A System of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Pseudocontractive Mappings in Hilbert Spaces
© Poom Kumam and Chaichana Jaiboon. 2010
Received: 2 April 2010
Accepted: 11 June 2010
Published: 4 July 2010
We introduce and analyze a new iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of a system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Furthermore, we prove new strong convergence theorems for a new iterative algorithm under some mild conditions. Finally, we also apply our results for solving convex feasibility problems in Hilbert spaces. The results obtained in this paper improve and extend the corresponding results announced by Qin and Kang (2010) and the previously known results in this area.
Let be 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. Let be a mapping. In the sequel, we will use to denote the set of fixed points of , that is, .
It is well known that if is nonempty, bounded, closed, and convex and is a nonexpansive mapping on then is nonempty; see, for example, .
The class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of fixed points for strict pseudocontractions.
where is a strict pseudocontraction. Under appropriate restrictions on , it is proved the mapping is nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudocontractions.
The generalized 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). In 1997, Combettes and Hirstoaga  introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. Many authors have proposed some useful methods for solving the , and ; see, for instance, [5, 12–23].
where is the metric projection of onto , is a -inverse-strongly monotone mapping, is a sequence in , and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.20) converges weakly to some .
where is a -inverse-strongly monotone mapping, and are sequences in the interval , and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.21) converges strongly to some .
where is a -strict pseudocontraction mapping and and are sequences in They proved that under certain appropriate conditions over , , and , the sequences and both converge strongly to some , which solves some variational inequality problems (1.26).
In 2008, Ceng and Yao  introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem (1.8) in Hilbert spaces and obtained the strong convergence theorem which used the following condition.
: is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology. We note that the condition for the function : is a very strong condition. We also note that the condition does not cover the case and for each . Very recently, Wangkeeree and Wangkeeree  introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a -strict pseudocontraction mapping, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the condition for the sequences generated by these processes.
In the present paper, motivated and inspired by Qin and Kang , Peng and Yao , Plubtieng and Punpaeng , and Liu , we introduce a new general iterative scheme for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. The results in this paper extend and improve the corresponding recent results of Qin and Kang , Peng and Yao , Plubtieng and Punpaeng , and Liu  and many others.
Then is the maximal monotone and if and only if see .
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.7 (see ).
Let be a nonempty closed convex subset of a real Hilbert space and let be a -strict pseudocontraction mapping with a fixed point. Then is closed and convex. Define by for each . Then is nonexpansive such that .
Lemma 2.8 (see ).
Lemma 2.9 (see ).
By similar argument as in the proof of Lemma in , we have the following lemma appearing.
We remark that Lemma 2.10 is not a consequence of Lemma in , because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case .
Lemma 2.12 (see ).
Lemma 2.13 (see ).
3. Main Results
In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of a common solutions of the variational inequalities for inverse-strongly monotone mappings in a real Hilbert space.
Next, we will divide the proof into five steps.
We claim that the following statements hold:
where , , , and are sequences in , where , , , and and are positive sequences. Assume that the control sequences satisfy the condition (C1)–(C6) in Theorem 3.1 and . Then, converges strongly to a point , where
where is an integer and each is assumed to be the of solutions of equilibrium problem with the bifunction and the solution set of the variational inequality problem. There is a considerable investigation on in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [38, 39], computer tomography , and radiation therapy treatment planning .
The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.
The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper. The first author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute.
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