# The Solvability of a New System of Nonlinear Variational-Like Inclusions

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## Abstract

We introduce and study a new system of nonlinear variational-like inclusions involving Open image in new window - Open image in new window -maximal monotone operators, strongly monotone operators, Open image in new window -strongly monotone operators, relaxed monotone operators, cocoercive operators, Open image in new window -relaxed cocoercive operators, Open image in new window - Open image in new window -relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with Open image in new window - Open image in new window -maximal monotone operators and Banach contraction principle, we demonstrate the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper improve and extend some known results in the literature.

### Keywords

Hilbert Space Banach Space Positive Constant Point Theorem Fixed Point Theorem## 1. Introduction

It is well known that the resolvent operator technique is an important method for solving various variational inequalities and inclusions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In particular, the generalized resolvent operator technique has been applied more and more and has also been improved intensively. For instance, Fang and Huang [5] introduced the class of Open image in new window -monotone operators and defined the associated resolvent operators, which extended the resolvent operators associated with Open image in new window -subdifferential operators of Ding and Luo [3] and maximal Open image in new window -monotone operators of Huang and Fang [6], respectively. Later, Liu et al. [17] researched a class of general nonlinear implicit variational inequalities including the Open image in new window -monotone operators. Fang and Huang [4] created a class of Open image in new window -monotone operators, which offered a unifying framework for the classes of maximal monotone operators, maximal Open image in new window -monotone operators and Open image in new window -monotone operators. Recently, Lan [8] introduced a class of Open image in new window -accretive operators which further enriched and improved the class of generalized resolvent operators. Lan [10] studied a system of general mixed quasivariational inclusions involving Open image in new window -accretive mappings in Open image in new window -uniformly smooth Banach spaces. Lan et al. [14] constructed some iterative algorithms for solving a class of nonlinear Open image in new window -monotone operator inclusion systems involving nonmonotone set-valued mappings in Hilbert spaces. Lan [9] investigated the existence of solutions for a class of Open image in new window -accretive variational inclusion problems with nonaccretive set-valued mappings. Lan [11] analyzed and established an existence theorem for nonlinear parametric multivalued variational inclusion systems involving Open image in new window -accretive mappings in Banach spaces. By using the random resolvent operator technique associated with Open image in new window -accretive mappings, Lan [13] established an existence result for nonlinear random multi-valued variational inclusion systems involving Open image in new window -accretive mappings in Banach spaces. Lan and Verma [15] studied a class of nonlinear Fuzzy variational inclusion systems with Open image in new window -accretive mappings in Banach spaces. On the other hand, some interesting and classical techniques such as the Banach contraction principle and Nalder's fixed point theorems have been considered by many researchers in studying variational inclusions.

Inspired and motivated by the above achievements, we introduce a new system of nonlinear variational-like inclusions involving Open image in new window - Open image in new window -maximal monotone operators in Hilbert spaces and a class of Open image in new window - Open image in new window -relaxed cocoercive operators. By virtue of the Banach's fixed point theorem and the resolvent operator technique, we prove the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper generalize some known results in the field.

## 2. Preliminaries

In what follows, unless otherwise specified, we assume that Open image in new window is a real Hilbert space endowed with norm Open image in new window and inner product Open image in new window , and Open image in new window denotes the family of all nonempty subsets of Open image in new window for Open image in new window Now let's recall some concepts.

Definition 2.1.

Let Open image in new window be mappings.

*Lipschitz continuous*, if there exists a constant Open image in new window such that

*expanding,*if there exists a constant Open image in new window such that

*strongly monotone,*if there exists a constant Open image in new window such that

*strongly monotone,*if there exists a constant Open image in new window such that

*relaxed cocoercive,*if there exist nonnegtive constants Open image in new window and Open image in new window such that

*relaxed Lipschitz,*if there exists a constant Open image in new window such that

Definition 2.2.

Let Open image in new window be mappings. Open image in new window is called

*relaxed cocoercive*with respect to Open image in new window in the first argument, if there exist nonnegative constants Open image in new window such that

*cocoercive*with respect to Open image in new window in the second argument, if there exists a constant Open image in new window such that

*relaxed Lipschitz*with respect to Open image in new window in the third argument, if there exists a constant Open image in new window such that

*relaxed monotone*with respect to Open image in new window in the third argument, if there exists a constant Open image in new window such that

*Lipschitz continuous*in the first argument, if there exists a constant Open image in new window such that

Similarly, we can define the Lipschitz continuity of Open image in new window in the second and third arguments, respectively.

Definition 2.3.

*∖*Open image in new window , let Open image in new window be mappings. For each given Open image in new window and Open image in new window is said to be Open image in new window - Open image in new window -

*relaxed monotone,*if there exists a constant Open image in new window such that

Definition 2.4.

For Open image in new window *∖* Open image in new window , let Open image in new window be mappings. For any given Open image in new window and Open image in new window is said to be Open image in new window - Open image in new window -*maximal monotone,* if (B1) Open image in new window is Open image in new window - Open image in new window -relaxed monotone; (B2) Open image in new window for Open image in new window

Lemma 2.5 (see [8]).

