APPROXIMATION OF COMMON FIXED POINT OF FAMILIES OF NONLINEAR MAPPINGS WITH APPLICATIONS

doi:10.1016/S0252-9602(15)30051-5

摘要/Abstract

摘要：

It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gâteaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly Gâteaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.

关键词: nonexpansive mappings, reflexive real Banach spaces, fixed point, uniformly Gâ, teaux differentiable norm

Abstract:

Key words: nonexpansive mappings, reflexive real Banach spaces, fixed point, uniformly Gâ, teaux differentiable norm

中图分类号:

47H06

Eric U. OFOEDU, Charles E. ONYI. APPROXIMATION OF COMMON FIXED POINT OF FAMILIES OF NONLINEAR MAPPINGS WITH APPLICATIONS[J]. 数学物理学报(英文版), 2015, 35(5): 1225-1240.

Eric U. OFOEDU, Charles E. ONYI. APPROXIMATION OF COMMON FIXED POINT OF FAMILIES OF NONLINEAR MAPPINGS WITH APPLICATIONS[J]. Acta mathematica scientia,Series B, 2015, 35(5): 1225-1240.

参考文献

[1] Bauschke H H. The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J Math Anal Appl, 1996, 202:150-159
[2] Belluce L P, Kirk W A. Fixed point theorem for families of contraction mappings. Pacific J Math, 1996, 18:213-217
[3] Blum E, OettliW. From optimization and variational inequalities to equilibrum problems. The Mathematics Student, 1994, 63(1/4):123-145
[4] Browder F E. Nonexpansive nonlinear operators in Banach space. Proc Nat Acad Sci USA, 1965, 54(4):1041-1044
[5] Bruck R E. Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans Amer Math Soc, 1973, 179:251-262
[6] Bruck R E, Jr. A common fixed point theorem for a commuting family of nonexpansive mappings. Pacific J Math, 1974, 53:59-71
[7] Chang S S, Tan K K, Lee H W Joseph, Chan C K. On the convergnce of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. J Math Anal Appl, 2006, 313:273-283
[8] Chidume C E. Geometric Properties of Banach Spaces and Nonlinear Iterations. Lecture Notes in Mathematics, Vol 1965. Springer-Verlag, 2009
[9] Chidume C E, Chidume C O, Nwogbaga A P. Approximation methods for common fixed points of a countable family of nonself nonexpansive mappings. Nonlinear Analysis, 2009, 71(12, 15):164-175
[10] Chidume C E, Chidume C O. Iterative methods for common fixed points for a countable family of nonex-pansive mappings in uniformly convex spaces. Nonlinear Anal, 2009, 71(10):4346-4356
[11] Chidume C E, Ofoedu E U. A new iteration process for finite families of generalized Lipschitz pseudocontractive and generalized Lipschitz accretive mappings. Nonlinear Analysis; TMA, 2008, 69(4):1200-1207
[12] Chidume C E, Ofoedu E U. Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J Math Anal Appl, 2007, 333(1):128-141
[13] Chidume C E, Zegeye H, Prempeh E. Strong convergence theorems for a common fixed point of a finite family of nonexpansive mappings. Comm Appl Nonlinear Anal, 2004, 11(2):25-32
[14] Chidume C E, Zegeye H, Shahzad N. Convergence theorems for a common fixed point of finite family of nonself nonexpansive mappings. Fixed Point Theory Appl, 2005, (2):233-241
[15] Cioranescu I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Dordrecht:Kluwer Academic, 1990
[16] DeMarr R. Common fixed points for commuting contraction mappings. Pacific J Math, 1963, 13:1139-1141
[17] Jung J S. Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. J Math Anal Appl, 2005, 302:509-520
[18] Jung J S, Cho Y J, Agarwal R P. Iterative schemes with some control conditions for family of finite nonexpansive mappings in Banach spaces. Fixed Point Theory Appl, 2005, 2:125-135
[19] Kikkawa M, Takahashi W. Strong convergence theorems by viscocity approximation methods for a countable family of nonexpansive mappings. Taiwa J Math, 2008, 12(3):583-598
[20] Kato T. Nonlinear semi-groups and evolution equations. J Math Soc Japan, 1967, 19:508-520
[21] Lim T C. A fixed point theorem for families of nonexpansive mappings. Pacific J Math, 1974, 53:487-493
[22] Maingé P. Approximation methods for common fixed points of nonexpansive mappings in Hilbert space. J Math Anal Appl, 2007, 325:469-479
[23] Megginson R E. An Introduction to Banach Space Theory. New York:Springer-Verlag, 1998
[24] Ofoedu E U, Zegeye H. Further investigation on iteration processes for pseudocontractive mappings with application. Nonlinear Anal TMA, 2012, 75:153162
[25] Katchang P, Jitpeera T, Kumam P. Strong convergence theorems for solving generalized mixed equilibrum problems and general system of variational inequalities by the hybrid method. Nonlinear Analysis:Hybrid Systems, 2010, 4(4):838-852
[26] Suzuki T. Strong convergence of Krasnoselkii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl, 2005, 305:227-239
[27] Xu H K. Iterative algorithm for nonlinear operators. J London Math Soc, 2002, 66(2):1-17
[28] Xu Z B, Roach G F. Characteristic inequalities of uniformly smooth Banach spaces. J Math Anal Appl, 1991, 157:189-210
[29] Zegeye H, Ofoedu E U, Shahzad N. Convergence theorems for equilibrum problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl Math Comput, 2010, 216:3439-3449
[30] Zegeye H. An iterativee approximation method for a common fixed point of two pseudo-contractive map-pings. ISRN Math Anal, 2011, 14(2011):Article ID621901