- Research Article
- Open Access
An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces
© Ming Tian. 2011
- Received: 9 December 2010
- Accepted: 13 February 2011
- Published: 9 March 2011
We use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes.
- Hilbert Space
- Variational Inequality
- Iterative Method
- Equilibrium Problem
- Monotone Operator
denoted the set of solution by . Given a mapping , let for all , then if and only if , that is, is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem, see, for instance, [1, 2].
A mapping of into itself is nonexpansive if , for all . The set of fixed points of is denoted by . In 2007, Plubtieng and Punpaeng , S. Takahashi and W. Takahashi , and Tada and W. Takahashi  considered iterative methods for finding an element of .
where is a self-nonexpansive mapping on , is a contraction of into itself with and satisfies certain conditions, and is a strongly positive bounded linear operator on and converges strongly to a fixed-point of which is the unique solution to the following variational inequality:
Note that the class of -strict pseudo-contraction strictly includes the class of nonexpansive mapping, that is, is nonexpansive if and only if is 0-srictly pseudocontractive; it is also said to be pseudocontractive if . Clearly, the class of -strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions.
The set of fixed points of is denoted by . Very recently, by using the general approximation method, Qin et al.  obtained a strong convergence theorem for finding an element of . On the other hand, Ceng et al.  proposed an iterative scheme for finding an element of and then obtained some weak and strong convergence theorems. Based on the above work, Y. Liu  introduced two iteration schemes by the general iterative method for finding an element of .
Motivated and inspired by these facts, in this paper, we introduced two iteration methods by the hybrid iterative method for finding an element of , where is a -strictly pseudocontractive non-self mapping, and then obtained two strong convergence theorems.
Such a is called the metric projection of onto . It is known that is nonexpansive. Furthermore, for and , , for all .
holds for every with . In order to solve the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:
(A1) , for all ,
(A2) is monotone, that is, , for all ,
(A3)For all .
(A4) For each fixed , the function is convex and lower semicontinuous. Let us recall the following lemmas which will be useful for our paper.
Lemma 2.1 (see ).
Further, if , then the following hold:
(1) is single-valued,
(2) is firmly nonexpansive, that is,
(4) is nonempty, closed and convex.
Lemma 2.2 (see ).
If is a -strict pseudo-contraction, then the fixed-point set is closed convex, so that the projection is well difened.
Lemma 2.3 (see ).
Let be a -strict pseudo-contraction. Define by for each , then, as , T is nonexpansive mapping such that .
Lemma 2.4 (see ).
Lemma 2.5 (see ).
where is a sequence in (0,1) and is a sequence in , such that
(ii) or Then .
where , , and satisfy if and satisfy the following conditions:
(i) , ,
(ii) and ,
then converges strongly to a point which solves the variational inequality (3.4).
It follows that .
This is a contradiction. So, we get and .
Since , it follows from (3.27) that as . Next, we show that solves the variational inequality (3.4).
Since is monotone (i.e., , for all . This is due to the nonexpansivity of ).
where , if , , and satisfy the following conditions:
(i) , , , ,
(ii) and , ,
(iii) , and ,
then and converge strongly to a point which solves the variational inequality(3.4).
Since is bounded, there exists a subsequence of which converges weakly to .
where , , and .
It is easy to see that , , and by (3.66). Hence by Lemma 2.5, the sequence converges strongly to .
M. Tian was supported in part by the Science Research Foundation of Civil Aviation University of China (no. 2010kys02). He was also Supported in part by The Fundamental Research Funds for the Central Universities (Grant no. ZXH2009D021).
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