Open Access

General System of -Monotone Nonlinear Variational Inclusions Problems with Applications

Journal of Inequalities and Applications20092009:364615

https://doi.org/10.1155/2009/364615

Received: 25 September 2009

Accepted: 2 November 2009

Published: 3 November 2009

Abstract

We introduce and study a new system of nonlinear variational inclusions involving a combination of -Monotone operators and relaxed cocoercive mappings. By using the resolvent technique of the -monotone operators, we prove the existence and uniqueness of solution and the convergence of a new multistep iterative algorithm for this system of variational inclusions. The results in this paper unify, extend, and improve some known results in literature.

1. Introduction

Recently, Fang and Huang [1] introduced a new class of -monotone mappings in the context of solving a system of variational inclusions involving a combianation of -monotone and strongly monotone mappings based on the resolvent operator techniques. The notion of the -monotonicity has revitalized the theory of maximal monotone mappings in several directions, especially in the domain of applications. Verma [2] introduced the notion of -monotone mappings and its applications to the solvability of a system of variational inclusions involving a combination of -monotone and strongly monotone mappings. As Verma point out "the class of -monotone mappings generalizes -monotone mappings. On the top of that, -monotonicity originates from hemivariational inequalities, and emerges as a major contributor to the solvability of nonlinear variational problems on nonconvex settings." and as a matter of fact, some nice examples on -monotone (or generalized maximal monotone) mappings can be found in Naniewicz and Panagiotopoulos [3] and Verma [4]. Hemivariational inequalities—initiated and developed by Panagiotopoulos [5]—are connected with nonconvex energy functions and turned out to be useful tools proving the existence of solutions of nonconvex constrained problems. It is worthy noting that -monotonicity is defined in terms of relaxed monotone mappings—a more general notion than the monotonicity or strong monotonocity—which gives a significant edge over the -monotonocity. Very recently, Verma [6] studied the solvability of a system of variational inclusions involving a combination of -monotone and relaxed cocoercive mappings using resolvent operator techniques of -monotone mappings. Since relaxed cocoercive mapping is a generalization of strong monotone mappings, the main result in [6] is more general than the corresponding results in [1, 2].

Inspired and motivated by recent works in [1, 2, 6], the purpose of this paper is to introduce a new mathematical model, which is called a general system of -monotone nonlinear variational inclusion problems, that is, a family of -monotone nonlinear variational inclusion problems defined on a product set. This new mathematical model contains the system of inclusions in [1, 2, 6], the variational inclusions in [7, 8], and some variational inequalities in literature as special cases. By using the resolvent technique for the -monotone operators, we prove the existence and uniqueness of solution for this system of variational inclusions. We also prove the convergence of a multistep iterative algorithm approximating the solution for this system of variational inclusions. The result in this paper unifies, extends, and improves some results in [1, 2, 68] and the references therein.

2. Preliminaries

We suppose that is a real Hilbert space with norm and inner product denoted by and , respectively, denotes the family of all the nonempty subsets of . If be a set-valued operator, then we denote the effective domain of as follows:

(2.1)

Now we recall some definitions needed later.

Definition 2.1 (see [2, 6, 7]).

Let be a single-valued operator and let be a set-valued operator. is said to be

(i) -relaxed monotone, if there exists a constant such that
(2.2)

(ii) -monotone with a constant if

(a) is -relaxed monotone,

(b) is maximal monotone for   (i.e., , for all ).

Remark 2.2.

If ,   , then the definition of -monotonicity is that of -monotonicity in [1, 8]. It is easy to know that if ( the identity map on ), then the definition of -monotone operators is that of maximal monotone operators. Hence, the class of -monotone operators provides a unifying frameworks for classes of maximal monotone operators, -monotone operators. For more details about the above definitions, please refer to [18] and the references therein.

It follow from [3, Lemma ] we know that if is a reflexive Banach space with its dual, and be -strongly monotone and is a locally Lipschitz such that is -relaxed monotone, then is -monotone with a constant .

Definition 2.3 (see [1, 7, 8]).

