Viscosity Approximation Methods for Generalized Mixed Equilibrium Problems and Fixed Points of a Sequence of Nonexpansive Mappings
© Wei-You Zeng et al. 2008
Received: 17 July 2008
Accepted: 11 November 2008
Published: 19 November 2008
We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.
Equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization, which has been extended and generalized in many directions using novel and innovative techniques; see [1–8]. Inspired and motivated by the research and activities going in this fascinating area, we introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems.
Such a mapping is called the -mapping generated by and , see .
The purpose of this paper is to develop an iterative algorithm for finding a common element of set of solutions of GMEP (1.2) and set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. The result presented in this paper improves and extends the main result of S. Takahashi and W. Takahashi .
We denote by the set of fixed points of a self-mapping on , that is, . It is well known that if is nonempty, bounded, closed, and convex and is nonexpansive, then is nonempty; see . Let be a sequence of nonexpansive mappings of into itself, where is a nonempty closed convex subset of a real Hilbert space . Given a sequence in , we define a sequence of self-mappings on by (1.4). Then we have the following lemmas which are important to prove our results.
Lemma 2.1 (see ).
Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, for every and the limit exists.
Lemma 2.2 (see ).
We denote for weak convergence and for strong convergence. A function is called weakly sequentially continuous at , if as for each sequence in converging weakly to . The function is called weakly sequentially continuous on if it is weakly sequentially continuous at each point of .
Lemma 2.3 (see ).
Lemma 2.4 (see ).
the auxiliary problem (2.8) has a unique solution;
We also need the following lemmas for our main results.
Lemma 2.5 (see ).
Lemma 2.7 (see ).
Lemma 2.8 (see ).
3. Iterative Algorithm and Convergence Theorem
Let be a nonempty closed convex subset of a real Hilbert space , a multivalued mapping, a contraction mapping with constant , and an -mapping generated by and , where sequence is nonexpansive. Let be a sequence in and a sequence in . We can develop Algorithm 3.1 for finding a common element of a set of fixed points of -mapping and a set of solutions of GMEP(1.2).
Let be a nonempty closed convex bounded subset of a real Hilbert space , a multivalued -Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying conditions and :
Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an -strongly convex function with constant which satisfies conditions and with . Let be an -mapping generated by and and , where sequence is nonexpansive. Let , and be sequences generated by Algorithm 3.1, where is a sequence in and in satisfying the following conditions:
Theorem 3.2 improves and extends the main results of S. Takahashi and W. Takahashi .
Let be a nonempty closed convex bounded subset of a real Hilbert space , a Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying the conditions and :
The authors would like to thank the referees very much for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005).
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