Algorithm for Solving a New System of Generalized Nonlinear Quasi-Variational-Like Inclusions in Hilbert Spaces

Abstract

We introduce and study a new system of generalized nonlinear quasi-variational-like inclusions with -cocoercive operator in Hilbert spaces. We suggest and analyze a class of iterative algorithms for solving the system of generalized nonlinear quasi-variational-like inclusions. An existence theorem of solutions for the system of generalized nonlinear quasi-variational-like inclusions is proved under suitable assumptions which show that the approximate solutions obtained by proposed algorithms converge to the exact solutions.

1. Introduction

Variational inclusion problems are important generalization of classical variational inequalities and have wide applications to many fields including mechanics, physics, optimization and control, nonlinear programming, economics, and engineering sciences; see, for example, [1]. For these reasons, various variational inclusions have been intensively studied in recent years. Many efficient ways have been studied to find solutions for variational inclusions. Those methods include the projection method and its various forms, linear approximation, descent and Newton’s method, and the method based on auxiliary principle technique. The method based on the resolvent operator technique is a generalization of the projection method and has been widely used to solve variational inclusions. For details, we refer to see [2–19].

Recently, Fang and Huang, Kazmi and Khan, and Lan et al. investigated several resolvent operators for generalized operators such as -monotone [3, 17], -accretive [4], -maximal relaxed accretive [14], -monotone [5], -accretive [13], -proximal point [8], and -accretive [9] operators. Very recently, Zou and Huang [19] introduced and studied -accretive operators, Kazmi et al. [10–12] introduced and studied generalized -accretive operators and --proximal point mapping, and Xu and Wang [18] introduced and studied -monotone operators. Ahmad et al. [2, 8] introduced and studied -cocoercive operators, showed some properties of the resolvent operator for the -cocoercive operators, and obtained an application for solving variational inclusions in Hilbert spaces. They also gave some examples to illustrate their results.

Inspired and motivated by the researches going on in this area, we introduce and discuss a new system of generalized nonlinear quasi-variational-like inclusions involving -cocoercive operators in Hilbert spaces. By using the resolvent operators associated with -cocoercive operators due to Ahmad et al. [2], we prove that the approximate solutions obtained by the iterative algorithms converge to the exact solutions of our system of generalized nonlinear quasi-variational-like inclusions. Our results can be viewed as an extension and generalization of some known results in the literature.

2. Preliminaries

Throughout this 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 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 [20]). A mapping 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.

Definition 2. Let be set-valued mapping. Then, is said to be -cocoercive if there exists a constant such that

Definition 3 (see [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 -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 4 (see [2]). Let  and  be the single-valued mappings. Then, the set-valued mapping is said to be -cocoercive with respect to and (or simply -cocoercive in the sequel), if(i) is cocoercive;(ii), for every .

Example 5. Let with the usual inner product. Let be defined by
Suppose that is defined by
Then, it is easy to check that is -cocoercive with respect to and -relaxed cocoercive with respect to .
Let , where is the identity mapping. Then, is -cocoercive mapping with respect to and .

Example 6. Let , where denotes the space of all real symmetric matrices. Let , for all and . Then, for , we have but because is not the square of any real symmetric matrix. Hence, is not -cocoercive with respect to and .

Proposition 7 (see [2]). Let be -cocoercive with respect to and -relaxed cocoercive with respect to ; is -expansive, is -Lipschitz continuous, and , . Let be an -cocoercive operator. If the inequality holds for all , then , where

Theorem 8 (see [2]). Let be -cocoercive with respect to and -relaxed cocoercive with respect to ; is -expansive, is -Lipschitz continuous, and   and . Let M be an -cocoercive operator with respect to and . Then, the operator is single-valued.

Definition 9 (see [2]). 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

Theorem 10 (see [2]). Let be -cocoercive with respect to and -relaxed cocoercive with respect to , is -expansive, is -Lipschitz continuous, and , with . Let be an -cocoercive operator with respect to and . Then, the resolvent operator is -Lipschitz continuous; that is,

3. The System of Generalized Nonlinear Quasi-Variational-Like Inclusions and Iterative Algorithm

Let and be real Hilbert spaces. Let , , , , ,, , , and be the single-valued mappings. Let and be the set-valued mappings, let be a set-valued mapping such that, for each , is a -cocoercive operator with respect to and , and let   be a set-valued mapping such that, for each , is an -cocoercive operator with respect to and . Assume that for each , and for each. Then, we consider the following system of generalized nonlinear quasi-variational-like inclusions.

