A New Iteration Method for Nonexpansive Mappings and Monotone Mappings in Hilbert Spaces

Abstract

We introduce a new composite iterative scheme by the viscosity approximation method for nonexpansive mappings and monotone mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of set of fixed points of nonexpansive mapping and the set of solutions of variational inequality for an inverse-strongly monotone mappings, which is a solution of a certain variational inequality. Our results substantially develop and improve the corresponding results of [Chen et al. 2007 and Iiduka and Takahashi 2005]. Essentially a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings is provided.

1. Introduction

Let be a real Hilbert space and a nonempty closed convex subset of . Recall that a mapping is a contraction on if there exists a constant such that We use to denote the collection of mappings verifying the above inequality. That is . A mapping is called nonexpansive if ; see [1, 2] for the results of nonexpansive mappings. We denote by the set of fixed points of ; that is,

Let be the metric projection of onto . A mapping of into is called monotone if for , . The variational inequality problem is to find a such that

(1.1)

for all ; see [3–6]. The set of solutions of the variational inequality is denoted by . A mapping of into is called inverse-strongly monotone if there exists a positive real number such that

(1.2)

for all ; see [7–9]. For such a case, is called -inverse-strongly monotone.

In 2005, Iiduka and Takahashi [10] introduced an iterative scheme for finding a common point of the set of fixed points of a nonexapnsive mapping and the set of solutions of the variational inequality for an inverse-strong monotone mapping as follows. For an -inverse-strongly monotone mapping of to and a nonexpansive mapping of into itself such that , , , and ,

(1.3)

for every . They proved that the sequence generated by (1.3) converges strongly to under the conditions on and for some with ,

(1.4)

On the other hand, the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping was proposed by Moudafi [11]. In 2004, in order to extend Theorem 2.2 of Moudafi [11] to a Banach space setting, Xu [12] considered the the following explicit iterative process. For nonexpansive mappings, and ,

(1.5)

Moreover, in [12], he also studied the strong convergence of generated by (1.5) as in either a Hilbert space or a uniformly smooth Banach space and showed that the strong is a solution of a certain variational inequality.

In 2007, Chen et al. [13] considered the following iterative scheme as the viscosity approximation method of (1.3). For an -inverse-strongly-monotone mapping of to and a nonexpansive mapping of into itself such that , , , , and ,

(1.6)

and showed that the sequence generated by (1.6) converges strongly to a point in under condition (1.4) on and , which is a solution of a certain variational inequality.

In this paper, motivated by above-mentioned results, we introduce a new composite iterative scheme by the viscosity approximation method. For an -inverse-strongly monotone mapping of to and a nonexpansive mapping of into itself such that , , , , and ,

(1.7)

If , then the iterative scheme (1.7) reduces to the iterative scheme (1.6). Under condition (1.4) on the sequences and and appropriate condition on sequence , we show that the sequence generated by (1.7) converges strongly to a point in , which is a solution of a certain variational inequality. Using this result, we also obtain a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Moreover, we investigate the problem of finding a common point of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. The main results develop and improve the corresponding results of Chen et al. [13] and Iiduka and Takahashi [10]. We point out that the iterative scheme (1.7) is a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings.

2. Preliminaries and Lemmas

Let be a real Hilbert space with inner product and norm , and a closed convex subset of . We write to indicate that the sequence converges weakly to . implies that converges strongly to . For every point , there exists a unique nearest point in , denoted by , such that

(2.1)

for all . is called the metric projection of to . It is well known that satisfies

(2.2)

for every . Moreover, is characterized by the properties

(2.3)

(2.4)

for all . In the context of the variational inequality problem, this implies that

(2.5)

We state some examples for inverse-strongly monotone mappings. If , where is a nonexpansive mapping of into itself and is the identity mapping of , then is -inverse-strongly monotone and . A mapping of into is called strongly monotone if there exists a positive real number such that

(2.6)

for all . In such a case, we say that is -strongly monotone. If is -strongly monotone and -Lipschitz continuous, that is, for all , then is -inverse-strongly monotone.

If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that for all and ,

(2.7)

So, if , then is a nonexpansive mapping of into . The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda [14].

Proposition 2.1.

Let be a bounded closed convex subset of a real Hilbert space and an -inverse-strongly monotone mapping of into . Then, is nonempty.

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be an inverse-strongly monotone mapping of into and let be the normal cone to at , that is, , and define

(2.8)

Then is maximal monotone and if and only if ; see [15, 16].

We need the following lemmas for the proof of our main results.

Lemma 2.2 (see [17]).

Let be a sequence of nonnegative real numbers satisfying

(2.9)

where and satisfy the following conditions:

(i) and or, equivalently,

(ii) or

Then .

Lemma 2.3 (see [1], demiclosedness principle).

Let be a real Hilbert space, a nonempty closed convex subset of , and a nonexpansive mapping. Then the mapping is demiclosed on , where is the identity mapping; that is, in and imply that and .

Lemma 2.4.

In a real Hilbert space , there holds the following inequality:

(2.10)

for all

3. Main Results

In this section, we introduce a new composite iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings and prove a strong convergence of this scheme.

Theorem 3.1.

Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of to and a nonexpansive mapping of into itself such that , and . Let be a sequence generated by

(3.1)

where , , and . If , and satisfy the following conditions:

(i); ;

(ii) for all and for some ;

(iii) for some with ;

(iv); ; ,

then converges strongly to , which is a solution of the following variational inequality:

(3.2)

Proof.

