The Solvability of a New System of Nonlinear Variational-Like Inclusions
- Zeqing Liu^{1},
- Min Liu^{1},
- Jeong Sheok Ume^{2}Email author and
- Shin Min Kang^{3}
DOI: 10.1155/2009/609353
© Zeqing Liu et al. 2009
Received: 23 November 2008
Accepted: 1 April 2009
Published: 5 May 2009
Abstract
We introduce and study a new system of nonlinear variational-like inclusions involving - -maximal monotone operators, strongly monotone operators, -strongly monotone operators, relaxed monotone operators, cocoercive operators, -relaxed cocoercive operators, - -relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with - -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.
Definition 2.2.
Similarly, we can define the Lipschitz continuity of in the second and third arguments, respectively.
Definition 2.3.
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 .
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.
which was studied by Fang and Huang [4] with the assumption that is -monotone for .
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.
Theorem 3.2.
then the problem (2.13) possesses a unique solution in .
Proof.
By Lemma 3.1, we derive that is a unique solution of the problem (2.13). This completes the proof.
Theorem 3.3.
then the problem (2.13) possesses a unique solution in .
Theorem 3.4.
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].
Declarations
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).
Authors’ Affiliations
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