A Shrinking Projection Method for Generalized Mixed Equilibrium Problems, Variational Inclusion Problems and a Finite Family of Quasi-Nonexpansive Mappings
© Wiyada Kumam et al. 2010
Received: 21 March 2010
Accepted: 29 June 2010
Published: 14 July 2010
The purpose of this paper is to consider a shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S.Takahashi and W. Takahashi (2008), and many others.
Throughout this paper, we assume that is a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . We denote weak convergence and strong convergence by notations and , respectively.
Recall that the following definitions.
It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous. It is easy to see that if any constant is in , then the mapping is nonexpansive, where is the identity mapping on
A set-valued mapping is called monotone if for all , and implying . A monotone mapping; is maximal if its graph 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 all imply .
(R2)The resolvent operator is 1-inverse strongly monotone; see , that is,
(R3)The solution of problem (1.3) is a fixed point of the operator for all ; see also , that is,
The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problems as special cases (see, e.g., [3–8]). Some methods have been proposed to solve the generalized mixed equilibrium problems, generalized equilibrium problems and equilibrium problems; see, for instance, [9–22].
In 2010, Katchang and Kumam  introduced an 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 mappings, and inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space.
In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed shrinking projection method under some suitable conditions. The results obtained in this paper extend and improve several recent results in this area.
We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
(B2)C is bounded set.
Lemma 2.6 (see ).
3. Main Results
In this section, we will introduce an iterative scheme by using shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space.
Such a mapping is called the K-mapping generated by and ; see .
We have the following crucial Lemma 3.1 and Lemma 3.2 concerning -mapping which can be found in . Now we only need the following similar version in Hilbert spaces.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasi-nonexpansive mappings and -Lipschitz mappings of into itself with and let be real numbers such that for every , and . Let be the K-mapping generated by and . Then, the followings hold:
Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasi-nonexpansive mappings and -Lipschitz mappings of into itself and sequences in such that Moreover, for every , let and be the K-mappings generated by and , and and , respectively. Then, for every , we have
Now we study the strong convergence theorem concerning the shrinking projection method.
Next, we will divide the proof into six steps.
From Theorem 3.3, we can obtain the following results.
We can obtain the desired conclusion from Theorem 3.3 immediately.
Next, we consider another class of important nonlinear mappings: strict pseudocontractions.
Now, we get the following result.
By using Theorem 3.5, it is easy to obtain the desired conclusion.
The authors would like to express their thank to the referees for helpful suggestions. The first author was supported by the National Research Council of Thailand and the Faculty of Science and Technology RMUTT Research Fund. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute. The third author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044.
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