Let and for every . Let . Since is nonexpansive and from (2.5), we have

(3.3)

Similarly we have .

We divide the proof into several steps.

Step 1.

We show that is bounded. In fact, since

(3.4)

we have

(3.5)

By induction, we get

(3.6)

This implies that is bounded and so , , , , and are bounded. Moreover, since and , and are also bounded. By condition (i), we also obtain

(3.7)

Step 2.

We show that . From (3.1), we have

(3.8)

Simple calculations show that

(3.9)

Since

(3.10)

for every we have

(3.11)

for every , where and .

On the other hand, from (3.1) we have

(3.12)

Also, simple calculations show that

(3.13)

Since

(3.14)

for every it follows that

(3.15)

Substituting (3.11) into (3.15), we derive

(3.16)

where and . From conditions (i) and (iv), it is easy to see that

(3.17)

Applying Lemma 2.2 to (3.16), we have

(3.18)

By (3.11), we also have that as .

Step 3.

We show that and . Indeed, it follows that

(3.19)

which implies that

(3.20)

Obviously, by (3.7) and Step 2, we have as . This implies that

(3.21)

By (3.7) and (3.21), we also have

(3.22)

Step 4.

We show that . To this end, let . Then, by convexity of , we have

(3.23)

So we obtain

(3.24)

Since and by condition (i) and (3.21), we have . Moreover, from (2.2) we obtain

(3.25)

and so

(3.26)

And hence

(3.27)

Then we have

(3.28)

Since , and , we get . Also by (3.21), we have

(3.29)

Step 5.

We show that for , where is a solution of the variational inequality

(3.30)

To this end, choose a subsequence of such that

(3.31)

Since is bounded, there exists a subsequence of which converges weakly to . We may assume without loss of generality that . Since by Steps 4 and 5, we have . Then we can obtain . Indeed, let us first show that . Let

(3.32)

Then is maximal monotone. Let . Since and , we have

(3.33)

On the other hand, from , we have and hence

(3.34)

Therefore we have

(3.35)

Hence we have as . Since is maximal monotone, we have and hence .

On the another hand, by Steps 3 and 4, . So, by Lemma 2.3, we obtain and hence . Then by (3.30) we have

(3.36)

Thus, from (3.7) we obtain

(3.37)

Step 6.

We show that for , where is a solution of the variational inequality

(3.38)

Indeed, from Lemma 2.4, we have

(3.39)

where

(3.40)

and . It is easily seen that , , and . Thus by Lemma 2.2, we obtain . This completes the proof.

Remark 3.2.

• (1) Theorem 3.1 improves the corresponding results in Chen et al. [13] and Iiduka and Takahashi [10]. In particular, if and is constant in (3.1), then Theorem 3.1 reduces to Theorem 3.1 of Iiduka and Takahashi [10].

• (2) We obtain a new composite iterative scheme for a nonexpansive mapping if in Theorem 3.1 as follows (see also Jung [18]):

  

  (3.41)

As a direct consequence of Theorem 3.1, we have the following result.

Corollary 3.3.

Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of to such that , and . Let be a sequence generated by

(3.42)

where , , and . If , , and satisfy the following conditions:

(i); ,

(ii) for all and for some ,

(iii) for some with ,

(iv); ; ,

then converges strongly to , which is a solution of the following variational inequality:

(3.43)

4. Applications

In this section, as in [10, 13], we obtain two theorems in a Hilbert space by using Theorem 3.1.

A mapping is called strictly pseudocontractive if there exists with such that

(4.1)

for every . If , then is nonexpansive. Put , where is a strictly pseudocontractive mapping with . Then is -inverse-strongly monotone; see [7]. Actually, we have, for all ,

(4.2)

On the other hand, since is a real Hilbert space, we have

(4.3)

Hence we have

(4.4)

Using Theorem 3.1, we first get a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping.

Theorem 4.1.

Let be a closed convex subset of a real Hilbert space . Let be an -strictly pseudocontractive mapping of into itself and a nonexpansive mapping of into itself such that , and . Let be a sequence generated by

(4.5)

where , , and . If , , and satisfy the conditions:

(i); ,

(ii) for all and for some ,

(iii) for some with ,

(iv); ; ,

then converges strongly to , which is a solution of the following variational inequality:

(4.6)

Proof.

Put . Then is -inverse-strongly monotone. We have and . Thus, the desired result follows from Theorem 3.1.

Using Theorem 3.1, we also have the following result.

Theorem 4.2.

Let be a real Hilbert space . Let be an -inverse-strongly monotone mapping of into itself and a nonexpansive mapping of into itself such that , and . Let be a sequence generated by

(4.7)

where , , and . If , , and satisfy the conditions:

(i); ,

(ii) for all and for some ,

(iii) for some with ,

(iv); ; ,

then converges strongly to , which is a solution of the following variational inequality:

(4.8)

Proof.

We have . So, putting , by Theorem 3.1, we obtain the desired result.

Remark 4.3.

If in Theorems 4.1 and 4.2, then Theorems 4.1 and 4.2 reduce to Chen et al. [13, Theorems 4.1 and 4.2]. Theorems 4.1 and 4.2 also extend in Iiduka and Takahashi [10, Theorems 4.1 and 4.2] to the viscosity methods in composite iterative schemes.