Strong Convergence of Monotone Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansive Mappings

Abstract

We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping in a Banach space by using monotone hybrid iteration method. By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Banach space.

1. Introduction

Let be a real Banach space and let be the dual space of Let be a maximal monotone operator from to . It is well known that many problems in nonlinear analysis and optimization can be formulated as follows. Find a point satisfying

(1.1)

We denote by the set of all points such that Such a problem contains numerous problems in economics, optimization, and physics and is connected with a variational inequality problem. It is well known that the variational inequalities are equivalent to the fixed point problems. There are many authors who studied the problem of finding a common element of the fixed point of nonlinear mappings and the set of solutions of a variational inequality in the framework of Hilbert spaces see; for instance, [1–11] and the reference therein.

A well-known method to solve problem (1.1) is called the proximal point algorithm: and

(1.2)

where and are the resovents of . Many researchers have studies this algorithm in a Hilbert space; see, for instance, [12–15] and in a Banach space; see, for instance, [16, 17].

In 2005, Matsushita and Takahashi [18] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space : chosen arbitrarily,

(1.3)

where is the duality mapping on , . They proved that generated by (1.3) converges strongly to a fixed point of under condition that .

In 2008, Su et al. [19] modified the CQ method (1.3) for approximation a fixed point of a closed hemi-relatively nonexpansive mapping in a Banach space. Their method is known as the monotone hybrid method defined as the following. chosen arbitrarily, then

(1.4)

where is the duality mapping on , They proved that generated by (1.4) converges strongly to a fixed point of under condition that .

Note that the hybrid method iteration method presented by Matsushita and Takahashi [18] can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping.

Very recently, Inoue et al. [20] proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method.

Theorem 1.1 (Inoue et al. [20]).

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of Let be a monotone operator satisfying and let for all . Let be a relatively nonexpansive mapping such that . Let be a sequence generated by and

(1.5)

for all , where is the duality mapping on , and for some . If , then converges strongly to , where is the generalized projection from onto .

Employing the ideas of Inoue et al. [20] and Su et al. [19], we modify iterations (1.4) and (1.5) to obtain strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemi-relatively nonexpansive mappings in a Banach space. The results of this paper modify and improve the results of Inoue et al. [20], and some others.

2. Preliminaries

Throughout this paper, all linear spaces are real. Let and be the sets of all positive integers and real numbers, respectively. Let be a Banach space and let be the dual space of . For a sequence of and a point the weak convergence of to and the strong convergence of to are denoted by and , respectively.

Let be a Banach space. Then the duality mapping from into is defined by

(2.1)

Let be the unit sphere centered at the origin of . Then the space is said to be smooth if the limit

(2.2)

exists for all . It is also said to be uniformly smooth if the limit exists uniformly in . A Banach space is said to be strictly convex if whenever and . It is said to be uniformly convex if for each , there exists such that whenever and . We know the following (see, [21]):

(i)if in smooth, then is single valued;

• (ii) if is reflexive, then is onto;

• (iii) if is strictly convex, then is one to one;

(iv)if is strictly convex, then is strictly monotone;

(v)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

Let be a smooth strictly convex and reflexive Banach space and let be a closed convex subset of Throughout this paper, define the function by

(2.3)

Observe that, in a Hilbert space , (2.3) reduces to , for all . It is obvious from the definition of the function that for all ,

(1)

(2),

(3)

Following Alber [22], the generalized projection from onto is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem

(2.4)

Existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping In a Hilbert space, is the metric projection of onto .

Let be a closed convex subset of a Banach space and let be a mapping from into itself. We use to denote the set of fixed points of that is, Recall that a self-mapping is hemi-relatively nonexpansive if and for all and .

A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . We denote the set of all asymptotic fixed points of by . A hemi-relative nonexpansive mapping is said to be relatively nonexpansive if . The asymptotic behavior of a relatively nonexpansive mapping was studied in [23].

Recall that an operator in a Banach space is call closed, if and , then .

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

Lemma 2.1 (Kamimura and Takahashi [13]).

Let be a uniformly convex and smooth Banach space and let and be two sequences in such that either or is bounded. If , then .

Lemma 2.2 (Matsushita and Takahashi [18]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space and let be a relatively hemi-nonexpansive mapping from into itself. Then is closed and convex.

Lemma 2.3 (Alber [22], Kamimura and Takahashi [13]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space, and let . Then, if and only if for all .

Lemma 2.4 (Alber [22], Kamimura and Takahashi [13]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space. Then

(2.5)

Let be a smooth, strictly convex, and reflexive Banach space, and let be a set-valued mapping from to with graph , domain and range We denote a set-valued operator from to by is said to be monotone of A monotone operator is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. It is known that a monotone mapping is maximal if and only if for for every implies that . We know that if is a maximal monotone operator, then is closed and convex; see [19] for more details. The following result is well known.

Lemma 2.5 (Rockafellar [24]).

Let be a smooth, strictly convex, and reflexive Banach space and let be a monotone operator. Then is maximal if and only if for all

Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying

(2.6)

Then we can define the resolvent by

(2.7)

We know that consists of one point. For , the Yosida approximation is defined by for all .

