A Hybrid Method for Monotone Variational Inequalities Involving Pseudocontractions
© Yonghong Yao et al. 2011
Received: 25 November 2010
Accepted: 24 January 2011
Published: 20 February 2011
We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point with the property such that , where , are two pseudocontractive self-mappings of a closed convex subset of a Hilbert space with the set of fixed points . Assume the solution set of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element . Our results improve and extend a recent result of (Lu et al. 2009).
If the mapping is a monotone operator, then we say that is monotone. It is well known that if is Lipschitzian and strongly monotone, then for small enough , the mapping is a contraction on and so the sequence of Picard iterates, given by ( ) converges strongly to the unique solution of the . Hybrid methods for solving the variational inequality were studied by Yamada , where he assumed that is Lipschitzian and strongly monotone.
In this paper, we will extend Lu et al.'s result to a general case. We will further study the strong convergence of the algorithm (1.4) for solving VI (1.2) under the assumption that the mappings are all pseudocontractive. As far as we know, this appears to be the first time in the literature that the solutions of the monotone variational inequalities of kind (1.2) are investigated in the framework that feasible solutions are fixed points of a pseudocontractive mapping .
We denote by the set of fixed points of ; that is, . Note that is always closed and convex (but may be empty). However, for VI (1.2), we always assume . It is not hard to find that is a pseudocontraction if and only if satisfies one of the following two equivalent properties:
Below is the so-called demiclosedness principle for pseudocontractive mappings.
Lemma 2.1 (see ).
Let be a closed convex subset of a Hilbert space . Let be a Lipschitz pseudocontraction. Then, is a closed convex subset of , and the mapping is demiclosed at 0; that is, whenever is such that and , then .
We also need the following lemma.
Lemma 2.2 (see ).
3. Main Results
We divide our details proofs into several lemmas as follows. Throughout, we assume all conditions of Theorem 3.1 are satisfied.
First, we note that the solution of the variational inequality VI (3.3) is unique.
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