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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 917857, 11 pages
http://dx.doi.org/10.1155/2012/917857
Research Article

Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 25 October 2012; Accepted 8 November 2012

Academic Editor: Yongfu Su

Copyright © 2012 Hongjie Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mapping T and the solution sets of zero of a maximal monotone mapping and an α-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.

1. Introduction

Let be a real Hilbert space with inner product , and let be a nonempty closed convex subset of . We denote by the set of fixed point of . Then, a mapping is said to be nonexpansive if for all . The mapping is said to be firmly nonexpansive if for all ; see, for instance, Browder [1] and Goebel and Kirk [2]. The mapping is said to be firmly nonspreading [3] if for all . Iemoto and Takahashi [4] proved that is nonspreading if and only if for all . It is not hard to know that a nonspreading mapping is deduced from a firmly nonexpansive mapping; see [5, 6], and a firmly nonexpansive mapping is a nonexpansive mapping.

Many studies have been done for structuring the fixed point of nonexpansive mapping . In 1953, Mann [7] introduced the iteration as follows: a sequence defined by where the initial guess is arbitrary and is a real sequence in . It is known that under appropriate settings, the sequence converges weakly to a fixed point of . However, even in a Hilbert space, Mann iteration may fail to converge strongly, for example see [8].

Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern [9] proposed the following so-called Halpern iteration: where are arbitrary and is a real sequence in which satisfies , and . Then, converges strongly to a fixed point of ; see [9, 10].

In 1975, Baillon [11] first introduced the nonlinear ergodic theorem in Hilbert space as follows: converges weakly to a fixed point of for some .

Recently, in the case when is a nonexpansive mapping, is an -inverse strongly monotone mapping, and is a maximal monotone operator, Takahashi et al. [12] proved a strong convergence theorem for finding a point of , where is the set of fixed points of and is the set of zero points of .

In 2011, Manaka and Takahashi [13] for finding a point of the set of fixed points of and the set of zero points of in a Hilbert space, they introduced an iterative scheme as follows: where is a nonspreading mapping, is an -inverse strongly monotone mapping, and is a maximal monotone operator such that ; and are sequences which satisfy and . Then they proved that converges weakly to a point .

Motivated by above authors, we generalize and modify the iterative algorithms (1.5) and (1.6) for finding a common element of the set of fixed points of a nonspreading mapping and the set of zero points of monotone operator ( is an -inverse strongly monotone mapping, and is a maximal monotone operator). First, we prove that the sequence generated by our iterative method is weak convergence under the property conditions. Then, we prove that the strong convergence in a Hilbert space. As expected, we get some weak and strong convergence theorems about the common element of the set of fixed points of a nonspreading mapping and the set of zero points of an -inverse strongly monotone mapping and a maximal monotone operator in a Hilbert space.

2. Preliminaries

Let be a real Hilbert space with inner product , and let be a nonempty closed convex subset of . A set-valued mapping is said to be monotone if for any and and , it holds that

A monotone operator on is said to be maximal if has no monotone extension, that is, its graph is not properly contained in the graph of any other monotone operator on . For a maximal monotone operator on and , we may define a single-valued operator , which is called the resolvent of for . Let be a maximal monotone operator on , and let . For a constant , the mapping is said to be an -inverse strongly monotone if for any for all ,

Remark 2.1. It is not hard to know that if is an -inverse strongly monotone mapping, then it is Lipschitzian and hence uniformly continuous. Clearly, the class of monotone mappings include the class of an -inverse strongly monotone mappings.

Remark 2.2. It is well known that if is a nonexpansive mapping, then is -inverse strongly monotone, where is the identity mapping on ; see, for instance, [14]. It is known that the resolvent is firmly nonexpansive and for all .
For a single-valued mapping , a point is called a fixed point of if . For a multivalued mapping , a point is called a fixed point of if . The set of fixed points of is denoted by .
Let be a uniformly convex real Banach space, be a nonempty closed convex subset of . A multivalued mapping is said to be as follows.(i)Contraction if there exists a constant such that (ii)Nonexpansive if (iii)Quasinonexpansive if and
It is well known that every nonexpansive multivalued mapping with is multivalued quasinonexpansive. But there exist multivalued quasi-nonexpansive mappings that are not multivalued nonexpansive. It is clear that if is a quasi-nonexpansive multivalued mapping, then is closed.
A Banach space is said to satisfy Opials condition if whenever is a sequence in which converges weakly to , then

Lemma 2.3 (Manaka and Takahashi [13]). Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let . Let be an -inverse strongly monotone mapping of into , and let be a maximal monotone operator on such that the domain of is included in . Let be the resolvent of for any . Then, the following hold(i)if , then ;(ii)for any , if and only if .

Lemma 2.4 (Schu [15]). Suppose that is a uniformly convex Banach space and for all positive integers . Also suppose that and are two sequences of such that , , and hold for some . Then, .

Lemma 2.5 (Liu [16] and Xu [17]). Let be a sequence of nonnegative real numbers satisfying the property as follows where , , and satisfy the restrictions as follows(i),(ii),(iii).
Then, converges to zero as .

3. Strong Convergence Theorem

In this section, we prove the strong convergence theorems for finding a common element in common set of the fixed sets of a nonspreading mapping and the solution sets of zero of a maximal monotone operator and an -inverse strongly monotone operator and in a Hilbert space.

Theorem 3.1. Let be a nonempty convex closed subset of a real Hilbert space , let be an -inverse strongly monotone, let be maximal monotone, let be the resolvent of for any , and let be a nonspreading mapping. Assume that . We define where is sequences in such that . There exists such that for each . Then, converges strongly to , and is the metric projection of onto .

Proof. First, we prove that is bounded and exists for each . In fact, from Lemma 2.3, we have , together with (3.1) and is an -inverse strongly monotone, we get that
From the definition of and is nonspreading mapping, we obtain that
Together with (3.1), we have that
Hence, we get that for all . This means that is bounded, so is bounded. From is nonspreading, (3.3), and (3.2), we get that , , and are all bounded.
Since is bounded, there exists a subsequence of such that exists. Since is bounded, there exists a subsequence of such that as . Now, we prove that . First, we prove that . Since , replacing by , we have . Together with and is bounded, we obtain that , so we have .
Let . Since is nonspreading, we have that for all and ,
Summing these inequalities from to and dividing by , we have
Replacing by , we have
Since and are bounded, we have that as . Putting , we have
Hence, .
Next, we prove that . From (3.2) and (3.3) we have that
We rewrite above inequality as follows:
Replacing by , we have
Together with , and since exists, we obtain that
Since is firmly nonexpansive, and from (3.2), we have that
This means that
Together with (3.1) and (3.3), we have
Therefore, we have
Replacing by , we have
Since exists, from (3.14) and , we obtain
Since is Lipschitz continuous, we also obtain
By the definition of and (3.1), we have that
Since is monotone, so for , we have that and hence
Replacing by , we have that
Since is an -inverse strongly monotone, we have This means that as . From (3.20) and , we get that , together with (3.25), we have that
Since is maximal monotone, so . That is, .
Now, we prove that as . Without loss of generality, we may assume that there exists a subsequence of such that
Since is the metric projection of onto and , we have
This implies that
From (2.1), (3.1), and (3.3), we have
From Lemma 2.5 and (3.30), we have
This means that as .

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