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Journal of Calculus of Variations
Volume 2013 (2013), Article ID 461371, 8 pages
http://dx.doi.org/10.1155/2013/461371
Research Article

Algorithm for Solving a New System of Generalized Variational Inclusions in Hilbert Spaces

Department of Applied Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh 202002, India

Received 23 March 2013; Accepted 29 April 2013

Academic Editor: Raouf Boucekkine

Copyright © 2013 Shamshad Husain and Sanjeev Gupta. 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 introduce and study a new system of generalized variational inclusions involving -cocoercive and relaxed -cocoercive operators, which contain the systems of variational inclusions and the systems of variational inequalities, variational inclusions, and variational inequalities as special cases. By using the resolvent technique for the -cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces. An example is given to justify the main result. Our results can be viewed as a generalization of some known results in the literature.

1. Introduction

Variational inclusions have been widely studied in recent years. The theory of variational inclusions includes variational, quasi-variational, variational-like inequalities as special cases. Various kinds of iterative methods have been studied to solve the variational inclusions. Among these methods, the resolvent operator technique to study the variational inclusions has been widely used by many authors. For details, we refer to [115]. For applications of variational inclusions, see [16].

Fang and Huang, Lan, Cho, and Verma, and kazmi investigated several resolvent operators for generalized operators such as -monotone, -monotone, -accretive, -accretive, -accretive, -monotone, -accretive, -proximal, and --proximal mappings. For further details, we refer to [26, 810, 13] and the references therein. Very recently, Zou and Huang [15] introduced and studied -accretive operators, Xu and Wang [14] introduced and studied -monotone operators, and Ahmad et al. [1] introduced and studied -cocoercive operators.

Inspired and motivated by researches going on in this area, we introduce and study a new system of generalized variational inclusions in Hilbert spaces. By using the resolvent operator technique for the -cocoercive operator, we develop a new class of iterative algorithms to solve a class of relaxed cocoercive variational inclusions associated with -cocoercive operators in Hilbert space. For illustration of Definitions 2, 5 and main result Theorem 19 Examples 4, 6, and 20 are given, respectively. Our results can be viewed as a refinement and improvement of Bai and Yang [2], Huang and Noor [17], and Noor et al. [11].

2. Preliminaries

Throughout this paper, we suppose that is a real Hilbert space endowed with a norm and an inner product , respectively. is the family of all the nonempty subsets of .

In the sequel, let us recall some concepts.

Definition 1 (see [18, 19]). 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 (see [1]). 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

Definition 3. A multivalued mapping is said to be -cocoercive if there exists a constant such that

Example 4 (see [1]). Let with usual inner product. Let be defined by such that . Suppose that is defined by
Then is -cocoercive with respect to and -relaxed cocoercive with respect to .

Definition 5 (see [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 6 (see [1]). Let , and be the same as in Example 4, and let be defined by , for all . Then is cocoercive and , for all ; that is, is -cocoercive with respect to and .

Proposition 7 (see [1]). Let be -cocoercive with respect to and -relaxed cocoercive with respect to . A is -expansive, is -Lipschitz continuous, and , . Let be -cocoercive operator. If the following inequality holds for all , then , where

Theorem 8 (see [1]). Let be a -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 . Then the operator is single-valued.

Definition 9 (see [1]). Let be a -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

Theorem 10 (see [1]). Let be a -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 . Then resolvent operator is −Lipschitz continuous; that is,

3. A New System of Generalized Variational Inclusions

In this section, we will introduce a new system of generalized variational inclusions involving -cocoercive operators.

Let be a real Hilbert space whose inner product and norm are denoted by , , respectively. Let be a closed and convex set in . Let , and be single-valued mappings. Let be a set-valued mapping such that is -cocoercive operator with respect to and and be a continuous function. We consider the system of generalized variational inclusions of finding such that Special Cases. (I) If are univariate mappings, problem (16) is equivalent to finding , such that  which appears to be a new one.

(II) If , problem (16) is equivalent to finding , such that  which appear to be a new one.

(III) If , , , and , problem (17) is equivalent to finding , such that  which is known as the variational inclusion problem or finding the zero of the sum of two (more) cocoercive operators. It is well known that a wide class of linear and nonlinear problems can be studied via variational inclusion problems.

(IV) We note that if , the subdifferential of a proper, convex and lower semicontinuous function, then the system of variational inclusions (16) is equivalent to finding such that  or equivalently the problem of finding such that  which is called the system of mixed general variational inequalities involving four different nonlinear operators. The problem of type (21) is studied in [7].

(V) If is univariate operator and , , and , problem (21) is equivalent to finding , such that  which is known as the mixed general variational inequality or variational inequality of the second type. For the applications and numerical methods for solving the mixed variational inequalities, see [12].

