009 ), 1-7
##[15]
T.-X. Li, A. Zada, Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ., 2016 (2016 ), 1-8
##[16]
T. Miura, S. Miyajima, S.-E. Takahasi, A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl., 286 (2003), 136-146
##[17]
Z. Moszner, Stability has many names, Aequationes Math., 90 (2016), 983-999
##[18]
M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat., 13 (1993), 259-270
##[19]
M. Obloza , Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.- Dydakt. Prace Mat., 14 (1997), 141-146
##[20]
D. Otrocol, V. Ilea, Ulam stability for a delay differential equation, Cent. Eur. J. Math., 11 (2013), 1296-1303
##[21]
T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
##[22]
C. Tunç, E. Biçer, Hyers-Ulam-Rassias stability for a first order functional differential equation, J. Math. Fundam. Sci., 47 (2015), 143-153
##[23]
S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London (1960)
##[24]
J.-R.Wang, M. Fečkan, Y. Zhou, On the stability of first order impulsive evolution equations, Opuscula Math., 34 (2014), 639-657
##[25]
B. Xu, J. Brzdęk, W.-N. Zhang, Fixed point results and the Hyers-Ulam stability of linear equations of higher orders, Pacific J. Math., 273 (2015), 483-498
##[26]
A. Zada, S. Faisal, Y.-J. Li , On the Hyers-Ulam stability of first-order impulsive delay differential equations, J. Funct. Spaces, 2016 (2016 ), 1-6
##[27]
A. Zada, O. Shah, R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput., 271 (2015), 512-518
]
Existence for boundary value problems of two-term Caputo fractional differential equations
Existence for boundary value problems of two-term Caputo fractional differential equations
en
en
This paper is concerned with a class of boundary value problem of nonlinear fractional differential equation \(^cD^\alpha u(t)-a^cD^\beta u(t)+f(t,u(t))=0\). This equation may be regarded as an extension of Bagley-Torvik equations. Some new existence and
uniqueness results are obtained by using standard Banach contraction principle and Krasnoselskii’s fixed point theorem.
511
520
Badawi Hamza Elbadawi
Ibrahim
School of Mathematical Sciences
Yangzhou University
P. R. China
badawi.12@hotmail.com
Qixiang
Dong
School of Mathematical Sciences
Yangzhou University
P. R. China
qxdongyz@outlook.com;qxdong@yzu.edu.cn
Zhenbin
Fan
School of Mathematical Sciences
Yangzhou University
P. R. China
zbfan@yzu.edu.cn
Fractional derivative
differential equation
boundary value problem.
Article.16.pdf
[
[1]
B. Ahmed, S. K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Difference Equ., 2011 (2011), 1-11
##[2]
T. S. Aleroev, On a boundary value problem for a fractional-order differential operator, (Russian) translated from Differ. Uravn., 34 (1998), 123, Differential Equations, 34 (1998), 1-126
##[3]
R. L. Bagley, P. J. Torvik , On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298
##[4]
Z.-B. Bai, H. Lü , Positive solutions for a boundary value problem of nonlinear fractional differential equation , J. Math. Anal. Appl., 311 (2005), 495-505
##[5]
M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 71 (2009), 2391-2396
##[6]
M. Benchohra, N. Hamidi, Fractional order differential inclusions on the half-line, Surv. Math. Appl., 5 (2010), 99-111
##[7]
K. Diethelm , The analysis of fractional differential equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, (2004), Springer-Verlag, Berlin, Springer-Verlag, Berlin (2010)
##[8]
K. Diethelm, A. D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, Sci. Comput. Chem. Eng., II, Springer Berlin Heidelberg, (1999), 217-224
##[9]
Q.-X. Dong, G.-X. Wu, J. Li, A boundary value problem for a class of fractional differential equations in Banach spaces, (Chinese) Pure Appl. Math. (Xi’an), 29 (2013), 1-10
##[10]
A. Granas, J. Dugundji, Fixed point theory , Springer Monographs in Mathematics, Springer-Verlag, New York (2003)
##[11]
N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann- Liouville fractional derivatives, Rheol. Acta, 45 (2006), 765-772
##[12]
H. Jafari, V. Daftardar-Gejji , Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput., 180 (2006), 700-706
##[13]
E. R. Kaufmann, K. D. Yao, Existence of solutions for a nonlinear fractional order differential equation, Electron. J. Qual. Theory Differ. Equ., 2009 (2009 ), 1-9
##[14]
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)
##[15]
P. A. Kilbas, J. J. Trujillo, Differential equations of fractional order: methods, results and problems, II , Appl. Anal., 81 (2002), 435-493
##[16]
M. A. Krasnoselskii , Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.), 10 (1955), 123-127
##[17]
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), 1-10
##[18]
F. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103 (1995), 7180-7186
##[19]
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)
##[20]
S. Ntouyas, Existence results for first order boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions, J. Fract. Calc. Appl., 3 (2012), 1-14
]
Two kinds of breather solitary wave and rogue wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation
Two kinds of breather solitary wave and rogue wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation
en
en
In this paper, the (3+1)-dimensional Kadomtsev-Petviashvili equation is investigated. Two kinds of periodic breather solitary
wave and rogue wave solutions are obtained by using the two-wave method and the homoclinic breather limit approach with
the aid of Maple. Deflection of rogue wave varying with the seed solution \(u_0\) is investigated.
521
527
Zhenhui
Xu
School of Science
Southwest University of Science and Technology
P. R. China
xuzhenhui19@163.com
Hanlin
Chen
School of Science
Southwest University of Science and Technology
P. R. China
Zhengde
Dai
School of Mathematics and Physics
Yunnan University
P. R. China
(3+1)-dimensional Kadomtsev-Petviashvili equation
homoclinic breather limit approach
two-wave method
rational breather solutions
rogue wave.
Article.17.pdf
[
[1]
N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schrödinger equation, Phys. Rev. E, 80 (2009), 2137-2145
##[2]
N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace, Phys. Lett. A, 373 (2009), 675-678
##[3]
C.-L. Bai, C.-J. Bai, H. Zhao, A generalized variable-coefficient algebraic method exactly solving (3 + 1)-dimensional Kadomtsev-Petviashvilli equation, Commun. Theor. Phys. (Beijing), 44 (2005), 821-826
##[4]
Y. V. Bludov, V. V. Konotop, N. Akhmediev, Matter rogue waves, Phys. Rev. A, 80 (2009), 1-033610
##[5]
H.-L. Chen, Z.-H. Xu, Z.-D. Dai, Rogue wave for the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation, Abstr. Appl. Anal., 2014 (2014 ), 1-7
##[6]
Z.-D. Dai, J. Liu, D.-L. Li, Applications of HTA and EHTA to YTSF equation, Appl. Math. Comput., 207 (2009), 360-364
##[7]
C.-Q. Dai, G.-Q. Zhou, J.-F. Zhang, Controllable optical rogue waves in the femtosecond regime, Phys. Rev. E, 85 (2012), 1-016603
##[8]
K. Dysthe, H. E. Krogstad, P. Müller, Oceanic rogue waves, Annu. Rev. Fluid Mech., 40 (2008), 287–310,40, Annual Reviews, Palo Alto, CA (2008)
##[9]
C. Kharif, E. Pelinovsky, A. Slunyaey, Rogue waves in the ocean, Advances in Geophysical and Environmental Mechanics and Mathematics, Springer-Verlag, Berlin (2009)
##[10]
Z.-Y. Ma, S.-H. Ma, Analytical solutions and rogue waves in (3+ 1)-dimensional nonlinear Schrödinger equation, Chin. Phys. B, 21 (2012), 1-030507
##[11]
D. R. Solli, C. Ropers, B. Jalali , Active control of rogue waves for stimulated supercontinuum generation, Phys. Rev. Lett., 101 (2008), 1-233902
##[12]
L. Stenflo, M. Marklund, Rogue waves in the atmosphere, J. Plasma Phys., 76 (2010), 293-306
##[13]
L.-Y. Wang, S.-Y. Lou, Some special types of solitary wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvilli equation, Commun. Theor. Phys. (Beijing), 33 (2000), 683-686
##[14]
R.-Q. Wu, Bilinear Bäklund transformation and explicit solutions for nonlinear evolution equation, Chin. Phys. B, 19 (2010), 1-040304
##[15]
Z.-H. Xu, H.-L. Chen, Z.-D. Dai , Rogue wave for the (2 + 1)-dimensional Kadomtsev-Petviashvili equation, Appl. Math. Lett., 37 (2014), 34-38
##[16]
Z.-H. Xu, H.-L. Chen, M.-R. Jiang, Z.-D. Dai, W. Chen, Resonance and deflection of multi-soliton to the (2 + 1)- dimensional Kadomtsev-Petviashvili equation, Nonlinear Dynam., 78 (2014), 461-466
##[17]
Z.-H. Xu, D.-Q. Xian, H.-L. Chen, New periodic solitary-wave solutions for the Benjiamin Ono equation, Appl. Math. Comput., 215 (2010), 4439-4442
##[18]
Z.-Y. Yan, Nonautonomous rogons in the inhomogeneous nonlinear Schrödinger equation with variable coefficients, Phys. Lett. A, 374 (2010), 672-681
##[19]
X.-J. Yang, A new integral transform method for solving steady heat-transfer problem, Therm. Sci., 20 (2016), 639-642
##[20]
X.-J. Yang, A new integral transform with an application in heat-transfer problem, Therm. Sci., 20 (2016), 677-681
]
Applications of a novel integral transform to partial differential equations
Applications of a novel integral transform to partial differential equations
en
en
In this paper, we establish and perfect the dualities among the Laplace transform (LT), Laplace-Carson transform (LCT),
Sumudu transform (ST), and a novel integral transform (NIT). In addition, some novel properties of the NIT are explored and
the NIT is applied to solve some partial differential equations (PDEs).
528
534
Xin
Liang
State Key Laboratory for Geomechanics and Deep Underground Engineering
China University of Mining and Technology
P. R. China
xliang@cumt.edu.cn
Feng
Gao
State Key Laboratory for Geomechanics and Deep Underground Engineering
School of Mechanics and Civil Engineering
China University of Mining and Technology
China University of Mining and Technology
P. R. China
P. R. China
jsppw@sohu.com
Ya-Nan
Gao
State Key Laboratory for Geomechanics and Deep Underground Engineering
School of Mechanics and Civil Engineering
China University of Mining and Technology
China University of Mining and Technology
P. R. China
P. R. China
yngao@cumt.edu.cn
Xiao-Jun
Yang
School of Mechanics and Civil Engineering
China University of Mining and Technology
P. R. China
dyangxiaojun@163.com
Laplace transform
Laplace-Carson transform
Sumudu transform
partial differential equations.
Article.18.pdf
[
[1]
M. A. Asiru, Sumudu transform and the solution of integral equations of convolution type, Internat. J. Math. Ed. Sci. Tech., 32 (2001), 906-910
##[2]
A. Atangana, Extension of the Sumudu homotopy perturbation method to an attractor for one-dimensional Keller-Segel equations, Appl. Math. Model., 39 (2015), 2909-2916
##[3]
A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948-956
##[4]
A. Atangana, D. Baleanu, Modelling the advancement of the impurities and the melted oxygen concentration within the scope of fractional calculus, Int. J. Nonlin. Mech., 67 (2014), 278-284
##[5]
R. J. Beerends, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie, Fourier and Laplace transforms, Translated from the 1992 Dutch edition by Beerends, Cambridge University Press, Cambridge (2003)
##[6]
F. B. M. Belgacem, A. A. Karaballi, S. L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Math. Probl. Eng., 58 (2003), 103-118
##[7]
L. Debnath, D. Bhatta, Integral transforms and their applications, Third edition, CRC Press, Boca Raton, FL (2015)
##[8]
C. Donolato, Analytical and numerical inversion of the Laplace-Carson transform by a differential method, Comput. Phys. Comm., 145 (2002), 298-309
##[9]
H. Eltayeb, A. Kılıçman, A note on solutions of wave, Laplace’s and heat equations with convolution terms by using a double Laplace transform, Appl. Math. Lett., 21 (2008), 1324-1329
##[10]
U. Graf, Applied Laplace transforms and z-transforms for scientists and engineers, A computational approach using a Mathematica package, With 1 CD-ROM (Windows and LINUX), Birkhäuser Verlag, Basel (2004)
##[11]
M.-J. Kang, J.-K. Jeon, H.-J. Han, S.-M. Lee, Analytic solution for American strangle options using Laplace–Carson transforms, Commun. Nonlinear Sci. Numer. Simul., 47 (2017), 292-307
##[12]
A. Kılıçman, H. Eltayeb, R. R. Ashurov , Further analysis on classifications of PDE(s) with variable coefficients, Appl. Math. Lett., 23 (2010), 966-970
##[13]
V. A. Kudinov, Approximate solutions of nonstationary junction heat-exchange problems for laminar fluid flow in channels, J. Eng. Phy. Thermophys., 51 (1986), 1326-1331
##[14]
A. D. Polyanin, V. F. Zaitsev, Handbook of exact solutions for ordinary differential equations, CRC Press, Boca Raton, FL (1995)
##[15]
G. K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Internat. J. Math. Ed. Sci. Tech., 24 (1993), 35-43
##[16]
S. Weerakoon , Application of Sumudu transform to partial differential equations, Internat. J. Math. Ed. Sci. Tech., 25 (1994), 277-283
##[17]
X.-J. Yang, A new integral transform with an application in heat-transfer problem, Therm. Sci., 20 (2016), 677-681
##[18]
X.-J. Yang, A new integral transform operator for solving the heat-diffusion problem, Appl. Math. Lett., 64 (2017), 193-197
##[19]
X.-J. Yang, J. T. 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
]
Existence of periodic solutions for a class of discrete systems with classical or bounded (\(\phi_1,\phi_2\))-Laplacian
Existence of periodic solutions for a class of discrete systems with classical or bounded (\(\phi_1,\phi_2\))-Laplacian
en
en
In this paper, we investigate the existence of periodic solutions for the nonlinear discrete system with classical or bounded
(\(\phi_1,\phi_2\))-Laplacian: \[
\begin{cases}
\Delta\phi_1(\Delta u_1(t-1))+\nabla_{u_1}F(t,u_1(t),u_2(t))=0,\\
\Delta\phi_2(\Delta u_2(t-1))+\nabla_{u_2}F(t,u_1(t),u_2(t))=0.
\end{cases}
\]
By using the saddle point theorem, we obtain that system with classical (\(\phi_1,\phi_2\))-Laplacian has at least one periodic solution
when F has (p, q)-sublinear growth, and system with bounded (\(\phi_1,\phi_2\))-Laplacian has at least one periodic solution when \(F\) has
(\(p,q\))-sublinear growth. By using the least action principle, we obtain that system with classical or bounded (\(\phi_1,\phi_2\))-Laplacian has at
least one periodic solution when \(F\) has a growth like Lipschitz condition.
535
559
Haiyun
Deng
Department of Mathematics, Faculty of Science
Kunming University of Science and Technology
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
Department of Mathematics, Faculty of Science
Kunming University of Science and Technology
P. R. China
Discrete systems
periodic solutions
saddle point theorem
(\(\phi_1،\phi_2\))-Laplacian
the least action principle.
