Hybrid Methods for Equilibrium Problems and Fixed Points Problems of a Countable Family of Relatively Nonexpansive Mappi
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Research Article Hybrid Methods for Equilibrium Problems and Fixed Points Problems of a Countable Family of Relatively Nonexpansive Mappings in Banach Spaces Somyot Plubtieng and Wanna Sriprad Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Somyot Plubtieng, [email protected] Received 1 August 2009; Accepted 19 November 2009 Academic Editor: Tomonari Suzuki Copyright q 2010 S. Plubtieng and W. Sriprad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of a countable family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Moreover, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. Our result improve and extend the corresponding results announced by Takahashi and Zembayashi 2008 and 2009, and many others.
1. Introduction Let E be a real Banach space and E∗ the dual space of E. Let C be a nonempty closed convex subset of E and f a bifunction from C × C to R, where R denotes the set of real numbers. The equilibrium problem is to find p ∈ C such that f p, y ≥ 0,
∀y ∈ C.
1.1
The set of solutions of 1.1 is denoted by EPf. Given a mapping T : C → E∗ , let fx, y T x, y − x for all x, y ∈ C. Then, p ∈ EPf if and only if T p, y − p ≥ 0 for all y ∈ C, that is, p is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduced to find a solution of 1.1. Some methods have been proposed to solve the
2
Fixed Point Theory and Applications
equilibrium problem; see, for instance, Blum and Oettli 1, Combettes and Hirstoaga 2, and Moudafi 3. Recall that a mapping S : C → C is said to be nonexpansive if Sx − Sy ≤ x − y,
∀x, y ∈ C.
1.2
We denote by FS the set of fixed points of S. If a Banach space E is uniformly convex, C ⊂ E is bounded, closed and convex, and S is a nonexpansive mapping of C into itself, then FS is nonempty; see 4 for more details. Recently, many authors studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces, respectively; see, for instance, 5–13 and the references therein. A popular method is the hybrid projection method developed by Nakajo and Takahashi 14, Kamimura and Takahashi 15, and Martinez-Yanes and Xu 16; see also 5, 17–20 and references therein. Recently Takahashi et al. 21 introduced an alterative projection method, which is called the shrinking projection method, and they showed several stro
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