(VI) If is an indicator function of a closed convex set in , then problem (21) is equivalent to finding , such that  which is called the system of general variational inequalities. Such type of problem is studied in [20].

(VII) If , then problem (21) is equivalent to finding such that  which can be viewed as a generalization of the system considered and studied in [17, 21].

(VIII) If is the indicator function of a closed convex set , then problem (22) is equivalent to finding such that  which is known as the general variational inequality introduced and studied by Noor [22, 23] in 1988. This shows that the system of generalized variational inclusions (16) is more general and includes several classes of variational inclusions/inequalities and related optimization problems as special cases. For the recent applications, numerical methods, and formulations of variational inequalities and variational inclusions, see [124] and the references therein.

We now show that the system of generalized variational inclusions (16) is equivalent to the fixed-point problem, and this is the motivation of our next result.

Lemma 11. Let be -cocoercive operator. Then is a solution of problem (16) if and only if satisfies the following: where and .

Proof. The conclusion can be drawn directly from the definition of resolvent operators and .
This equivalent formulation is used to suggest and analyze a number of iterative methods for solving the system of generalized variational inclusions (16). To do so, one rewrites the equations in the following form:
Based on Lemma 11, we construct the following iterative algorithm for solving (16).

Algorithm 12. For a given , compute the sequences and from the iterative schemes: where .

If , then Algorithm 12 reduces to Algorithm 13.

Algorithm 13. For a given , compute the sequences and from the iterative schemes:

For suitable and appropriate choice of the operators , ,, , , , , , and spaces, one can obtain a wide class of iterative methods for solving different classes of variational inclusions and related optimization problems. This shows that Algorithm 12 is quite flexible and general and includes various known and new algorithms for solving variational inequalities and related optimization problems as special cases.

Definition 14. A mapping is called p-strongly monotone in the first variable if there exists a constant such that, for all ,

Definition 15. A mapping is called relaxed -cocoercive if there exists a constant such that, for all ,

Definition 16. A mapping is called relaxed -cocoercive in the first variable if there exist constants , such that, for all ,

The class of relaxed -cocoercive mappings is more general than the class of strongly monotone mappings.

Definition 17. A mapping is called -Lipschitz continuous in the first variable if there exists a constant such that, for all ,

Lemma 18 (see [24]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in with , , and then .

Theorem 19. Let be a real Hilbert space. Suppose that , are single-valued mappings and is a set-valued mapping such that is -cocoercive operator with respect to and . Assume that(i) is -cocoercive with respect to , -relaxed cocoercive with respect to , and ;(ii) is -expansive, is -Lipschitz continuous, and ;(iii) is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to ;(iv) is relaxed -cocoercive and -Lipschitz continuous in the first variable;(v) is relaxed -cocoercive and -Lipschitz continuous in the first variable;(vi) is relaxed -cocoercive and -Lipschitz continuous;(vii) is relaxed -cocoercive and -Lipschitz continuous;(viii) and ;(ix) and , where and , . Then the iterative sequences and generated by Algorithm 12 converge strongly to and , respectively, and is a solution of problem (16).

Proof. To prove the result, we need first to evaluate for all . From (28), and the Lipschitz continuity of the resolvent operator , we have
By the assumption that is relaxed -cocoercive and -Lipschitz continuous in the first variable, we obtain that where . By the assumption that is relaxed -cocoercive and -Lipschitz continuous, we arrive at where . Now, we estimate where and .
Substituting (37)–(39) into (36) yields where .
Next we estimate
By the assumption that is relaxed -cocoercive and -Lipschitz continuous in the first variable, we see that where . From the proof of (38), we can obtain that where .
Now, we estimate where and . Substituting (42)–(44) into (41) yields where . Since , we observe that Substituting (46) into (40) yields
Noticing condition (ix) and applying Lemma 18 to (47), we get the desired conclusion easily. This completes the proof.

Example 20. Let with usual inner product. Let be defined by Suppose that is defined by Then, it is easy to cheek the following.(i) is -cocoercive with respect to , for , 8, and -relaxed cocoercive with respect to , for , 2.(ii) is -expansive, for , and is -Lipschitz continuous, for .(iii) is -Lipschitz continuous with respect to , for , and -Lipschitz continuous with respect to , for .
Let be defined by Then, it is easy to verify the following.(iv) is relaxed -cocoercive and -Lipschitz continuous.(v) is relaxed -cocoercive, for , and -Lipschitz continuous, for .
Let be defined by Then, it is easy to verify the following.(vi) is relaxed -cocoercive, for , and -Lipschitz continuous, for .(vii) is relaxed -cocoercive, for , and -Lipschitz continuous, for .(viii)Clearly, for the constants obtained in (i) to (vii) above, the conditions of Theorem 19 are satisfied for the inclusion system (16) for , .

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