- Research Article
- Open Access

# Iterative Methods for Variational Inequalities over the Intersection of the Fixed Points Set of a Nonexpansive Semigroup in Banach Spaces

- Issa Mohamadi
^{1}Email author

**2011**:620284

https://doi.org/10.1155/2011/620284

© Issa Mohamadi. 2011

**Received:**8 November 2010**Accepted:**19 November 2010**Published:**25 November 2010

The Erratum to this article has been published in Fixed Point Theory and Applications 2011 2011:187439

## Abstract

This paper presents a framework of iterative methods for finding specific common fixed points of a nonexpansive self-mappings semigroup in a Banach space. We prove, with appropriate conditions, the strong convergence to the solution of some variational inequalities.

## Keywords

- Banach Space
- Variational Inequality
- Nonexpansive Mapping
- Strong Convergence
- Regular Semigroup

## 1. Introduction

where is an arbitrarily fixed constant and is the projection of onto . This alternative equivalence has been used to study the existence theory of the solution and to develop several iterative type algorithms for solving variational inequalities. But the fixed point formulation in (1.4) involves the projection , which may not be easy to compute, due to the complexity of the convex set . So, projection methods and their variant forms can be implemented for solving variational inequalities.

Assume that is -strongly monotone and -Lipschitzian on . Take a fixed number and a sequence in satisfying the following conditions:

,

,

.

Yamada [1] proved that the sequence converges strongly to a unique solution of . Xu and Kim [3] further considered and studied the hybrid steepest-descent algorithm (1.6). Their major contribution is that the strong convergence of (1.6) holds with condition being replaced by the following condition:

.

It is clear that condition is strictly weaker than condition , coupled with conditions and . Moreover, includes the important and natural choice for whereas does not. For more related results, see [4, 5].

Let be a Banach space we recall that a nonexpansive semigroup is a family of self-mappings of satisfies the following conditions:

(i) for ,

(ii) for and ,

- (iv)
for each is nonexpansive. that is,

The problem is to find some fixed point in . For this, so many algorithms have been developed and under some restrictions partial answers have been obtained [6–11].

for . With some appropriate assumptions, we prove the strong convergence of (1.8), (1.9), and (1.10) to the unique solution of the variational inequality in , where is the single-valued normalized duality mapping from into .

Our main purpose is to improve some of the conditions and results in the mentioned papers, especially those of Song and Xu [11].

## 2. Preliminaries

*Gateaux differentiable norm*(or is said to be

*smooth*), if the limit

exists for each
, and
is said to have a *uniformly Gateaux differentiable* norm if for each
, the limit (2.1) converges uniformly for
. Further,
is said to be *uniformly smooth* if the limit (2.1) exists uniformly for
.

where denotes the generalized duality pairing. It is well known if is smooth then any duality mapping on is single valued, and if has a uniformly Gateaux differentiable norm, then the duality mapping is norm to weak* uniformly continuous on bounded sets.

Recall that a Banach space
is said to be *strictly convex* if
and
implies
. In a strictly convex Banach space
, we have that if
for
and
, then
.

*uniformly asymptotically regular*on if for all and any bounded subset of , we have

The nonexpansive semigroup is an example of uniformly asymptotically regular operator semigroup [11].

*mean*on if and only if

*a Banach limit*if . We know that if is a Banach limit, then

A discussion on these and related concepts can be found in [12].

We make use of the following well-known results throughout the paper.

Lemma 2.1 (see [12, Lemma 4.5.4]).

for all .

Lemma 2.2 (see [13]).

for all . Assume also that . Then, the following results hold:

(i)if (where ), then is bounded,

then .

Lemma 2.3.

where , for all , and is a bounded sequence in .

## 3. Implicit Iterative Method

for a constant .

The following lemma will be be used to show the convergence of (1.8) and (1.9).

Lemma 3.1.

is a contraction on for every .

Proof.

and for , we have . That is, is a contraction, and the proof is complete.

In the following theorem, which is the main result in this section, we establish the strong convergence of the sequence defined by (1.8).

Theorem 3.2.

Proof.

We divide the proof into several steps.

Step 1.

and so, .

Step 2.

therefore, is bounded and so is .

Step 3.

This yields . Hence, there exists a subsequence of such as that converges strongly to ; that is, is sequentially compact.

Step 4.

Cosequently, is the unique solution of .

Step 5.

It yields that , which proves the uniqness of . Thus, itself converges strongly to . This completes the proof.

## 4. Explicit Iterative Method

In this section, we will present our result of the strong convergence of (1.9), but first, we need to prove, with different approach, the following lemma.

Lemma 4.1.

*μ*, and be as those in Theorem 3.2. If , and there exists a bounded sequence such that

Proof.

This completes the proof.

Next, we prove the strong convergence of explicit iteration scheme (1.9).

Theorem 4.2.

Let be a real Banach space with a uniformly Gateaux differentiable norm, and let be a nonexpansive semigroup from into itself. Let also defined by (1.9) satisfies the following conditions:

(i) ,

(ii) for all .

