%0 Journal Article
%T Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi- -Asymptotically Nonexpansive Mappings
%A Siwaporn Saewan
%A Poom Kumam
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/357120
%X We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results. 1. Introduction and Preliminaries The convex feasibility problem (CFP) is the problem of computing points laying in the intersection of a finite family of closed convex subsets , of a Banach space This problem appears in various fields of applied mathematics. The theory of optimization [1], Image Reconstruction from projections [2], and Game Theory [3] are some examples. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [4]. The advantage of a Hilbert space is that the projection onto a closed convex subset of is nonexpansive. So projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [5] deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in a uniformly convex Banach space. It is known that if is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space, then the generalized projection (see, Alber [6] or Kamimura and Takahashi [7]) from onto is relatively nonexpansive, whereas the metric projection from onto is not generally nonexpansive. We note that the block iterative method is a method which is often used by many authors to solve the convex feasibility problem (CFP) (see, [8, 9], etc.). In 2008, Plubtieng and Ungchittrakool [10] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let be a nonempty closed convex subset of a real Banach space with and being the dual space of . Let be a bifunction of into and a monotone mapping. The generalized equilibrium problem, denoted by , is to find such that The set of solutions for the problem (1.1) is denoted by , that is If , the problem (1.1) reducing into the equilibrium problem for , denoted by , is to find such that If , the problem (1.1) reducing into the
%U http://www.hindawi.com/journals/aaa/2010/357120/