Iterative Algorithms for New General Systems of Set-Valued Variational Inclusions Involving -Maximal Relaxed Monotone Operators

Abstract

We introduce and study a class of new general systems of set-valued variational inclusions involving -maximal relaxed monotone operators in Hilbert spaces. By using the general resolvent operator technique associated with -maximal relaxed monotone operators, we construct some new iterative algorithms for finding approximation solutions to the general system of set-valued variational inclusion problem and prove the convergence of this algorithm. Our results improve and extend some known results.

1. Introduction

It is well known that variational inequalities and variational inclusions, which have been extended and generalized in different directions by using novel and innovative techniques and ideas, provide mathematical models to some problems arising in economics, mechanics, engineering science, and other pure and applied sciences. Among these methods, the resolvent operator technique is very important. See, for example, [1–17] and the references therein.

Recently, Huang and Fang [18] introduced a system of order complementarity problems and established some existence results for the system using fixed-point theory. Verma [19] introduced and studied some systems of the system variational inequalities and developed some iterative algorithms for approximating the solutions of the systems of variational inequalities. Cho et al. [20] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. The authors also proved some existence and uniqueness theorems of solutions for the system and also constructed an iterative algorithm for approximating the solution of the system of nonlinear variational inequalities. Further, Fang et al. [1], Yan et al. [2], Fang and Huang [3], and Cao [4] considered some new systems of variational inclusions involving -monotone operators and -monotone operators in Hilbert space, respectively. Using the corresponding resolvent operator associated with -monotone operators and -monotone operators, the authors proved the existence of solutions for these new systems of variational inclusions and constructed a new algorithm for approximating the solution of this system and discussed the convergence of the sequence of iterations generated by the algorithm.

Very recently, Lan et al. [5] and Peng and Zhao [7] introduced and studied a new system of nonlinear -monotone multivalued variational inclusions in Hilbert spaces, respectively. By using the concept and properties of -monotone operators and the resolvent operator technique associated with -monotone operators due to Verma [8], the author constructed a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions associated with -monotone operators in Hilbert spaces and proved the existence of solutions for the nonlinear multivalued variational inclusion systems and the convergence of iterative sequences generated by the algorithm. For more details, see, for example, [1–5, 7, 8, 10–19, 21–25].

On the other hand, Lan [6] first introduced a new concept of -monotone (the so-called -maximal relaxed monotone [9]) operators, which generalizes the -monotonicity, -monotonicity, and other existing monotone operators as special cases, and studied some properties of -monotone operators and defined resolvent operators associated with -monotone operators.

Inspired and motivated by the above works, the purpose of this paper is to consider the following new general system of set-valued variational inclusions involving relative -maximal monotone operators in Hilbert spaces: find and such that where is a given positive integer, , , and are single-valued operators, is a set-valued operator, is -maximal relaxed monotone, and and .

Some special cases of the problem (1) had been studied by many authors. Here, we mention some of them as follows.

Case 1. If , then the problem (1) reduces to the problem of finding and and such that The problem (2) is called a nonlinear set-valued variational inclusion system problem, which was considered and studied by Agarwal and Verma [9].

Case 2. When and , for all , , where is proper, convex, and lower semicontinuous functional and denotes the subdifferential operator of , then problem (1) becomes the following system of set-valued mixed variational inequalities: find and and such that
If , the identity operator, then the problem (3) reduces to the following problem of finding such that which is called the system of nonlinear variational inequalities considered by Cho et al. [20]. Some specializations of problem (4) are dealt by Kim and Kim [21].

Case 3. If and , then the problem (1) reduces to finding such that which was considered by Fang et al. [1].

In brief, the problem (1) is the most general and unifying system form, so long as, for appropriate and suitable choices of positive integer and operators , , , , and for , one can know that the problem (1) includes a number of known general problems of variational character, including variational inequality (system) problems and variational inclusion (system) problems as special cases. For more details, see [1–5, 7–25] and the reference therein.

Furthermore, in this paper, we will construct some new iterative algorithms to approximate the solution of the general system of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms in Hilbert spaces.

2. Preliminaries

Thereafter, let , be real Hilbert spaces endowed with the norm and inner product . Let and denote the family of all the nonempty subsets of and the family of all closed subsets of , respectively.

In order to get the main results of the paper, we need the following concepts and lemmas.

Definition 1. Let be a single-valued operator. Then the map is said to be (i)-strongly monotone, if there exists a constant such that (ii)-Lipschitz continuous, if there exists a constant such that

Definition 2. Let and be single-valued operators; let be set-valued operator. Then (i) is said to be -Lipschitz continuous, if there exists a constant such that (ii) is said to be -monotone, if (iii) is said to be strictly -monotone, if is -monotone and (iv) is said to be -strongly monotone, if there exists a constant such that (v) is said to be -relaxed monotone, if there exists a constant such that (vi) is said to be -maximal relaxed monotone, if is -relaxed monotone and

Definition 3. For , let be a Hilbert space and let be a single-valued operator; then nonlinear operator is said to be (i)-strongly monotone with respect to in the th argument, if there exist constants such that, for , , (ii)-relaxed cocoercive with respect to (or relative -relaxed cocoercive) in the th argument, if there exist constants such that, for , (iii)-Lipschitz continuous in the th argument, if there exists constant such that, for , ,

Remark 4. When , then Definition 3 reduces to the corresponding concept of the relative strong monotonicity, relative relaxed cocoercive, and Lipschitz continuity.