Article.19.pdf
[
[1]
Y.-H. Ding, Variational methods for strongly indefinite problems, Interdisciplinary Mathematical Sciences, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2007)
##[2]
X.-L. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problem involving the p(x)-Laplacian, J. Math. Anal. Appl., 334 (2007), 248-260
##[3]
Z.-M. Guo, J.-S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506-515
##[4]
Z.-M. Guo, J.-S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430
##[5]
T.-S. He, W. Chen, Periodic solutions of second order discrete convex systems involving the p-Laplacian, Appl. Math. Comput., 206 (2008), 124-132
##[6]
X.-F. He, P. Chen, Homoclinic solutions for second order discrete p-Laplacian systems, Adv. Difference Equ., 2011 (2011), 1-16
##[7]
Q. Jiang, C.-L. Tang, Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 327 (2007), 380-389
##[8]
C. Li, Z.-Q. Ou, C.-L. Tang, Periodic solutions for non-autonomous second-order differential systems with (q, p)-Laplacian, Electron. J. Differential Equations, 2014 (2014 ), 1-13
##[9]
Y.-K. Li, T.-W. Zhang, Infinitely many periodic solutions for second-order (p, q)-Laplacian differential systems, Nonlinear Anal., 74 (2011), 5215-5221
##[10]
X.-Y. Lin, X.-H. Tang , Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59-72
##[11]
J.-Q. Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372-385
##[12]
W.-D. Lu, Variational methods in differential equations, Science Press, Beijing, China (2002)
##[13]
M.-J. Ma, Z.-M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67 (2007), 1737-1745
##[14]
J. Mawhin, Periodic solutions of second order nonlinear difference systems with \(\phi\)-Laplacian: a variational approach, Nonlinear Anal., 75 (2012), 4672-4687
##[15]
J. Mawhin, Periodic solutions of second order Lagrangian difference systems with bounded or singular \(\phi\)-Laplacian and periodic potential, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1065-1076
##[16]
J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer- Verlag, New York (1989)
##[17]
D. Paşca, C.-L. Tang, Some existence results on periodic solutions of nonautonomous second-order differential systems with (q, p)-Laplacian, Appl. Math. Lett., 23 (2010), 246-251
##[18]
D. Paşca, C.-L. Tang, Some existence results on periodic solutions of ordinary (q, p)-Laplacian systems, J. Appl. Math. Inform., 29 (2011), 39-48
##[19]
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)
##[20]
B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal., 74 (2011), 7446-7454
##[21]
M. Schechter, Minimax systems and critical point theory, Birkhäuser Boston, Inc., Boston, MA (2009)
##[22]
C.-L. Tang, X.-P. Wu, Notes on periodic solutions of subquadratic second order systems, J. Math. Anal. Appl., 285 (2003), 8-16
##[23]
X.-H. Tang, X.-Y. Zhang, Periodic solutions for second-order discrete Hamiltonian systems, J. Difference Equ. Appl., 17 (2011), 1413-1430
##[24]
Y. Wang, X.-Y. Zhang, Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded (\(\phi_1,\phi_2\))-Laplacian, Adv. Difference Equ., 2014 (2014), 1-33
##[25]
Y.-F. Xue, C.-L. Tang, Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system, Nonlinear Anal., 67 (2007), 2072-2080
##[26]
X.-X. Yang, H.-B. Chen, Periodic solutions for autonomous (q, p)-Laplacian system with impulsive effects, J. Appl. Math., 2011 (2011), 1-19
##[27]
X.-Y. Zhang, Notes on periodic solutions for a nonlinear discrete system involving the p-Laplacian, Bull. Malays. Math. Sci. Soc., 37 (2014), 499-510
##[28]
X.-Y. Zhang, X.-H. Tang, Existence of solutions for a nonlinear discrete system involving the p-Laplacian, Appl. Math., 57 (2012), 11-30
##[29]
Q.-F. Zhang, X.-H. Tang, Q.-M. Zhang, Existence of periodic solutions for a class of discrete Hamiltonian systems, Discrete Dyn. Nat. Soc., 2011 (2011), 1-14
##[30]
X.-Y. Zhang, Y. Wang, Homoclinic solutions for a class of nonlinear difference systems with classical (\(\phi_1,\phi_2\))-Laplacian, Adv. Difference Equ., 2015 (2015), 1-24
##[31]
X.-Y. Zhang, L. Wang, Multiple periodic solutions for two classes of nonlinear difference systems involving classical (\(\phi_1,\phi_2\))-Laplacian, ArXiv, 2016 (2016), 1-20
]
Certain relations between Bessel and Whittaker functions related to some diagonal and block-diagonal \(3\times 3\)-matrices
Certain relations between Bessel and Whittaker functions related to some diagonal and block-diagonal \(3\times 3\)-matrices
en
en
The authors derive the matrix elements of the linear operators which appear under the representation of the group SO(2, 1)
and correspond to some diagonal or block-diagonal matrices belonging to the above group. Then, by applying these matrix
elements, that is, from a group theoretical point of view, the authors show how certain interesting integral and series representations
of the Whittaker function of the second kind and some formulas for the (basic and modified) Bessel functions can be
obtained. A special case of one of the results presented here is indicated to be also a special one of a known formula.
560
574
I. A.
Shilin
Department of Mathematics
Department of Energetics
Sholokhov Moscow State University for the Humanities
University of Economics and Energetics
Russia
Russia
ilyashilin@li.ru
J.
Choi
Department of Mathematics
Dongguk University
Republic of Korea
junesang@mail.dongguk.ac.kr
Whittaker function
Bessel functions
Macdonald function
semisimple Lie group SO(2، 1)
matrix elements of representation.
Whittaker function
Bessel functions
Macdonald function
semisimple Lie group SO(2، 1)
matrix elements of representation.
Article.20.pdf
[
[1]
I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger., Academic Press, Amsterdam (2007)
##[2]
A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev , Integrals and Series, Vol. 1: Elementary Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, New York (1986)
##[3]
I. A. Shilin, Double SO(2, 1)-integrals and formulas for Whittaker functions, Russian Math., 56 (2012), 47-56
##[4]
I. A. Shilin, J. S. Choi, Certain connections between the spherical and hyperbolic bases on the cone and formulas for related special functions, Integral Transforms Spec. Funct., 25 (2014), 374-383
##[5]
I. A. Shilin, J. Choi , Some connections between the spherical and parabolic bases on the cone expressed in terms of the Macdonald function, Abstr. Appl. Anal., 2014 (2014), 1-8
##[6]
I. A. Shilin, J. Choi, Transformations of bases related to the six-dimensional split orthogonal group and special functions, , (submitted), -
##[7]
H. M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers , Amsterdam (2012)
##[8]
N. J. Vilenkin, M. A. Sleinikova, Integral relations for the Whittakers functions and the representations of the threedimensional Lorentz group, Math. USSR Sb., 10 (1970), 173-180
]
Contraction principles in \(M_s\)-metric spaces
Contraction principles in \(M_s\)-metric spaces
en
en
In this paper, we give an interesting extension of the partial S-metric space which was introduced [N. Mlaiki, Univers. J.
Math. Math. Appl., 5 (2014), 109–119] to the \(M_s\)-metric space. Also, we prove the existence and uniqueness of a fixed point for
a self-mapping on an \(M_s\)-metric space under different contraction principles.
575
582
N.
Mlaiki
Department of Mathematical Sciences
Prince Sultan University
nmlaiki@psu.edu.sa
N.
Souayah
Department of Natural Sciences, Community College
King Saud University
nsouayah@ksu.edu.sa
K.
Abodayeh
Department of Mathematical Sciences
Prince Sultan University
kamal@psu.edu.sa
T.
Abdeljawad
Department of Mathematical Sciences
Prince Sultan University
tabdeljawad@psu.edu.sa
Functional analysis
\(M_s\)-metric space
fixed point.
Article.21.pdf
[
[1]
T. Abdeljawad, Fixed points for generalized weakly contractive mappings in partial metric spaces, Math. Comput. Modelling , 54 (2011), 2923-2927
##[2]
T. Abdeljawad, Meir-Keeler \(\alpha\)-contractive fixed and common fixed point theorems, Fixed Point Theory Appl., 2013 (2013 ), 1-10
##[3]
T. Abdeljawad, H. Aydi, E. Karapınar , Coupled fixed points for Meir-Keeler contractions in ordered partial metric spaces, Math. Probl. Eng., 2012 (2012), 1-20
##[4]
T. Abdeljawad, E. Karapınar, K. Taş, A generalized contraction principle with control functions on partial metric spaces, Comput. Math. Appl., 63 (2012), 716-719
##[5]
I. Altun, A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory Appl., 2011 (2011 ), 1-10
##[6]
I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl., 157 (2010), 2778-2785
##[7]
M. Asadi, E. Karapınar, P. Salimi, New extension of p-metric spaces with some fixed-point results on M-metric spaces, J. Inequal. Appl., 2014 (2014), 1-9
##[8]
S. G. Matthews, Partial metric topology , Papers on general topology and applications, Flushing, NY, (1992), 183–197, Ann. New York Acad. Sci., 728 (1994), -
##[9]
N. Mlaiki, A contraction principle in partial S-metric spaces, Univers. J. Math. Math. Appl., 5 (2014), 109-119
##[10]
N. Mlaiki, Common fixed points in complex S-metric space, Adv. Fixed Point Theory, 4 (2014), 509-524
##[11]
N. Mlaiki , \(\alpha-\psi\)-contractive mapping on S-metric space, Math. Sci. Lett., 4 (2015), 9-12
##[12]
N. Mlaiki, A. Zarrad, N. Souayah, A. Mukheimer, T. Abdeljawad, Fixed point theorems in \(M_b\)-metric spaces, J. Math. Anal., 7 (2016), 1-9
##[13]
A. Shoaib, M. Arshad, J. Ahmad, Fixed point results of locally contractive mappings in ordered quasi-partial metric spaces, Sci. World J., 2013 (2013), 1-8
##[14]
S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math., 11 (2014), 703-711
##[15]
N. Souayah, A fixed point in partial \(S_b\)-metric spaces, An. St. Univ. Ovidius Constanţa, 24 (2016), 351-362
##[16]
N. Souayah, N. Mlaiki, A fixed point in \(S_b\)-metric spaces, J. Math. Comput. Sci., 16 (2016), 131-139
##[17]
O. Valero, On Banach fixed point theorems for partial metric spaces, Appl. Gen. Topol., 6 (2005), 229-240
]
On some inequalities for generalized s-convex functions and applications on fractal sets
On some inequalities for generalized s-convex functions and applications on fractal sets
en
en
The authors present some new inequalities of generalized Hermite-Hadamard’s type for the class of functions whose second
local fractional derivatives of order \(\alpha\) in absolute value at certain powers are generalized s-convex functions in the second sense.
Moreover, some applications are given.
583
594
Adem
Kilicman
Department of Mathematics and Institute for Mathematical Research
University Putra Malaysia
Malaysia
akilic@upm.edu.my
Wedad
Saleh
Department of Mathematics
Putra University of Malaysia (UPM)
Malaysia
wed_10_777@hotmail.com
s-convex functions
fractal space
local fractional derivative.
Article.22.pdf
[
[1]
A. Atangana, S. B. Belhaouari , Solving partial differential equation with space- and time-fractional derivatives via homotopy decomposition method, Math. Probl. Eng., 2013 (2013 ), 1-9
##[2]
A. Atangana, E. F. Doungmo-Goufo, Solution of diffusion equation with local derivative with new parameter, Therm. Sci., 19 (2015), 231-238
##[3]
D. Baleanu, H. M. Srivastava, X.-J. Yang, Local fractional variational iteration algorithms for the parabolic Fokker-Planck equation defined on Cantor sets, Prog. Fract. Differ. Appl., 1 (2015), 1-11
##[4]
S. S. Dragomir, On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., 5 (2001), 775-788
##[5]
S. S. Dragomir, S. Fitzpatrick, The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math., 32 (1999), 687-696
##[6]
M. Grinblatt, J. T. Linnainmaa, Jensen’s inequality, parameter uncertainty, and multi-period investment, Rev. Asset Pric. Stud., 1 (2011), 1-34
##[7]
L. Hörmander, Notions of convexity, Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA (1994)
##[8]
J. Hua, B.-Y. Xi, F. Qi, Inequalities of Hermite-Hadamard type involving an s-convex function with applications, Appl. Math. Comput., 246 (2014), 752-760
##[9]
A. Kılıçman, W. Saleh, Notions of generalized s-convex functions on fractal sets, J. Inequal. Appl., 2015 (2015 ), 1-16
##[10]
A. Kılıçman, W. Saleh, Some generalized Hermite-Hadamard type integral inequalities for generalized s-convex functions on fractal sets, Adv. Difference Equ., 2015 (2015 ), 1-15
##[11]
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)
##[12]
H.-X. Mo, X. Sui, Generalized s-convex functions on fractal sets, Abstr. Appl. Anal., 2014 (2014 ), 1-8
##[13]
H.-X. Mo, X. Sui, Hermite-Hadamard type inequalities for generalized s-convex functions on real linear fractal set \(R^\alpha(0 < 1)\), ArXiv, 2015 (2015 ), 1-10
##[14]
H.-X. Mo, X. Sui, D.-Y. Yu, Generalized convex functions on fractal sets and two related inequalities, Abstr. Appl. Anal., 2014 (2014 ), 1-7
##[15]
M. E. Özdemir, Ç . Yıldız, A. O. Akdemir, E. Set, On some inequalities for s-convex functions and applications, J. Inequal. Appl., 2013 (2013 ), 1-11
##[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
##[17]
J. J. Ruel, M. P. Ayres, Jensen’s inequality predicts effects of environmental variation, Trends Ecol. Evol., 14 (1999), 361-366
##[18]
M. Z. Sarikaya, T. Tunc, H. Budak, On generalized some integral inequalities for local fractional integrals, Appl. Math. Comput., 276 (2016), 316-323
##[19]
X.-J. Yang, Local fractional integral transforms, Prog. Nonlinear Sci., 4 (2011), 1-225
##[20]
X.-J. Yang, Advanced local fractional calculus and its applications, World Science Publ., New York (2012)
##[21]
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
##[22]
A.-M. Yang, X.-J. Yang, Z.-B. Li, Local fractional series expansion method for solving wave and diffusion equations on Cantor sets, Abstr. Appl. Anal., 2013 (2013 ), 1-5
]
Approximate controllability of impulsive Hilfer fractional differential inclusions
Approximate controllability of impulsive Hilfer fractional differential inclusions
en
en
In this paper, firstly by utilizing the theory of operators semigroup, probability density functions via impulsive conditions,
we establish a new \(PC_{1-\nu}\)-mild solution for impulsive Hilfer fractional differential inclusions. Secondly we prove the existence
of mild solutions for the impulsive Hilfer fractional differential inclusions by using fractional calculus, multi-valued analysis and
the fixed-point technique. Then under some assumptions, the approximate controllability of associated system are formulated
and proved. An example is provided to illustrate the application of the obtained theory
595
611
Jun
Du
School of Mathematical Sciences
Department of Applied Mathematics
Anhui University
Huainan Normal University
P. R. China
P. R. China
djwlm@163.com
Wei
Jiang
School of Mathematical Sciences
Anhui University
P. R. China
jiangwei89018@126.com
Azmat Ullah Khan
Niazi
School of Mathematical Sciences
Anhui University
P. R. China
Approximate controllability
impulsive system
Hilfer fractional differential inclusions
multivalued maps
fixed point theorem
semigroup theory.