Let Open image in new window be a real Hilbert space, Open image in new window be a mapping, Open image in new window be a Open image in new window - Open image in new window -strongly monotone mapping and Open image in new window be a Open image in new window - Open image in new window -maximal monotone mapping. Then the generalized resolvent operator Open image in new window is singled-valued for Open image in new window .

Lemma 2.6 (see [8]).

Let Open image in new window be a real Hilbert space, Open image in new window be a Open image in new window -Lipschitz continuous mapping, Open image in new window be a Open image in new window - Open image in new window -strongly monotone mapping, and Open image in new window be a Open image in new window - Open image in new window -maximal monotone mapping. Then the generalized resolvent operator Open image in new window is Open image in new window -Lipschitz continuous for Open image in new window .

*∖*Open image in new window , assume that Open image in new window are single-valued mappings, Open image in new window satisfies that for each given Open image in new window is Open image in new window - Open image in new window -maximal monotone, where Open image in new window is Open image in new window - Open image in new window -strongly monotone and Open image in new window We consider the following problem of finding Open image in new window such that

where Open image in new window for Open image in new window and Open image in new window . The problem (2.13) is called the system of nonlinear variational-like inclusions problem.

Special cases of the problem (2.13) are as follows.

which was studied by Fang and Huang [4] with the assumption that Open image in new window is Open image in new window -monotone for Open image in new window .

which was studied in Shim et al. [19].

It is easy to see that the problem (2.13) includes a number of variational and variational-like inclusions as special cases for appropriate and suitable choice of the mappings Open image in new window for Open image in new window .

## 3. Existence and Uniqueness Theorems

In this section, we will prove the existence and uniqueness of solution of the problem (2.13).

Lemma 3.1.

where Open image in new window , for all Open image in new window .

Theorem 3.2.

*∖*Open image in new window let Open image in new window be Lipschitz continuous with constant Open image in new window , Open image in new window be Lipschitz continuous with constants Open image in new window respectively, Open image in new window be Lipschitz continuous in the first, second and third arguments with constants Open image in new window respectively, let Open image in new window be Open image in new window -relaxed cocoercive with respect to Open image in new window in the first argument, and Open image in new window -relaxed Lipschitz with respect to Open image in new window in the third argument, Open image in new window be Open image in new window - Open image in new window -relaxed cocoercive, Open image in new window be Open image in new window -strongly monotone, Open image in new window be Open image in new window -Lipschitz continuous and Open image in new window - Open image in new window -strongly monotone, and Open image in new window be Open image in new window -relaxed Lipschitz, Open image in new window satisfy that for each fixed Open image in new window is Open image in new window - Open image in new window -maximal monotone, Open image in new window and

then the problem (2.13) possesses a unique solution in Open image in new window .

Proof.

By Lemma 3.1, we derive that Open image in new window is a unique solution of the problem (2.13). This completes the proof.

Theorem 3.3.

*∖*Open image in new window let Open image in new window be all the same as in Theorem 3.2, Open image in new window be Open image in new window -expanding, Open image in new window be Lipschitz continuous in the first, second and third arguments with constants Open image in new window respectively, and Open image in new window be Open image in new window -relaxed cocoercive with respect to Open image in new window in the first argument, be Open image in new window -cocoercive with respect to Open image in new window in the second argument, be Open image in new window -relaxed Lipschtz with respect to Open image in new window in the third argument. If there exist constants Open image in new window and Open image in new window such that (3.3) and (3.4), but

then the problem (2.13) possesses a unique solution in Open image in new window .

Theorem 3.4.

*∖*Open image in new window let Open image in new window be all the same as in Theorem 3.2, Open image in new window be Lipschitz continuous in the first, second and third arguments with constants Open image in new window respectively, and Open image in new window be Open image in new window -relaxed cocoercive with respect to Open image in new window in the first argument, be Open image in new window -relaxed Lipschitz with respect to Open image in new window in the second argument, be Open image in new window -relaxed monotone with respect to Open image in new window in the third argument. If there exist constants Open image in new window and Open image in new window such that (3.3) and (3.4), but

then the problem (2.13) possesses a unique solution in Open image in new window .

Remark 3.5.

In this paper, there are three aspects which are worth of being mentioned as follows:

(1)Theorem 3.2 extends and improves in [4, Theorem 3.1] and in [19, Theorem 4.1];

(2)the class of Open image in new window - Open image in new window -relaxed cocoercive operators includes the class of Open image in new window -relaxed cocoercive operators in [8] as a special case;

(3)the class of Open image in new window - Open image in new window -maximal monotone operators is a generalization of the classes of Open image in new window -subdifferential operators in [3], maximal Open image in new window -monotone operators in [6], Open image in new window -monotone operators in [5] and Open image in new window -monotone operators in [4].

## Notes

### Acknowledgments

This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2009A419) and the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00042).

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