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Journal of Mathematics
Volume 2013 (2013), Article ID 738491, 10 pages
http://dx.doi.org/10.1155/2013/738491
Research Article

Generalized -Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

1Department of Applied Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh 202002, India
2Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat 395 007, India

Received 18 March 2013; Revised 4 May 2013; Accepted 8 May 2013

Academic Editor: Kaleem R. Kazmi

Copyright © 2013 Shamshad Husain 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 investigate a new class of cocoercive operators named generalized -cocoercive operators in Hilbert spaces. We prove that generalized -cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized -cocoercive operators. Some examples are given to justify the definition of generalized -cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized -cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

1. Introduction

Variational inclusions, as the generalization of variational inequalities, have been widely studied in recent years. One of the most interesting and important problems in the theory of variational inclusions include variational, quasi-variational, and variational-like inequalities as special cases. For applications of variational inclusions, see, for example, [1]. Various kinds of iterative methods have been studied to solve the variational inclusions. Among these methods, the resolvent operator technique for the study of variational inclusions has been widely used by many authors. For details, we refer to [216].

Recently Fang and Huang, Kazmi and Khan, and Lan et al. investigated several resolvent operators for generalized operators such as -monotone [5], -accretive [6], -proximal point [11], -accretive [12], -monotone [7], -accretive [13], and mappings. Very recently, Zou and Huang [16] introduced and studied -accretive operators, Kazmi et al. [810] introduced and studied generalized -accretive operators, --proximal point mapping, Xu and Wang [15] introduced and studied -monotone operators, and Ahmad et al. [2] introduced and studied -cocoercive operators.

Motivated by the recent work going in this direction, we consider a class of cocoercive operators called generalized --cocoercive, a natural generalization of monotone (accretive) operators in Hilbert (Banach) spaces. For details, we refer to [2, 57, 1316]. We prove that generalized --cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized --cocoercive operators. Further, we consider the generalized set-valued variational-like inclusion problem involving generalized --cocoercive operator in Hilbert spaces. Using new a resolvent operator technique, we prove the existence of solutions and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as an extension and generalization of some known results [2, 15, 16]. For illustration of Definitions 2 and 7 and examples 2.3 and 3.2 are given, respectively.

2. Preliminaries

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

In the sequel, let us recall some concepts.

Definition 1 (see [4, 17]). Let and be two mappings. Then, is said to be(i)-monotone if (ii)--strongly monotone if there exists a constant such that (iii)--cocoercive if there exists a constant such that (iv)--relaxed cocoercive if there exists a constant such that (v)-Lipschitz continuous, if there exists a constant such that (vi)-expansive, if there exists a constant such that if , then it is expansive;(vii) is said to be -Lipschitz continuous, if there exists a constant such that

If , for all , then definitions (i) to (iv) reduce to the Definitions of monotonicity, strong monotonicity [18], cocoercivity [19], and relaxed cocoercive, respectively.

Definition 2. Let , , and be the single-valued mappings. Then,(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 --strongly monotone 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 (v) is said to be -Lipschitz continuous with respect to if there exists a constant such that (vi) is said to be -Lipschitz continuous with respect to if there exists a constant such that

If , for all and , then Definitions (i)-(ii) are reduced to the definition of cocoercivity and relaxed cocoercive [2], respectively, and (iii) reduces to strong accretivity [16].

Example 3. Let with usual inner product. Let be defined by for all scalers with and for all .
Suppose that . Then, is --cocoercive with respect to , --relaxed cocoercive with respect to , --strongly monotone with respect to , and is -Lipschitz continuous.

Indeed, let for any , that is, is --cocoercive with respect to . Consider that is, is --relaxed cocoercive with respect to . In addition, that is, is --strongly monotone with respect to . Moreover, that is, is -Lipschitz continuous.

