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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 643828, 21 pages
http://dx.doi.org/10.1155/2012/643828
Research Article

On a System of Nonlinear Variational Inclusions with -Monotone Operators

1Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China
2Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 15 August 2012; Accepted 11 October 2012

Academic Editor: Yongfu Su

Copyright © 2012 Zeqing Liu et al. 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.

Abstract

This paper is concerned mainly with the existence and iterative approximation of solutions for a system of nonlinear variational inclusions involving the strongly -monotone operators in Hilbert spaces. The results presented in this paper extend, improve, and unify many known results in the literature.

1. Introduction

Recently, a few authors introduced and studied several classes of systems of nonlinear variational inequalities and inclusions in Hilbert spaces and established the existence of solutions or the approximate solutions for these systems of nonlinear variational inequalities and inclusions [119]. Using projection methods, Liu et al. [12], Verma [1417], Rhoades and Verma [18], and Wu et al. [19] suggested some iterative algorithms for approximating the solutions of several classes of systems of variational inequalities involving relaxed monotone operators, strongly monotone operators, relaxed cocoercive operators, and pseudocontractive operators, respectively. Applying the resolvent operator techniques, Liu et al. [11] and Nie et al. [13] discussed the existence and uniqueness of solutions and suggested iterative algorithms for a system of general quasivariational-like inequalities and a system of nonlinear variational inequalities, respectively, and gave the convergence analysis for the iterative algorithms. Utilizing the resolvent operator method associated with -monotone operators, Fang et al. [5] investigated the existence and uniqueness of solutions for a class of system of variational inclusions in Hilbert spaces, constructed an iterative algorithm for approximating the solution of this system of variational inclusions, and discussed the convergence of iterative sequences generated by the algorithm.

Motivated and inspired by the results [119], in this paper, we introduce two new classes of strictly -monotone and strongly -monotone operators and investigate the existence and Lipschitz continuity of the resolvent operators with respect to the strictly -monotone and strongly -monotone operators. We introduce also a class of system of nonlinear variational inclusions involving strongly -monotone operators in Hilbert spaces and suggest two new iterative algorithms for approximating solutions of the system of nonlinear variational inclusions by using the resolvent operator technique associated with -monotone operators. The convergence criteria of iterative sequences generated by the algorithms are established. The results presented in this paper extend, improve, and unify some known results in the literature.

This paper is organized as follows. In Section 2, we recall and introduce some definitions, notation, and a lemma. In Section 3, we study properties of the resolvent operators with respect to strictly -monotone and strongly -monotone operators, respectively, in Hilbert spaces. In Section 4, we introduce a new class of system of nonlinear variational inclusions in Hilbert spaces and use the resolvent operator technique with respect to strongly -monotone operators to investigate the existence and uniqueness of solution and suggest two new iterative algorithms for the system of nonlinear variational inclusions. The convergence criteria of the sequences generated by the iterative algorithms are also given under certain conditions.

2. Preliminaries and Lemmas

In this section, we recall and introduce some notation, definitions, and a lemma, which will be used in this paper. Let be a real Hilbert space endowed with an inner product and norm , respectively. stands for the identity mapping on and . Let be a set-valued operator. The graph of , denoted by , is defined as follows:

Definition 2.1. Let and be operators. The operator is said to be(1)monotone if (2)-monotone if (3)strictly monotone if (4)strictly -monotone if (5)strongly monotone if there exists a constant satisfying (6)strongly -monotone if there exists a constant satisfying (7)maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) if is monotone (resp., strictly monotone, strongly monotone) and for any ;(8)maximal -monotone (resp., maximally strictly -monotone, maximally strongly -monotone) if is -monotone (resp., strictly -monotone, strongly -monotone) and for any ;(9)-monotone (resp., strictly -monotone, strongly -monotone) if is monotone (resp., strictly monotone, strongly monotone) and for any ;(10)-monotone (resp., strictly -monotone, strongly -monotone) if is -monotone (resp., strictly -monotone, strongly -monotone) and for any .

