- Research Article
- Open Access
- Published:

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

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 276859 (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.

## 1. Introduction

Let be a nonempty closed and convex subset of a real Hilbert space with inner product and norm , and let be a nonlinear mapping. The variational inequality problem is formulated as finding a point such that

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]).

It is well known that if is an -Lipschitz continuous and -strongly monotone, that is, satisfies the following conditions:

where and are fixed positive numbers, then (1.1) has a unique solution. It is also known that (1.1) is equivalent to the fixed-point equation

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

Let be a finite family of nonexpansive self-mappings of . For finding an element , Xu and Ori introduced in [4] the following implicit iteration process. For and , the sequence is generated as follows:

The compact expression of the method is the form

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 .

Further, Zeng and Yao introduced in [5] the following implicit method. For an arbitrary initial point , the sequence is generated as follows:

The scheme is written in a compact form as

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.

Clearly, from we have that as . To obtain strong convergence without the condition , in this paper we propose the following implicit algorithm:

where are defined by

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.

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 , such that

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

Proof..

By using Lemma 2.2 with , that is, , we have that

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

Next, we show that is bounded. Indeed, for a fixed point , we have that for , and hence

Therefore,

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

Put

Then,

Moreover,

Thus,

Further, for the sake of simplicity, we put and prove that

as for .

Let be an arbitrary sequence converging to zero as and . We have to prove that , where are defined by (2.5) with and . Let be a subsequence of such that

Let be a subsequence of such that

From (2.6) and Lemma 2.1, it implies that

Hence,

By Lemma 2.1,

Without loss of generality, we can assume that for some . Then, we have

This together with (2.13) implies that

It means that as for .

Next, we show that as . In fact, in the case that we have . So, as . Further, since

and , we have that . Therefore, from

it follows that as . On the other hand, since

we obtain that as . Now, from

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

Let be any sequence of converging weakly to as . Then, , for and also converges weakly to . By Lemma 2.3, we have and from (2.8), it follows that

Since , by replacing by in the last inequality, dividing by and taking in the just obtained inequality, we obtain

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

Recall that a mapping is called a -strictly pseudocontractive if there exists a constant such that

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.

Let be fixed numbers. Then, with , a nonexpansive mapping, for , and hence

So, we have the following result.

Theorem 3.1 ..

Let be a real Hilbert space and a mapping such that for some constants , is -Lipschitz continuous and -strongly monotone. Let be -strictly pseudocontractive self-maps of such that . Let and let , such that

Then, the net defined by

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 ..

Let be a real Hilbert space and a mapping such that for some constants , is -Lipschitz continuous and -strongly monotone. Let be -strictly pseudocontractive self-maps of such that . Let , where , , and let , such that

Then, the net , defined by

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

## References

- [1]
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. - [2]
Glowinski R:

*Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics*. Springer, New York, NY, USA; 1984:xv+493. - [3]
Zeidler E:

*Nonlinear Functional Analysis and Its Applications. III*. Springer, New York, NY, USA; 1985:xxii+662. - [4]
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-100105317 - [5]
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.2005.08.028 - [6]
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.2008.05.002 - [7]
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.2006.06.055 - [8]
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. - [9]
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. - [10]
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.2007.05.032 - [11]
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.2006.08.036

## Acknowledgment

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

## Author information

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Buong, N., Quynh Anh, N. An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces.
*Fixed Point Theory Appl* **2011, **276859 (2011). https://doi.org/10.1155/2011/276859

Received:

Accepted:

Published:

### Keywords

- Hilbert Space
- Variational Inequality
- Nonexpansive Mapping
- Inequality Problem
- Real Hilbert Space