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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 Applications20112011:276859

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

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.

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
(1.1)

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 [13]).

It is well known that if is an -Lipschitz continuous and -strongly monotone, that is, satisfies the following conditions:
(1.2)
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
(1.3)

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:
(1.4)
The compact expression of the method is the form
(1.5)

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:
(1.6)
The scheme is written in a compact form as
(1.7)

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:
(1.8)
where are defined by
(1.9)

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
(2.1)

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
(2.2)

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
(2.3)
Therefore,
(2.4)

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

Put
(2.5)
Then,
(2.6)
Moreover,
(2.7)
Thus,
(2.8)
Further, for the sake of simplicity, we put and prove that
(2.9)

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
(2.10)
Let be a subsequence of such that
(2.11)
From (2.6) and Lemma 2.1, it implies that
(2.12)
Hence,
(2.13)
By Lemma 2.1,
(2.14)
Without loss of generality, we can assume that for some . Then, we have
(2.15)
This together with (2.13) implies that
(2.16)

It means that as for .

Next, we show that as . In fact, in the case that we have . So, as . Further, since
(2.17)
and , we have that . Therefore, from
(2.18)
it follows that as . On the other hand, since
(2.19)
we obtain that as . Now, from
(2.20)

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
(2.21)
Since , by replacing by in the last inequality, dividing by and taking in the just obtained inequality, we obtain
(2.22)

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
(3.1)

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
(3.2)

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
(3.3)
Then, the net defined by
(3.4)

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
(3.5)
Then, the net , defined by
(3.6)

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

(1)
Vietnamese Academy of Science and Technology, Institute of Information Technology
(2)
Department of Information Technology, Thai Nguyen National University

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Copyright

© Nguyen Buong and Nguyen Thi Quynh Anh. 2011

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