Article.23.pdf
[
[1]
N. Abada, M. Benchohra, H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations, 246 (2009), 3834-3863
##[2]
T. Abdeljawad , On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66
##[3]
R. P. Agarwal, Y. Zhou, Y.-Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095-1100
##[4]
D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus. Models and numerical methods, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2012)
##[5]
A. E. Bashirov, N. I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 37 (1999), 1808-1821
##[6]
M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York (2006)
##[7]
A. Debbouche, D. F. M. Torres, Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions, Appl. Math. Comput., 243 (2014), 161-175
##[8]
K. Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin (1992)
##[9]
K. Diethelm, The analysis of fractional differential equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, Springer-Verlag, Berlin (2010)
##[10]
Z.-B. Fan, G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709-1727
##[11]
K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616-1626
##[12]
H.-B. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344-354
##[13]
R. Hilfer, Fractional calculus and regular variation in thermodynamics, Applications of fractional calculus in physics, World Sci. Publ., River Edge, NJ (2000)
##[14]
R. Hilfer, Y. Luchko, Z . Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299-318
##[15]
M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces , De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin (2001)
##[16]
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)
##[17]
V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems, Cambridge Scientific Publ., Cambridge, UK (2009)
##[18]
Z.-H. Liu, X.-W. Li, On the controllability of impulsive fractional evolution inclusions in Banach spaces, J. Optim. Theory Appl., 156 (2013), 167-182
##[19]
Z.-H. Liu, X.-W. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920-1933
##[20]
J. A. Machado, C. Ravichandran, M. Rivero, J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory Appl., 2013 (2013 ), 1-16
##[21]
N. I. Mahmudov, M. A. Mckibben, On the approximate controllability of fractional evolution equations with generalized Riemann-Liouville fractional derivative, J. Funct. Spaces, 2015 (2015 ), 1-9
##[22]
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)
##[23]
J. J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680-690
##[24]
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)
##[25]
K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces, Nonlinear Anal., 75 (2012), 2701-2712
##[26]
R. Sakthivel, R. Ganesh, S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput., 225 (2013), 708-717
##[27]
X.-B. Shu, Y.-J. Shi , A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465-476
##[28]
V. E. Tarasov, Fractional dynamics, Applications of fractional calculus to dynamics of particles, fields and media, Nonlinear Physical Science, Springer, Heidelberg; Higher Education Press, Beijing (2010)
##[29]
J.-R. Wang, M. Fečkan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8 (2011), 345-361
##[30]
C. Xiao, B. Zeng, Z.-H. Liu, Feedback control for fractional impulsive evolution systems, Appl. Math. Comput., 268 (2015), 924-936
##[31]
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)
##[32]
M. Yang, Q.-R. Wang, Approximate controllability of Riemann-Liouville fractional differential inclusions, Appl. Math. Comput., 274 (2016), 267-281
##[33]
Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077
##[34]
Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl., 11 (2010), 4465-4475
##[35]
Y. Zhou, F. Jiao, J. Li , Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal., 71 (2009), 3249-3256
]
Topological degree and applications to elliptic problems with discontinuous nonlinearity
Topological degree and applications to elliptic problems with discontinuous nonlinearity
en
en
We develop a topological degree theory for a class of locally bounded weakly upper semicontinuous set-valued operators of
generalized (\(S_+\)) type in real reflexive separable Banach spaces, based on the Berkovits-Tienari degree. The method of approach
is to use elliptic super-regularization by means of certain compact embeddings, instead of the Galerkin method. Applying the
degree theory, we tackle an elliptic boundary value problem with discontinuous nonlinearity.
612
624
In-Sook
Kim
Department of Mathematics
Sungkyunkwan University
Republic of Korea
iskim@skku.edu
Set-valued operators of (\(S_+\)) type
degree theory
p-Laplacian.
Article.24.pdf
[
[1]
J. Berkovits, On the degree theory for nonlinear mappings of monotone type, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 58 (1986), 1-58
##[2]
J. Berkovits, Extension of the Leray-Schauder degree for abstract Hammerstein type mappings, J. Differential Equations, 234 (2007), 289-310
##[3]
J. Berkovits, M. Tienari, Topological degree theory for some classes of multis with applications to hyperbolic and elliptic problems involving discontinuous nonlinearities, Dynam. Systems Appl., 5 (1996), 1-18
##[4]
F. E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 1-39
##[5]
F. E. Browder, B. A. Ton, Nonlinear functional equations in Banach spaces and elliptic super-regularization, Math. Z., 105 (1968), 177-195
##[6]
K.-C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math., 33 (1980), 117-146
##[7]
A. Granas, Sur la notion du degré topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, (French) Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 7 (1959), 191-194
##[8]
I.-S. Kim, J.-H. Bae, Elliptic boundary value problems with discontinuous nonlinearities, J. Nonlinear Convex Anal., 17 (2016), 27-38
##[9]
I.-S. Kim, S.-J. Hong, A topological degree for operators of generalized (\(s_+\)) type, Fixed Point Theory Appl., 2015 (2015 ), 1-16
##[10]
J. Leray, J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. Ec. Norm., 51 (1934), 45-78
##[11]
T.-W. Ma, Topological degrees of set-valued compact fields in locally convex spaces, Dissertationes Math. Rozprawy Mat., 92 (1972), 1-43
##[12]
D. O’Regan, Y. J. Cho, Y.-Q. Chen, Topological degree theory and applications, Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, FL (2006)
##[13]
W. Rudin, Functional analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw- Hill, Inc., New York (1991)
##[14]
I. V. Skrypnik, Nonlinear elliptic equations of higher order, (Russian) Gamoqeneb. Math. Inst. Sem. Mosen. Anotacie., 7 (1973), 51-52
##[15]
I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Translated from the 1990 Russian original by Dan D. Pascali, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI (1994)
##[16]
E. Zeidler, Nonlinear functional analysis and its applications, I, Fixed-point theorems, Translated from the German by Peter R. Wadsack, Springer-Verlag, New York (1985)
##[17]
E. Zeidler, Nonlinear functional analysis and its applications, II/B, Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron, Springer-Verlag, New York (1990)
]
Existence of solutions for fractional Schrödinger equation with asymptotically periodic terms
Existence of solutions for fractional Schrödinger equation with asymptotically periodic terms
en
en
In this paper, we investigate the following nonlinear fractional Schr¨odinger equation
\[(-\Delta)^su + V(x)u = f(x, u), x \in \mathbb{R}^N,\]
where \(s \in (0, 1),N > 2\) and \((-\Delta)^s\) is fractional Laplacian operator. We prove that the problem has a non-trivial solution under
asymptotically periodic case of \(V\) and \(f\) at infinity. Moreover, the nonlinear term \(f\) does not satisfy any monotone condition and
Ambrosetti-Rabinowitz condition.
625
636
Da-Bin
Wang
Department of Applied Mathematics
Lanzhou University of Technology
People’s Republic of China
wangdb96@163.com
Man
Guo
Department of Applied Mathematics
Lanzhou University of Technology
People’s Republic of China
guoman615@163.com
Wen
Guan
Department of Applied Mathematics
Lanzhou University of Technology
People’s Republic of China
mathguanw@163.com
asymptotically periodic
Fractional Schrödinger equation
variational method.
Article.25.pdf
[
[1]
C. O. Alves, M. A. S. Souto, S. H. M. Soares, Schrödinger -Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 377 (2011), 584-592
##[2]
A. Ambrosetti, M. Badiale, S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300
##[3]
B. Barrios, E. Colorado, A. de Pablo, U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2011), 6133-6162
##[4]
H. Berestycki, P.-L. Lions, Nonlinear scalar field equations, II, Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375
##[5]
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (1996)
##[6]
G. M. Bisci, V. D. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008
##[7]
G. M. Bisci, V. D. Rădulescu, R. Servadei, Variational methods for nonlocal fractional problems, With a foreword by Jean Mawhin,/ Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (2016)
##[8]
X. Cabré, J.-G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093
##[9]
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260
##[10]
X. Chang, Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity , Nonlinearity, 26 (2013), 479-494
##[11]
M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 54 (2012), 1-7
##[12]
R. de Marchi, Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A , 145 (2015), 745-757
##[13]
M. del Pino, P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32
##[14]
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573
##[15]
Y.-H. Ding, F.-H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30 (2007), 231-249
##[16]
S. Dipierro, G. Palatucci, E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68 (2013), 201-216
##[17]
P. Felmer, A. Quaas, J.-G. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262
##[18]
J. Giacomoni, P. K. Mishra, K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal., 5 (2016), 57-74
##[19]
L. Jeanjean, K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21 (2004), 287-318
##[20]
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305
##[21]
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 1-7
##[22]
L. Li, V. Rădulescu, D. Repovš, Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness, Int. J. Nonlinear Sci. Numer. Simul., 17 (2016), 325-333
##[23]
G.-B. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776
##[24]
H. F. Lins, E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905
##[25]
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)
##[26]
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291
##[27]
M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., 44 (1991), 491-502
##[28]
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\), J. Math. Phys., 54 (2013), 1-17
##[29]
S. Secchi, On fractional Schrödinger equations in \(\mathbb{R}^N\)without the Ambrosetti-Rabinowitz condition, Topol. Methods Nonlinear Anal., 47 (2016), 19-41
##[30]
X.-D. Shang, J.-H. Zhang , Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207
##[31]
E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth , Calc. Var. Partial Differential Equations, 39 (2010), 1-33
##[32]
E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949
##[33]
A. Szulkin, T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, Int. Press, Somerville, MA, (2010), 597-632
##[34]
J.-G. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41
##[35]
K.-M. Teng, Multiple solutions for a class of fractional Schrödinger equations in \(\mathbb{R}^N\), Nonlinear Anal. Real World Appl., 21 (2015), 76-86
##[36]
H. Weitzner, G. M. Zaslavsky, Some applications of fractional equations , Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281
##[37]
M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA (1996)
##[38]
H. Zhang, J.-X. Xu, F.-B. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in \(\mathbb{R}^N\), J. Math. Phys., 56 (2015), 1-13
##[39]
X. Zhang, B.-L. Zhang, D. Repovš, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48-68
##[40]
X. Zhang, B.-L. Zhang, M.-Q. Xiang, Ground states for fractional Schrödinger equations involving a critical nonlinearity, Adv. Nonlinear Anal., 5 (2016), 293-314
]
Projection and contraction methods for constrained convex minimization problem and the zero points of maximal monotone operator
Projection and contraction methods for constrained convex minimization problem and the zero points of maximal monotone operator
en
en
In this paper, we introduce a new iterative scheme for the constrained convex minimization problem and the set of zero
points of the maximal monotone operator problem, based on the projection and contraction methods. The core idea is to build
the corresponding iterative algorithms by constructing reasonable error metric function and profitable direction to assure that
the distance form the iteration points generated by the algorithms to a point of the solution set is strictly monotone decreasing.
Under suitable conditions, new convergence theorems are obtained, which are useful in nonlinear analysis and optimization.
The main advantages of the method presented are its simplicity, robustness, and ability to handle large problems with any
start point. As an application, we apply our algorithm to solve the equilibrium problem, the constrained convex minimization
problem and the split feasibility problem, the split equality problem in Hilbert spaces.
637
646
Yujing
Wu
Tianjin Vocational Institute
P. R. China
xiaomi20062008@163.com
Luoyi
Shi
Department of Mathematics
Tianjin Polytechnic University
P. R. China
shiluoyi@tjpu.edu.cn
Fixed point
constrained convex minimization
maximal monotone operator
resolvent
variational inequality
split equality problem.
Article.26.pdf
[
[1]
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453
##[2]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120
##[3]
X.-J. Cai, G.-Y. Gu, B.-S. He, On the \(O(1/t)\) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators, Comput. Optim. Appl., 57 (2014), 339-363
##[4]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Extragradient-projection method for solving constrained convex minimization problems, Numer. Algebra Control Optim., 1 (2011), 341-359
##[5]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239
##[6]
P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136
##[7]
Q. L. Dong, J. Yang, H. B. Yuan, The projection and contraction algorithm for solving variational inequality problems in Hilbert spaces, , (to appear in J. Nonlinear Convex Anal. ), -
##[8]
J. Eckstein, D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318
##[9]
J. Eckstein, B. F. Svaiter, A family of projective splitting methods for the sum of two maximal monotone operators, Math. Program., 111 (2008), 1173-1199
##[10]
H. W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1996)
##[11]
S. D. Flåm, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming, 78 (1997), 29-41
##[12]
K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
##[13]
K. Geobel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1984)
##[14]
B.-S. He, A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim., 35 (1997), 69-76
##[15]
P.-L. Lions, Une méthode itérative de résolution d’une inéquation variationnelle, (French) Israel J. Math., 31 (1978), 204-208
##[16]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121
##[17]
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
##[18]
X. L. Qin, Y. J. Cho, S. M. Kang, Convergence analysis on hybrid projection algorithms for equilibrium problems and variational inequality problems, Math. Model. Anal., 14 (2009), 335-351
##[19]
B. Qu, N.-H. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665
##[20]
F. Schöpfer, T. Schuster, A. K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems, 24 (2008), 1-20
##[21]
L. Y. Shi, R.-D. Chen, Y.-J. Wu, Iterative algorithms for finding the zeroes of sums of operators, J. Inequal. Appl., 2014 (2014 ), 1-16
##[22]
L. Y. Shi, R.-D. Chen, Y.-J.Wu, Strong convergence of iterative algorithms for the split equality problem, J. Inequal. Appl., 2014 (2014), 1-19
##[23]
D. Sun, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl., 91 (1996), 123-140
##[24]
W. Takahashi, Totsu kaiseki to fudoten kinji, (Japanese) [[Convex analysis & approximation of fixed points]] Sūrikaiseki Shiriizu [Mathematical Analysis Series], Yokohama Publishers, Yokohama (2000)
##[25]
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
##[26]
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446
##[27]
H.-K. Xu, A variable Krasnoselski-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034
##[28]
H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 105-128
##[29]
H.-K. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., 150 (2011), 360-378
##[30]
Q.-Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266
##[31]
Q.-Z. Yang, J.-L. Zhao, Generalized KM theorems and their applications, Inverse Problems, 22 (2006), 833-844
]
Convergence analysis of a novel iteration algorithm for solving split feasibility problems
Convergence analysis of a novel iteration algorithm for solving split feasibility problems
en
en
In this paper, our aim is to construct a convergence theorem in Banach spaces via the following Ishikawa recursive algorithm
\[
\begin{cases}
x_{n+1}=(1-\alpha_n)x_n+\alpha_nT_ny_n,\\
y_n=(1-\beta_n)x_n+\beta_nT_nx_n,
\end{cases}
\]
where \(\{\alpha_n\}\), \(\{\beta_n\}\) are sequences in \([0, 1]\) and \(\{T_n\}\) is a sequence of nonexpansive mappings. Moreover, we also apply these results
to solve a split feasibility problem.
647
655
Qinwei
Fan
School of Science
Xi’an Polytechnic University
China
qinweifan@126.com
Split feasibility problem
fixed point
nonexpansive mapping
weak convergence.
