- Research Article
- Open Access
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}
https://doi.org/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.
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 [1–20]. 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
References
- Ansari QH, Yao J-C: A fixed point theorem and its applications to a system of variational inequalities. Bulletin of the Australian Mathematical Society 1999,59(3):433–442. 10.1017/S0004972700033116MathSciNetView ArticleMATHGoogle Scholar
- Cho YJ, Qin X: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Analysis: Theory, Methods & Applications 2008,69(12):4443–4451. 10.1016/j.na.2007.11.001MathSciNetView ArticleMATHGoogle Scholar
- Ding XP, Luo CL: Perturbed proximal point algorithms for general quasi-variational-like inclusions. Journal of Computational and Applied Mathematics 2000,113(1–2):153–165. 10.1016/S0377-0427(99)00250-2MathSciNetView ArticleMATHGoogle Scholar
- Fang Y-P, Huang N-J, Thompson HB: A new system of variational inclusions with -monotone operators in Hilbert spaces. Computers & Mathematics with Applications 2005,49(2–3):365–374. 10.1016/j.camwa.2004.04.037MathSciNetView ArticleMATHGoogle Scholar
- Fang Y-P, Huang N-J: -monotone operator and resolvent operator technique for variational inclusions. Applied Mathematics and Computation 2003,145(2–3):795–803. 10.1016/S0096-3003(03)00275-3MathSciNetView ArticleMATHGoogle Scholar
- Huang N-J, Fang Y-P: A new class of general variational inclusions involving maximal -monotone mappings. Publicationes Mathematicae Debrecen 2003,62(1–2):83–98.MathSciNetMATHGoogle Scholar
- Huang N-J, Fang Y-P: Fixed point theorems and a new system of multivalued generalized order complementarity problems. Positivity 2003,7(3):257–265. 10.1023/A:1026222030596MathSciNetView ArticleMATHGoogle Scholar
- Lan H-Y: -accretive mappings and set-valued variational inclusions with relaxed cocoercive mappings in Banach spaces. Applied Mathematics Letters 2007,20(5):571–577. 10.1016/j.aml.2006.04.025MathSciNetView ArticleMATHGoogle Scholar
- Lan H-Y: New proximal algorithms for a class of -accretive variational inclusion problems with non-accretive set-valued mappings. Journal of Applied Mathematics & Computing 2007,25(1–2):255–267. 10.1007/BF02832351MathSciNetView ArticleMATHGoogle Scholar
- Lan H-Y: Stability of iterative processes with errors for a system of nonlinear -accretive variational inclusions in Banach spaces. Computers & Mathematics with Applications 2008,56(1):290–303. 10.1016/j.camwa.2007.12.007MathSciNetView ArticleMATHGoogle Scholar
- Lan H-Y: Nonlinear parametric multi-valued variational inclusion systems involving -accretive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(5–6):1757–1767. 10.1016/j.na.2007.07.021View ArticleMathSciNetMATHGoogle Scholar
- Lan H-Y: A stable iteration procedure for relaxed cocoercive variational inclusion systems based on -monotone operators. Journal of Computational Analysis and Applications 2007,9(2):147–157.MathSciNetMATHGoogle Scholar
- Lan H-Y: Nonlinear random multi-valued variational inclusion systems involving -accretive mappings in Banach spaces. Journal of Computational Analysis and Applications 2008,10(4):415–430.MathSciNetMATHGoogle Scholar
- Lan H-Y, Kang JI, Cho YJ: Nonlinear -monotone operator inclusion systems involving non-monotone set-valued mappings. Taiwanese Journal of Mathematics 2007,11(3):683–701.MathSciNetMATHGoogle Scholar
- Lan H-Y, Verma RU: Iterative algorithms for nonlinear fuzzy variational inclusion systems with -accretive mappings in Banach spaces. Advances in Nonlinear Variational Inequalities 2008,11(1):15–30.MathSciNetMATHGoogle Scholar
- Liu Z, Ume JS, Kang SM: On existence and iterative algorithms of solutions for mixed nonlinear variational-like inequalities in reflexive Banach spaces. Dynamics of Continuous, Discrete & Impulsive Systems. Series B 2007,14(1):27–45.MathSciNetMATHGoogle Scholar
- Liu Z, Kang SM, Ume JS: The solvability of a class of general nonlinear implicit variational inequalities based on perturbed three-step iterative processes with errors. Fixed Point Theory and Applications 2008, Article ID 634921, 2008:-13.Google Scholar
- Qin X, Shang M, Su Y: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3897–3909. 10.1016/j.na.2007.10.025MathSciNetView ArticleMATHGoogle Scholar
- Shim SH, Kang SM, Huang NJ, Cho YJ: Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions. Journal of Inequalities and Applications 2000,5(4):381–395. 10.1155/S1025583400000205MathSciNetMATHGoogle Scholar
- Zeng L-C, Ansari QH, Yao J-C: General iterative algorithms for solving mixed quasi-variational-like inclusions. Computers & Mathematics with Applications 2008,56(10):2455–2467. 10.1016/j.camwa.2008.05.016MathSciNetView ArticleMATHGoogle Scholar
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