Definition 5. Let be a single-valued operator, let be a strictly -monotone operator, and let be an -maximal relaxed monotone operator. Then general resolvent operator is defined by where is a constant.

Lemma 6 (see [6]). Let be a -Lipschitz continuous operator, let be an -strongly monotone operator, and let be an -maximal relaxed monotone operator. Then general resolvent operator is -Lipschitz continuous; that is, where .

Next, we define the Hausdorff pseudometric as follows: Note that if is restricted to closed bounded subsets of the family , then the Hausdorff pseudometric reduces to Hausdorff metric defined by

Definition 7. A set-valued operator is said to be --Lipschitz continuous if there exists a constant such that

Lemma 8. Let be a constant and ; then function , for , is nonnegative and strictly decreases and . Further, if , then .

Proof. Since is linear function, the conclusions immediately hold.

3. Iterative Algorithm and Convergence

In this section, we first prove the equivalence between the problem (1) and the problem of finding the fixed points of the general resolvent operator associated with -maximal relaxed monotone operators. This equivalence is quite general and very important from a numerical point of view. Then, by using the equivalence, some new iterative algorithms for finding the approximation solutions of the problem (1) are analyzed. Further, the convergence criteria for the algorithms are also discussed.

Lemma 9. Let and ; then (denoted by ) is a solution of the problem (1) if and only if satisfy where and is a constant, for .

Proof. Let satisfy the relation (22). By Definition 5 of general resolvent operator, the equality (22) holds if and only if that is, where . Hence are the solution of the problem (1). This completes the proof.

By using formula (22) and Nadler [26], we can develop the following new iterative algorithms.

Algorithm 10. Consider the following.
Step  1. Choose and for .
Step  2. Let for all and , where and are nonnegative constants such that and .
Step  3. Choose such that where is the Hausdorff pseudometric on .
Step  4. If and satisfy (25) to sufficient accuracy, stop. Otherwise, set and return to Step 2.

Algorithm 11. Consider the following.
Step  1. Choose and , for .
Step  2. Let for all and , where and are nonnegative constants such that .
Step  3. Choose such that where is the Hausdorff pseudometric on .
Step  4. If and satisfy (27) to sufficient accuracy, stop. Otherwise, set and return to Step 2.

Algorithm 12. Consider the following.
Step  1. Choose and , for .
Step  2. Let for all and , where is a nonnegative constant such that .
Step  3. Choose such that where is the Hausdorff pseudometric on .
Step  4. If and satisfy (29) to sufficient accuracy, stop. Otherwise, set and return to Step 2.

Remark 13. Let ; then Algorithms 10–12 reduce to Algorithms  4.1–4.3 of Agarwal and Verma [9], respectively.

Now, we provide the main results concerning problem (1) with respect to Algorithms 10–12.

Theorem 14. For , let be -Lipschitz continuous operator, let be -Lipschitz continuous and -strongly monotone operator, and let be -maximal relaxed monotone operator. Suppose that is --Lipschitz continuous for and and nonlinear operator is -relaxed cocoercive with respect to in the th argument and -Lipschitz continuous in the th argument for . If there exists constant , for , such that where for , then the problem (1) admits a solution (in short, ), where and , for any , and sequences and generated by Algorithm 10 strongly converge to and , respectively.

Proof. For , applying Algorithm 10 and Lemma 6, we have
Since is -Lipschitz continuous, is -relaxed cocoercive with respect to in the th argument, and is -Lipschitz continuous in the -th argument for , then we have By --Lipschitz continuity of and (26), we get It follows from (32)–(34) that which implies that where By condition (31), we know sequence is monotonely decreasing and as . Therefore, Since , for , we get . By Lemma 8, we have . From (36), it follows that are Cauchy sequences and there exists such that as , for .
Next, we show that as for and .
It follows from (34) that are also Cauchy sequences. Hence, there exists such that as , for . Furthermore, Since is closed, we have , for .
Using continuity, that is, and satisfy (22) and so, in light of Lemma 9, , is a solution to problem (1). This completes the proof.

Remark 15. If , then Theorem 14 reduces to Theorem  4.2 in [9].

From Theorem 14, we have the following results.

Corollary 16. For and , assume that , , , , and are the same as in Theorem 14. Let be -strongly monotone with respect to in the th argument and -Lipschitz continuous in the th argument for and . If there exist constants , for , such that where , for , then problem (1) admits a solution , where, for any , and and sequences , generated by Algorithm 11 strongly converge to and , respectively.

Corollary 17. For and , let , , , , and be the same as in Corollary 16 and let be --Lipschitz continuous. If condition (40) in Corollary 16 holds, then problem (1) admits a solution , where and , and sequences , , generated by Algorithm 12 strongly converge to and , for any ,  , respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Cultivation Project of Sichuan University of Science & Engineering (2011PY01) and the Open Research Fund of Artificial Intelligence Key Laboratory of Sichuan Province (2012RYY04).