- Research Article
- Open Access
On the Convergence for an Iterative Method for Quasivariational Inclusions
© Y. Li and C.Wu. 2010
- Received: 27 September 2009
- Accepted: 13 December 2009
- Published: 12 January 2010
We introduce an iterative algorithm for finding a common element of the set of solutions of quasivariational inclusion problems and of the set of fixed points of strict pseudocontractions in the framework Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.
- Variational Inequality
- Nonexpansive Mapping
- Real Hilbert Space
- Variational Inequality Problem
- Lipschitz Continuous Mapping
Recall the following definitions.
Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. See, for example, [1–6] and the references therein.
Recently, Marino and Xu  studied the following iterative scheme:
which is the optimality condition for the minimization problem (1.5).
(6)Recall also that a set-valued mapping is called monotone if for all , and imply The monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping.
In this paper, we use to denote the solution of the problem (1.14). A number of problems arising in structural analysis, mechanics, and economic can be studied in the framework of this class of variational inclusions.
Next, we consider two special cases of the problem (1.14).
For finding a common element of the set of fixed points of a nonexpansive mapping and of the set of solutions to the variational inequality (1.16), Iiduka and Takahashi  proved the following theorem.
Recently, Zhang et al.  considered the problem (1.14). To be more precise, they proved the following theorem.
In this paper, motivated by the research work going on in this direction, see, for instance, [2, 3, 7–21], we introduce an iterative method for finding a common element of the set of fixed points of a strict pseudocontraction and of the set of solutions to the problem (1.14) with multivalued maximal monotone mapping and relaxed -cocoercive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.
In order to prove our main results, we need the following conceptions and lemmas.
Definition 1.1 (see ).
Let be a multivalued maximal monotone mapping. Then the single-valued mapping defined by for all , is called the resolvent operator associated with , where is any positive number and is the identity mapping.
Lemma 1.2 (see ).
Lemma 1.3 (see ).
Lemma 1.4 (see ).
Lemma 1.5 (see ).
Lemma 1.6 (see ).
Let be a closed convex subset of a strictly convex Banach space . Let and be two nonexpansive mappings on . Suppose that is nonempty. Then a mapping on defined by , where , for is well defined and nonexpansive and holds.
Lemma 1.7 (see ).
Lemma 1.8 (see ).
The strong monotonicity of (see [2, Lemma ]) implies that and the uniqueness is proved. Below we use to denote the unique solution of (2.2).
On the other hand, we have
Corollary 2.2 improves Theorem 2.1 of Zhang et al.  in the following sense:
(1)from nonexpansive mappings to strict pseudocontractions;
The authors are extremely grateful to the referee for useful suggestions that improved the contents of the paper. This work was supported by Important Science and Technology Research Project of Henan province, China (092102210134).
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