Let , be two single-valued operators. is said to be

(i)monotone if
(2.3)
(ii)strictly monotone if is monotone and
(2.4)
(iii) -strongly monotone if there exists a constant such that
(2.5)
(iv) -Lipschitz continuous if there exists a constant such that
(2.6)
(v) -strongly monotone with respect to if there exists a constant such that
(2.7)

Definition 2.4 . (see [2]).

Let be a -strongly monotone operator and let be an -monotone operator. Then the resolvent operator is defined by
(2.8)

We also need the following result obtained by Verma [2].

Lemma 2.5.

Let be a -strongly monotone operator and let be an -monotone operator. Then, the resolvent operator is Lipschitz continuous with constant for , that is,
(2.9)

One needs the following new notions.

Definition 2.6.

Let be Hilbert spaces and denote the norm of , also let and be two single-valued mappings:

(i) is said to be -Lipschitz continuous in the first argument if there exists a constant such that
(2.10)
(ii) is said to be monotone with respect to in the first argument if
(2.11)
(iii) is said to be -strongly monotone with respect to in the first argument if there exists a constant such that
(2.12)
(iv) is said to be -cocoercive with respect to in the first argument if there exists a constant such that
(2.13)
(v) is said to be -relaxed cocoercive with respect to in the first argument if there exists a constant such that
(2.14)
(vi) is said to be -relaxed cocoercive with respect to in the first argument if there exists a constant such that
(2.15)

In a similar way, we can define the Lipschitz continuity and the strong monotonicity (monotonicity), relaxed cocoercivity (cocoercivity) of with respect to in the th argument .

3. A System of Set-Valued Variational Inclusions

In this section, we will introduce a new system of nonlinear variational inclusions in Hilbert spaces. In what follows, unless other specified, for each , we always suppose that is a Hilbert space with norm denoted by , , are single-valued mappings, and is a nonlinear mapping. We consider the following problem of finding such that for each ,
(3.1)

Below are some special cases of (3.1).

If , then (3.1) becomes the following problem of finding such that
(3.2)

However, (3.2) is called a system of set-valued variational inclusions introduced and researched by Fang and Huang [1, 9] and Verma [2, 6].

If , then (3.1) becomes the following variational inclusion with an -monotone operator, which is to find such that
(3.3)

problem (3.3) is introduced and studied by Fang and Huang [8]. It is easy to see that the mathematical model (2) studied by Verma [7] is a variant of (3.3).

4. Existence of Solutions and Convergence of an Iterative Algorithm

In this section, we will prove existence and uniqueness of solution for (3.1). For our main results, we give a characterization of the solution of (3.1) as follows.

Lemma 4.1.

For , let be a strictly monotone operator and let be an -monotone operator. Then is a solution of (3.1) if and only if for each ,
(4.1)

where is a constant.

Proof.

It holds that is a solution of (3.1)
(4.2)

Let .

Theorem 4.2.

For , let be -strongly monotone and let -Lipschitz continuous, be an -monotone operator with a constant , let be a single-valued mapping such that is -relaxed cocoercive monotone with respect to and -Lipschitz continuous in the th argument, is -Lipschitz continuous in the th arguments for each . Suppose that there exist constants such that
(4.3)

Then, (3.1) admits a unique solution.

Proof.

For and for any given , define a single-valued mapping by
(4.4)

for any .

For any , it follows from (4.4) and Lemma 2.5 that for ,
(4.5)
For , since is -Lipschitz continuous, is -relaxed cocoercive with respected to and -Lipschitz continuous in the th argument, we have
(4.6)
For , since is -Lipschitz continuous in the th arguments ( ), we have
(4.7)
It follows from (4.5)–(4.7) that for each ,
(4.8)
Hence,
(4.9)
where
(4.10)
Define on by , for all . It is easy to see that is a Banach space. For any given , define by
(4.11)

for all .

By (4.3), we know that , it follows from (4.9) that
(4.12)
This shows that is a contraction operator. Hence, there exists a unique , such that
(4.13)
that is, for ,
(4.14)

By Lemma 4.1, is the unique solution of (3.1). This completes this proof.