Definition 2.2. Let and be operators. The operator is called(1)Lipschitz continuous if there exists a constant satisfying (2)monotone if (3)-monotone if (4)strongly monotone if there exists a constant satisfying (5)strongly -monotone if there exists a constant satisfying (6)relaxed Lipschitz if there exists a constant satisfying

Definition 2.3. Let and be operators. The operator is said to be(1)strongly monotone with respect to and in the first argument if there exists a constant satisfying (2)Lipschitz continuous in the first argument if there exists a constant satisfying
Similarly, we could define the strong monotonicity of with respect to and in the second argument and the Lipschitz continuity of in the second argument.

Remark 2.4. For , the definition of the -monotone (resp., strictly -monotone, strongly -monotone) operator reduces to the definition of the maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) operator.

Remark 2.5. Notice that is maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) if and only if is monotone (resp., strictly monotone, strongly monotone) and there is no other monotone (resp., strictly monotone, strongly monotone) operator whose graph contains strictly the graph of .

Lemma 2.6 (see [20]). Let , and be nonnegative sequences satisfying where , and . Then .

3. The Properties of Strictly -Monotone Operators and Strongly -Monotone Operators

In this section, we discuss some properties of the set-valued strictly -monotone operators and set-valued strongly -monotone operators, respectively, dealing with a -monotone operator in Hilbert spaces.

Theorem 3.1. Let be a real Hilbert space and let and be operators such that is -monotone and is strictly -monotone. Then(i)is maximally strictly -monotone;(ii) is a single-valued operator for each .

Proof. (i) Since is strictly -monotone, it follows that is strictly -monotone. Suppose that there exists a strictly -monotone set-valued operator satisfying Graph Graph, that is, there exists GraphGraph such that Notice that for any . Thus there exists Graph satisfying Suppose that . Equation (3.2) means that . Therefore, which is impossible. Suppose that . It follows from (3.1), (3.2), and the -monotonicity of and strict -monotonicity of that which is a contradiction. Hence is maximally strictly -monotone.
(ii) Suppose that there exists some such that contains at least two different elements and . Since is strictly -monotone, is -monotone, and , , it follows that which is a contradiction. Consequently, the operator is single valued. This completes the proof.

Definition 3.2. Let be a real Hilbert space and let and be operators such that is -monotone and is strictly -monotone. Then for each , the resolvent operator is defined by

Theorem 3.3. Let be a real Hilbert space and let be Lipschitz continuous with constant . Assume that is -monotone and is strongly -monotone with constant . Then for every , the resolvent operator is Lipschitz continuous with constant .

Proof. Since is strongly -monotone with constant , it follows that is strictly -monotone. Let be in . In view of and and the strong monotonicity of , we deduce that This leads to which yields that This completes the proof.

Theorem 3.4. Let be a real Hilbert space and let be Lipschitz continuous with constant . Assume that is strongly -monotone with constant and is strongly -monotone with constant . Then for every , the resolvent operator is Lipschitz continuous with constant .

Proof. As in the proof of Theorem 3.3, by the strong monotonicity of and , we infer that for any which means that This completes the proof.

The following example shows that Theorem 3.3 is different from Lemma 2.2 in [5].

Example 3.5. Let with the usual norm. Define single-valued and set-valued operators and by It is clear that is Lipschitz continuous with constant 2 and which yields that is -monotone. On the other hand, for any , there exist satisfying that is, is not strongly -monotone. Now we claim that is strongly -monotone. Let be in with . We have to consider the following five cases.

Case 1. Suppose that . It follows that

Case 2. Suppose that . For any , we get that

Case 3. Suppose that . Clearly, we have

Case 4. Suppose that . For each , we deduce that

Case 5. Suppose that . It is easy to verify that Hence is strongly -monotone with constant .

Let . In order to show that , we need only to verify that . Assume that . We know that . Assume that . Define a function by Notice that the function is continuous and It follows that there exists such that , that is, Assume that . Define by Obviously, is continuous and which implies that there exists with . Consequently, Therefore, . Thus is strongly -monotone. It follows from Theorem 3.3 that the resolvent operator is Lipschitz continuous with constant . However, we cannot invoke Lemma 2.2 in [5] to prove the Lipschitz continuity of the resolvent operator because is not strongly -monotone.

