- Research Article
- Open Access
Strong Convergence Theorems for a Generalized Equilibrium Problem with a Relaxed Monotone Mapping and a Countable Family of Nonexpansive Mappings in a Hilbert Space
© Shenghua Wang et al. 2010
- Received: 15 March 2010
- Accepted: 20 June 2010
- Published: 8 July 2010
We introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space and then prove that the sequence converges strongly to a common element of the two sets. Using this result, we prove several new strong convergence theorems in fixed point problems, variational inequalities, and equilibrium problems.
- Hilbert Space
- Variational Inequality
- Iterative Method
- Equilibrium Problem
- Monotone Mapping
then is called a solution of the variational inequality. The set of all solutions of the variational inequality is denoted by . It is known that is closed and convex. Recently Takahashi and Toyoda  introduced an iterative method for finding an element of ; see also . On the other hand, Plubtieng and Punpaeng  introduced an iterative method for finding an element of ; see also .
where is a constant; see . In the case of for all , is said to be relaxed -monotone. In the case of for all and , where and , is said to be -monotone; see [11–13]. In fact, in this case, if , then is a -strongly monotone mapping. Moreover, every monotone mapping is relaxed - monotone with for all and .
where is a relaxed - monotone mapping, is a -inverse-strongly monotone mapping, and is a countable family of nonexpansive mappings such that , , and , , and are three control sequences. We prove that defined by (1.14) converges strongly to . Using the main result in this paper, we also prove several new strong convergence theorems for finding the elements of , , , and , respectively, where is a nonexpansive mapping.
Definition 2.1 (see ).
Let be a Banach space with the dual space and let be a nonempty subset of . Let and be two mappings. The mapping is said to be -hemicontinuous if, for any fixed , the function defined by is continuous at .
Let be a Hilbert space and let be a nonempty closed convex subset of . Let be an -hemicontinuous and relaxed - monotone mapping. Let be a bifunction from to satisfying (A1) and (A4). Let and . Assume that
Definition 2.3 (see ).
Lemma 2.4 (see ).
Next we use the concept of KKM mapping to prove two basic lemmas for our main result. The idea of the proof of the next lemma is contained in the paper of Fang and Huang .
Let be a real Hilbert space and be a nonempty bounded closed convex subset of . Let be an -hemicontinuous and relaxed - monotone mapping, and let be a bifunction from to satisfying (A1) and (A4). Let . Assume that
Then problem (2.6) is solvable.
This completes the proof.
Then, the following holds:
Finally, we prove that is closed and convex. Indeed, Since every firm nonexpansive mapping is nonexpansive, we see that is nonexpansive from (2). On the other hand, since the set of fixed points of every nonexpansive mapping is closed and convex, we have that is closed and convex from (2) and (3). This completes the proof.
In this section, we prove a strong convergence theorem which is our main result.
Let be a nonempty bounded closed convex subset of a real Hilbert space and let be a bifunction satisfying (A1), (A2), (A3), and (A4). Let be an -hemicontinuous and relaxed - monotone mapping, let be a -inverse-strongly monotone mapping, and let be a countable family of nonexpansive mappings such that . Assume that the conditions (i)–(iv) of Lemma 2.6 are satisfied. Let and assume that is a strictly decreasing sequence. Assume that with some and with some . Then, for any , the sequence generated by (1.14) converges strongly to . In particular, if contains the origin 0, taking , then the sequence generated by (1.14) converges strongly to the minimum norm element in .
We split the proof into following steps.
Then, we obtain the desired result by Theorem 3.1.
The novelty of this paper lies in the following aspects.
(i)A new general equilibrium problem with a relaxed monotone mapping is considered.
This work was supported by the Natural Science Foundation of Hebei Province (A2010001482).
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