- Research Article
- Open Access
A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems
© P. Katchang and P. Kumam 2010
- Received: 27 October 2009
- Accepted: 16 March 2010
- Published: 31 May 2010
The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Maximal Monotone
- Real Hilbert Space
where is the zero vector in . The set of solutions of problem (1.2) is denoted by .
which is called the mixed quasivariational inequality (see ).
A set-valued mapping is called monotone if, for all , and imply that . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if, for , for every implies that .
The set of solutions of (1.8) is denoted by EP(F). Given a mapping , let for all . Then if and only if for all that is, is a solution of the variational inequality. 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 (1.8). Some methods have been proposed to solve the equilibrium problem (see [8–21]).
They proved that if and satisfy appropriate conditions, then converges strongly to , where .
where is a potential function for , that is, , for all .
where . Such a mapping is called the W-mapping generated by and . Nonexpansivity of each ensures the nonexpansivity of . Moreover, in [24, Lemma ], it is shown that .
The concept of -mappings was introduced in [25, 26]. It is now one of the main tools in studying convergence of iterative methods for approaching common fixed points of nonlinear mappings; more recent progresses can be found in [24, 27, 28] and the references cited therein.
where is monotone and Lipschitz continuous mapping and is an inverse-strongly monotone mapping. They proved the weak convergence theorem if the sequences and of parameters satisfy appropriate conditions.
In this paper, motivated by the above results and the iterative schemes considered by Zhang et al. in , Peng et al. in , Peng and Yao in , and Plubtieng and Sriprad in , we present a new general iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mapping and inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Zhang et al. , Peng et al. , Peng and Yao , Plubtieng and Sriprad , and some authors.
for all .
So, if then is a nonexpansive mapping from into itself.
holds for every with .
For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, , and the set C.
(A1) for all
(A2) is monotone, that is, for all
(A4)For each is convex and lower semicontinuous.
(A5)For each is weakly upper semicontinuous.
(B2)C is a bounded set.
We need the following lemmas for proving our main result.
Lemma 2.1 (Peng and Yao ).
for all . Then, the following hold.
(1)For each .
(2) is single valued.
(3) is firmly nonexpansive, that is, for any
(5) is closed and convex.
Lemma 2.2 (Xu ).
where is a sequence in and is a sequence in such that
Lemma 2.3 (Osilike and Igbokwe ).
Lemma 2.4 (Colao et al. ).
Lemma 2.5 (Suzuki ).
Let and be bounded sequences in a Banach space and let be a sequence in with Suppose that for all integers and Then,
Lemma 2.6 (Marino and Xu ).
Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .
Lemma 2.7 (Brézis ).
Let be a maximal monotone mapping and be a Lipschitz continuous mapping. Then the mapping is a maximal monotone mapping.
Lemma 2.7 implies that is closed and convex if is a maximal monotone mapping and is a Lipschitz continuous mapping.
Lemma 2.9 (Zhang et al. ).
In this section, we prove 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 a mixed equilibrium problem, and the set of solutions of a variational inclusion problem for an inverse-strongly monotone mapping in a real Hilbert space.
for every , where and satisfy the following conditions:
(i) and ,
(iv) for all .
Therefore, is bounded. We also obtain that and , are all bounded.
where is an approximate constant such that , .
Dividing by t, we get . From (A3) and the weakly lower semicontinuity of , we have for all and hence
which is a contradiction. Thus, we obtain .
It follows from the maximal monotonicity of that , that is, . This implies that .
where . By (3.65), we get . Hence by Lemma 2.2 to (3.66), we conclude that . This completes the proof.
Using Theorem 3.1, we obtain the following corollaries.
for every , where and satisfy the conditions (i)–(iii) in Theorem 3.1. Then converges strongly to
Taking for , and , in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.
Equivalently, one has
From Theorem 3.1 put ; then . So we have and . The conclusion of Corollary 3.3 can be obtained from Theorem 3.1 immediately.
where is a convex and lower semicontinuous functional defined convex subset of a Hilbert space . We denote by the set of solutions of (4.1). Let be a bifunction defined by . We consider the equilibrium problem (1.8); it is obvious that . Therefore, from Theorem 3.1, we give the following corollary.
for every , where and satisfy the following conditions:
(i) and .
(iv) for all .
From Theorem 3.1 put and . The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.
The authors would like to thank the Centre of Excellence in Mathematics, under the Commission on Higher Education, Ministry of Education, Thailand. Mr. Phayap Katchang was supported by King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT for Ph.D. Program at KMUTT. Moreover, the authors are also very grateful to Professor Yeol Je Cho and Professor Jong Kyu Kim for the hospitality.
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