Article.27.pdf
[
[1]
H. Attouchi, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984)
##[2]
R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32 (1979), 107-116
##[3]
C. Byren, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453
##[4]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2014), 103-120
##[5]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239
##[6]
S.-S. Chang, J. K. Kim, Y. J. Cho, J. Y. Sim, Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014 ), 1-12
##[7]
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
##[8]
S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013 ), 1-14
##[9]
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
##[10]
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
##[11]
D. V. Hieu, L. D. Muu, P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms, 73 (2016), 197-217
##[12]
S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150
##[13]
J. K. Kim, S. Y. Cho, X.-L. Qin, Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 2041-2057
##[14]
J. K. Kim, Salahuddin, A system of nonconvex variational inequalities in Banach spaces, Commun. Optim. Theory, 2016 (2016), 1-19
##[15]
M. A. Krasnosel’skii, Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.), 10 (1955), 123-127
##[16]
S.-T. Lv, Some results on a two-step iterative algorithm in Hilbert spaces, J. Nonlinear Funct. Anal., 2015 (2015 ), 1-10
##[17]
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510
##[18]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121
##[19]
X.-L. Qin, S.-S. Chang, Y. J. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl., 11 (2010), 2963-2972
##[20]
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
##[21]
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
##[22]
B. Qu, B.-H. Liu, N. Zheng, On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218-223
##[23]
S. Rathee, Ritika, \(\delta\)-convergence theorems for Mann and Ishikawa iteration procedures with errors in CAT(0) spaces, Commun. Optim. Theory, 2013 (2013 ), 1-11
##[24]
S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276
##[25]
H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138
##[26]
H.-K. Xu, A variable Krasnoselskiı-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034
##[27]
Q.-Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266
##[28]
Y.-H. Yao, R.-D. Chen, H.-Y. Zhou , Iterative process for certain nonlinear mappings in uniformly smooth Banach spaces, Nonlinear Funct. Anal. Appl., 10 (2005), 651-664
##[29]
C.-J. Zhang, J.-L. Li, B.-Q. Liu, Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Comput. Math. Appl., 61 (2011), 262-276
##[30]
J.-L. Zhao, Q.-Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799
##[31]
F. Zhao, L. Yang, Hybrid projection methods for equilibrium problems and fixed point problems of infinite family of multivalued asymptotically nonexpansive mappings, J. Nonlinear Funct. Anal., 2016 (2016 ), 1-13
]
Time effect on the dynamical behavior of a life energy system dynamic model
Time effect on the dynamical behavior of a life energy system dynamic model
en
en
This article is concerned with a life energy system dynamic model with two different delays. A set of sufficient criteria
which ensures the local stability and the existence of Hopf bifurcation for the model is derived. Some explicit formulas which
determine the nature of Hopf bifurcations are obtained by means of the normal form theory and center manifold theorem. Our
analytical findings are supported by numerical experiments. Finally, a brief conclusion is included.
656
670
Changjin
Xu
Guizhou Key Laboratory of Economics System Simulation
Guizhou University of Finance and Economics
P. R. China
xcj403@126.com
Peiluan
Li
School of Mathematics and Statistics
Henan University of Science and Technology
P. R. China
lpllpl_lpl@163.com
Life energy system model (LESM)
delay
stability
Hopf bifurcation.
Article.28.pdf
[
[1]
J.-D. Cao, M. Xiao, Stability and Hopf bifurcation in a simplified BAM neural network with two time delays, IEEE Trans. Neural Netw., 18 (2007), 416-430
##[2]
Z.-S. Cheng, J.-D. Cao, Hybrid control of Hopf bifurcation in complex networks with delays, Neurocomputing, 131 (2014), 164-170
##[3]
J. Hale, Theory of functional differential equations, Second edition. Applied Mathematical Sciences, Springer-Verlag, New York-Heidelberg (1977)
##[4]
B. D. Hassard, N. D. Kazarinoff, Y.-H. Wan, Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge-New York (1981)
##[5]
J.-Q. Hu, J.-D. Cao, T. Hayat, Stability and Hopf bifurcation analysis for an energy resource system, Nonlinear Dynam., 78 (2014), 219-234
##[6]
X. Huang, How do simple energy activities comprise complex behavior of life systems?: A conceptual synthesis and decomposition of the energy structure of life systems, Ecol. Model., 165 (2003), 79-89
##[7]
X. Huang, On the energy trait control phenomena in ecosystems, Ecol. Model., 211 (2008), 36-46
##[8]
X. Huang, Y.-G. Zu, The LES population model: essentials and relationship to the Lotka-Volterra model, Ecol. Model., 143 (2001), 215-225
##[9]
S. E. Jøgensen, Use of models as experimental tool to show that structural changes are accompanied by increased energy, Ecol. Model., 41 (1988), 117-126
##[10]
S. A. L. M. Kooijman, Dynamic energy and mass budgets in biological systems, Cambridge Univ. Press, Cambridge (2000)
##[11]
S. A. L. M. Kooijman, T. Andersen, B. W. Kooi, Dynamic energy budget representations of stoichiometric constraints on population dynamics, Ecol., 85 (2004), 1230-1243
##[12]
S. A. L. M. Kooijman, B. W. Kooi, T. G. Hallam, The application of mass and energy conservation laws in physiologically structured population models of heterotrophic organisms, J. Theor. Biol., 197 (1999), 371-392
##[13]
S. A. L. M. Kooijman, T. A. Troost, Quantitative steps in the evolution of metabolic organisation as specified by the dynamic energy budget theory, Biol. Rev., 82 (2007), 113-142
##[14]
Y. Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA (1993)
##[15]
M.-X. Liao, C.-J. Xu, X.-H. Tang, Stability and Hopf bifurcation for a competition and cooperation model of two enterprises with delay, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3845-3856
##[16]
X.-H. Lin, H. Wang, Stability analysis of delay differential equations with two discrete delays, Can. Appl. Math. Q., 20 (2012), 519-533
##[17]
R. L. Lindeman , Seasonal food-cycle dynamics in a senescent lake, Am. Midl. Nat., 26 (1941), 636-673
##[18]
R. L. Lindeman, The trophicdynamic aspect of ecology , Ecol., 23 (1942), 399-417
##[19]
E. B. Muller, S. A. L. M. Kooijman, P. J. Edmunds, F. J. Doyle, R. M. Nisbe, Dynamic energy budgets in syntrophic symbiotic relationships between heterotrophic hosts and photoautotrophic symbionts, J. Theor. Biol., 259 (2009), 44-57
##[20]
H .T. Odum, Systems ecology, John Wiley & Sons, New York, NY (1983)
##[21]
S.-G. Ruan, J.-J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874
##[22]
M. Xiao, J.-D. Cao, W.-X. Zheng, Bifurcation with regard to combined interaction parameter in a life energy system dynamic model of two components with multiple delays, J. Franklin Inst., 348 (2011), 2647-2669
##[23]
W.-Y. Xu, J.-D. Cao, M. Xiao, Bifurcation analysis and control in exponential RED algorithm, Neurocomputing, 129 (2014), 232-245
##[24]
W.-Y. Xu, J.-D. Cao, M. Xiao, Bifurcation analysis of a class of (n + 1)-dimension internet congestion control systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1-17
##[25]
W.-Y. Xu, T. Hayat, J.-D. Cao, M. Xiao, Hopf bifurcation control for a fluid flow model of internet congestion control systems via state feedback, IMA J. Math. Control Inform., 33 (2016), 69-93
##[26]
C.-J. Xu, Y.-S. Wu, Bifurcation and control of chaos in a chemical system, Appl. Math. Model., 39 (2015), 2295-2310
##[27]
C.-R. Zhang, Y.-G. Zu, B.-D. Zheng, Stability analysis in a two-dimensional life energy system model with delay, Ecol. Model., 193 (2006), 691-702
]
Coupled fixed point results for (\(\varphi,G\))-contractions of type (b) in b-metric spaces endowed with a graph
Coupled fixed point results for (\(\varphi,G\))-contractions of type (b) in b-metric spaces endowed with a graph
en
en
The purpose of this paper is to present some existence results for coupled fixed points of generalized contraction type
operators in b-metric spaces endowed with a directed graph. Our results generalize the results obtained by Gnana Bhaskar
and Lakshmikantham in [T. Gnana Bhaskar, V. Lakshmikantham, Nonlinear Anal., 65 (2006), 1379–1393]. Data dependence,
well-posednes and Ulam-Hyres stability of the fixed point problem are also studied.
671
683
Cristian
Chifu
Faculty of Business
Babeş-Bolyai University Cluj-Napoca
Romania
Cristian.Chifu@tbs.ubbcluj.ro
Gabriela
Petrusel
Faculty of Business
Babeş-Bolyai University Cluj-Napoca
Romania
Gabi.Petrusel@tbs.ubbcluj.ro
Fixed point
coupled fixed point
b-metric space
connected graph.
Article.29.pdf
[
[1]
I. Beg, A. R. Butt, S. Radojević, The contraction principle for set valued mappings on a metric space with a graph, Comput. Math. Appl., 60 (2010), 1214-1219
##[2]
V. Berinde, Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, ”Babe-Bolyai” Univ., Cluj-Napoca, (1993), 3-9
##[3]
V. Berinde, Sequences of operators and fixed points in quasimetric spaces, Studia Univ. Babeş-Bolyai Math., 41 (1996), 23-27
##[4]
V. Berinde, Contracţii generalizate şi aplicaţii, (Romanian) [[Generalized contractions and applications]] Colecia Universitaria (Baia Mare) [University Collection], Editura Cub Press 22, Baia Mare (1997)
##[5]
I. C. Chifu, G. Petruşel, Generalized contractions in metric spaces endowed with a graph, Fixed Point Theory Appl., 2012 (2012 ), 1-9
##[6]
I. C. Chifu, G. Petruşel, New results on coupled fixed point theory in metric spaces endowed with a directed graph, Fixed Point Theory Appl., 2014 (2014 ), 1-13
##[7]
S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263-276
##[8]
T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393
##[9]
G. Gwóźdź-Łukawska, J. Jachymski, IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J. Math. Anal. Appl., 356 (2009), 453-463
##[10]
J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. , 71 (2009), 3403-3410
##[11]
J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359-1373
##[12]
J. Jachymski, J. Klima, Around Perov’s fixed point theorem for mappings on generalized metric spaces, Fixed Point Theory, 17 (2016), 367-380
##[13]
M. Jleli, B. Samet, C. Vetro, F. Vetro, Fixed points for multivalued mappings in b-metric spaces, Abstr. Appl. Anal., 2015 (2015 ), 1-7
##[14]
J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239
##[15]
J. J. Nieto, R. L. Pouso, R. Rodríguez-López, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc., 135 (2007), 2505-2517
##[16]
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
##[17]
D. O’Regan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl., 341 (2008), 1241-1252
##[18]
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
##[19]
C. Vetro, F. Vetro, Metric or partial metric spaces endowed with a finite number of graphs: a tool to obtain fixed point results, Topology Appl., 164 (2014), 125-137
]
Formal balls in fuzzy quasi-metric spaces
Formal balls in fuzzy quasi-metric spaces
en
en
The notions of Yoneda completeness and Smyth completeness on fuzzy quasi-metric spaces are introduced and their
relationship with other types of completeness including sequentially Yoneda completeness and bicompleteness are investigated.
Then we use the standard Yoneda completeness to characterize the order-theoretical properties of the poset \((BX,\sqsubseteq_M )\) of formal
balls in a fuzzy quasi-metric space \((X,M,\wedge)\). The results show that if \((BX,\sqsubseteq_M )\) is a dcpo, then \((X,M,\wedge)\) is standard complete
and conversely, \((BX,\sqsubseteq_M )\) forms a dcpo provided that \((X,M,\wedge)\) is standard Yoneda complete. Particularly, in a fuzzy metric
space, we clarify three types of completeness which can be characterized by the directed completeness of the related poset of
formal balls.
684
698
You
Gao
College of Mathematics and Econometrics
Hunan University
China
gaoyoumath@126.com
Qingguo
Li
College of Mathematics and Econometrics
Hunan University
China
liqingguoli@aliyun.com
Lankun
Guo
College of Mathematics and Computer Science
Hunan Normal University
China
Jialiang
Xie
College of Science
Jimei University
China
Fuzzy quasi-metric space
Yoneda complete
standard Yoneda complete
Smyth complete
formal ball.
Article.30.pdf
[
[1]
M. Ali-Akbari, B. Honari, M. Pourmahdian, M. M. Rezaii, The space of formal balls and models of quasi-metric spaces, Math. Structures Comput. Sci., 19 (2009), 337-355
##[2]
F. Castro-Company, S. Romaguera, P. Tirado, The bicompletion of fuzzy quasi-metric spaces, Fuzzy Sets and Systems, 166 (2011), 56-64
##[3]
A. George, P. Veeramani , On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399
##[4]
J. Goubault-Larrecq, Non-Hausdorff topology and domain theory, [On the cover: Selected topics in point-set topology] New Mathematical Monographs, Cambridge University Press, Cambridge (2013)
##[5]
V. Gregori, J. A. Mascarell, A. Sapena, On completion of fuzzy quasi-metric spaces, Topology Appl., 153 (2005), 886-899
##[6]
V. Gregori, S. Morillas, B. Roig, Fuzzy quasi-metrics for the Sorgenfrey line, Fuzzy Sets and Systems, 222 (2013), 98-107
##[7]
V. Gregori, S. Morillas, A. Sapena, Examples of fuzzy metrics and applications, Fuzzy Sets and Systems, 170 (2011), 95-111
##[8]
V. Gregori, S. Romaguera, Fuzzy quasi-metric spaces, Appl. Gen. Topol., 5 (2004), 129-136
##[9]
V. Gregori, S. Romaguera, A. Sapena, A characterization of bicompletable fuzzy quasi-metric spaces, Fuzzy Sets and Systems, 152 (2005), 395-402
##[10]
J. Gutiérrez García, M. A. de Prada Vicente, Hutton [0, 1]-quasi-uniformities induced by fuzzy (quasi)-metric spaces, Fuzzy Sets and Systems, 157 (2006), 755-766
##[11]
M. Kostanek, P. Waszkiewicz , The formal ball model for Q-categories , Math. Structures Comput. Sci., 21 (2011), 41-64
##[12]
I. Kramosil, J. Michálek, Fuzzy metrics and statistical metric spaces , Kybernetika (Prague), 11 (1975), 336-344
##[13]
I. Mardones-Pérez, M. A. de Prada Vicente, Fuzzy pseudometric spaces vs fuzzifying structures, Fuzzy Sets and Systems, 267 (2015), 117-132
##[14]
I. Mardones-Pérez, M. A. de Prada Vicente, An application of a representation theorem for fuzzy metrics to domain theory, Fuzzy Sets and Systems, 300 (2016), 72-83
##[15]
D. Mihet, Fuzzy quasi-metric versions of a theorem of Gregori and Sapena, Iran. J. Fuzzy Syst., 7 (2010), 59-64
##[16]
J. Miñana, A. Šostak, Fuzzifying topology induced by a strong fuzzy metric, Fuzzy Sets and Systems, 300 (2016), 24-39
##[17]
D. Qiu, W.-Q. Zhang, C. Li, Extension of a class of decomposable measures using fuzzy pseudometrics, Fuzzy Sets and Systems, 222 (2013), 33-44
##[18]
D. Qiu, W.-Q. Zhang, C. Li, On decomposable measures constructed by using stationary fuzzy pseudo-ultrametrics, Int. J. Gen. Syst., 42 (2013), 395-404
##[19]
L. A. Ricarte, Topological and computational models for fuzzy metric spaces via domain theory, Ph.D. thesis, University Politècnica de València (2013)
##[20]
L. A. Ricarte, S. Romaguera , A domain-theoretic approach to fuzzy metric spaces, Topology Appl., 163 (2014), 149-159
##[21]
J. Rodríguez-López, S. Romaguera, J. M. Sánchez-Aívarez, The Hausdorff fuzzy quasi-metric, Fuzzy Sets and Systems, 161 (2010), 1078-1096
##[22]
S. Romaguera, A. Sapena, O. Valero, Quasi-uniform isomorphisms in fuzzy quasi-metric spaces, bicompletion and D- completion, Acta Math. Hungar., 114 (2007), 49-60
##[23]
S. Romaguera, O. Valero, Domain theoretic characterisations of quasi-metric completeness in terms of formal balls, Math. Structures Comput. Sci., 20 (2010), 453-472
##[24]
A. Savchenko, M. Zarichnyi, Fuzzy ultrametrics on the set of probability measures, Topology, 48 (2009), 130-136
##[25]
Y.-H. Shen, D. Qiu, W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Lett., 25 (2012), 138-141
##[26]
J.-Y. Wu, Y.-L. Yue, Formal balls in fuzzy partial metric spaces, Iran. J. Fuzzy Syst., (2016), -
##[27]
J.-L. Xie, Q.-G. Li, S.-L. Chen, Y. Gao, On pseudo-metric spaces induced by \(\sigma-\bot\)-decomposable measures, Fuzzy Sets and Systems, 289 (2016), 33-42
##[28]
Y.-L. Yue, F.-G. Shi, On fuzzy pseudo-metric spaces, Fuzzy Sets and Systems, 161 (2010), 1105-1116
]
Contraction mapping principle in partially ordered quasi metric space concerning to w-distances
Contraction mapping principle in partially ordered quasi metric space concerning to w-distances
en
en
The fixed point theorems in various contraction mappings have been provided by many researchers. Some of them used
certain functions in mapping to guarantee the existence of fixed point. The purpose of this paper is to present some fixed
point result on contraction mapping in partially ordered quasi-metric space that applying a w-distance. The generalized altering
distance function on the mapping plays a role in theorems. The results extend some well-known results in the references. We
also improve these new results to the common fixed point.
