ALGORITHMS FOR A SYSTEM OF GENERALIZED MIXED EQUILIBRIUM PROBLEMS AND A COUNTABLE FAMILY OF SOME NONLINEAR MULTI-VALUED NONEXPANSIVE-TYPE MAPS

Abstract

Abstract: In this paper, relaxed iterative algorithms of Krasnoselskii-type and Halpern-type that approximate a solution of a system of a generalized mixed equilibrium problem and a common fixed point of a countable family of totally quasi-φ-asymptotically nonexpansive multi-valued maps are constructed. Strong convergence of the sequence generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with Kadec-Klee property. Furthermore, several applications of our theorems are also presented. Finally, our theorems are significant improvements on several important recent results for this class of nonlinear problems.

Key words: Halpern-type algorithm, Krasnoselskii-type algorithm, system of generalized mixed equilibrium problems, totally quasi-φ-asymptotically nonexpansive multi-valued maps, equally continuous maps

Ogonnaya Michael ROMANUS, Ukamaka Victoria NNYABA, Monday Ogudu NNAKWE. ALGORITHMS FOR A SYSTEM OF GENERALIZED MIXED EQUILIBRIUM PROBLEMS AND A COUNTABLE FAMILY OF SOME NONLINEAR MULTI-VALUED NONEXPANSIVE-TYPE MAPS[J].Acta mathematica scientia,Series B, 2018, 38(6): 1805-1820.

Trendmd

References

[1] Moudafi A, Thera M. Proximal and dynamical approaches to equilibrium problems//Lecture Notes in Economics and Mathematics Systems. Vol 477. Springer, 1999:187-201
[2] Barbagallo A. Existence and regularity of solutions to nonlinear degenerate evolutionary variational inequalities with applications to dynamic network equilibrium problems. Appl Math Comput, 2009, 208:1-13
[3] Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Mathematics Student, 1994, 63(1/4):123-145
[4] Stampacchia G. Formes bilineaires coercitives sur les ensembles convexes. CR Acad Sci Paris, 1964, 258:4413-4416
[5] Cioranescu I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Dordrecht, The Netherlands:Kluwer Academic Publishers, 1990
[6] Fan K. A minimax inequality and applications//Shisha O, ed. Inequality Ⅲ. New York:Academic Press, 1972:103-113
[7] Ceng L C, Yao J C. A hybrid iterative scheme for mixed equilibriun problems and fixed point problem. J Comput Appl Math, 2008, 214:186-201
[8] Ceng L C, Guu S M, Hu H Y, Yao J C. Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings. Comput Math Appl, 2011, 61:2468-2479
[9] Feng Q, Su Y, Yan F. Modified hybrid block iterative algorithm for uniformly quasi-φ-nonexpansive mappings. Abs Appl Anal, 2012, 2012, doi:10.1155/2012/215261
[10] Plubtieng S, Ungchittrakool K. Hybrid iterative method for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl, 2008, 19:Article ID 583082, doi:10.1155/2008/58308
[11] Chang S S, Wang L, Tang Y K, Zhao Y H, Ma Z L. Strong convergence theorems of nonlinear operator equations for countable family of multi-valued total quasi-φ-asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl, 2012, 2012:69
[12] Chang S S, Joseph Lee H W, Chan C K, Zhang W B. A modified halpern-type iteration algorithm for totally quasi-φ-asymptotically nonexpansive mappings with applications. Appl Maths Compu, 2012, 218:6489-6497
[13] Qian S. Strong convergence theorem for totally quasi-φ-asymptotically nonexpansive multi-valued mappings under relaxed conditions. Fixed Point Theory Appl, 2015, 2015:213, doi:10.1186/s13663-015-0452-9
[14] Zhang S S. Generalized mixed equilibrium problem in Banach spaces. Appl Math Mech, 2009, 30(9):1105-1112
[15] Homaeipour S, Razani A. Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces. Fixed Point Theory Appl, 2011, 2011:73, doi:10.1186/1687-1812-2011-73
[16] Takahashi S, Takahashi W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal, 2008, 69:1025-1033
[17] Dafermos S, Nagurney A. A network formulation of market equilibrium problems and variational inequalities. Oper Res Lett, 1984, 3:247-250
[18] Su T V. Second-order optimality conditions for vector equilibrium problems. J Nonlinear Funct Anal, 2015, 2015:31 pages
[19] Cholamjiak W, Cholamjiak P, Suantai S. Convergence of iterative schemes for solving fixed point problems for multi-valued nonself mappings and equilibrium problems. J Nonlinear Sci Appl, 2015, 8:1245-1256
[20] Deng W Q. A relaxed hybrid shrinking iteration approach to solving generalized mixed equilibrium problems for totally quasi-φ-asymptotically nonexpansive mappings. Fixed Point Theory Appl, 2014, 2014:63
[21] Qin X, Cho S Y, Kang S M. Strong convergence of shrinking projection methods for quasi-nonexpansive mappings and equilibrium problems. J Comput Appl Math, 2010, 234:750-760
[22] Cho Y J, Qin X, Kang S M. Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal, 2009, 71:4203-4214
[23] Haugazeau Y. Sur les Inequations Variationnelles et la Minimization de Fonctionnelles Convexes[D]. Paris:University of Paris, 1968
[24] Alber Ya. Metric and generalized projection operators in Banach spaces:properties and applications//Kartsatos A G, ed. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. New York:Marcel Dekker, 1996:15-50
[25] Alber Ya, Ryazantseva I. Nonlinear Ill Posed Problems of Monotone Type. London, UK:Springer, 2006
[26] Wang Z M, Su Y F, Wang D X, Dong Y C. A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces. J Comput Appl Math, 2011, 235:2364-2371
[27] Wang Z M, Zhang X. Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems. J Non Funct Anal, 2014, 2014:Article ID 15
[28] Qin X, Cho X Y. Convergence analysis of a monotone projection algorithm in reflexive Banach spaces. Acta Math Sci, 2017, 37B(2):488-502
[29] Zhao F, Yang L. Hybrid projection methods for equilibrium problems and fixed point problems of infinite family of multivalued asymptotically nonexpansive mappings. J Nonlinear Funct Anal, 2016, 2016:Article ID 22
[30] Zhang S, Wang L, Zhao Y. Multi-valued totally quasi-φ-asymptotically nonexpansive semi-groups and strong convergence theorems in Banach spaces. Acta Math Sci, 2013, 33B(2):589-599
[31] Su Y, Shang M, Wang D. Strong convergence of monotone CQ algorithm for relatively nonexpansive mappings. Banach J Math Anal, 2008, 2(1):1-10