4. A System of Nonlinear Variational Inclusions

In this section, we investigate a new class of system of nonlinear variational inclusions involving strongly -monotone operators in Hilbert spaces and suggest two new iterative algorithms for approximating solutions of the system of nonlinear variational inclusions by using the resolvent operator technique.

Let be single-valued operators and set-valued operators and for . We now consider the following system of nonlinear variational inclusions:

find such that

It is easy to see that the system of nonlinear variational inclusions (4.1) includes a lot of variational inequalities, quasivariational inequalities, variational-like inequalities, variational inclusions, and systems of variational inequalities, quasivariational inequalities, variational-like inequalities, and variational inclusions in [116] and the references therein as special cases.

Now we use the resolvent operator technique to establish the equivalence between the existence of solutions for the system of nonlinear variational inclusions (4.1) and the existence of fixed points for the single-valued operator defined by (4.3) below.

Lemma 4.1. Let be a real Hilbert space. Let and be constants, and be single-valued and set-valued operators, respectively, such that is -monotone and is strictly -monotone for . Then the following statements are pairwise equivalent.(a)The system of nonlinear variational inclusions (4.1) has a solution .(b)There exists satisfying (c)The single-valued operator defined by where has a fixed point .

Proof. It follows from Theorem 3.1, (4.3), and (4.4) that is a solution of the system of nonlinear variational inclusions (4.1) if and only if This completes the proof.

Based on Lemma 4.1, we suggest the following new iterative algorithms for the system of nonlinear variational inclusions (4.1).

Algorithm 4.2 (The Mann iteration method with errors). For any , compute by where and are sequences in , and are sequences in introduced to take into account possible in inexact computation with .

Algorithm 4.3 (The implicit Mann iteration method with errors). For any , compute by where and are sequences in , and and are sequences in introduced to take into account possible in inexact computation with .
Now we investigate those conditions under which the approximate solutions generated by Algorithms 4.2 and 4.3 converge strongly to the exact solutions of the system of nonlinear variational inclusions (4.1).

Theorem 4.4. Let be a real Hilbert space and let be Lipschitz continuous with constants , respectively, -monotone, strongly monotone with constant , relaxed Lipschitz with constant for . Let be strongly -monotone with constant for . Assume that is Lipschitz continuous with constants and in the first and second arguments, respectively, is strongly monotone with constant with respect to and in the first argument for . Let If the following condition is fulfilled: then the system of nonlinear variational inclusions (4.1) has a unique solution . Moreover, if then for each , the sequences and generated by Algorithm 4.2 converge strongly to and , respectively.

Proof. First of all we prove that the operator defined by (4.3) is a contraction on the Banach space with norm for . It follows from (4.9) that such that Put . By the Lipschitz continuity and various monotonicity of for , (4.4) and (4.9), we deduce that In light of (4.3) and (4.8)–(4.14), we infer that where that is, is a contraction on . Therefore, the operator has a unique fixed point , which is the unique solution of the system of nonlinear variational inclusions (4.1) by Lemma 4.1. It follows that In view of Algorithm 4.2 and (4.17), we deduce that for any By virtue of (4.18), we have where by (4.10). Lemma 2.6 and (4.19) ensure that the sequences and generated by Algorithm 4.2 converge strongly to and , respectively. This completes the proof.

Theorem 4.5. Let for be as in Theorem 4.4. If (4.9) is satisfied, then the system of nonlinear variational inclusions (4.1) has a unique solution . Moreover, if hold, then for each , the sequences and generated by Algorithm 4.3 converge strongly to and , respectively.

Proof. It follows from Theorem 4.4 that the system of nonlinear variational inclusions (4.1) has a unique solution . Now we prove that Algorithm 4.3 is well defined. Let be a fixed integer. For each , define and by As in the proof of Theorem 4.4, we infer that for any which yield that (4.21) and (4.25) ensure that is a contraction on . Thus has a unique fixed point . Consequently, for each , there exists a unique element satisfying That is, Algorithm 4.3 is well defined. Next we show the convergence of the sequences