699
712
Rahma
Zuhra
School of Mathematical Sciences, Faculty of Science and Technology
Department of Mathematics, Faculty of Mathematics and Natural Sciences
University Kebangsaan Malaysia
Syiah Kuala University
Malaysia
Indonesia
r4hm4@siswa.ukm.edu.my
Mohd Salmi Md
Noorani
School of Mathematical Sciences, Faculty of Science and Technology
University Kebangsaan Malaysia
Malaysia
msn@ukm.edu.my
Fawzia
Shaddad
Department of Mathematics
Sana’a University
Yemen
fzsh99@gmail.com
Fixed point
w-distance
a generalized altering distance function
common fixed point.
Article.31.pdf
[
[1]
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
##[2]
R. P. Agarwal, M. A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., 87 (2008), 109-116
##[3]
C. Alegre, J. Marín, S. Romaguera, A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces, Fixed Point Theory Appl., 2014 (2014), 1-8
##[4]
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
##[5]
L. Ćirić, N. Cakić, M. Rajović, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2008 (2008), 1-11
##[6]
P. N. Dutta, B. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl., 2008 (2008), 1-8
##[7]
M. Eshaghi Gordji, H. Baghani, G. H. Kim, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Discrete Dyn. Nat. Soc., 2012 (2012), 1-8
##[8]
L. Gholizadeh, R. Saadati, W. Shatanawi, S. M. Vaezpour, Contractive mapping in generalized, ordered metric spaces with application in integral equations, Math. Probl. Eng., 2011 (2011), 1-14
##[9]
R. H. Haghi, S. Rezapour, N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal., 74 (2011), 1799-1803
##[10]
J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations,/ Nonlinear Anal.,/ 72 (2010), 1188–1197., Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72 (2010), 1188-1197
##[11]
D. Ilić, V. Rakočević, Common fixed points for maps on metric space with w-distance, Appl. Math. Comput., 199 (2008), 599-610
##[12]
M. Imdad, F. Rouzkard , Fixed point theorems in ordered metric spaces via w-distances, Fixed Point Theory Appl., 2012 (2012), 1-17
##[13]
G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly, 83 (1976), 261-263
##[14]
O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996), 381-391
##[15]
M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), 1-9
##[16]
A. Latif, S. A. Al-Mezel, Fixed point results in quasimetric spaces, Fixed Point Theory Appl., 2011 (2011), 1-8
##[17]
S. H. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal., 39 (2000), 881-889
##[18]
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
##[19]
B. E. Rhoades, H. K. Pathak, S. N. Mishra, Some weakly contractive mapping theorems in partially ordered spaces and applications, Demonstratio Math., 45 (2012), 621-636
##[20]
F. Rouzkard, M. Imdad, D. Gopal , Some existence and uniqueness theorems on ordered metric spaces via generalized distances, Fixed Point Theory Appl., 2013 (2013), 1-20
##[21]
R. Saadati, S. M. Vaezpoura, P. Vetro, B. E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Math. Comput. Modelling, 52 (2010), 797-801
##[22]
F. Shaddad, M. S. M. Noorani, S. M. Alsulami, H. Akhadkulov, Coupled point results in partially ordered metric spaces without compatibility, Fixed Point Theory Appl., 2014 (2014 ), 1-18
##[23]
W. A. Shatanawi, K. K. Abodaye, A. Bataihah, Fixed point theorem through \(\Omega\)-Distance of Suzuki type contraction condition, Gazi Univ. J. Sci., 29 (2016), 129-133
##[24]
W. Shatanawi, A. Bataihah, A. Pitea, Fixed and common fixed point results for cyclic mappings of \(\Omega\)-distance, J. Nonlinear Sci. Appl., 9 (2016), 727-735
##[25]
W. Shatanawi, A. Pitea, Fixed and coupled fixed point theorems of \(\Omega\)-distance for nonlinear contraction, Fixed Point Theory Appl., 2013 (2013 ), 1-16
##[26]
W. Shatanawi, A. Pitea, \(\Omega\)-distance and coupled fixed point in G-metric spaces, Fixed Point Theory Appl.,, 2013 (2013 ), 1-15
##[27]
W. Shatanawi, M. Postolache, Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces, Fixed Point Theory Appl., 2013 (2013 ), 1-17
##[28]
Y.-F. Su, Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl., 2014 (2014 ), 1-15
##[29]
F.-F. Yan, Y.-F. Su, Q.-S. Feng, A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl., 2012 (2012 ), 1-13
]
A nonstandard numerical scheme for a predator-prey model with Allee effect
A nonstandard numerical scheme for a predator-prey model with Allee effect
en
en
In this paper, we present a Lotka-Volterra predator-prey model with Allee effect. This system with general functional
response has an Allee effect on prey population. A nonstandard finite difference scheme is constructed to transform the continuous
time predator-prey model with Allee effect into the discrete time model. We use the Schur-Cohn criteria which deal
with coefficients of the characteristic polynomial for determining the stability of discrete time system. The proposed numerical
schemes preserve the positivity of the solutions with positive initial conditions. The new discrete-time model shows dynamic
consistency with continuous-time model.
713
723
Mevlude Yakit
Ongun
Department of Mathematics
Suleyman Demirel University
Turkey
mevludeyakit@sdu.edu.tr
Nihal
Ozdogan
Vocational School of Higher Education
Suleyman Demirel University
Turkey
nihalozdogan@sdu.edu.tr
Allee effect
stability analysis
nonstandard finite difference scheme
predator-prey model.
Article.32.pdf
[
[1]
H. N. Agiza, E. M. ELabbasy, H. EL-Metwally, A. A. Elsadany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl., 10 (2009), 116-129
##[2]
R. Anguelov, J. M. S. Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Partial Differential Equations, 17 (2001), 518-543
##[3]
A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454
##[4]
N. Bairagi, M. Biswas, A predator-prey model with Beddington-DeAngelis functional response: a non-standard finitedifference method, J. Difference Equ. Appl., 4 (2016), 529-541
##[5]
F. Berezovskaya, G. Karev, R. Arditi, Parametric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43 (2001), 221-246
##[6]
Q.-Y. Bie, Q.-R. Wang, Z.-A. Yao, Cross-diffusion induced instability and pattern formation for a Holling type-II predatorprey model, Appl. Math. Comput., 247 (2014), 1-12
##[7]
C. Çelik, O. Duman, Allee effect in a discrete-time predator-prey system , Chaos Solitons Fractals, 40 (2009), 1956-1962
##[8]
D. T. Dimitrov, H. V. Kojouharov, Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems, Appl. Math. Lett., 18 (2005), 769-774
##[9]
D. T. Dimitrov, H. V. Kojouharov , Positive and elementary stable nonstandard numerical methods with applications to predator-prey models, J. Comput. Appl. Math., 189 (2006), 98-108
##[10]
D. T. Dimitrov, H. V. Kojouharov, Nonstandard numerical methods for a class of predator-prey models with predator interference, Proceedings of the Sixth Mississippi State–UBA Conference on Differential Equations and Computational Simulations, Electron. J. Differ. Equ. Conf., Southwest Texas State Univ., San Marcos, TX, 15 (2007), 67-75
##[11]
D. T. Dimitrov, H. V. Kojouharov, Nonstandard finite-difference methods for predator-prey models with general functional response, Math. Comput. Simulation, 78 (2008), 1-11
##[12]
S. N. Elaydi, An introduction to difference equations, Second edition, Undergraduate Texts in Mathematics, Springer- Verlag, New York (1999)
##[13]
A. Gkana, L. Zachilas, Incorporating prey refuge in a prey-predator model with a Holling type I functional response: random dynamics and population outbreaks, J. Biol. Phys., 39 (2013), 587-606
##[14]
C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5-60
##[15]
S. Jang, S. Elaydi, Difference equations from discretization of a continuous epidemic model with immigration of infectives, Can. Appl. Math. Q., 11 (2003), 93-105
##[16]
H. Jansen, E. H. Twizell, An unconditionally convergent discretization of the SEIR model, Math. Comput. Simulation, 58 (2002), 147-158
##[17]
Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406
##[18]
J. M. S. Lubuma, A. Roux, An improved theta-method for systems of ordinary differential equations, Dedicated to Professor Ronald E. Mickens on the occasion of his 60th birthday, J. Difference Equ. Appl., 9 (2003), 1023-1035
##[19]
H. Merdan, Stability analysis of a Lotka-Volterra type predator-prey system involving Allee effects, ANZIAM J., 52 (2010), 139-145
##[20]
R. E. Mickens, Nonstandard finite difference models of differential equations, World Scientific Publishing Co., Inc., River Edge, NJ (1994)
##[21]
J. D. Murray, Mathematical biology , Second edition, Biomathematics, Springer-Verlag, Berlin (1993)
##[22]
M. Y. Ongun, D. Arslan, R. Garrappa , Nonstandard finite difference schemes for a fractional-order Brusselator system, Adv. Difference Equ., 2013 (2013), 1-13
##[23]
M. Y. Ongun, I. Turhan, A numerical comparison for a discrete HIV infection of \(CD4^+T\)-Cell model derived from nonstandard numerical scheme, J. Appl. Math., 2013 (2013 ), 1-9
##[24]
W. Piyawong, E. H. Twizell, A. B. Gumel , An unconditionally convergent finite-difference scheme for the SIR model, Appl. Math. Comput., 146 (2003), 611-625
##[25]
M. Sen, M. Banarjee, A. Morozov, Bifurcation analysis of a ratio-dependent preypredator model with the Allee effect, Ecol. Complex., 11 (2013), 12-27
##[26]
S. Sharma, G. P. Samanta, A ratio-dependent predator-prey model with Allee effect and disease in prey, J. Appl. Math. Comput., 47 (2015), 345-364
##[27]
R.-Z. Yang, J.-J. Wei, Stability and bifurcation analysis of a diffusive prey-predator system in Holling type III with a prey refuge, Nonlinear Dynam., 79 (2015), 631-646
##[28]
S.-R. Zhou, Y.-F. Liu, G. Wang, The stability of predatorprey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23-31
]
On numerical solutions of time-fraction generalized Hirota Satsuma coupled KdV equation
On numerical solutions of time-fraction generalized Hirota Satsuma coupled KdV equation
en
en
In this study, we obtain the approximate soliton solution of the fractional generalized Hirota-Satsuma coupled Kortewegde
Vries equation (GHS-cKdV) within the homotopy analysis method (HAM). Numerical results are successfully compared
with other solutions obtained by the differential transform method (DTM) and the homotopy perturbation method (HPM). The
numerical results indicate that the only few terms are sufficient to get the correct solutions. Also, the results are given by tables
and figures.
724
733
Ebru Cavlak
Aslan
Department of Mathematics, Science Faculty
Firat University
Turkey
ebrucavlak@hotmail.com
Mustafa
Inc
Department of Mathematics, Science Faculty
Firat University
Turkey
minc@firat.edu.tr
Maysaa’ Mohamed Al
Qurashi
Department of Mathematics
King Saud University
Saudi Arabia
maysaa@ksu.edu.sa
Dumitru
Baleanu
Department of Mathematics
Cankaya University
Turkey
dumitru@cankaya.edu.tr
HAM
fractional partial differential equation(FPD)
HS-cKdV equation
time-fractional GHS-cKdV equation.
Article.33.pdf
[
[1]
R. Abazari, M. Abazari , Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 619-629
##[2]
S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Appl. Math. Model., 32 (2008), 2706-2714
##[3]
S. Abbasbandy, E. Shivanian, Series solution of the system of integro-differential equations, Z. Naturforsch. A, 64 (2009), 811-818
##[4]
D. Baleanu, O. P. Agrawal, Fractional Hamilton formalism within Caputo’s derivative, Czechoslovak J. Phys., 56 (2006), 1087-1092
##[5]
Z. Z. Ganji, D. D. Ganji, Y. Rostamiyan, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique, Appl. Math. Model., 33 (2009), 3107-3113
##[6]
C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math., 27 (1974), 97-133
##[7]
A. K. Golmankhaneh, A. Golmankhaneh, D. Baleanu, On nonlinear fractional Klein–Gordon equation, Signal Process., 91 (2011), 446-451
##[8]
A. K. Golmankhaneh, N. A. Porghoveh, D. Baleanu, Mean square solutions of second-order random differential equations by using homotopy analysis method, Rom. Rep. Phys., 65 (2013), 350-362
##[9]
R. Hirota, Exact solution of the Korteweg—de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194
##[10]
R. Hirota, J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407-408
##[11]
M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method, Phys. Lett. A, 365 (2007), 412-415
##[12]
S.-J. Liao, Beyond perturbation, Introduction to the homotopy analysis method, CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, FL (2003)
##[13]
Y.-P. Liu, Z.-B. Li, The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation, Chaos Solitons Fractals, 39 (2009), 1-8
##[14]
J.-C. Liu, H. Li, Approximate analytic solutions of time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation, Abstr. Appl. Anal., 2013 (2013 ), 1-11
##[15]
H. Jafari, S. Das, H. Tajadodi, Solving a multi-order fractional differential equation using homotopy analysis method, J. King Saud Univ. Sci., 23 (2011), 151-155
##[16]
H. Jafari, S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2006-2012
##[17]
S. Momani, Z. Odibat, A. Alawneh, Variational iteration method for solving the space- and time-fractional KdV equation, Numer. Methods Partial Differential Equations, 24 (2008), 262-271
##[18]
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)
##[19]
M. Shateri, D. D. Ganji, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by a new analytical technique, Int. J. Differ. Equ., 2010 (2010 ), 1-10
##[20]
X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Sci., NewYork, USA (2012)
##[21]
X.-J. Yang, D. Baleanu, Y. Khan, S. T. Mohyud-Din, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian J. Phys., 59 (2014), 36-48
##[22]
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)
##[23]
X.-J. Yang, H. M. Srivastava, J.-H. He, D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Phys. Lett. A, 377 (2013), 1696-1700
]
Convergence theorems for two finite families of some generalized nonexpansive mappings in hyperbolic spaces
Convergence theorems for two finite families of some generalized nonexpansive mappings in hyperbolic spaces
en
en
We propose and analyze a one-step explicit iterative algorithm for two finite families of mappings satisfying condition (C)
in hyperbolic spaces. Our results are new and generalize several recent results in uniformly convex Banach spaces and CAT(0)
spaces, simultaneously.
