Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 609353, 9 pages
doi:10.1155/2009/609353

Copyright © 2009 Zeqing Liu 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 introduce and study a new system of nonlinear variational-like inclusions involving s-(G,η)-maximal monotone operators, strongly monotone operators, η-strongly monotone operators, relaxed monotone operators, cocoercive operators, (λ,ξ)-relaxed cocoercive operators, (ζ,φ,ϱ)-g-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with s-(G,η)-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.

1. Introduction

It is well known that the resolvent operator technique is an important method for solving various variational inequalities and inclusions [1–20]. 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 -monotone operators and defined the associated resolvent operators, which extended the resolvent operators associated with -subdifferential operators of Ding and Luo [3] and maximal -monotone operators of Huang and Fang [6], respectively. Later, Liu et al. [17] researched a class of general nonlinear implicit variational inequalities including the -monotone operators. Fang and Huang [4] created a class of -monotone operators, which offered a unifying framework for the classes of maximal monotone operators, maximal -monotone operators and -monotone operators. Recently, Lan [8] introduced a class of -accretive operators which further enriched and improved the class of generalized resolvent operators. Lan [10] studied a system of general mixed quasivariational inclusions involving -accretive mappings in -uniformly smooth Banach spaces. Lan et al. [14] constructed some iterative algorithms for solving a class of nonlinear -monotone operator inclusion systems involving nonmonotone set-valued mappings in Hilbert spaces. Lan [9] investigated the existence of solutions for a class of -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 -accretive mappings in Banach spaces. By using the random resolvent operator technique associated with -accretive mappings, Lan [13] established an existence result for nonlinear random multi-valued variational inclusion systems involving -accretive mappings in Banach spaces. Lan and Verma [15] studied a class of nonlinear Fuzzy variational inclusion systems with -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 --maximal monotone operators in Hilbert spaces and a class of --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 is a real Hilbert space endowed with norm and inner product , and denotes the family of all nonempty subsets of for Now let's recall some concepts.Definition 2.1. Let be mappings.

(1) is said to be Lipschitz continuous, if there exists a constant such that (2.1) (2) is said to be - expanding, if there exists a constant such that (2.2) (3) is said to be - strongly monotone, if there exists a constant such that (2.3) (4) is said to be -- strongly monotone, if there exists a constant such that (2.4) (5) is said to be -- relaxed cocoercive, if there exist nonnegtive constants and such that (2.5) (6) is said to be - relaxed Lipschitz, if there exists a constant such that (2.6)Definition 2.2. Let be mappings. is called (1) - relaxed cocoercive with respect to in the first argument, if there exist nonnegative constants such that (2.7) (2) - cocoercive with respect to in the second argument, if there exists a constant such that (2.8) (3) - relaxed Lipschitz with respect to in the third argument, if there exists a constant such that (2.9) (4) - relaxed monotone with respect to in the third argument, if there exists a constant such that (2.10) (5) Lipschitz continuous in the first argument, if there exists a constant such that (2.11) Similarly, we can define the Lipschitz continuity of in the second and third arguments, respectively.Definition 2.3. For ∖ , let be mappings. For each given and is said to be -- relaxed monotone, if there exists a constant such that (2.12)Definition 2.4. For ∖ , let be mappings. For any given and is said to be -- maximal monotone, if (B1) is --relaxed monotone; (B2) for Lemma 2.5 (see [8]). Let be a real Hilbert space, be a mapping, be a - -strongly monotone mapping and be a - -maximal monotone mapping. Then the generalized resolvent operator is singled-valued for .

Lemma 2.6 (see [8]). Let be a real Hilbert space, be a -Lipschitz continuous mapping, be a - -strongly monotone mapping, and be a - -maximal monotone mapping. Then the generalized resolvent operator is -Lipschitz continuous for .

For and ∖ , assume that are single-valued mappings, satisfies that for each given is --maximal monotone, where is --strongly monotone and We consider the following problem of finding such that (2.13) where for and . The problem (2.13) is called the system of nonlinear variational-like inclusions problem.

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

If for each , then the problem (2.13) collapses to finding such that (2.14) which was studied by Fang and Huang [4] with the assumption that is -monotone for.

If and , for all for , then the problem (2.13) reduces to finding such that (2.15) 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 for .

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. Let and be two positive constants. Then is a solution of the problem (2.13) if and only if satisfies that (3.1) where , for all .

Theorem 3.2. For ∖ let be Lipschitz continuous with constant be Lipschitz continuous with constants respectively, be Lipschitz continuous in the first, second and third arguments with constants respectively, let be -relaxed cocoercive with respect to in the first argument, and -relaxed Lipschitz with respect to in the third argument, be - -relaxed cocoercive, be -strongly monotone, be -Lipschitz continuous and - -strongly monotone, and be -relaxed Lipschitz, satisfy that for each fixed is - -maximal monotone, and (3.2) If there exist positive constants , and such that (3.3) (3.4) where (3.5) then the problem (2.13) possesses a unique solution in .

Proof. For any , define (3.6) For each it follows from Lemma 2.6 that (3.7) Because is -strongly monotone, and are Lipschitz continuous, and is -relaxed Lipschitz, we deduce that

Since are all Lipschitz continuous, is -relaxed cocoercive with respect to -relaxed Lipschitz with respect to , and is Lipschitz continuous in the first, second and third arguments, respectively, we infer that In terms of (3.7)–(3.12), we obtain that (3.13) Similarly, we deduce that (3.14) Define on by for any It is easy to see that is a Banach space. Define by (3.15) By virtue of (3.3), (3.4), (3.13) and (3.14), we achieve that and (3.16) which means that is a contractive mapping. Hence, there exists a unique such that That is, (3.17) By Lemma 3.1, we derive that is a unique solution of the problem (2.13). This completes the proof.

Theorem 3.3. For ∖ let be all the same as in Theorem 3.2, be -expanding, be Lipschitz continuous in the first, second and third arguments with constants respectively, and be -relaxed cocoercive with respect to in the first argument, be -cocoercive with respect to in the second argument, be -relaxed Lipschtz with respect to in the third argument. If there exist constants and such that (3.3) and (3.4), but (3.18) then the problem (2.13) possesses a unique solution in .

Theorem 3.4. For ∖ let be all the same as in Theorem 3.2, be Lipschitz continuous in the first, second and third arguments with constants respectively, and be -relaxed cocoercive with respect to in the first argument, be -relaxed Lipschitz with respect to in the second argument, be -relaxed monotone with respect to in the third argument. If there exist constants and such that (3.3) and (3.4), but (3.19) then the problem (2.13) possesses a unique solution in .

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 --relaxed cocoercive operators includes the class of -relaxed cocoercive operators in [8] as a special case; (3)the class of --maximal monotone operators is a generalization of the classes of -subdifferential operators in [3], maximal -monotone operators in [6], -monotone operators in [5] and -monotone operators in [4].

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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