Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems

Fixed Point Theory and Applications, Jan 2014

Using Cesàro means of a mapping, we modify the progress of Mann’s iteration in hybrid method for asymptotically nonexpansive mappings in Hilbert spaces. Under suitable conditions, we prove that the iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. We also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces. MSC:47H10, 47J25, 90C33.

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Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems

Fixed Point Theory and Applications Iterative algorithms based on hybrid method and Cesro mean of asymptotically nonexpansive mappings for equilibrium problems Jingxin Zhang 0 Yunan Cui 1 0 Mathematics and Applied Mathematics, Harbin University of Commerce , Harbin, 150028 , P.R. China 1 Department of Mathematics, Harbin University of Science and Technology , Harbin, 150080 , P.R. China Using Cesro means of a mapping, we modify the progress of Mann's iteration in hybrid method for asymptotically nonexpansive mappings in Hilbert spaces. Under suitable conditions, we prove that the iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. We also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces. MSC: 47H10; 47J25; 90C33 strong convergence theorem; asymptotically nonexpansive mapping; hybrid method; equilibrium problem; Cesro means 1 Introduction Let H be a real Hilbert space with the inner product , and the norm a nonempty closed convex subset of H. A mapping T : C C is said to be asymptotilimn kn = such that T nx T ny kn x y , x, y C; when kn , T is called nonexpansive. The concept of asymptotically nonexpansive mapping was introduced by Goebel and known that if T : H H is asymptotically nonexpansive, then F(T ) is nonempty convex. In , Mann [] introduced the iteration as follows: a sequence {xn} defined by In an infinite-dimensional Hilbert space, Mann iteration could conclude only weak convergence []. Attempts to modify the Mann iteration method (.) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [] proposed the following modification of Mann iteration method for a nonexpansive mapping T in a Hilbert n = ( n) kn (diam C) as n . where PK denotes the metric projection from H onto a closed convex subset K of H. The above method is also called CQ method or hybrid method. In , Kim and Xu [] adapted the iteration (.) in a Hilbert space. More precisely, they introduced the following iteration process for asymptotically nonexpansive mappings: They proved that {xn} converges in norm to PF(T)x under some conditions. Several authors (see [, ]) have studied the convergence of hybrid method. Baillon [] first proved that the following Cesro mean iterative sequence weakly converges to a fixed point of a nonexpansive mapping in Hilbert spaces: Tnx = n + i= Shimizu and Takahashi [] proved a strong convergence theorem of the above iteration for an asymptotically nonexpansive mapping in Hilbert spaces. Let C be a nonempty closed convex subset of a real Hilbert space H, let f : C C R be a functional, where R is the set of real numbers. The equilibrium problem is to find x C such that f (x, y) , y C. The set of solutions of (.) is denoted by EP(f ). Given a mapping T : C X*, let f (x, y) = Tx, y x , x, y C, then z EP(f ) if and only if Tz, y z , y C, i.e., z is the solution of the variational inequality. There are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an equilibrium problem. So, equilibrium problems provide us with a systematic framework to study a wide class of problems arising in financial economics, optimization and operation research etc., which motivates the extensive concern. See, for example, []. In recent years, equilibrium problems have been deeply and thoroughly researched. See, for example, []. Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, []. In , Jitpeera, Katchang, and Kumam [] found a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a -inverse strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesro mean approximation method. Motivated by the above-mentioned results, in this paper we introduce the following iteration process for asymptotically nonexpansive mappings T with C a closed convex bounded subset of a real Hilbert space: n = ( n) Ln (diam C) as n . We shall prove that the above iterative sequence {xn} converges strongly to a fixed point of T under some proper conditions. In addition, we also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces. We will use the notation for weak convergence and for strong convergence. 2 Preliminaries x y = x y x y, y x + ( )y = x + ( ) y ( ) x y for all x, y H and [, ]. It is also known that H satisfies () Opials condition, that is, for any sequence {xn} with xn (...truncated)


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Jingxin Zhang, Yunan Cui. Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems, Fixed Point Theory and Applications, 2014, pp. 16, Volume 2014, Issue 1, DOI: 10.1186/1687-1812-2014-16