734
743
Safeer Hussain
Khan
Department of Mathematics, Statistics and Physics
Qatar University
State of Qatar
safeer@qu.edu.qa
Hafiz
Fukhar-ud-din
Department of Mathematics and Statistics
King Fahd University of Petroleum and Minerals
Saudi Arabia
hfdin@kfupm.edu.sa
Hyperbolic space
one-step iterative algorithm
nonexpansive mapping
condition (C)
common fixed point
strong convergence
\(\Delta\)-convergence.
Article.34.pdf
[
[1]
M. Abbas, S. H. Khan, Some \(\Delta\)-convergence theorems in CAT(0) spaces, Hacet. J. Math. Stat., 40 (2011), 563-569
##[2]
H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, With a foreword by Hédy Attouch, CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, Springer, New York (2011)
##[3]
H. Fukhar-ud-Din, Existence and approximation of fixed points in convex metric spaces, Carpathian J. Math., 30 (2014), 175-185
##[4]
H. Fukhar-ud-din, Common fixed points of two finite families of nonexpansive mappings by iterations, Carpathian J. Math., 31 (2015), 325-331
##[5]
H. Fukhar-ud-din, One step iterative scheme for a pair of nonexpansive mappings in a convex metric space, Hacet. J. Math. Stat., 44 (2015), 1023-1031
##[6]
B. Gunduz, S. Akbulut, Strong and \(\Delta\)-convergence theorems in hyperbolic spaces, Miskolc Math. Notes, 14 (2013), 915-925
##[7]
S. H. Khan, N. Hussain, Convergence theorems for nonself asymptotically nonexpansive mappings, Comput. Math. Appl., 55 (2008), 2544-2553
##[8]
S. H. Khan, A. Rafiq, N. Hussain , A three-step iterative scheme for solving nonlinear \(\varphi\)-strongly accretive operator equations in Banach spaces, Fixed Point Theory Appl., 2012 (2012 ), 1-10
##[9]
S. H. Khan, I. Yildirim, M. Ozdemir, Convergence of an implicit algorithm for two families of nonexpansive mappings, Comput. Math. Appl., 59 (2010), 3084-3091
##[10]
W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696
##[11]
U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc., 357 (2005), 89-128
##[12]
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510
##[13]
K. Menger, Untersuchungen über allgemeine Metrik, (German) Math. Ann., 100 (1928), 75-163
##[14]
J. P. Penot, Fixed point theorems without convexity, Analyse non convexe, Proc. Colloq., Pau, (1977), Bull. Soc. Math. France Mém., 60 (1979), 129-152
##[15]
S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 15 (1990), 537-558
##[16]
T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 341 (2008), 1088-1095
##[17]
W. Takahashi, A convexity in metric space and nonexpansive mappings, I, Kōdai Math. Sem. Rep., 22 (1970), 142-149
##[18]
W. Takahashi, T. Tamura , Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal., 5 (1998), 45-56
##[19]
K.-K. Tan, H.-K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308
]
Some identities of degenerate Daehee numbers arising from certain differential equations
Some identities of degenerate Daehee numbers arising from certain differential equations
en
en
In this paper, we introduce the degenerate Daehee numbers and study a family of differential equations associated with
the generating function of these numbers. From those differential equations, we will be able to obtain some new and interesting
combinatorial identities involving the degenerate Daehee numbers and generalized harmonic numbers.
744
751
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
Degenerate Daehee numbers
differential equation
generalized harmonic numbers.
Article.35.pdf
[
[1]
A. Bayad, T. Kim, Identities for Apostol-type Frobenius-Euler polynomials resulting from the study of a nonlinear operator, Russ. J. Math. Phys., 23 (2016), 164-171
##[2]
L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel), 7 (1956), 28-33
##[3]
L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15 (1979), 51-88
##[4]
D. V. Dolgy, D. S. Kim, T. Kim, On the Korobov polynomials of the first kind, (Russian) Mat. Sb., 208 (2017), 65-79
##[5]
D. V. Dolgy, D. S. Kim, T. Kim, T. Mansour, Degenerate poly-Bernoulli polynomials of the second kind, J. Comput. Anal. Appl., 21 (2016), 954-966
##[6]
B. S. El-Desouky, A. Mustafa, New results on higher-order Daehee and Bernoulli numbers and polynomials, Adv. Difference Equ., 2016 (2016), 1-21
##[7]
T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory, 132 (2012), 2854-2865
##[8]
D. S. Kim, T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. (Ruse), 7 (2013), 5969-5976
##[9]
D. S. Kim, T. Kim, A note on degenerate poly-Bernoulli numbers and polynomials, Adv. Difference Equ., 2015 (2015), 1-8
##[10]
D. S. Kim, T. Kim, Identities arising from higher-order Daehee polynomial bases, Open Math., 13 (2015), 196-208
##[11]
D. S. Kim, T. Kim, Some identities for Bernoulli numbers of the second kind arising from a non-linear differential equation, Bull. Korean Math. Soc., 52 (2015), 2001-2010
##[12]
T. Kim, D. S. Kim, A note on nonlinear Changhee differential equations, Russ. J. Math. Phys., 23 (2016), 88-92
##[13]
T. Kim, D. S. Kim, K.-W. Hwang, J.-J. Seo, Some identities of Laguerre polynomials arising from differential equations, Adv. Difference Equ., 2016 (2016), 1-9
##[14]
T. Kim, D. S. Kim, H. I. Kwon, D. V. Dolgy, J.-J. Seo, Degenerate falling factorial polynomials, Adv. Stud. Contemp. Math., 26 (2016), 481-499
##[15]
T. Kim, D. S. Kim, T. Mansour, J.-J. Seo, Linear differential equations for families of polynomials, J. Inequal. Appl., 2016 (2016 ), 1-8
##[16]
N. M. Korobov , On some properties of special polynomials, (Russian) Proceedings of the IV International Conference ''Modern Problems of Number Theory and its Applications'' (Russian), Tula, (2001), Chebyshevskiı Sb., 1 (2001), 40-49
##[17]
H. I. Kwon, T. Kim, J.-J. Seo, A note on degenerate Changhee numbers and polynomials, Proc. Jangjeon Math. Soc., 18 (2015), 295-305
##[18]
J. G. Lee, L.-C. Jang, J.-J. Seo, S.-K. Choi, H. I. Kwon, On Appell-type Changhee polynomials and numbers, Adv. Difference Equ., 2016 (2016 ), 1-10
##[19]
E.-J. Moon, J.-W. Park, S.-H. Rim, A note on the generalized q-Daehee numbers of higher order, Proc. Jangjeon Math. Soc., 17 (2014), 557-565
##[20]
H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. (Kyungshang), 18 (2009), 41-48
##[21]
J.-J. Seo, S. H. Rim, T. Kim, S. H. Lee, Sums products of generalized Daehee numbers, Proc. Jangjeon Math. Soc., 17 (2014), 1-9
##[22]
Y. Simsek, Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math., 26 (2016), 555-566
##[23]
A. V. Ustinov, Korobov polynomials and umbral analysis, (Russian) Chebyshevskiı Sb., 4 (2003), 137-152
##[24]
N. L. Wang, H.-L. Li, Some identities on the higher-order Daehee and Changhee numbers, Pure Appl. Math. J., 4 (2015), 33-37
]
Cloud hybrid methods for solving split equilibrium and fixed point problems for a family of countable quasi-Lipschitz mappings and applications
Cloud hybrid methods for solving split equilibrium and fixed point problems for a family of countable quasi-Lipschitz mappings and applications
en
en
The purpose of this article is to introduce a new multidirectional hybrid shrinking projection iterative algorithm (or called
cloud hybrid shrinking projection iterative algorithm) for solving the common element problems which consist of a generalized
split equilibrium problems and fixed point problems for a family of countable quasi-Lipschitz mappings in the framework of
Hilbert spaces. It is proved that under appropriate conditions, the sequence generated by the multidirectional hybrid shrinking
projection method, converges strongly to some point which is the common fixed point of a family of countable quasi-Lipschitz
mappings and the solution of the generalized split equilibrium problems. This iteration algorithm can accelerate the convergence
speed of iterative sequence. The main results were also applied to solve split variational inequality problem and split optimization
problems. Meanwhile, the main results were also used for solving common problems which consist of a generalized split
equilibrium problems and fixed point problems for asymptotically nonexpansive mappings. The results of this paper improve
and extend the previous results given in the literature.
752
770
Yongchun
Xu
Department of Mathematics, College of Science
Hebei North University
China
hbxuyongchun@163.com
Yanxia
Tang
Department of Mathematics, College of Science
Hebei North University
China
sutang2016@163.com
Jinyu
Guan
Department of Mathematics
Tianjin Polytechnic University
China
sutang2016@163.com
Yongfu
Su
Department of Mathematics
Tianjin Polytechnic University
China
tjsuyongfu@163.com
Hybrid shrinking projection
split equilibrium problem
fixed point
quasi-Lipschitz mapping
split variational inequality
split optimization problem.
Article.36.pdf
[
[1]
R. P. Agarwal, J.-W. Chen, Y. J. Cho, Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces, J. Inequal. Appl., 2013 (2013 ), 1-16
##[2]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145
##[3]
A. Bnouhachem, Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems, Scientific World J., 2014 (2014 ), 1-12
##[4]
Y. Censor, A. Gibali, S. Reich, Algorithm for split variational inequality problems, Numer. Algorithms., 59 (2012), 301-323
##[5]
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
##[6]
S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013 ), 1-14
##[7]
P. L. Combette, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 61 (2005), 117-136
##[8]
S. D. Fl°am, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming, 78 (1997), 29-41
##[9]
J.-Y. Guan, Y.-X. Tang, P.-C. Ma, Y.-C. Xu, Y.-F. Su, Non-convex hybrid algorithm for a family of countable quasi-Lipschitz mappings and application, Fixed Point Theory Appl., 2015 (2015 ), 1-11
##[10]
Z.-H. He, The split equilibrium problem and its convergence algorithms, J. Inequal. Appl., 2012 (2012 ), 1-15
##[11]
H. Iiduka, W. Takahashi , Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350
##[12]
I. Inchan, Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces, Int. J. Math. Anal. (Ruse), 2 (2008), 1135-1145
##[13]
P. Katchang, P. Kumam, A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space, J. Appl. Math. Comput., 32 (2010), 19-38
##[14]
K. Kazmi, S. H. Rizvi, terative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egyptian Math. Soc., 21 (2013), 44-51
##[15]
J. K. Kim, Y. M. Nam, J. Y. Sim, Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonxpansive type mappings, Nonlinear Anal., 71 (2009), 1-2839
##[16]
T.-H. Kim, H.-K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 64 (2006), 1140-1152
##[17]
P.-K. Lin, K.-K. Tan, H.-K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal., 24 (1995), 929-946
##[18]
H. Piri, R. Yavarimehr, Solving systems of monotone variational inequalities on fixed point sets of strictly pseudocontractive mappings, J. Nonlinear Funct. Anal., 2016 (2016 ), 1-18
##[19]
S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 336 (2007), 455-469
##[20]
X.-L. Qin, M.-J. Shang, Y.-F. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal., 69 (2008), 3897-3909
##[21]
A. Tada, W. Takahashi , Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl., 113 (2007), 359-370
##[22]
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
##[23]
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
##[24]
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
]
Approximating common fixed points of total asymptotically nonexpansive mappings in CAT(0) spaces
Approximating common fixed points of total asymptotically nonexpansive mappings in CAT(0) spaces
en
en
We introduce and study convergence of a one-step iterative algorithm for a finite family of total asymptotically nonexpansive
mappings on a CAT(0) space. Our results are new in Hilbert spaces as well as CAT(0) spaces; in particular, an analogue of
Rhoades weak convergence theorem [B. E. Rhoades, Bull. Austral. Math. Soc., 62 (2000), 307–310] is established both for
\(\Delta\)-convergence and strong convergence in CAT(0) spaces.
771
779
Hafiz
Fukhar-ud-din
Department of Mathematics and Statistics
King Fahd University of Petroleum and Minerals
Saudi Arabia
hfdin@kfupm.edu.sa
Abdul Rahim
Khan
Department of Mathematics and Statistics
King Fahd University of Petroleum and Minerals
Saudi Arabia
arahim@kfupm.edu.sa
Nawab
Hussain
Department of Mathematics
King Abdul Aziz University
Saudi Arabia
nhusain@kau.edu.sa
CAT(0) space
total asymptotically nonexpansive mapping
one-step iterative algorithm
common fixed point
\(\Delta\)-convergence
strong convergence.
Article.37.pdf
[
[1]
M. Abbas, S. H. Khan, Some \(\Delta\)-convergence theorems in CAT(0) spaces, Hacet. J. Math. Stat., 40 (2011), 563-569
##[2]
M. Abbas, S. H. Khan, J. K. Kim, A new one-step iterative process for common fixed points in Banach spaces, J. Inequal. Appl., 2008 (2008), 1-10
##[3]
Y. I. Alber, C. E. Chidume, H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2006 (2006), 1-20
##[4]
A. Alotaibi, V. Kumar, N. Hussain, Convergence comparison and stability of Jungck-Kirk-type algorithms for common fixed point problems, Fixed Point Theory Appl., 2013 (2013 ), 1-30
##[5]
M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1999)
##[6]
C. E. Chidume, A. U. Bello, P. Ndambomve, Strong and \(\Delta\)-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr. Appl. Anal., 2014 (2014 ), 1-6
##[7]
S. Dhompongsa, A. Kaewkhao, B. Panyanak, On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces, Nonlinear Anal., 75 (2012), 459-468
##[8]
S. Dhompongsa, B. Panyanak, On \(\Delta\)-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572-2579
##[9]
H. Fukhar-ud-din, Existence and approximation of fixed points in convex metric spaces, Carpathian J. Math., 30 (2014), 175-185
##[10]
K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174
##[11]
N. Hussain, W. Takahashi, Weak and strong convergence theorems for semigroups of mappings without continuity in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 769-783
##[12]
S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150
##[13]
M. A. Khamsi, A. R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal., 74 (2011), 4036-4045
##[14]
A. R. Khan, M. A. Khamsi, H. Fukhar-ud-din, Strong convergence of a general iteration scheme in CAT(0) spaces, Nonlinear Anal., 74 (2011), 783-791
##[15]
A. R. Khan, V. Kumar, N. Hussain, Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput., 231 (2014), 521-535
##[16]
G. E. Kim, T. H. Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Comput. Math. Appl., 42 (2001), 1565-1570
##[17]
W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696
##[18]
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-610
##[19]
B. E. Rhoades, Finding common fixed points of nonexpansive mappings by iteration, Bull. Austral. Math. Soc., 62 (2000), 307–310; Corrigendum, Bull. Austral. Math. Soc., 63 (2001), 345-346
##[20]
H. F. Senter, W. G. Dotson Jr., Approximating fixed points of nonexpansive mappings , Proc. Amer. Math. Soc., 44 (1974), 375-380
##[21]
K.-K. Tan, H.-K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308
##[22]
I. Uddin, J. J. Nieto, J. Ali, One-step iteration scheme for multivalued nonexpansive mappings in CAT(0) spaces, Mediterr. J. Math., 13 (2016), 1211-1225
##[23]
H. Y. Zhou, Y. J. Cho, S. M. Kang, A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2007 (2007), 1-10
]
New Geraghty type contractions on metric-like spaces
New Geraghty type contractions on metric-like spaces
en
en
Very recently, Fulga and Proca [A. Fulga, A. M. Proca, Abstr. Appl. Anal., (In press)] considered new Geraghty type
contraction mappings and established a fixed point theorem for such mappings in complete metric spaces. In this paper, we
prove the analogous result in the class of metric-like spaces which generalizes the main result of Karapinar et al. [E. Karapınar, H.