Definition 4. A set-valued mapping is said to be --relaxed monotone if there exists a constant such that

Definition 5. A set-valued mapping is said to be -Lipschitz continuous, if there exists a constant such that

Definition 6. Let be the set-valued mappings. A mapping is said to be(i)-Lipschitz continuous in the first argument with respect to , if there exists a constant such that (ii)-Lipschitz continuous in the second argument with respect to , if there exists a constant such that

3. Generalized --Cocoercive Operators

Definition 7. Let , , and be the single-valued mappings. Let be --cocoercive with respect to , --relaxed cocoercive with respect to , and --strongly monotone with respect to . Then, the set-valued mapping is said to be a generalized --cocoercive with respect to the mappings , and , if(i) is --relaxed monotone;(ii), for all .

Example 8. Let , and be the same as in Example 3, and let be defined by , for all .

We claim that is --relaxed monotone mapping. Indeed, for any , we have Furthermore, is also a generalized --cocoercive operator since for any .

Remark 9. If , is -strongly monotone, and is -relaxed monotone, then generalized --cocoercive operator reduces to --monotone operator introduced and studied by Xu and Wang [15].

Proposition 10. Let set-valued mapping be a generalized --cocoercive operator with respect to the mappings , and . If is -expansive, is -Lipschitz continuous, and , with , then the following inequality holds for all and implies , where

Proof. Suppose on the contrary that there exists such that Since is a generalized --cocoercive, we know that holds for all , and so there exists such that Now, Setting in (27), and then from the resultant, (28) and --relaxed monotonicity of , we obtain Since is --cocoercive with respect to , --relaxed cocoercive with respect to , --strongly monotone with respect to , and is -expansive, is -Lipschitz continuous, thus (30) becomes which gives since . By (27), we have , a contradiction. This completes the proof.

Theorem 11. Let set-valued mapping be a generalized --cocoercive operator with respect to the mappings , and . If is -expansive, is -Lipschitz continuous, and , with , then is single-valued.

Proof. For any given , let . It follows that Since is --relaxed monotone, we have Since is --cocoercive with respect to , --relaxed cocoercive with respect to , --strongly monotone with respect to , and is -expansive, is -Lipschitz continuous, thus (33) becomes since . Hence, it follows that . This implies that , and so is single-valued.

Definition 12. Let set-valued mapping be a generalized --cocoercive operator with respect to the mappings , and . If is -expansive, is -Lipschitz continuous, and , and , then the resolvent operator is defined by

Now, we prove that the resolvent operator defined by (35) is Lipschitz continuous.

Theorem 13. Let set-valued mapping be a generalized --cocoercive operator with respect to the mappings , and . If is -expansive, is -Lipschitz continuous, is -Lipschitz continuous, and , with , then the resolvent operator is -Lipschitz continuous, that is,

Proof. Let be any given points. It follows from (35) that Let and .
Since is --relaxed monotone, we have which implies Further, we have and hence, that is, This completes the proof.

4. An Application of --Cocoercive Operators for Solving Generalized Variational Inclusions

In this section, we will show that under suitable assumptions, the generalized --cocoercive operator can also play important roles for solving the variational inclusion problem in Hilbert spaces.

Let , , , , and be the single-valued mappings, and be the set-valued mappings such that is generalized --cocoercive with respect to , and and range dom . Then we consider the problem to find such that

The problem (43) is called generalized set-valued variational-like inclusion problem. The problem of type (43) was introduced and studied by Chidume et al. [3] by applying -proximal mapping. If and , for all , and , where is a set-valued mapping. Then, problem (43) reduces to the problem of finding such that The problem of type (44) was studied by Ahmad et al. [2] by applying -cocoercive operators.

If , , and , for all , then problem (43) reduces to the problem of finding such that The problem of type (45) was studied by Verma [14] in the setting of Banach spaces when is -maximal-relaxed accretive.

Lemma 14. The , where is a solution of the problem (43), if and only if satisfies the following relation: where and is a constant.

Proof. By using the definitions of resolvent operators , the conclusion follows directly.

Using Lemma 14 and using the technique of Chidume et al. [3] and Nadler [20], we develop an iterative algorithm for finding the approximate solution of problem (43) as follows.

Algorithm 15. Let , , , , , and be such that, for each , , where is the set-valued mappings defined by where is a set-valued mapping such that, is generalized --cocoercive with respect to the mappings , and .
For given , take and . Let Hence, there exists such that . Since and , then, by Nadler’s result [20], there exist and such that Let Hence, there exists such that . By induction, we can define iterative sequences , , , and as follows: for all , and is a constant.

