A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings
© C. Jaiboon and P. Kumam. 2009
Received: 25 December 2008
Accepted: 4 May 2009
Published: 9 June 2009
We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for -inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area.
One can see that the variational inequality (1.1) is equivalent to a fixed point problem. The variational inequality has been extensively studied in literature; see, for instance, [2–6]. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems. Recall the following.
A set-valued mapping is called monotone if for all , and imply . 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 . Let be a monotone map of into and let be the normal cone to at , that is, and define
The set of solutions of (1.13) is denoted by . Given a mapping let for all . Then, if and only if for all Numerous problems in physics, saddle point problem, fixed point problem, variational inequality problems, optimization, and economics are reduced to find a solution of (1.13). Some methods have been proposed to solve the equilibrium problem; see, for instance, [10–16]. Recently, 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.
In 1976, Korpelevich  introduced the following so-called extragradient method:
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [21–24] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :
Furthermore, for finding approximate common fixed points of an infinite countable family of nonexpansive mappings under very mild conditions on the parameters. Wangkeeree  introduced an iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.13) and the set of common fixed points of a countable family of nonexpansive mappings on . Starting with an arbitrary initial , define a sequence recursively by
where and are sequences in . It is proved that under certain appropriate conditions imposed on and , the sequence generated by (1.22) strongly converges to the unique solution , where which extend and improve the result of Kumam .
Definition 1.2 (see ).
On the other hand, Colao et al.  introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.13) and the set of common fixed points of infinitely many nonexpansive mappings on . Starting with an arbitrary initial , define a sequence recursively by
where is a sequence in . It is proved  that under certain appropriate conditions imposed on and , the sequence generated by (1.24) strongly converges to , where is an equilibrium point for and is the unique solution of the variational inequality (1.20), that is, .
In this paper, motivated by Wangkeeree , Plubtieng and Punpaeng , Marino and Xu , and Colao, et al. , we introduce a new iterative scheme in a Hilbert space which is mixed bythe iterative schemes of (1.18), (1.19), (1.22), and (1.24) as follows.
where is the sequence generated by (1.23), and and satisfying appropriate conditions. We prove that the sequences , , and generated by the above iterative scheme (1.25) converge strongly to a common element of the set of solutions of the equilibrium problem (1.13), the set of common fixed points of infinitely family nonexpansive mappings, and the set of solutions of variational inequality (1.1) for a -inverse-strongly monotone mapping in Hilbert spaces. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree , Plubtieng and Punpaeng , Marino and Xu , Colao, et al. , and many others.
We now recall some well-known concepts and results.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
The following lemma appears implicitly in .
Lemma 2.3 (see ).
The following lemma was also given in .
Lemma 2.4 (see ).
For each , let the mapping be defined by (1.23). Then we can have the following crucial conclusions concerning . You can find them in . Now we only need the following similar version in Hilbert spaces.
Lemma 2.5 (see ).
Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for every . Then, for every and , the limit exists.
Lemma 2.6 (see ).
Lemma 2.7 (see ).
Lemma 2.8 (see ).
Lemma 2.9 (see ).
3. Main Results
In this section, we prove the strong convergence theorem for infinitely many nonexpansive mappings in a real Hilbert space.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), let be an infinitely many nonexpansive of into itself, and let be an -inverse-strongly monotone mapping of into such that . Let be a contraction of into itself with and let be a strongly positive linear bounded operator on with coefficient and . Let , , and be sequences generated by (1.25), where is the sequence generated by (1.23), , are three sequences in and is a real sequence in satisfying the following conditions:
We will divide the proof into five steps.
Combining (3.27) and (3.28), we have
We claim that the following statements hold:
which together with (3.38) gives
Corollary 3.2 ([28, Theorem 3.1]).
4. Application for Optimization Problem
It is obvious that , where denotes the set of solution of equilibrium problem (4.2). In addition, it is easy to see that satisfies the conditions (A 1)–(A 4) in Section 1. Therefore, from the Corollary 3.6, we know the following iterative sequence defined by
In fact, the solution of optimization problem (4.4) is named the minimum norm point on the closed convex set . From iterative algorithm (4.4) we obtain the following iterative algorithm (4.5), and is defined by
The first author was supported by the Faculty of Applied Liberal Arts RMUTR Research Fund and King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT. The second author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5180034. Moreover, the authors would like to thank Professor Somyot Plubiteng for providing valuable suggestions, and they also would like to thank the referee for the comments.
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