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Abstract and Applied Analysis
Volume 2011, Article ID 261534, 12 pages
http://dx.doi.org/10.1155/2011/261534
Research Article

-Cocoercive Operator and an Application for Solving Generalized Variational Inclusions

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 32023, Taiwan
3Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan
4Department of Applied Mathematics, National Sun-Yat Sen University, Kaohsiung 804, Taiwan

Received 15 April 2011; Accepted 25 June 2011

Academic Editor: Ngai-Ching Wong

Copyright © 2011 Rais Ahmad 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

The purpose of this paper is to introduce a new -cocoercive operator, which generalizes many existing monotone operators. The resolvent operator associated with -cocoercive operator is defined, and its Lipschitz continuity is presented. By using techniques of resolvent operator, a new iterative algorithm for solving generalized variational inclusions is constructed. Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm. For illustration, some examples are given.

1. Introduction

Various concepts of generalized monotone mappings have been introduced in the literature. Cocoercive mappings which are generalized form of monotone mappings are defined by Tseng [1], Magnanti and Perakis [2], and Zhu and Marcotte [3]. The resolvent operator techniques are important to study the existence of solutions and to develop iterative schemes for different kinds of variational inequalities and their generalizations, which are providing mathematical models to some problems arising in optimization and control, economics, and engineering sciences. In order to study various variational inequalities and variational inclusions, Fang and Huang, Lan, Cho, and Verma investigated many generalized operators such as -monotone [4], -accretive [5], -accretive [6], -monotone[7, 8], -accretive mappings [9]. Recently, Zou and Huang [10] introduced and studied -accretive operators and Xu and Wang [11] introduced and studied -monotone operators.

Motivated and inspired by the excellent work mentioned above, in this paper, we introduce and discuss new type of operators called -cocoercive operators. We define resolvent operator associated with -cocoercive operators and prove the Lipschitz continuity of the resolvent operator. We apply -cocoercive operators to solve a generalized variational inclusion problem. Some examples are constructed for illustration.

2. Preliminaries

Throughout the paper, we suppose that is a real Hilbert space endowed with a norm and an inner product , is the metric induced by the norm (resp., ) is the family of all nonempty (resp., closed and bounded) subsets of , and is the Hausdorff metric on defined by where .

Definition 2.1. A mapping is said to be (i)Lipschitz continuous if there exists a constant such that (ii)monotone if (iii)strongly monotone if there exists a constant such that (iv) -expansive if there exists a constant such that if , then it is expansive.

Definition 2.2. A mapping is said to be cocoercive if there exists a constant such that

Note 1. Clearly is -Lipschitz continuous and also monotone but not necessarily strongly monotone and Lipschitz continuous (consider a constant mapping). Conversely, strongly monotone and Lipschitz continuous mappings are cocoercive, and it follows that cocoercivity is an intermediate concept that lies between simple and strong monotonicity.

Definition 2.3. A multivalued mapping is said to be cocoercive if there exists a constant such that

Definition 2.4. A mapping is said to be relaxed cocoercive if there exists a constant such that

Definition 2.5. Let and be the mappings. (i) is said to be cocoercive with respect to if there exists a constant such that (ii) is said to be relaxed cocoercive with respect to if there exists a constant such that (iii) is said to be -Lipschitz continuous with respect to if there exists a constant such that (iv) is said to be -Lipschitz continuous with respect to if there exists a constant such that

Example 2.6. Let with usual inner product. Let be defined by Suppose that is defined by Then is -cocoercive with respect to and -relaxed cocoercive with respect to since which implies that That is, is -cocoercive with respect to . which implies that that is, is -relaxed cocoercive with respect to .

3. -Cocoercive Operator

In this section, we define a new -cocoercive operator and discuss some of its properties.

Definition 3.1. Let be three single-valued mappings. Let be a set-valued mapping. is said to be -cocoercive with respect to mappings and (or simply -cocoercive in the sequel) if is cocoercive and , for every .

Example 3.2. Let , , , and be the same as in Example 2.6, and let be define by . Then it is easy to check that is cocoercive and , that is, is -cocoercive with respect to and .

Remark 3.3. Since cocoercive operators include monotone operators, hence our definition is more general than definition of -monotone operator [10]. It is easy to check that -cocoercive operators provide a unified framework for the existing -monotone, -monotone operators in Hilbert space and -accretive, -accretive operators in Banach spaces.
Since -cocoercive operators are more general than maximal monotone operators, we give the following characterization of -cocoercive operators.

