# A Hybrid Method for Monotone Variational Inequalities Involving Pseudocontractions

- Yonghong Yao
^{1}, - Giuseppe Marino
^{2}Email author and - Yeong-Cheng Liou
^{3}

**2011**:180534

https://doi.org/10.1155/2011/180534

© Yonghong Yao et al. 2011

**Received: **25 November 2010

**Accepted: **24 January 2011

**Published: **20 February 2011

## Abstract

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).

## 1. Introduction

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 [1], where he assumed that is Lipschitzian and strongly monotone.

where are two nonexpansive mappings with the set of fixed points . Let denote the set of solutions of VI (1.2) and assume that is nonempty.

We next briefly review some literatures in which the involved mappings and are all nonexpansive.

Note that Lu et al.'s regularization (1.4) does no longer depend on . Related work can also be found in [4–9].

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 .

## 2. Preliminaries

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:

(b) is monotone on : for all .

Below is the so-called demiclosedness principle for pseudocontractive mappings.

Lemma 2.1 (see [10]).

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]).

## 3. Main Results

Below is our main result of this paper which displays the behavior of the net as and successively.

Theorem 3.1.

Hence, for each null sequence in , there exists another null sequence in , such that the sequence in norm as .

We divide our details proofs into several lemmas as follows. Throughout, we assume all conditions of Theorem 3.1 are satisfied.

Lemma 3.2.

For each fixed , the net is bounded.

Proof.

It follows that for each fixed , is bounded, so are the nets , , and .

We will use to denote possible constant appearing in the following.

Lemma 3.3.

Proof.

Consequently, the weak convergence of to actually implies that strongly. This has proved the relative norm compactness of the net as .

Hence, we conclude that the entire net converges in norm to as .

Lemma 3.4.

Proof.

Lemma 3.5.

The net which solves the variational inequality (3.3).

Proof.

First, we note that the solution of the variational inequality VI (3.3) is unique.

It turns out that solves VI (3.3). By uniqueness, we have . This is sufficient to guarantee that in norm, as . The proof is complete.

## Authors’ Affiliations

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