H. Alsulami, M. Noorwali, Fixed Point Theory Appl., 2015 (2015), 22 pages]. We give some examples illustrating the presented
result where [E. Karapınar, H. H. Alsulami, M. Noorwali, Fixed Point Theory Appl., 2015 (2015), 22 pages] is not applicable. An
application is also provided.
780
788
Hassen
Aydi
Department of Mathematics, College of Education of Jubail
Department of Medical Research, China Medical University Hospital
University of Dammam
China Medical University
Saudi Arabia
Taiwan
hmaydi@uod.edu.sa
Abdelbasset
Felhi
Department of Mathematics and Statistics, College of Sciences
King Faisal University
Saudi Arabia
afelhi@kfu.edu.sa
Hojjat
Afshari
Faculty of Basic Science
University of Bonab
Iran
hojat.afshari@bonabu.ac.ir
Fixed point
Geraghty type contraction
metric-like space.
Article.38.pdf
[
[1]
C. T. Aage, J. N. Salunke, The results on fixed points in dislocated and dislocated quasi-metric space, Appl. Math. Sci. (Ruse), 59 (2008), 2941-2948
##[2]
S. A. Al-Mezel, C.-M. Chen, E. Karapınar, V. Rakočević, Fixed point results for various \(\alpha\)-admissible contractive mappings on metric-like spaces, Abstr. Appl. Anal., 2014 (2014 ), 1-15
##[3]
H. H. Alsulami, E. Karapınar, H. Piri, Fixed points of modified F-contractive mappings in complete metric-like spaces, J. Funct. Spaces, 2015 (2015 ), 1-9
##[4]
A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl., 2012 (2012 ), 1-10
##[5]
H. Aydi, A. Felhi, S. Sahmim, Fixed points of multivalued nonself almost contractions in metric-like spaces, Math. Sci. (Springer), 9 (2015), 103-108
##[6]
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
##[7]
H. Aydi, E. Karapınar, S. Rezapour, A generalized Meir-Keeler-type contraction on partial metric spaces, Abstr. Appl. Anal., 2012 (2012 ), 1-10
##[8]
R. D. Daheriya, R. Jain, M. Ughade, Some fixed point theorem for expansive type mapping in dislocated metric space, ISRN Math. Anal., 2012 (2012 ), 1-5
##[9]
A. Fulga, A. M. Proca, Fixed points for \(\varphi_E\)-Geraghty contractions, Abstr. Appl. Anal., (In press), -
##[10]
R. George, R. Rajagopalan, S. Vinayagam, Cyclic contractions and fixed points in dislocated metric spaces, Int. J. Math. Anal. (Ruse), 9 (2013), 403-411
##[11]
V. Gupta, W. Shatanawi, N. Mani, Fixed point theorems for (\(\psi,\beta\))-Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations , J. Fixed Point Theory Appl., 2016 (2016 ), 1-17
##[12]
P. Hitzler, A. K. Seda, Dislocated topologies, J. Electr. Eng., 51 (2000), 3-7
##[13]
N. Hussain, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Fixed points of contractive mappings in b-metric-like spaces, Scientific World J., 2014 (2014 ), 1-15
##[14]
A. Isufati , Fixed point theorems in dislocated quasi-metric space, Appl. Math. Sci. (Ruse), 5 (2010), 217-233
##[15]
E. Karapınar, H. H. Alsulami, M. Noorwali, Some extensions for Geragthy type contractive mappings, Fixed Point Theory Appl., 2015 (2015 ), 1-22
##[16]
E. Karapınar, H. Aydi, A. Felhi, S. Sahmim, Hausdorff metric-like, generalized Nadler’s Fixed Point Theorem on Metric- Like Spaces and application, Miskolc Math. Notes, (), -
##[17]
E. Karapınar, P. Salimi, Dislocated metric space to metric spaces with some fixed point theorems, Fixed Point Theory Appl. , 2013 (2013 ), 1-19
##[18]
P. S. Kumari, Some fixed point theorems in generalized dislocated metric spaces, Math. Theory Model., 1 (2011), 16-22
##[19]
P. S. Kumari, V. V. Kumar, I. Rambhadra Sarma, Common fixed point theorems on weakly compatible maps on dislocated metric spaces , Math. Sci. (Springer), 6 (2012), 1-5
##[20]
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
##[21]
I. R. Sarma, P. S. Kumari, On dislocated metric spaces, Int. J. Math. Arch., 1 (2012), 72-77
##[22]
W. Shatanawi, M. S. MD. Noorani, H. Alsamir, A. Bataihah, Fixed and common fixed point theorems in partially ordered quasi-metric spaces, J. Math. Computer Sci., 16 (2016), 516-528
##[23]
R. Shrivastava, Z. K. Ansari, M. Sharma , Some results on fixed points in dislocated and dislocated quasi-metric spaces, J. Adv. Stud. Topol., 3 (2012), 25-31
##[24]
M. Shrivastava, K. Qureshi, A. D. Singh, A fixed point theorem for continuous mapping in dislocated quasi-metric spaces, Int. J. Theor. Appl. Sci., 4 (2012), 39-40
##[25]
K. Zoto, E. Hoxha, A. Isufati, Some new results in dislocated and dislocated quasi-metric spaces, Appl. Math. Sci. (Ruse), 6 (2012), 3519-3526
]
Generalized mixed equilibria, variational inequalities and constrained convex minimization
Generalized mixed equilibria, variational inequalities and constrained convex minimization
en
en
In this paper, we introduce one multistep relaxed implicit extragradient-like scheme and another multistep relaxed explicit
extragradient-like scheme for finding a common element of the set of solutions of the minimization problem for a convex and
continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems
and the set of solutions of a finite family of variational inequalities for inverse strongly monotone mappings in a real Hilbert
space. Under suitable control conditions, we establish the strong convergence of these two multistep relaxed extragradient-like
schemes to the same common element of the above three sets, which is also the unique solution of a variational inequality
defined over the intersection of the above three sets.
789
804
Lu-Chuan
Ceng
Department of Mathematics
Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities
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
Convex minimization problem
generalized mixed equilibrium problem
variational inequality
inverse-strongly monotone mapping.
Article.39.pdf
[
[1]
A. E. Al-Mazrooei, B. A. Bin Dehaish, A. Latif, J.-C. Yao, On general system of variational inequalities in Banach spaces, J. Nonlinear Convex Anal., 16 (2015), 639-658
##[2]
A. S. M. Alofi, A. Latif, A. E. Al-Mazrooei, 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
##[3]
A. Bnouhachem, Q. H. Ansari, J.-C. Yao, Strong convergence algorithm for hierarchical fixed point problems of a finite family of nonexpansive mappings, Fixed Point Theory, 17 (2016), 47-62
##[4]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120
##[5]
L.-C. Ceng, Q. H. Ansari, A. Petruşel, J.-C. Yao, Approximation methods for triple hierarchical variational inequalities (I), Fixed Point Theory, 16 (2015), 67-90
##[6]
L.-C. Ceng, Q. H. Ansari, A. Petruşel, J.-C. Yao, Approximation methods for triple hierarchical variational inequalities (II)., Fixed Point Theory, 16 (2015), 237-259
##[7]
L.-C. Ceng, Q. H. Ansari, S. Schaible, Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems, J. Global Optim., 53 (2012), 69-96
##[8]
L.-C. Ceng, Q. H. Ansari, S. Schaible, J.-C. Yao, Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces, Fixed Point Theory, 12 (2011), 293-308
##[9]
L.-C. Ceng, Q. H. Ansari, M.-M. Wong, J.-C. Yao, Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems, Fixed Point Theory, 13 (2012), 403-422
##[10]
L.-C. Ceng, S.-M. Guu, J.-C. Yao, A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 61 (2011), 2447-2455
##[11]
L.-C. Ceng, S.-M. Guu, J.-C. Yao, Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method, Fixed Point Theory Appl., 2011 (2011), 1-22
##[12]
L.-C. Ceng, S. Plubtieng, M.-M. Wong, J.-C. Yao, System of variational inequalities with constraints of mixed equilibria, variational inequalities, and convex minimization and fixed point problems, J. Nonlinear Convex Anal., 16 (2015), 385-421
##[13]
L.-C. Ceng, C.-F. Wen, C. Liou, Multi-step iterative algorithms with regularization for triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions, and convex minimization, J. Inequal. Appl., 2014 (2014 ), 1-47
##[14]
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
##[15]
L.-C. Ceng, J.-C. Yao, A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Anal., 72 (2010), 1922-1937
##[16]
S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013 ), 1-14
##[17]
K. Goebel, W. A. Kirk, Topics in metric fixed point theory , Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
##[18]
G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) ´Ekonom. i Mat. Metody, 12 (1976), 747-756
##[19]
G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-56
##[20]
G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-346
##[21]
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
##[22]
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877-898
##[23]
N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 3641-3645
##[24]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256
##[25]
H.-K. Xu, T.-H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl., 119 (2003), 185-201
##[26]
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
##[27]
L.-C. Zeng, J.-C. Yao, Modified combined relaxation method for general monotone equilibrium problems in Hilbert spaces, J. Optim. Theory Appl., 131 (2006), 469-483
]
Furstenberg families and chaos on uniform limit maps
Furstenberg families and chaos on uniform limit maps
en
en
Let (\(f_n\)) be a given sequence of continuous selfmaps of a compact metric space \(X\) which converges uniformly to a continuous
selfmap \(f\) of a compact metric space \(X\), and let \(F, F_1\), and \(F_2\) be given Furstenberg families. In this paper, we obtain an equivalence
condition for the uniform limit map \(f\) to be \(F\)-transitive or weakly \(F\)-sensitive or \(F\)-sensitive or \((F_1, F_2)\)-sensitive and a necessary
condition for the uniform limit map \(f\) to be weakly \(F\)-sensitive or \(F\)-sensitive or \((F_1, F_2)\)-sensitive. Our results extend and
improve some existing ones.
805
816
Risong
Li
School of Science
Guangdong Ocean University
People’s Republic of China
gdoulrs@163.com
Yu
Zhao
School of Science
Guangdong Ocean University
People’s Republic of China
datom@189.cn
Hongqing
Wang
School of Science
Guangdong Ocean University
People’s Republic of China
wanghq3333@126.com
Furstenberg families
transitivity
F-transitivity
sensitivity
weak F-sensitivity
F-sensitivity
\((F_1،F_2)\)-sensitivity
Article.40.pdf
[
[1]
C. Abraham, G. Biau, B. Cadre, Chaotic properties of mappings on a probability space, J. Math. Anal. Appl., 266 (2002), 420-431
##[2]
R. Abu-Saris, K. Al-Hami, Uniform convergence and chaotic behavior, Nonlinear Anal., 65 (2006), 933-937
##[3]
E. Akin, Recurrence in topological dynamics. Furstenberg families and Ellis actions, The University Series in Mathematics, Plenum Press, New York (1997)
##[4]
J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334
##[5]
J. S. Cánovas, Li-Yorke chaos in a class of nonautonomous discrete systems, J. Difference Equ. Appl., 17 (2011), 479-486
##[6]
J. Dvořáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4649-4652
##[7]
A. Fedeli, A. Le Donne, A note on the uniform limit of transitive dynamical systems, Bull. Belg. Math. Soc. Simon Stevin, 16 (2009), 59-66
##[8]
E. Glasner, B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075
##[9]
L.-F. He, X.-H. Yan, L.-S. Wang, Weak-mixing implies sensitive dependence, J. Math. Anal. Appl., 299 (2004), 300-304
##[10]
R.-S. Li, A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2815-2823
##[11]
R.-S. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45 (2012), 753-758
##[12]
R.-S. Li , A note on uniform convergence and transitivity, Chaos Solitons Fractals, 45 (2012), 759-764
##[13]
R.-S. Li , The large deviations theorem and ergodic sensitivity, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 819-825
##[14]
R.-S. Li, H.-Q.Wang, Erratnm to ''A note on uniform convergence and transitivity [Chaos, Solitons and Fractals 45 (2012), 759–764]'', Chaos Solitons Fractals, 59 (2014), 112-118
##[15]
H. Liu, E.-H. Shi, G.-F. Liao, Sensitivity of set-valued discrete systems, Nonlinear Anal., 71 (2009), 6122-6125
##[16]
T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126
##[17]
H. Román-Flores, Uniform convergence and transitivity, Chaos Solitons Fractals, 38 (2008), 148-153
##[18]
H.-Y. Wang, J.-C. Xiong, F. Tan, Furstenberg families and sensitivity, Discrete Dyn. Nat. Soc., 2010 (2010 ), 1-12
##[19]
K.-S. Yan, F.-P. Zeng, G.-R. Zhang, Devaney’s chaos on uniform limit maps, Chaos Solitons Fractalss, 44 (2011), 522-525
]
Generalized Srivastava's triple hypergeometric functions and their associated properties
Generalized Srivastava's triple hypergeometric functions and their associated properties
en
en
The main object of this paper is to introduce generalized Srivastava’s triple hypergeometric functions by using the generalized
Pochhammer symbol and investigate certain properties, for example, their various integral representations, derivative
formulas and recurrence relations. Various (known or new) special cases and consequences of the results presented here are also
considered.
817
827
Junesang
Choi
Department of Mathematics
Dongguk University
Republic of Korea
junesang@mail.dongguk.ac.kr
Rakesh Kumar
Parmar
Department of Mathematics
Government College of Engineering and Technology
India
rakeshparmar27@gmail.com
Gamma function
beta function
generalized Pochhammer symbol
generalized hypergeometric function
extended Appell functions
generalized Srivastava’s triple hypergeometric functions
Whittaker function
Bessel and modified Bessel functions.
