Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space
© S. Plubtieng and K. Sombut. 2010
Received: 26 January 2010
Accepted: 15 April 2010
Published: 24 May 2010
We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. The result obtained in this paper improves and extends the recent ones announced by Moudafi (2009),Iemoto and Takahashi (2009), and many others. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.
In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see .
where and satisfies the assumptions and and proved that in case is a Banach space, and is closed, and is continuous, then the convergence of to a point implies that . Recently, Dotson  proved that a Mann iteration process was applied to the approximation of fixed points of quasi-nonexpansive mappings in Hilbert space and in uniformly convex and strictly convex Banach spaces.
which is called a general system of variational inequalities where and are two constants. Moreover, if we add up the requirement that , then problem (1.10) reduces to the classical variational inequality .
Then, they proved that the iterative sequence converges strongly to a common element of the set of fixed points of a nonexpansive mapping and a general system of variational inequalities with inverse-strongly monotone mappings under some parameters controlling conditions.
In this paper, we introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping.
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.7 (see ).
Lemma 2.8 (see ).
The following lemma was also given in .
Lemma 2.9 (see ).
We note that Lemma 2.9 is equivalent to the following lemma.
Then, the following results hold:
Lemma 2.11 (see ).
By a similar argument as in the proof of Proposition 2.1 in , we obtain the desired result.
We note from Lemma 2.11 that the mapping is nonexpansive. Moreover, if is a closed bounded convex subset of , then the solution of problem (1.8) always exists. Throughout this paper, we denote the set of solutions of (1.8) by
3. Main Result
In this section, we prove a weak convergence theorem for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the system of mixed equilibrium problems.
The authors would like to thank the referee for the insightful comments and suggestions. The first author is thankful to the Thailand Research Fund for financial support under Grant BRG5280016. Moreover, the second author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant CHE-Ph.D-THA-SUP/86/2550, Thailand.
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