- Research Article
- Open Access
Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings
© Atid Kangtunyakarn. 2010
Received: 7 October 2010
Accepted: 2 November 2010
Published: 21 November 2010
We introduce a new method for a system of generalized equilibrium problems, system of variational inequality problems, and fixed point problems by using -mapping generated by a finite family of nonexpansive mappings and real numbers. Then, we prove a strong convergence theorem of the proposed iteration under some control condition. By using our main result, we obtain strong convergence theorem for finding a common element of the set of solution of a system of generalized equilibrium problems, system of variational inequality problems, and the set of common fixed points of a finite family of strictly pseudocontractive mappings.
Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be a nonlinear mapping, and let be a bifunction. A mapping of into itself is called nonexpansive if for all . We denote by the set of fixed points of (i.e., ). Goebel and Kirk  showed that is always closed convex, and also nonempty provided has a bounded trajectory.
The set of solutions of (1.2) is denoted by . Many problems in physics, optimization, and economics are seeking some elements of , see [2, 3]. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, [2–4]. In 2005, Combettes and Hirstoaga  introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem.
The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mapping (see [6, 7]).
The ploblem of finding a common element of and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance. Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed point problem of nonexpansive mappings, see [8–10].
where is an -inverse strongly monotone mapping of into with positive real number , and , , , and proved strong convergence of the scheme (1.7) to , where in the framework of a Hilbert space, under some suitable conditions on , , and bifunction .
where is a contraction mapping and is -mapping generated by infinite family of nonexpansive mappings and infinite real number. Under suitable conditions of these parameters they proved strong convergence of the scheme (1.8) to , where .
In this section, we collect and give some useful lemmas that will be used for our main result in the next section.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
The following lemma appears implicitly in .
Lemma 2.6 (see ).
Lemma 2.7 (see ).
In 2009, Kangtunyakarn and Suantai  defined a new mapping and proved their lemma as follows.
Let be a nonempty closed convex subset of strictly convex. Let be a finite family of nonexpanxive mappings of into itself with , and let , , where , , for all , for all . Let be the mapping generated by and . Then .
Let be a nonempty closed convex subset of Banach space. Let be a finite family of nonexpansive mappings of into itself and , , where , and such that as for and Moreover, for every , let and be the -mappings generated by and and and , respectively. Then for every .
Lemma 2.11 (see ).
3. Main Result
We will divide our proof into 6 steps.
To prove strong convergence theorem in this section, we need definition and lemma as follows.
Lemma 4.2 (see ).
The authors would like to thank Professor Dr. Suthep Suantai for his suggestion in doing and improving this paper.
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