Abstract and Applied Analysis

Volume 2013 (2013), Article ID 757986, 6 pages

http://dx.doi.org/10.1155/2013/757986

Research Article

A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions

Wei Xu¹ and Yuanheng Wang²

¹Tongji Zhejiang College, Zhejiang 314000, China

²Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China

Received 6 November 2012; Accepted 2 February 2013

Academic Editor: Claudia Timofte

Copyright © 2013 Wei Xu and Yuanheng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with a new iterative algorithm with a strongly positive operator A for a k-strict pseudo-contraction T and a non-self-Lipschitzian mapping S in Hilbert spaces. Under certain appropriate conditions, the sequence converges strongly to a fixed point of T, which solves some variational inequality. The results here improve and extend some recent related results.

1. Introduction

Let be a closed convex subset of Hilbert space with inner product and norm , be a nonlinear mapping. The fixed point set of is denoted by ; that is, . Fixed point problem is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others.

Recall that a mapping is said to be nonexpansive if for all . A mapping is said to be strongly positive, if there exists a constant such that for all . In 2000, Moudafi [1] investigated the fixed point problem of nonexpansive mapping with viscosity approximation method. Let be a contraction on ; that is, there exists a constant such that for all ; define a sequence by

where is an arbitrary starting point in and is a sequence in . In 2004 Xu [2] proved that if the parameter satisfies some approximate conditions, the sequence generated by (1) converges strongly to not only a fixed point of but also the unique solution of the variational inequality

In 2010, Tian [3] considered a general hybrid steepest-descent method:

where is a Lipschitzian and strongly monotone operator. Under certain conditions, he proved that the sequence generated by (3) converges strongly to the unique solution of the variational inequality

On the other hand, Marino and Xu [4] introduced the following iterative scheme:

where is a strongly positive bounded linear operator. It was proven that under certain conditions on the parameters, the sequence generated by (5) converges strongly to the unique solution of the variational inequality

It is well known that a typical convex minimization is that of minimizing a quadratic function on the sets of the fixed points of a nonexpansive mapping:

where is a given point of . The solution is also the optimality condition for the minimization problem

where is a potential function for ; that is, . Some authors investigated each iterative method for nonexpansive mappings for solving convex minimization problems and got some convergence results; see, for example [5–7].

In 2011, Ceng et al. [8] introduced a general iterative algorithm with strongly positive operators for nonexpansive mappings:

and proved that under certain conditions on the parameters the sequence generated by (9) converges strongly to a fixed point of , which also solves the variational inequality

Recently the problems of the approximation of the common fixed points of nonexpansive mappings were extended to the case of a family of finite or infinite pseudo-contractions; see, for example, [9–11].

Motivated and inspired by the above research works, we consider some fixed point problems with non-self mappings and introduce a new general iterative algorithm with strongly positive operators for -strict pseudo-contractions which is a wider map class then the nonexpansive map class

where is the metric projection, is a non-self-Lipschitzian mapping, is a -strict pseudo-contraction, and is a strongly positive bounded linear operator. Under certain conditions on the parameters, we prove that the sequence generated by (11) converges strongly to a fixed point of , which solves the variational inequality

2. Preliminaries

In this section, we recall some useful definitions and lemmas for the proof of the main results.

Definition 1. A mapping is said to be -Lipschitzian, if there exists a constant such that

A mapping is said to be -strict pseudo-contraction, if there exists a constant   such that

It is clear that a Lipschitzian map is a contractive map when and is a nonexpansive map when . If ; then a -strict pseudo-contraction map is a nonexpansive map.

Definition 2. A mapping is said to be the metric projection, if for any , there exists a unique nearest point in C denoted by such that

And it is well known that if is a nonempty closed convex subset of , then the exists (e.g., see [12]).

Lemma 3 (see [13]). Let and be any points. There holds

And if and only if there holds

and if and only if there holds the relation

Lemma 4 (see [9], Demiclosedness princple). Let be a nonempty closed convex subset of a real Hilbert space and let be a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then ; in particular if , then .

Lemma 5 (see [14]). Let be a number in and . Let be a t-Lipschitzian and -strongly monotone operator on a Hilbert space. Associate with a nonexpansive mapping and define the mapping by

Then is a contraction provided ; that is,

Lemma 6 (see [4]). Assume that is a strongly positive bounded linear operator on a Hilbert space with coefficient and ; then .

Lemma 7 (see [15]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strict pseudo-contractive mapping. Let and be two nonnegative real numbers such that ; then

Lemma 8 (see [16]). Let be a Hilbert space and a nonempty convex subset of . Let be a -strict pseudo-contractive mapping. Define a mapping for all . Then as , is a nonexpansive mapping such that .

Lemma 9 (see [17]). Let be a sequence of nonnegative real numbers satisfying the following relation: , where (i) , ; (ii) or ; then .

3. Main Results

In this section, we prove the strong convergence results on the iterative algorithm for -strict pseudo-contractions.

Theorem 10. Let be a nonempty closed convex subset of a real Hilbert space , a non-self-L-Lipschitzian mapping, and a -strict pseudo-contractive mapping such that . Let be a t-Lipschitzian and -strongly monotone mapping and a -strongly positive bounded linear operator. For a given , let the sequences and generated by (11), where , , , satisfy the following conditions:

(i), , ; (ii), , , , ;(iii), , , .

Then the sequence converges strongly to a fixed point of , which solves the variational inequality

Proof. The proof is divided into five steps.

Step  1. We first show that the sequences , are bounded. Take , own to be a -strict pseudo-contractive mapping, and define . By Lemma 8   is nonexpansive and ; therefore :

Thus we immediately get that is a -Lipschitzian mapping. Then we estimate :

On the other hand, notice that , ; without loss of generality, we may assume that ; thus

Together with (24), we have

By conditions (ii) and (iii), we get that is bounded, and so are ,,,.

Step  2. Now we prove that as . Denote :

where is a constant such that

By the conditions (i), (ii), and (iii) and Lemma 9, we get as .

Step  3. Now we prove that as :

On the other hand,

Thus we have as . Observe that

we immediately get as .

Step  4. Now we show that , where is the unique solution of the variational inequality. Take a subsequence of such that

Observe that the sequence is bounded; without loss of generality we may assume that . By Lemma 4, we get . Therefore by Lemma 3, we have

Step  5. Next we prove that as :

Notice that

thus

By the conditions (ii), (iii) and Lemma 9, we conclude that as , which solves the variational inequality . This completes the proof.

Remark 11. The iterative algorithm in Theorem 10 here is a new approximating method, and Lemma 7 plays a key role in the proof of the main results which makes the proof simple.

Remark 12. The results in this paper improve and extend some recent related results. For example, Theorem 10 here improves and extends Theorem  3.2 in [8] in the following ways:

(i)the nonexpansive mapping in [8] is extended to the case of -strict pseudo-contractions ;(ii)the self-contraction in [8] is extended to the case of a (possiblly non-self) Lipschitzian mapping .

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work is partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696, Y6110270) and the National Natural Science Foundation (11071169, 11271330).

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