Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces
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Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces Muhammad Aqeel Ahmad Khan1* , Hafiz Fukhar-ud-din1,2 and Abdul Rahim Khan2 Dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday *
Correspondence: [email protected] 1 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan Full list of author information is available at the end of the article
Abstract In this paper, we propose and analyze an explicit type algorithm for finding a common element of the set of solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two countable families of total quasi-ϕ -asymptotically nonexpansive mappings in a Banach space E. As an application of our result, we suggest a framework for finding a common solution of a finite family of generalized equilibrium problems and common zeros of two finite families of maximal monotone operators on E. MSC: 47H05; 47H10; 47H15; 47J25; 49M05 Keywords: total quasi-ϕ -asymptotically nonexpansive mapping; asymptotic fixed point; equilibrium problem; variational inequality problem; maximal monotone operator; Mosco convergence
1 Introduction and preliminaries Let E be a real Banach space with the norm · and E∗ be its dual. Let C be a nonempty subset of E and T : C → C be a mapping. We denote by F(T) = {x ∈ C : x = Tx} the set of fixed points of T. We symbolize weak convergence and strong convergence of a sequence {xn } in E as xn x and xn → x, respectively. Let f : C × C → R (the set of reals) be a bifunction and A : C → E∗ be a nonlinear mapping. A generalized equilibrium problem is to find the set GEP(f ) = x ∈ C : f (x, y) + Ax, y – x ≥ for all y ∈ C ,
(.)
where · , · stands for the duality product. Note that: (i) if A ≡ , then problem (.) reduces to the following equilibrium problem EP(f ): find x ∈ C such that f (x, y) ≥ for all y ∈ C; (ii) if f ≡ , then problem (.) reduces to the classical variational inequality problem VI(C, A): find x ∈ C such that Ax, y – x ≥ for all y ∈ C. © 2014 Khan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Khan et al. Fixed Point Theory and Applications 2014, 2014:59 http://www.fixedpointtheoryandapplications.com/content/2014/1/59
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The equilibrium problem provides a unified approach to finding a solution of a large number of problems arising in physics, optimization, economics and fixed point problems []. Moreover, the generalized equilibrium problem addresses monotone inclusion problems, variational inequality problems, minimization problems and vector equilibrium problem [–]. Since an algorithmic construction plays a key role in solving nonlinear equations in various fields of investigation, num
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