An Iterative Scheme with a Countable Family of Nonexpansive Mappings for Variational Inequality Problems in Hilbert Spaces

Abstract

We introduce a new iterative scheme with a countable family of nonexpansive mappings for the variational inequality problems in Hilbert spaces and prove some strong convergence theorems for the proposed schemes.

1. Introduction

Let be a Hilbert space and be a nonempty closed convex subset of . Let be a nonlinear mapping. The classical variational inequality problem (for short, ) is to find a point such that

(1.1)

This variational inequality was initially studied by Kinderlehrer and Stampacchia [1]. Since then, many authors have introduced and studied many kinds of the variational inequality problems (inclusions) and applied them to many fields.

It is well known that, if is a strongly monotone and Lipschitzian mapping on , then the has a unique solution (see [2]).

Let be a mapping. Recall that a mapping is nonexpansive if

(1.2)

The set of fixed points of is denoted by . Recently, the iterative methods for nonexpansive mappings and some kinds of nonlinear mappings have been applied to solve the convex minimization problems (see [3–7]).

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on :

(1.3)

where is the fixed point set of a nonexpansive mapping on H, is a given point in and is a strongly positive operator, that is, there is a constant such that

(1.4)

Recently, for solving the variational inequality on , Marino and Xu [8] introduced the following general iterative scheme:

(1.5)

where is a strongly positive linear bounded operator on , is a contraction on and .

More precisely, they gave the following result.

Theorem 1 MX (see [8, Theorem 3.4]).

Let be generated by algorithm (1.5) with the sequence satisfying the following conditions:

(C1),

(C2),

(C3) either or .

Then the scheme defined by (1.5) converges strongly to an element which is the unique solution of the variational inequality for short, :

(1.6)

Let be a contraction with coefficient and let be two strongly positive linear bounded operators with coefficients and , respectively.

Motivated and inspired by the iterative sheme (1.5), Ceng et al. [9] introduced the following so-called hybrid viscosity-like approximation algorithms with variable parameters for nonexpansive mappings in Hilbert spaces.

Theorem 1 CGYone (see [9, Theorem 3.1]).

Let and . Let be a sequence in and be a sequence in . Starting with an arbitrary initial guess , generate a sequence by the following iterative scheme:

(1.7)

Assume that

(i),

(ii),

(iii)either or ,

(iv).

Then the scheme defined by (1.7) converges strongly to an element which is the unique solution of the variational inequality for short, :

(1.8)

Theorem 1 CGYtwo (see [9, Theorem 3.2]).

Let and . Let be a sequence in and be a sequence in . Starting with an arbitrary initial guess , generate a sequence by the following iterative scheme:

(1.9)

Assume that

(i),

(ii),

(iii)either or ,

(iv).

In addition, assume that

(1.10)

Then the scheme defined by (1.9) converges strongly to an element which is the unique solution of the variational inequality for short, :

(1.11)

In this paper, motivated and inspired by the above research results, we introduce a new iterative process with a countable family of nonexpansive mappings for the variational inequality problem in Hilbert spaces.

More precisely, let be a Hilbert space and be a countable family of nonexpansive mappings from to such that . Let be a contraction with coefficient and be strongly positive linear bounded operators with coefficients and , respectively. Let and with . Take three fixed numbers , and such that , and . For any , generate the iterative scheme by

(1.12)

We prove that the iterative scheme defined by (1.12) strongly converges to an element which is the unique solution of the variational inequality for short, :

(1.13)

2. Preliminaries

Let be a Hilbert space and be a nonexpansive mapping of into itself such that . For all and , we have

(2.1)

and hence

(2.2)

Let be a sequence in a Hilbert space and let . Throughout this paper, and denote that strongly converges to and converges weakly to a point , respectively.

Lemma 2.1 (see [10]).

Let be a closed convex subset of a Hilbert space and be a nonexpansive mapping from into itself. Then is demiclosed at zero, that is,

(2.3)

The following lemma is an immediate consequence of the equality:

(2.4)

Lemma 2.2.

Let be a real Hilbert space. Then the following identity holds:

(2.5)

Lemma 2.3 (see [4, 11]).

Let , be the sequences of nonnegative real numbers and let . Suppose that is a sequence of real numbers such that

(2.6)

Assume that . Then the following results hold.

(1)If , where , then is a bounded sequence.

(2)If one has

(2.7)

then .

Lemma 2.4 (see [8]).

Let be a real Hilbert space, be a contraction with coefficient and be a strongly positive linear bounded operator with coefficient . Then, for any with ,

(2.8)

that is, is strongly monotone with coefficient .

Lemma 2.5.

Assume is a strongly monotone linear bounded operator on a Hilbert space with coefficient . Take a fixed number such that . Then .

Proof.

The proof method is mainly from the idea of Marino and Xu [8, Lemma 2.5]. It is known that the norm of a linear bounded self-adjoint operator on is as follows:

(2.9)

Now, for all with , we see that (here denotes zero point in )

(2.10)

This completes the proof.

