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Fixed Point Theory and Applications

, 2009:609353 | Cite as

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

  • Zeqing Liu
  • Min Liu
  • Jeong Sheok Ume
  • Shin Min Kang
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

(1) Open image in new window is said to be Lipschitz continuous, if there exists a constant Open image in new window such that
(2) Open image in new window is said to be Open image in new window -expanding, if there exists a constant Open image in new window such that
(3) Open image in new window is said to be Open image in new window -strongly monotone, if there exists a constant Open image in new window such that
(4) Open image in new window is said to be Open image in new window - Open image in new window -strongly monotone, if there exists a constant Open image in new window such that
(5) Open image in new window is said to be Open image in new window - Open image in new window -relaxed cocoercive, if there exist nonnegtive constants Open image in new window and Open image in new window such that
(6) Open image in new window is said to be Open image in new window -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

(1) Open image in new window -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
(2) Open image in new window -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
(3) Open image in new window -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
(4) Open image in new window -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
(5)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.

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 .

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.

Let Open image in new window and Open image in new window be two positive constants. Then Open image in new window is a solution of the problem (2.13) if and only if Open image in new window satisfies that

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

Theorem 3.2.

For Open image in new window 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
If there exist positive constants Open image in new window , and Open image in new window such that

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

Proof.

For each Open image in new window it follows from Lemma 2.6 that
Because Open image in new window is Open image in new window -strongly monotone, Open image in new window and Open image in new window are Lipschitz continuous, and Open image in new window is Open image in new window -relaxed Lipschitz, we deduce that
Since Open image in new window are all Lipschitz continuous, Open image in new window is Open image in new window -relaxed cocoercive with respect to Open image in new window , Open image in new window -relaxed Lipschitz with respect to Open image in new window , 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
Similarly, we deduce that
By virtue of (3.3), (3.4), (3.13) and (3.14), we achieve that Open image in new window and
which means that Open image in new window is a contractive mapping. Hence, there exists a unique Open image in new window such that Open image in new window That is,

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.

For Open image in new window 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.

For Open image in new window 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).

References

  1. 1.
    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/S0004972700033116MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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.2007.11.001MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)00250-2MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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.2004.04.037MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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)00275-3MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    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:1026222030596MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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.2006.04.025MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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/BF02832351MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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.2007.12.007MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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.2007.07.021CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    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
  14. 14.
    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
  15. 15.
    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
  16. 16.
    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
  17. 17.
    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
  18. 18.
    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.2007.10.025MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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
  20. 20.
    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.2008.05.016MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Zeqing Liu et al. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • Zeqing Liu
    • 1
  • Min Liu
    • 1
  • Jeong Sheok Ume
    • 2
  • Shin Min Kang
    • 3
  1. 1.Department of MathematicsLiaoning Normal UniversityDalian LiaoningChina
  2. 2.Department of Applied MathematicsChangwon National UniversityChangwonSouth Korea
  3. 3.Department of Mathematics and Research Institute of Natural ScienceGyeongsang National UniversityJinjuSouth Korea

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