# Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

- Songnian He
^{1}Email author and - Xiao-Lan Liang
^{1}

**2010**:673932

https://doi.org/10.1155/2010/673932

© S. He and X.-L. Liang. 2010

**Received: **30 September 2009

**Accepted: **13 January 2010

**Published: **24 January 2010

## Abstract

Let be a real Hilbert space and let be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality of finding a point such that , for all , where is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudocontractions. Strong convergence of the algorithms is proved.

## 1. Introduction

This is known as the variational inequality problem (i.e., initially introduced and studied by Stampacchia [1] in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics, optimization, operations research, and engineering sciences; see [1–6] and the references therein.

Yamada [7] proposed hybrid methods to solve , where is composed of fixed points of a nonexpansive mapping; that is, is of the form

where is a nonexpansive mapping (i.e., for all ), is Lipschitzian and strongly monotone.

He and Xu [8] proved that has a unique solution and iterative algorithms can be devised to approximate this solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . In the case where is the set of fixed points of a nonexpansive mapping, they invented a hybrid iterative algorithm to approximate the unique solution of and this extended the Yamada's results.

The main purpose of this paper is to continue our research in [8]. We assume that is a boundedly Lipschitzian and strongly monotone operator as in [8], but is the set of fixed points of a strict pseudo-contraction , or the set of common fixed points of finite strict pseudo-contractions . For the two cases of , we will design the hybrid iterative algorithms for solving and prove their strong convergence, respectively. Relative definitions are stated as below.

Let be a nonempty closed and convex subset of a real Hilbert space , and , then

Obviously, the nonexpansive mapping class is a proper subclass of the strict pseudo-contraction class and the Lipschitzian operator class is a proper subclass of the boundedly Lipschitzian operator class, respectively.

We will use the following notations:

(i) for weak convergence and for strong convergence,

## 2. Preliminaries

We need some facts and tools which are listed as lemmas below.

Lemma 2.1.

Let be a real Hilbert space. The following expressions hold:

Lemma 2.2 (see [9]).

If and satisfy the following conditions:

Lemma 2.3 (see [10]).

Let be a nonempty closed convex subset of a real Hilbert space and is a nonexpansive mapping. If a one has sequence in such that and then

Lemma 2.4 (see [11]).

Let be a nonempty closed convex subset of a real Hilbert space , if is a -strict pseudo-contraction, then the mapping is demiclosed at 0. That is, if is a sequence in such that and then

Lemma 2.5 (see [8]).

Assume that is a nonempty closed convex subset of a real Hilbert space , if is boundedly Lipschitzian and -strongly monotone, then variational inequality (1.1) has a unique solution.

Lemma 2.6.

Proof.

so is nonexpansive. is obvious.

Lemma 2.7.

where the constants and are such that and , respectively, and is defined as in Lemma 2.6 above. Then restricted to is a contraction.

Proof.

Therefore, restricted to that is a contraction with coefficient , where

Lemma 2.8 (see [11]).

Assume is a closed convex subset of a Hilbert space .

- (ii)

Lemma 2.9.

Proof.

## 3. Further Extension of Hybrid Iterative Algorithm

Yamada got the following result.

Theorem 3.1 (see [7]).

Assume that is a real Hilbert space, is nonexpansive such that and is -strongly monotone and -Lipschitzian. Fix a constant . Assume also that the sequence satisfies the following conditions:

then converges strongly to the unique solution of .

He and Xu [8] proved that has a unique solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . Using this result, they were able to relax the global Lipschitz condition on in Theorem 3.1 to the weaker bounded Lipschitz condition and invented a hybrid iterative algorithm to approximate the unique solution of . Their result extended the Yamada's above theorem.

We have the following result.

Theorem 3.2.

If the sequences and satisfy the following conditions:

then generated by (3.2) converges strongly to the unique solution of .

Proof.

It proves that . Therefore, for all . Thus is bounded. It is not difficult to verify that the sequences and are all bounded.

By (3.2) and Lemma 2.7, we have

Finally conditions (i)–(iii) and (3.14) allow us to apply Lemma 2.2 to the relation (3.13) to conclude that

## 4. Parallel Algorithm and Cyclic Algorithm

In this section, we discuss the parallel algorithm and the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite strict pseudo-contractions.

Since is a nonempty closed convex subset of , (4.1) has a unique solution. Throughout this section, is an arbitrary fixed point, , is the Lipschitz constant of on , the fixed constant satisfies , and the sequence belongs to .

Firstly we consider the parallel algorithm. Take a positive sequence such that and let

Using Lemma 2.8 and Thorem 3.2, the following conclusion can be deduced directly.

Theorem 4.1.

Suppose that and satisfy the same conditions as in Theorem 3.2. Then the sequence generated by the parallel algorithm (4.3) converges strongly to the unique solution of (4.1).

where the constant such that . Then we turn to defining the cyclic algorithm as follows:

We get the following result

Theorem 4.2.

If satisfies the following conditions:

then the sequence generated by (4.6) converges strongly to the unique solution of .

Proof.

- (1)

- (2)

## Declarations

### Acknowledgment

This research is supported by the Fundamental Research Funds for the Central Universities (GRANT:ZXH2009D021).

## Authors’ Affiliations

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