Proposition 3.4. Let be -cocoercive with respect to , -relaxed cocoercive with respect to , is -expansive, is -Lipschitz continuous, and , . Let be -cocoercive operator. If the following inequality holds for all , then , where

Proof. Suppose that there exists some such that Since is -cocoercive, we know that holds for every , and so there exists such that It follows from (3.3) and (3.4) that which gives since . By (3.4), we have . Hence and so .

Theorem 3.5. Let be a Hilbert space and a maximal monotone operator. Suppose that is a bounded cocoercive and semicontinuous with respect to and . Let be also -cocoercive with respect to and -relaxed cocoercive with respect to . The mapping is -expansive, and is -Lipschitz continuous. If and , then is -cocoercive with respect to and .

Proof. For the proof we refer to [10].

Theorem 3.6. Let be a -cocoercive with respect to and -relaxed cocoercive with respect to , is -expansive, and is -Lipschitz continuous, and . Let be an -cocoercive operator with respect to and . Then the operator is single-valued.

Proof. For any given , let . It follows that As is cocoercive (thus monotone), we have Since is -cocoercive with respect to and -relaxed cocoercive with respect to , is -expansive and is -Lipschitz continuous, thus (3.7) becomes since . Thus, we have and so is single-valued.

Definition 3.7. Let be -cocoercive with respect to and -relaxed cocoercive with respect to , is -expansive, is -Lipschitz continuous, and , . Let be an -cocoercive operator with respect to and . The resolvent operator is defined by

Now, we prove the Lipschitz continuity of resolvent operator defined by (3.9) and estimate its Lipschitz constant.

Theorem 3.8. Let be -cocoercive with respect to , -relaxed cocoercive with respect to , is -expansive, is -Lipschitz continuous, and , . Let be an -cocoercive operator with respect to and . Then the resolvent operator is -Lipschitz continuous, that is,

Proof. Let and be any given points in . It follows from (3.9) that This implies that For the sake of clarity, we take Since is cocoercive (hence monotone), we have which implies that Further, we have and so thus, This completes the proof.

4. Application of -Cocoercive Operators for Solving Variational Inclusions

We apply -cocoercive operators for solving a generalized variational inclusion problem.

We consider the problem of finding and such that where , and are the mappings. Problem (4.1) is introduced and studied by Huang [12] in the setting of Banach spaces.

Lemma 4.1. The , where , is a solution of the problem (4.1), if and only if is a solution of the following: where is a constant.

Proof. By using the definition of resolvent operator , the conclusion follows directly.

Based on (4.2), we construct the following algorithm.

Algorithm 4.2. For any , compute the sequences and by iterative schemes such that for all , and is a constant.

Theorem 4.3. Let be a real Hilbert space and the single-valued mappings. Let be a multi-valued mapping and the multi-valued -cocoercive operator. Assume that (i) is -Lipschitz continuous in the Hausdorff metric ;(ii) is -cocoercive with respect to and -relaxed cocoercive with respect to ;(iii) is -expansive; (iv) is -Lipschitz continuous; (v) is -Lipschitz continuous and -strongly monotone; (vi) is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to ; (vii) .

Then the generalized variational inclusion problem (4.1) has a solution with , and the iterative sequences and generated by Algorithm 4.2 converge strongly to and , respectively.

Proof. Since is -Lipschitz continuous, it follows from Algorithm 4.2 that for .
Using the -strong monotonicity of , we have which implies that Now we estimate by using the Lipschitz continuity of , Since is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to , is -Lipschitz continuous and using (4.4), (4.7) becomes or Using (4.9), (4.6) becomes where Let We know that and . From assumption (vii), it is easy to see that . Therefore, it follows from (4.10) that is a Cauchy sequence in . Since is a Hilbert space, there exists such that as . From (4.4), we know that is also a Cauchy sequence in , thus there exists such that and . By the continuity of , and and Algorithm 4.2, we have Now, we prove that . In fact, since , we have which implies that . Since , it follows that . By Lemma 4.1, we know that is a solution of problem (4.1). This completes the proof.

Acknowledgments

This work is partially done during the visit of the first author to National Sun-Yat Sen University, Kaohsiung, Taiwan. The first and second authors are supported by Department of Science and Technology, Government of India under Grant no. SR/S4/MS: 577/09. The fourth author is supported by Grant no. NSC 99-2221-E-037-007-MY3.

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