# An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

- Nguyen Buong
^{1}Email author and - NguyenThi Quynh Anh
^{2}

**2011**:276859

https://doi.org/10.1155/2011/276859

© Nguyen Buong and Nguyen Thi Quynh Anh. 2011

**Received: **17 December 2010

**Accepted: **7 March 2011

**Published: **15 March 2011

## Abstract

We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.

## Keywords

## 1. Introduction

Variational inequalities were initially studied by Kinderlehrer and Stampacchia in [1] and ever since have been widely investigated, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [1–3]).

where denotes the metric projection from onto and is an arbitrarily fixed positive constant.

where , for integer , with the mod function taking values in the set . They proved the following result.

Theorem 1.1.

Let be a real Hilbert space and a nonempty closed convex subset of . Let be nonexpansive self-maps of such that , where . Let and be a sequence in such that . Then, the sequence defined implicitly by (1.5) converges weakly to a common fixed point of the mappings .

They proved the following result.

Theorem 1.2.

Let be a real Hilbert space and a mapping such that for some constants , is -Lipschitz continuous and -strongly monotone. Let be nonexpansive self-maps of such that . Let , and let and satisfying the conditions: and , for some . Then, the sequence defined by (1.7) converges weakly to a common fixed point of the mappings . The convergence is strong if and only if .

Recently, Ceng et al. [6] extended the above result to a finite family of asymptotically self-maps.

denotes the identity mapping of , and the parameters for all satisfy the following conditions: as and .

## 2. Main Result

We formulate the following facts for the proof of our results.

Lemma 2.1 (see [7]).

(i) and for any fixed , (ii) , for all .

Put ; for any nonexpansive mapping of , we have the following lemma.

Lemma 2.2 (see [8]).

and for a fixed number , where .

Lemma 2.3 (Demiclosedness Principle [9]).

Assume that is a nonexpansive self-mapping of a closed convex subset of a Hibert space . If has a fixed point, then is demiclosed; that is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that .

Now, we are in a position to prove the following result.

Theorem 2.4.

Then, the net defined by (1.8)-(1.9) converges strongly to the unique element in (1.1).

Proof..

So, is a contraction in . By Banach's Contraction Principle, there exists a unique element such that for all .

that implies the boundedness of . So, are the nets .

and , it follows that . Similarly, we obtain that , for and as .

The uniqueness of in (1.1) guarantees that . Again, replacing in (2.8) by , we obtain the strong convergence for . This completes the proof.

## 3. Application

It is well known [10] that a mapping by with a fixed for all is a nonexpansive mapping and . Using this fact, we can extend our result to the case , where is -strictly pseudocontractive as follows.

So, we have the following result.

Theorem 3.1 ..

where , for , are defined by (3.2) and , converges strongly to the unique element in (1.1).

It is known in [11] that where with and for -strictly pseudocontractions . Moreover, is -strictly pseudocontractive with . So, we also have the following result.

Theorem 3.2 ..

where , , and , converges strongly to the unique element in (1.1).

## Declarations

### Acknowledgment

This work was supported by the Vietnamese National Foundation of Science and Technology Development.

## Authors’ Affiliations

## References

- Kinderlehrer D, Stampacchia G:
*An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics*.*Volume 88*. Academic Press, New York, NY, USA; 1980:xiv+313.MATHGoogle Scholar - Glowinski R:
*Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics*. Springer, New York, NY, USA; 1984:xv+493.View ArticleGoogle Scholar - Zeidler E:
*Nonlinear Functional Analysis and Its Applications. III*. Springer, New York, NY, USA; 1985:xxii+662.MATHView ArticleGoogle Scholar - Xu H-K, Ori RG:
**An implicit iteration process for nonexpansive mappings.***Numerical Functional Analysis and Optimization*2001,**22**(5–6):767–773. 10.1081/NFA-100105317MATHMathSciNetView ArticleGoogle Scholar - Zeng L-C, Yao J-C:
**Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings.***Nonlinear Analysis. Theory, Methods & Applications*2006,**64**(11):2507–2515. 10.1016/j.na.2005.08.028MATHMathSciNetView ArticleGoogle Scholar - Ceng L-C, Wong N-C, Yao J-C:
**Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption.***Computers & Mathematics with Applications*2008,**56**(9):2312–2322. 10.1016/j.camwa.2008.05.002MATHMathSciNetView ArticleGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MATHMathSciNetView ArticleGoogle Scholar - Yamada Y:
**The hybrid steepest-descent method for variational inequalities problems over the intesectionof the fixed point sets of nonexpansive mappings.**In*Inhently Parallel Algorithms in Feasibility and Optimization and Their Applications*. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, Holland; 2001:473–504.View ArticleGoogle Scholar - Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MATHMathSciNetView ArticleGoogle Scholar - Acedo GL, Xu H-K:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis. Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MATHMathSciNetView ArticleGoogle Scholar

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