Article.41.pdf
[
[1]
M. A. Chaudhry, S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994), 99-124
##[2]
M. A. Chaudhry, S. M. Zubair, On a class of incomplete gamma functions with applications, Chapman & Hall/CRC, Boca Raton, FL (2002)
##[3]
J.-S. Choi, R. K. Parmar, P. Chopra , Extended Mittag-Leffler function and associated fractional calculus operators, Georgian Math. J., (2017), -
##[4]
J.-S. Choi, R. K. Parmar, T. K. Pogány, Mathieu-type series built by (p, q)-extended Gaussian hypergeometric function, ArXiv , 2016 (2016 ), 1-9
##[5]
J.-S. Choi, A. K. Rathie, R. K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J., 36 (2014), 357-385
##[6]
A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, Vols. I, II, Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London (1953)
##[7]
Y. L. Luke, Mathematical functions and their approximations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1975)
##[8]
W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York (1966)
##[9]
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (Eds.), NIST handbook of mathematical functions , [With 1 CD-ROM (Windows, Macintosh and UNIX)], US Department of Commerce, National Institute of Standards and Technology, Washington, DC, (2010); Cambridge University Press, Cambridge, London and New York (2010)
##[10]
R. K. Parmar, Extended \(\tau\)-hypergeometric functions and associated properties, C. R. Math. Acad. Sci. Paris, 353 (2015), 421-426
##[11]
R. K. Parmar, T. K. Pogány, Extended Srivastava’s triple hypergeometric \(H_{A,p,q}\) function and related bounding inequalities, J. Contemp. Math. Anal., (In press). (2016)
##[12]
T. K. Pogány, R. K. Parmar, On p-extended Mathieu series, ArXiv, 2016 (2016 ), 1-8
##[13]
E. D. Rainville, Special functions, Reprint of 1960 first edition, Chelsea Publishing Co., Bronx, N.Y. (1971)
##[14]
L. J. Slater, Confluent hypergeometric functions, Cambridge University Press, New York (1960)
##[15]
L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge (1966)
##[16]
H. M. Srivastava, Hypergeometric functions of three variables, Gadot nita , 15 (1964), 97-108
##[17]
H. M. Srivastava , On transformations of certain hypergeometric functions of three variables, Publ. Math. Debrecen, 12 (1965), 65-74
##[18]
H. M. Srivastava, On the reducibility of certain hypergeometric functions, Univ. Nac. Tucumn Rev. Ser. A, 16 (1966), 7-14
##[19]
H.M. Srivastava, Relations between functions contiguous to certain hypergeometric functions of three variables, Proc. Nat. Acad. Sci. India Sect. A, 36 (1966), 377-385
##[20]
H. M. Srivastava, Some integrals representing triple hypergeometric functions, Rend. Circ. Mat. Palermo, 16 (1967), 99-115
##[21]
R. Srivastava, Some generalizations of Pochhammer’s symbol and their associated families of hypergeometric functions and hypergeometric polynomials, Appl. Math. Inf. Sci., 7 (2013), 2195-2206
##[22]
R. Srivastava, Some classes of generating functions associated with a certain family of extended and generalized hypergeometric functions, Appl. Math. Comput., 243 (2014), 132-137
##[23]
H. M. Srivastava, A. Çetinkaya, ˙I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput., 226 (2014), 484-491
##[24]
R. Srivastava, N. E. Cho , Some extended Pochhammer symbols and their applications involving generalized hypergeometric polynomials, Appl. Math. Comput., 234 (2014), 277-285
##[25]
H. M. Srivastava, J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier, Inc., Amsterdam (2012)
##[26]
H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York (1985)
##[27]
H. M. Srivastava, H. L. Manocha, A treatise on generating functions, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York (1984)
##[28]
H. M. Srivastava, R. K. Parmar, P. Chopra, A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions, Axioms, 1 (2012), 238-258
##[29]
H. M. Srivastava, R. K. Parmar, M. M. Joshi, Extended Lauricella and Appell functions and their associated properties, Adv. Stud. Contemp. Math., 25 (2015), 151-165
]
The method of almost convergence with operator of the form fractional order and applications
The method of almost convergence with operator of the form fractional order and applications
en
en
The purpose of this paper is twofold. First, basic concepts such as Gamma function, almost convergence, fractional order
difference operator and sequence spaces are given as a survey character. Thus, the current knowledge about those concepts
are presented. Second, we construct the almost convergent spaces with fractional order difference operator and compute dual
spaces which help us in the characterization of matrix mappings. After we characterize to the matrix transformations, we give
some examples. In this paper, the notation \(\Gamma(n)\) will be shown the Gamma function. For \(n\not\in \{0,-1,-2,...\}\), Gamma function is
defined by an improper integral \(\Gamma(n)=\int^\infty_0 e^{-t}t^{n-1}dt\) .
828
842
Murat
Kirişci
Department of Mathematical Education, Hasan Ali Yücel Education Faculty
Istanbul University
Turkey
mkirisci@hotmail.com;murat.kirisci@istanbul.edu.tr
Uğur
Kadak
Department of Mathematics
Gazi University
Turkey
ugurkadak@gmail.com
Gamma function
almost convergence
fractional order difference operator
matrix domain
dual spaces.
Article.42.pdf
[
[1]
B. Altay, F. Başar, Some Euler sequence spaces of nonabsolute type , translated from Ukraïn. Mat. Zh., 57 (2005), 3–17, Ukrainian Math. J., 57 (2005), 1-17
##[2]
B. Altay, F. Başar, Certain topological properties and duals of the domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336 (2007), 632-645
##[3]
B. Altay, F. Başar, M. Mursaleen, On the Euler sequence spaces which include the spaces \(l_p\) and \(l_\infty\), I, Inform. Sci., 176 (2006), 1450-1462
##[4]
A. F. Andersen, Comparison Theorems in the Theory of Cesaro Summability, Proc. London Math. Soc., 2 (1928), 39-71
##[5]
A. F. Andersen, Summation of nonintegral order, (Danish) Mat. Tidsskr. B., 1946 (1946), 33-52
##[6]
A. F. Andersen, On difference transformations, (Danish) Mat. Tidsskr. B., 1950 (1950 ), 110-122
##[7]
P. Baliarsingh , Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput., 219 (2013), 9737-9742
##[8]
P. Baliarsingh, S. Dutta, A unifying approach to the difference operators and their applications, Bol. Soc. Parana. Mat., 33 (2015), 49-57
##[9]
P. Baliarsingh, S. Dutta, On the classes of fractional order difference sequence spaces and their matrix transformations, App. Math. Comput., 250 (2015), 665-674
##[10]
F. Başar, Strongly-conservative sequence-to-series matrix transformations, Erc. Uni. Fen Bil. Derg., 5 (1989), 888-893
##[11]
F. Başar, Summability theory and its applications, With a foreword by M. Mursaleen, Edited by Rifat Çolak, Bentham Science Publishers, Ltd., Oak Park, IL (2012)
##[12]
F. Başar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, translated from Ukraïn. Mat. Zh., 55 (2003), 108–118, Ukranian Math. J., 55 (2003), 136-147
##[13]
F. Başar, M. Kirişci, Almost convergence and generalized difference matrix, Comput. Math. Appl., 61 (2011), 602-611
##[14]
F. Başar, I. Solak, Almost-coercive matrix transformations, Rend. Mat. Appl., 11 (1991), 249-256
##[15]
L. S. Bosanquet, Note on the Bohr-Hardy theorem, J. London Math. Soc., 17 (1942), 166-173
##[16]
M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl., 2012 (2012), 1-15
##[17]
M. Candan, Almost convergence and double sequential band matrix, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 354-366
##[18]
M. Candan, A new sequence space isomorphic to the space \(\ell(p)\) and compact operators, J. Math. Comput. Sci., 4 (2014), 306-334
##[19]
M. Candan, Domain of the double sequential band matrix in the spaces of convergent and null sequences, Adv. Difference Equ., 2014 (2014 ), 1-18
##[20]
M. Candan, K. Kayaduman, Almost convergent sequence space derived by generalized Fibonacci matrix and Fibonacci core, British J. Math. Comput. Sci., 7 (2015), 150-167
##[21]
S. Chapman, On Non-Integral Orders of Summability of Series and Integrals, Proc. London Math. Soc., 2 (1911), 369-409
##[22]
S. Demiriz, O. Duyar, On some new difference sequence spaces of fractional order, Int. J. Mod. Math. Sci., 13 (2015), 1-11
##[23]
S. Demiriz, E. E. Kara, M. Başarır, On the Fibonacci almost convergent sequence space and Fibonacci core, Kyungpook Math. J., 55 (2015), 355-372
##[24]
J. P. Duran , Infinite matrices and almost-convergence, Math. Z., 128 (1972), 75-83
##[25]
S. Dutta, P. Baliarsingh, A note on paranormed difference sequence spaces of fractional order and their matrix transformations, J. Egyptian Math. Soc., 22 (2014), 249-253
##[26]
C. Eizen, G. Laush, Infinite matrices and almost convergence, Math. Japon, 14 (1969), 137-143
##[27]
U. Kadak, Generalized lacunary statistical difference sequence spaces of fractional order, Int. J. Math. Math. Sci., 2015 (2015 ), 1-6
##[28]
U. Kadak, Weighted statistical convergence based on generalized difference operator involving (p, q)-gamma function and its applications to approximation theorems, J. Math. Anal. Appl., 448 (2017), 1633-1650
##[29]
U. Kadak, P. Baliarsingh , On certain Euler difference sequence spaces of fractional order and related dual properties, J. Nonlinear Sci. Appl., 8 (2015), 997-1004
##[30]
U. Kadak, N. L. Braha, H. M. Srivastava, Statistical weighted B-summability and its applications to approximation theorems, Appl. Math. Comput., 302 (2017), 80-96
##[31]
E. E. Kara, M. İlkhan, On some Banach sequence spaces derived by a new band matrix, British J. Math. Comput. Sci., 9 (2015), 141-159
##[32]
E. E. Kara, M. İlkhan, Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra, 64 (2016), 2208-2223
##[33]
A. Karaisa, F. Özger, Almost difference sequence spaces derived by using a generalized weighted mean, J. Comput. Anal. Appl., 19 (2015), 27-38
##[34]
A. Karaisa, F. Özger, On almost convergence and difference sequence spaces of order m with core theorems, Gen. Math. Notes, 26 (2015), 102-125
##[35]
K. Kayaduman, M. Şengönül, The spaces of Cesáro almost convergent sequences and core theorems, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 2265-2278
##[36]
J.P. King, Almost summable sequences, Proc. Amer. Math. Soc., 17 (1966), 1219-1225
##[37]
M. Kirişci, Almost convergence and generalized weighted mean, First International Conference on Analysis and Applied Mathematics, AIP Conf. Proc., 1470 (2012), 191-194
##[38]
M. Kirişci, On the spaces of Euler almost null and Euler almost convergent sequences, Commun. Fac. Sci. Univ. Ank. Sr. A1 Math. Stat., 62 (2013), 85-100
##[39]
M. Kirişci, Almost convergence and generalized weighted mean II, J. Inequal. Appl., 2014 (2014), 1-13
##[40]
M. Kirişci, A note on the some geometric properties of the sequence spaces defined by Taylor method, ArXiv, 2016 (2016 ), 1-11
##[41]
B. Kuttner, On differences of fractional order, Proc. London Math. Soc., 7 (1975), 453-466
##[42]
B. Kuttner, A limitation theorem for differences of fractional order, J. London Math. Soc., 43 (1968), 758-762
##[43]
G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math., 80 (1948), 167-190
##[44]
E. Malkowsky, Recent results in the theory of matrix transformations in sequence spaces, 4th Symposium on Mathematical Analysis and Its Applications, Arane. lovac, (1997), Mat. Vesnik, 49 (1997), 187-196
##[45]
H. I. Miller, C. Orhan, On almost convergent and statistically convergent subsequences, Acta Math. Hungar., 93 (2001), 135-151
##[46]
M. Mursaleen, On A-invariant mean and A-almost convergence, Anal. Math., 37 (2011), 173-180
##[47]
P. N. Ng, P. Y. Lee, Cesàro sequence spaces of non-absolute type, Comment. Math. Prace Mat., 20 (1977/78), 429-433
##[48]
I. I. Ogieveckiĭ, On the summability of series by Borel’s method of fractional order, (Ukrainian) Dopovidi Akad. Nauk Ukraïn. RSR, 1959 (1959 ), 815-818
##[49]
I. I. Ogieveckiĭ, On the theory of summability of series by Borel’s method of fractional order, II, (Ukrainian) Dopovidi Akad. Nauk Ukraïn. RSR, 1962 (1962 ), 719-722
##[50]
P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc., 36 (1972), 104-110
##[51]
J. A. Siddiqi, Infinite matrices summing every almost periodic sequence, Pacific J. Math., 39 (1971), 235-251
##[52]
A. Sönmez, Almost convergence and triple band matrix, Math. Comput. Model., 57 (2012), 2393-2402
##[53]
A. Wilansky, Summability through functional analysis, North-Holland Mathematics Studies, Notas de Matemtica [Mathematical Notes], North-Holland Publishing Co., Amsterdam (1984)
]
Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations
Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations
en
en
In the present paper, we consider the split variational inequality and fixed point problem that requires to find a solution
of a generalized variational inequality in a nonempty closed convex subset \(\mathcal{C}\) of a real Hilbert space \(\mathcal{H}\) whose image under a
nonlinear transformation is a fixed point of a pseudocontractive operator. An iterative algorithm is introduced to solve this split
problem and the strong convergence analysis is given.
843
854
Yonghong
Yao
Department of Mathematics
Tianjin Polytechnic University
China
yaoyonghong@aliyun.com
Yeong-Cheng
Liou
Department of Healthcare Administration and Medical Informatics; and Research Center of Nonlinear Analysis and Optimization and Center for Fundamental Science
Kaohsiung Medical University
Taiwan
simplex_liou@hotmail.com
Jen-Chih
Yao
Center for General Education
China Medical University
Taiwan
yaojc@mail.cmu.edu.tw
Split problem
variational inequality
fixed point
iterative algorithm
pseudocontractive mappings.
Article.43.pdf
[
[1]
M. Aslam Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199-277
##[2]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120
##[3]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642
##[4]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116-2125
##[5]
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
##[6]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239
##[7]
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
##[8]
F. Facchinei, J.-S. Pang, Finite-dimensional variational inequalities and complementarity problems, Vol. I, II, Springer Series in Operations Research, Springer-Verlag, New York (2003)
##[9]
R. Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag , New York (1984)
##[10]
Z.-H. He, W.-S. Du, On hybrid split problem and its nonlinear algorithms, Fixed Point Theory Appl., 2013 (2013 ), 1-20
##[11]
Z.-H. He, W.-S. Du, On split common solution problems: new nonlinear feasible algorithms, strong convergence results and their applications, Fixed Point Theory Appl., 2014 (2014 ), 1-16
##[12]
A. N. Iusem, An iterative algorithm for the variational inequality problem, Mat. Apl. Comput., 13 (1994), 103-114
##[13]
G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) ´Ekonom. i Mat. Metody, 12 (1976), 747-756
##[14]
P. E. Maingé , Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479
##[15]
B. Qu, N.-H. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665
##[16]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256
##[17]
H.-K. Xu, A variable Krasnoselskiı-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034
##[18]
H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 1-17
##[19]
Q.-Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266
##[20]
Y.-H. Yao, R.-D. Chen, H.-K. Xu, Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447-3456
##[21]
Y.-H. Yao, W. Jigang, Y.-C. Liou, Regularized methods for the split feasibility problem, Abstr. Appl. Anal., 2012 (2012 ), 1-13
##[22]
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
##[23]
Y.-H. Yao, Y.-C. Liou, J.-C. Yao, Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction, Fixed Point Theory Appl., 2015 (2015 ), 1-19
##[24]
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
##[25]
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
##[26]
Y.-H. Yao, M. Postolache, Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013 ), 1-12
##[27]
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
##[28]
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
##[29]
L. J. Zhang, J. M. Chen, Z. B. Hou, Viscosity approximation methods for nonexpansive mappings and generalized variational inequalities, (Chinese) Acta Math. Sinica (Chin. Ser.), 53 (2010), 691-698
##[30]
J.-L. Zhao, Q.-Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799
##[31]
H.-Y. Zhou, Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces, Nonlinear Anal., 70 (2009), 4039-4046
]