If , for all , then Algorithm 15 reduces to the following algorithm for solving the problem (44).

Algorithm 16. For any , and , compute the sequence and by the following: for all , and is a constant.

If , for all , then Algorithm 15 reduces to the following algorithm for solving the problem (45).

Algorithm 17. For any , compute the sequence by the following: for all , and is a constant.

Now, we prove the existence of a solution of problem (43) and the convergence of Algorithm 15.

Theorem 18. Let be a real Hilbert space. Let , , , and be single-valued mappings, and and be the set-valued mappings such that is generalized --cocoercive operator with respect to the mappings , and and range   dom , and, for each , let , where is defined by (47). Assume that(i) are -Lipschitz continuous with constants and , respectively;(ii) is -Lipschitz continuous, is -expansive, and is -Lipschitz continuous;(iii) is -Lipschitz continuous and -strongly monotone;(iv) is -Lipschitz continuous with respect to , -Lipschitz continuous with respect to , and -Lipschitz continuous with respect to ;(v) is -Lipschitz continuous in the first argument with respect to and -Lipschitz continuous in the second argument with respect to .
In addition, where and , , .
Then, generalized set-valued variational inclusion problem (43) has a solution , and the iterative sequences , , , and generated by Algorithm 15 converge strongly to , , , and , respectively.

Proof. Since are -Lipschitz continuous with constants and , respectively, it follows from (52) and (53) that for .
It follows from (51) and Theorem 13 that Since is -Lipschitz continuous with respect to , -Lipschitz continuous with respect to , and -Lipschitz continuous with respect to , and is -Lipschitz continuous, we have
Since is -Lipschitz continuous in the first argument with respect to and - Lipschitz continuous in the second argument with respect to , and are -Lipschitz continuous with constants and , respectively, we have
Using (59), (60) in (58), we have
Using the -strong monotonicity of , we have which implies that Combining (61) and (63), we have where Let
From (56), it is easy to see that . Therefore, (64) implies that is a Cauchy sequence in . Since is a Hilbert space, there exists such that as . From (57), and are also Cauchy sequences in ; thus, there exist such that and as . By the continuity of , and (51) of Algorithm 15, we have
Now, we prove that . In fact, since , we have which implies that . Since , it follows that . Similarly, it is easy to see that . By Lemma 14, is the solution of problem (43). This completes the proof.

Based on Lemma 14 and Algorithm 16, Theorem 18 reduced to the following result for solving problem (44).

Theorem 19. Let be a real Hilbert space. Let , , , and be single-valued mappings, and let and be the set-valued mappings such that is generalized --cocoercive operator with respect to the mappings , and and range   dom , and, for each , let , where is defined by (47). Assume that(i) is -Lipschitz continuous with constants ;(ii) is -Lipschitz continuous, is -expansive, and is -Lipschitz continuous;(iii) is -Lipschitz continuous and -strongly monotone;(iv) is -Lipschitz continuous with respect to , -Lipschitz continuous with respect to , and -Lipschitz continuous with respect to .
In addition, where and , , .
Then, generalized set-valued variational inclusion problem (44) has a solution , and the iterative sequences , , and generated by Algorithm 16 converge strongly to , and , respectively.

Based on Lemma 14 and Algorithm 17, Theorem 18 reduced to the following result for solving problem (45).

Theorem 20. Let be a real Hilbert space. Let , , , and be single-valued mappings, and let be the set-valued mappings such that is generalized --cocoercive operator with respect to the mappings , and and range   dom , and, for each , let , where is defined by (47). Assume that(i) is -Lipschitz continuous, is -expansive, and is -Lipschitz continuous;(ii) is -Lipschitz continuous and -strongly monotone;(iii) is -Lipschitz continuous with respect to , -Lipschitz continuous with respect to , and -Lipschitz continuous with respect to .
In addition, where and , , .
Then, generalized set-valued variational inclusion problem (45) has a solution , and the iterative sequence and generated by Algorithm 17 converge strongly to and , respectively.

Acknowledgments

The authors are grateful to the editor and referees for valuable comments and suggestions.

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