Proof.

Taking , , , and using Lemma 2.2, we conclude that is bounded and so is .

Taking , , and and using Lemma 4.1 together with Lemma 2.2 lead to , that is, in norm. This completes the proof.

Corollary 4.3.

Let be a real reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm. Let also be a nonexpansive semigroup from into itself such that . Assume that defined by (1.9) satisfies condition in Theorem 4.2, then condition holds.

Proof.

that is, , by uniqueness of . Thus, . This completes the proof.

Corollary 4.4.

Let be a real Banach space, and let be a nonexpansive uniformly asymptotically regular semigroup from into itself. If is defined by (1.9), where satisfies , then condition in Theorem 4.2 holds.

Proof.

and it completes the proof.

Remark 4.5.

Remark 4.6.

converges strongly to the variational inequality .

## 5. Modified Iterative Method

In this section, we show that the modified sequence defined by (1.10) also converges strongly to the solution of variational equality , but first, we need to prove the following lemma.

Lemma 5.1.

is a contraction on .

Proof.

Note that for , we conclude . That is, is a contraction and the proof is complete.

Theorem 5.2.

Let be a real Banach space with a uniformly Gateaux differentiable norm and a nonexpansive semigroup from into itself. Let also defined by (1.10) satisfies the following conditions:

(i) ,

(ii) for all .

Assume that is -strongly monotone and -Lipschitzian and a sequence of positive numbers that . Assume also that the sequences , where , and in satisfy the following control conditions:

,

does not take 0 as it's limit point.

Then, converges strongly to some fixed point , which is the unique solution in for the variational inequality .

Proof.

and from Lemma 2.2, we conclude that is bounded.

This completes the proof.

Remark 5.3.

In Theorem 5.2, if , then , and therefore we can remove , also turns to .

Remark 5.4.

We can easily see that under some restrictions all the strongly monotone and Lipschitzian nonlinear operators used in this paper are replaceable by strongly accretive and strictly pseudocontractive ones (see [15]).

## Notes

## Authors’ Affiliations

## References

- Yamada I:
**The hybrid steepest-descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings.**In*Inherently Parallel Algorithm for Feasibility and Optimization*. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, New York, NY, USA; 2001:473–504.View ArticleGoogle Scholar - Deutsch F, Yamada I:
**Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.***Numerical Functional Analysis and Optimization*1998,**19**(1–2):33–56.MATHMathSciNetView ArticleGoogle Scholar - Xu HK, Kim TH:
**Convergence of hybrid steepest-descent methods for variational inequalities.***Journal of Optimization Theory and Applications*2003,**119**(1):185–201.MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar - Saeidi S:
**Modified hybrid steepest-descent methods for variational inequalities and fixed points.***Mathematical and Computer Modelling*2010,**52**(1–2):134–142. 10.1016/j.mcm.2010.01.023MATHMathSciNetView ArticleGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*1997,**125**(12):3641–3645. 10.1090/S0002-9939-97-04033-1MATHMathSciNetView ArticleGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces.***Nonlinear Analysis: Theory, Methods and Applications*1998,**34**(1):87–99. 10.1016/S0362-546X(97)00682-2MATHMathSciNetView ArticleGoogle Scholar - Suzuki T:
**On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces.***Proceedings of the American Mathematical Society*2003,**131**(7):2133–2136. 10.1090/S0002-9939-02-06844-2MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**A strong convergence theorem for contraction semigroups in Banach spaces.***Bulletin of the Australian Mathematical Society*2005,**72**(3):371–379. 10.1017/S000497270003519XMATHMathSciNetView ArticleGoogle Scholar - Chen R, Song Y:
**Convergence to common fixed point of nonexpansive semigroups.***Journal of Computational and Applied Mathematics*2007,**200**(2):566–575. 10.1016/j.cam.2006.01.009MATHMathSciNetView ArticleGoogle Scholar - Song Y, Xu S:
**Strong convergence theorems for nonexpansive semigroup in Banach spaces.***Journal of Mathematical Analysis and Applications*2008,**338**(1):152–161. 10.1016/j.jmaa.2007.05.021MATHMathSciNetView ArticleGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis: Fixed Point Theory and Its Applications*. Yokohama Publishers, Yokohama, Japan; 2002.Google Scholar - Maingé P-E:
**Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**325**(1):469–479. 10.1016/j.jmaa.2005.12.066MATHMathSciNetView ArticleGoogle Scholar - Megginson RE:
*An Introduction to Banach Space Theory, Graduate Texts in Mathematics*.*Volume 183*. Springer, New York, NY, USA; 1998:xx+596.View ArticleMATHGoogle Scholar - Ceng L-C, Ansari QH, Yao J-C:
**Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces.***Numerical Functional Analysis and Optimization*2008,**29**(9–10):987–1033. 10.1080/01630560802418391MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.