Find with , such that

Next are some special cases of problem (21).(1)Let , be identity mappings, for each , and ; then problem (21) reduces to the following problem considered in [15]: (2),  , is an identity mapping, , , for each , ; then problem (21) reduces to the following problem considered in [16]:

For a suitable choice of the mappings , , , , , , , , , , , and the space , a number of known systems of quasi-variational inequalities, systems of variational inequalities, systems of quasi-variational inclusions, and variational inclusions can be obtained as special cases of the generalized nonlinear quasi-variational inclusion problem (21). We would like to mention that the problem of finding zero of the sum of two maximal monotone operators is also a special case of problem (21). Furthermore, these types of variational inclusion enable us to study many important problems arising in mathematical, physical, and engineering science in a general and unified framework.

Lemma 11. Let and be real Hilbert spaces. Let , ,,,, be the single-valued mappings. Let be a single-valued mapping such that is -cocoercive with respect to and -relaxed cocoercive with respect to ; is -expansive, is -Lipschitz continuous, and , . Let   be a single-valued mapping such that is -cocoercive with respect to and -relaxed cocoercive with respect to ; is -expansive, is -Lipschitz continuous, and , . Let and be the set-valued mappings, let be a set-valued mapping such that, for each , is a -cocoercive operator, and let   be a set-valued mapping such that, for each , is an -cocoercive operator. Assume that for each , and for each. Then, for any with , is a solution of the problem (21), if and only if where , , and are constants.

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

The preceding lemma allows us to suggest the following iterative algorithm for problem (21).

Algorithm 12. For with , , compute the sequences , , , as follows: for all , 1, 2,, and and are constants.

Definition 13. Let and be two set-valued mappings. A single-valued 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

Definition 14. Let and be two set-valued mappings. A single-valued 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

Definition 15. A set-valued mapping   is said to be -Lipschitz continuous, if there exists a constant such that

Theorem 16. Let and be real Hilbert spaces. Let , ,, ,, , and be the single-valued mappings. Let and be the set-valued mappings, let be a set-valued mapping such that, for each , is a -cocoercive operator with respect to and , and let be a set-valued mapping such that for each , is an -cocoercive operator with respect to and . Assume that for each , and for each. Assume that (i), are , -Lipschitz continuous in the Hausdorff metric , respectively;(ii) is -cocoercive with respect to and -relaxed cocoercive with respect to ;(iii) is -cocoercive with respect to and -relaxed cocoercive with respect to ;(iv), is , -expansive, respectively;(v), is , -Lipschitz continuous, respectively;(vi) is -Lipschitz continuous and -strongly monotone;(vii) is -Lipschitz continuous and -strongly monotone;(viii) is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to ;(ix) is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to ;(x) is -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument;(xi)is -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument;(xii)
Then, the iterative sequences , , , and , generated by Algorithm 12, converge strongly to , , , and respectively, and , , , is a solution of problem (21).

Proof. Since , are Lipschitz continuous with constants , , respectively, it follows from Algorithm 12 that for all .
Using the -strong monotonicity of , we have which implies that
Now we estimate by using Algorithm 12 and the Lipschitz continuity of as follows:
Since is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to , is -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument. is -Lipschitz continuous and using (32), (34), and (35) it becomes
Let where
Now we estimate by using Algorithm 12 and the Lipschitz continuity of as follows:
Since is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to , is -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument. is -Lipschitz continuous and using (32), (34), and (39) it becomes
In the light of (34), we have
Let where
From adding (37) and (42), we get where .
Letting , we obtain , where
By (31), , and (44)  and both are Cauchy sequences. Thus, there exists such that , , as . From the Lipschitz continuity of and and (32), , are also Cauchy sequences, and thus there exists such that , , as .
Now, we prove that and . In fact, since and , we have which implies that . Since , it follows that . Similarly, we have . By Lemma 11, it follows that , , , is a solution of problem (21), and this completes the proof.

Conflict of Interests

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