A Viscosity of Cesàro Mean Approximation Methods for a Mixed Equilibrium, Variational Inequalities, and Fixed Point Problems
© Thanyarat Jitpeera et al. 2011
Received: 6 September 2010
Accepted: 15 October 2010
Published: 20 October 2010
We introduce a new iterative method for finding 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 Cesàro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang (2009), Peng and Yao (2009), Shimizu and Takahashi (1997), and some authors.
Throughout this paper, we assume that is a real Hilbert space with inner product and norm are denoted by and , respectively and let be a nonempty closed convex subset of . A mapping is called if , for all . We use to denote the set of fixed points of , that is, . It is assumed throughout the paper that is a nonexpansive mapping such that . Recall that a self-mapping is a on if there exists a constant and such that
The set of solutions of (1.2) is denoted by . The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.2). Some methods have been proposed to solve the equilibrium problem (see [2–14]).
In this paper, motivated by the above results and the iterative schemes considered in [9, 18–20], we introduce a new iterative process below based on viscosity and Cesàro mean approximation method for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of the variational inequality problem for a -inverse-strongly monotone mapping and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we prove strong convergence theorems which are connected with [5, 21–24]. We extend and improve the corresponding results of Kumam and Katchang , Peng and Yao , Shimizu and Takahashi  and some authors.
Then is the maximal monotone and if and only if ; see .
(B2)C is a bounded set.
We need the following lemmas for proving our main results.
Lemma 2.1 (Peng and Yao ).
Lemma 2.2 (Xu ).
Lemma 2.3 (Osilike and Igbokwe ).
Lemma 2.4 (Suzuki ).
Lemma 2.5 (Marino and Xu ).
Lemma 2.6 (Bruck ).
3. Main Results
In this section, we show a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of mixed equilibrium problem and the set of solutions of a variational inequality problem for a -inverse-strongly monotone mapping in a real Hilbert space by using the viscosity of Cesàro mean approximation method.
Using Theorem 3.1, we obtain the following corollaries.
4. Applications to Optimization Problem
The authors would like to thank the Faculty of science KMUTT. Poom Kumam was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5380044.
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