- Research Article
- Open Access
Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problems
© Y.-A. Chen and Y.-P. Zhang. 2010
- Received: 27 December 2009
- Accepted: 1 June 2010
- Published: 24 June 2010
We introduce an iterative scheme by the viscosity iterative method for finding a common element of the solution set of an equilibrium problem, the solution set of the variational inequality, and the fixed points set of infinitely many nonexpansive mappings in a Hilbert space. Then we prove our main result under some suitable conditions.
- Hilbert Space
- Real Number
- Variational Inequality
- Convex Subset
- Equilibrium Problem
Recently, S. Takahashi and W. Takahashi  introduced an iterative scheme for finding a common element of the solution set of (1.1) and the fixed points set of a nonexpansive mapping in a Hilbert space. If is bifunction which satisfies the following conditions:
then they proved the following strong convergence theorem.
Theorem A (see ).
Such a mapping is called the -mapping generated by and (see ).
where and are sequences in and are sequences in , is a fixed contractive mapping with contractive coefficient , is an -inverse-strongly monotone mapping of to , is a bifunction which satisfies conditions , and is generated by (1.8). Then we proved that the sequences and converge strongly to , where .
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Then, the following holds:
Lemma 2.5 (Opial's theorem ).
Let be a sequence of nonexpansive self-mappings on , where is a nonempty, closed and convex subset of a real Hilbert space . Given a sequence in , one defines a sequence of self-mappings on generated by (1.8). Then one has the following results.
Lemma 2.6 (see ).
Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and is a sequence in for some . Then, for every and the limit exists.
Lemma 2.9 (see ).
Let be a Hilbert space. Let be a nonempty, closed, and convex subset of . Let be a bifunction which satisfies conditions , an -inverse-strongly monotone mapping of to , a contraction of into itself, and a sequence of nonexpansive self-mappings on such that . Suppose that , and are sequences in , and and are sequences in which satisfies the following conditions:
for all So is a contraction by Banach contraction principle . Since is a complete space, there exists a unique element such that .
Using Theorem 3.1, we prove the following theorem.
The author would like to express his thanks to Professor Simeon Reich, Technion-Israel Institute of Technology, Israel, and the anonymous referees for their valuable comments and suggestions on a previous draft, which resulted in the present version of the paper. This work was supported by the Natural Science Foundation of China (10871217) and Grant KJ080725 of the Chongqing Municipal Education Commission.
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