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

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 [1–3]).

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).