Lemma 2.6 (Kohsaka and Takahashi [25]).

Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying

(2.8)

Let and let and be the resolvent and the Yosida approximation of , respectively. Then, the following hold:

(i)

(ii);

(iii)

Lemma 2.7 (Kamimura and Takahashi [13]).

Let be a uniformly convex and smooth Banach space and let . Then there exists a strictly increasing, continuous and convex function such that and

(2.9)

for all , where .

3. Main Results

In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space by using the monotone hybrid iteration method.

Theorem 3.1.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a monotone operator satisfying and let for all . Let be a closed hemi-relatively nonexpansive mapping such that . Let be a sequence generated by

(3.1)

for all , where is the duality mapping on and for some . If , then converges strongly to , where is the generalized projection from onto

Proof.

We first show that and are closed and convex for each . From the definition of and it is obvious that is closed and is closed and convex for each . Next, we prove that is convex.

Since

(3.2)

is equivalent to

(3.3)

which is affine in , and hence is convex. So, is a closed and convex subset of for all . Let Put for all Since and are hemi-relatively nonexpansive mappings, we have

(3.4)

So, for all , which implies that . Next, we show that for all . We prove that by induction. For , we have . Assume that for some . Because is the projection of onto by Lemma 2.3, we have

(3.5)

Since , we have

(3.6)

This together with definition of implies that and hence for all . So, we have that for all . This implies that is well defined. From definition of we have . So, from , we have

(3.7)

Therefore, is nondecreasing. It follows from Lemma 2.4 and that

(3.8)

for all . Therefore, is bounded. Moreover, by definition of , we know that and are bounded. So, the limit of exists. From we have that for any positive integer,

(3.9)

This implies that . Since is bounded, there exists such that . Using Lemma 2.7, we have, for with

(3.10)

where is a continuous, strictly increasing, and convex function with . Then the properties of the function yield that is a Cauchy sequence in . So there exists such that . In view of and definition of , we also have

(3.11)

It follows that Since is uniformly convex and smooth, we have from Lemma 2.1 that

(3.12)

So, we have Since is uniformly norm-to-norm continuous on bounded sets, we have

(3.13)

On the other hand, we have

(3.14)

This follows

(3.15)

From (3.13) and we obtain that

Since is uniformly norm-to-norm continuous on bounded sets, we have

(3.16)

From

(3.17)

we have

(3.18)

From (3.4), we have

(3.19)

Using and Lemma 2.6, we have

(3.20)

It follows that

(3.21)

From (3.13) and we have

Since is uniformly convex and smooth, we have from Lemma 2.1 that

(3.22)

From we have

(3.23)

Since and we have . Since is a closed operator and , is a fixed point of . Next, we show . Since is uniformly norm-to-norm continuous on bounded sets, from (3.22) we have

(3.24)

From , we have

(3.25)

Therefore, we have

(3.26)

For , from the monotonicity of , we have for all . Letting , we get . From the maximality of , we have . Finally, we prove that . From Lemma 2.4, we have

(3.27)

Since and we get from Lemma 2.4 that

(3.28)

By the definition of , it follows that and , whence . Therefore, it follows from the uniqueness of the that .

As direct consequences of Theorem 3.1, we can obtain the following corollaries.

Corollary 3.2.

Let be a uniformly convex and uniformly smooth Banach space. Let be a maximal monotone operator with and let for all . Let be a sequence generated by and

(3.29)

for all where is the duality mapping on , and for some . Then converges strongly to , where is the generalized projection from onto .

Proof.

Putting , and in Theorem 3.1, we obtain Corollary 3.2.

Let be a Banach space and let be a proper lower semicontinuous convex function. Define the subdifferential of as follows:

(3.30)

for each . Then, we know that is a maximal monotone operator; see [21] for more details.

Corollary 3.3 (Su et al. [19, Theorem  3.1]).

Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of and let be a closed hemi-relatively nonexpansive mapping from into itself such that . Let be a sequence generated by

(3.31)

for all , where is the duality mapping on and . If   then converges strongly to , where is the generalized projection from onto .

Proof.

Set in Theorem 3.1, where is the indicator function; that is,

(3.32)

Then, we have that is a maximal monotone operator and for , in fact, for any and , we have from Lemma 2.3 that

(3.33)

So, we obtain the desired result by using Theorem 3.1.

Since every relatively nonexpansive mapping is a hemi-relatively one, the following theorem is obtained directly from Theorem 3.1.

Theorem 3.4.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of Let be a monotone operator satisfying and let for all Let be a closed relatively nonexpansive mapping such that . Let be a sequence generated by

(3.34)

for all , where is the duality mapping on and for some . If then converges strongly to , where is the generalized projection from onto .

Corollary 3.5 (Su et al. [19, Theorem  3.2]).

Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of and let be a closed relatively nonexpansive mapping from into itself such that . Let be a sequence generated by

(3.35)

for all , where is the duality mapping on and . If   , then converges strongly to , where is the generalized projection from onto .

Proof.

Set in Theorem 3.4, where is the indicator function. So, from Theorem 3.4, we obtain the desired result.