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

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.

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

(3.8)
(3.9)

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

(3.10)
(3.11)
(3.12)

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].

References

  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/S0004972700033116

    Article  MathSciNet  MATH  Google Scholar 

  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.2007.11.001

    Article  MathSciNet  MATH  Google Scholar 

  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)00250-2

    Article  MathSciNet  MATH  Google Scholar 

  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.2004.04.037

    Article  MathSciNet  MATH  Google Scholar 

  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)00275-3

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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:1026222030596

    Article  MathSciNet  MATH  Google Scholar 

  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.2006.04.025

    Article  MathSciNet  MATH  Google Scholar 

  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/BF02832351

    Article  MathSciNet  MATH  Google Scholar 

  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.2007.12.007

    Article  MathSciNet  MATH  Google Scholar 

  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.2007.07.021

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. 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.2007.10.025

    Article  MathSciNet  MATH  Google Scholar 

  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/S1025583400000205

    MathSciNet  MATH  Google Scholar 

  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.2008.05.016

    Article  MathSciNet  MATH  Google Scholar 

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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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Correspondence to Jeong Sheok Ume.

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Liu, Z., Liu, M., Ume, J.S. et al. The Solvability of a New System of Nonlinear Variational-Like Inclusions. Fixed Point Theory Appl 2009, 609353 (2009). https://doi.org/10.1155/2009/609353

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