A Method for a Solution of Equilibrium Problem and Fixed Point Problem of a Nonexpansive Semigroup in Hilbert's Spaces
© Nguyen Buong and Nguyen Dinh Duong. 2011
Received: 3 October 2010
Accepted: 13 January 2011
Published: 15 February 2011
We introduce a viscosity approximation method for finding a common element of the set of solutions for an equilibrium problem involving a bifunction defined on a closed, convex subset and the set of fixed points for a nonexpansive semigroup on another one in Hilbert's spaces.
Let be a nonempty, closed, and convex subset of a real Hilbert space . Denote the metric projection from onto by . Let be a nonexpansive mapping on , that is, and , for all . We use to denote the set of fixed points of , that is, .
Denote the set of solutions of (1.1) by . We also know  that is a closed convex subset in .
In the case that , , , and , a nonexpansive mapping on , for all , (1.2) is the fixed point problem of a nonexpansive mapping. In 2000, Moudafi  proved the following strong convergence theorem.
The strong convergence of (1.12)-(1.13) and its corollaries are showed in the next section.
2. Main Results
We formulate the following facts needed in the proof of our results.
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Then, the following statements hold:
Lemma 2.6 (see ).
Lemma 2.7 (Demiclosedness Principle ).
Lemma 2.8 (see ).
Now, we are in a position to prove the following result.
Let and be two nonempty, closed, convex subsets in a real Hilbert space . Let be a bifunction from to satisfying conditions (A1)–(A4) with replaced by , let be a nonexpansive semigroup on such that and let be a contraction of into itself. Then, and generated by (1.12)-(1.13) converge strongly to , where .
As is bounded, there exists a subsequence of the sequence which converges weakly to . From (2.37), we also have that converges weakly to . Since and and , are two closed convex subsets in , we have that .
where is defined by (1.13) and , , , and satisfy conditions (i)–(v). This algorithm is different from Yao and Noor's algorithm (1.6), in which for all . It likes completely the Plubtieng and Punpaeng's algorithm (1.8), but converges under a new condition on .
This work was supported by the Vietnamese National Foundation of Science and Technology Development.
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