Corollary 4.3.

For , let be -strongly monotone and -Lipschitz continuous, let be an -monotone operator, let be a single-valued mapping such that is -strongly monotone with respect to and -Lipschitz continuous in the th argument, is -Lipschitz continuous in the th arguments for each . Suppose that there exist constants such that
(4.15)

Then, problem (3.1) admits a unique solution.

Remark 4.4.

Theorem 4.2 and Corollary 4.3 unify, extend, and generalize the main results in [1, 2, 68].

5. Iterative Algorithm and Convergence

In this section, we will construct some multistep iterative algorithm for approximating the unique solution of (3.1) and discuss the convergence analysis of these Algorithms.

Lemma 5.1 (see [8, 9]).

Let and be two real sequences of nonnegative numbers that satisfy the following conditions:

(1) and ,

(2)

then converges to 0 as .

Algorithm 5.2.

For , let be the same as in Theorem 4.2. For any given , define a multistep iterative sequence by
(5.1)
where
(5.2)

Theorem 5.3.

For , let be the same as in Theorem 4.2. Assume that all the conditions of theorem 4.1 hold. Then generated by Algorithm 5.2 converges strongly to the unique solution of (3.1).

Proof.

By Theorem 4.2, problem (3.1) admits a unique solution , it follows from Lemma 4.1 that for each ,
(5.3)
It follows from (4.3), (5.1) and (5.3) that for each ,
(5.4)
For , since is -Lipschitz continuous, is -relaxed cocoercive with respected to , and -Lipschitz is continuous in the th argument, we have
(5.5)
For , since is -Lipschitz continuous in the th arguments ( ), we have
(5.6)
It follows from (5.4)–(5.6) that for ,
(5.7)
Hence,
(5.8)
where
(5.9)

It follows from hypothesis (4.3) that .

Let . Then, (5.8) can be rewritten as By (5.2), we know that , it follows from Lemma 5.1 that
(5.10)

Therefore, converges to the unique solution of (3.1). This completes the proof.

Declarations

Acknowledgment

This study was supported by grants from National Natural Science Foundation of China (project no. 70673012, no. 70741028 and no. 90924030), China National Social Science Foundation (project no. 08CJY026).

Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Chongqing Normal University
(2)
Management School, Shanghai University

References

  1. Fang YP, Huang NJ: -monotone operators and system of variational inclusions. Communications on Applied Nonlinear Analysis 2004,11(1):93–101.MathSciNetMATHGoogle Scholar
  2. Verma RU: -monotonicity and applications to nonlinear variational inclusion problems. Journal of Applied Mathematics and Stochastic Analysis 2004,17(2):193–195.View ArticleMathSciNetMATHGoogle Scholar
  3. Naniewicz Z, Panagiotopoulos PD: Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 188. Marcel Dekker, New York, NY, USA; 1995.Google Scholar
  4. Verma RU: Nonlinear variational and constrained hemivariational inequalities involving relaxed operators. Zeitschrift für Angewandte Mathematik und Mechanik 1997,77(5):387–391. 10.1002/zamm.19970770517View ArticleMathSciNetMATHGoogle Scholar
  5. Panagiotopoulos PD: Hemivariational Inequalities and Their Applications in Mechanics and Engineering. Springer, New York, NY, USA; 1993.View ArticleMATHGoogle Scholar
  6. Verma RU: General system of -monotone nonlinear variational inclusion problems with applications. Journal of Optimization Theory and Applications 2006,131(1):151–157. 10.1007/s10957-006-9133-5MathSciNetView ArticleMATHGoogle Scholar
  7. Verma RU: General nonlinear variational inclusion problems involving -monotone mappings. Applied Mathematics Letters 2006,19(9):960–963. 10.1016/j.aml.2005.11.010MathSciNetView ArticleMATHGoogle Scholar
  8. 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
  9. Fang Y-P, Huang N-J: 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

Copyright

© J.-W. Peng and L.-J. Zhao. 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.