Remark 2.6.

Lemma 2.5 still holds if is a strongly positive linear bounded operator (see [8, Lemma 2.5]). That is, Lemma 2.5 in this section and Lemma 2.5 in [8] both hold when is a strongly monotone linear bounded operator or a strongly positive linear bounded one because an operator on a Hilbert space is strongly monotone linear if and only if it is strongly positive linear.

In fact, if is a strongly monotone linear operator with coefficient on a Hilbert space , then, for all ,

(2.11)

which shows that is strongly positive linear. Assume that is a strongly positive linear operator with coefficient on . Then, for all ,

(2.12)

which shows that is strongly monotone and linear.

3. Main Results

Let be a Hilbert space and be a nonempty closed and convex subset of . Let be a contraction with coefficient . Let be strongly positive linear bounded operator with coefficient and , respectively. Take a fixed number such that . Then, from Lemma 2.4, it follows that is strongly monotone with coefficient . For any fixed numbers and , we have , which can be seen easily from the following:

(3.1)

(3.2)

Moreover, observe that

(3.3)

which implies that is Lipschitzian with coefficient .

On the other hand, from Lemma 2.4, it follows that

(3.4)

which implies that is strongly monotone with coefficient . Hence the variational inequality (for short, )

(3.5)

has the unique solution.

Let be a nonexpansive mapping. Take two fixed numbers and such that and and, for all , define a mapping by

(3.6)

Then we have the following results.

Lemma 3.1.

If , then is a contraction with coefficient , where , that is,

(3.7)

Proof.

From Lemma 2.5 and Remark 2.6, it follows that, for all ,

(3.8)

This completes the proof.

Let be a countable family of nonexpansive mappings from into itself such that . Since each is closed and convex, then is closed and convex.

Throughout this paper, let be a contraction with coefficient . Let be strongly positive linear bounded mapping with coefficient and , respectively. Take a fixed number such that . Suppose that , (assuming that such that ( is nonempty), with and with .

Now, we can rewrite the iterative scheme (1.12) as follows:

(3.9)

where . Then, by Lemma 3.1, for all , we have

(3.10)

where .

Lemma 3.2.

If is strictly decreasing, then the scheme defined by (3.9) is bounded.

Proof.

Since , it follows from (3.10) that, for all ,

(3.11)

By induction, we obtain

(3.12)

Hence is bounded and so are and for each . This completes the proof.

Lemma 3.3.

If is strictly decreasing and the following conditions hold:

(3.13)

then

Proof.

By the iterative scheme (3.9), we have

(3.14)

and hence

(3.15)

where is a constant. Since , there exists a constant such that for all . Therefore, we have

(3.16)

Put . Since is a strictly decreasing sequence and , we have . By Lemma 2.3, it follows that as . This completes the proof.

Lemma 3.4.

If is strictly decreasing and the following conditions hold:

(3.17)

then

Proof.

By the iterative scheme (3.9), we have

(3.18)

that is,

(3.19)

Hence, for any , we get

(3.20)

Since each is nonexpansive, it follows from (2.2) that

(3.21)

Hence, combining (3.21) with (3.20), it follows that

(3.22)

which implies that

(3.23)

Since each and , then we have

(3.24)

that is,

(3.25)

Since and are both bounded, there exists a constant such that

(3.26)

By Lemma 3.3 and the assumption condition , it follows that

(3.27)

This completes the proof.

Finally, we give the main result in this paper.

Theorem 3.5.

If is strictly decreasing and the following conditions hold:

(3.28)

then the scheme defined by (3.9) converges strongly to an element which is the unique solution of the variational inequality :

(3.29)

Proof.

First, we prove that .

To prove this, we pick a subsequence of such that

(3.30)

Without loss of generality, we may, further, assume that for some . From Lemmas 2.1 and 3.3, it follows that for each and so . Since is the unique solution of the problem , we obtain

(3.31)

It follows from Lemma 2.2 and (3.10) that

(3.32)

where is a constant such that for all . Since and , by Lemma 2.3, we conclude that the scheme converges strongly to . This completes the proof.

Remark 3.6.

() For each , a simple example on control parameters is and , where is a constant in .

() We obtain the desired results without any assumptions on the family . Foe example, in Theorem CGY2, the authors gave the strong condition (1.10).

Remark 3.7.

() If in (3.9), then we have the following iterative scheme:

(3.33)

and the scheme defined by (3.33) converges strongly to an element which is the unique solution of the variational inequality ():

(3.34)

() If and in (3.33), then we have the following iterative scheme:

(3.35)

and the scheme defined by (3.35) converges strongly to an element which is the unique solution of the variational inequality ():

(3.36)

() Furthermore, if and in (3.35), then we have the following iterative scheme:

(3.37)

and the scheme defined by (3.37) converges strongly to an element which is the unique solution of the variational inequality (), which is Stampacchia's variational inequality:

(3.38)