Journal of Applied Mathematics

Volume 2012 (2012), Article ID 560248, 15 pages

doi:10.1155/2012/560248

Research Article

System of Nonlinear Set-Valued Variational Inclusions Involving a Finite Family of -Accretive Operators in Banach Spaces

Prapairat Junlouchai and Somyot Plubtieng

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 15 February 2012; Accepted 27 March 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Prapairat Junlouchai and Somyot Plubtieng. 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.

We study a new system of nonlinear set-valued variational inclusions involving a finite family of -accretive operators in Banach spaces. By using the resolvent operator technique associated with a finite family of -accretive operators, we prove the existence of the solution for the system of nonlinear set-valued variational inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions for this system.

1. Introduction

Variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in pure and applied sciences which include work on differential equations, control problems, mechanics, general equilibrium problems in transportation and economics. In 1994, Hassouni and Moudafi [1] introduced and studied a class of variational inclusions and developed a perturbed algorithm for finding approximate solutions of the variational inclusions. In 1996, Adly [2] obtained some important extensions and generalizations of the results in [1] for nonlinear variational inclusions. Recently, Ding [3] introduced and studied a class of generalized quasivariational inclusions and Kazmi [4] introduced and studied another class of quasivariational inclusions in the same year. In [5, 6], Ansari et al. introduced the system of vector equilibrium problems and they proved the existence of solutions for such problems (see also in [7–9]). In 2004, Verma [10] studied nonlinear variational inclusion problems based on the generalized resolvent operator technique involving -monotone mapping. For existence result and approximating solution of the system of set-valued variational inclusions and the class of nonlinear relaxed cocoercive variational inclusions, we refer the reader to Yan et al. [11], Plubtieng and Sriprad [12], Verma [13] and Cho et al. [14].

Very recently, Verma [15] introduced and studied approximation solvability of a general class of nonlinear variational inclusion problems based on -resolvent operator technique in a Hilbert space. On the other hand, Zou and Huang [16] studied the Lipschitz continuity of resolvent operator for the -accretive operator in Banach spaces. Moreover, they also applied these new concepts to solve a variational-like inclusion problem. One year later, Zou and Huang [17] introduced and studied a new class of system of variational inclusions involving -accretive operator in Banach spaces. By using the resolvent operator technique associated with -accretive operator, they proved the existence of the solution for the system of inclusions. Moreover, they also develop a step-controlled iterative algorithm to approach the unique solution.

In this paper, we introduce a new system of nonlinear set-valued variational inclusions involving a finite family of -accretive operators in Banach spaces. By using the resolvent operators technique associated with a finite family of -accretive operator, we prove the existence of the solution for the system of nonlinear set-valued variational inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions of this system.

2. Preliminaries

Let be a real Banach space with dual space the dual pair between and and and denote the family of all the nonempty subsets of and the family of all closed subsets of , respectively. The generalized duality mapping is defined by

where is a constant. It is known that, in general, for all and is single-valued if is strictly convex. In the sequel, we always assume that is a real Banach space such that is single-valued.

The modulus of smoothness of is the function defined by

A Banach space is called uniformly smooth if

is called -uniformly smooth if there exists a constant such that

Note that is single valued if is uniformly smooth. In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu [18] proved the following result.

Definition 2.1. Let be two single-valued mappings and two single-valued mappings.

(i) is said to be accretive if

(ii) is said to be strictly accretive if is accretive and

if and only if ;(iii) is said to be -strongly accretive with respect to if there exists a constant such that

(iv) is said to be -relaxed accretive with respect to if there exists a constant such that

(v) is said to be -Lipschitz continuous with respect to if there exists a constant such that

(vi) is said to be -Lipschitz continuous if there exists a constant such that

(vii) is said to be strongly accretive with respect to if there exists a constant such that

Definition 2.2. Let be single-valued mapping. Let be a set-valued mapping.

(i) is said to be -Lipschitz continuous if there exists a constant such that

(ii) is said to be accretive if

(iii) is said to be -accretive if

(iv) is said to be strictly -accretive if is -accretive and equality holds if and only if ;(v) is said to be -strongly -accretive if there exists a positive constant such that

(vi) is said to be -relaxed -accretive if there exists a positive constant such that

Definition 2.3. Let , be three single-valued mappings. Let be a set-valued mapping. is said to be -accretive with respect to and (or simply -accretive in the sequel), if is accretive and for every .

Lemma 2.4. Let be a real uniformly smooth Banach space. Then is -uniformly smooth if and only if there exists a constant such that for all

Lemma 2.5 (see[16]). Let be -strongly accretive with respect to , -relaxed accretive with respect to , and . Let be an -accretive operator with respect to and . Then, the operator is single valued. Based on Lemma 2.4, one can define the resolvent operator as follows.

Definition 2.6. Let be defined as in Definition 2.3. Let be -strongly accretive with respect to , -relaxed accretive with respect to , and . Let be an -accretive operator with respect to and . The resolvent operator is defined by

where is a constant.

Lemma 2.7 (see [16]). Let be defined as in Definition 2.3. Let be -strongly accretive with respect to , -relaxed accretive with respect to , and . Suppose that is an -accretive operator. Then resolvent operator defined by (2.18) is Lipschitz continuous. That is,

We define a Hausdorff pseudometric by

for any given . Note that if the domain of is restricted to closed bounded subsets, then is the Hausdorff metric.

Lemma 2.8 (see [19]). Let and be two real sequences of nonnegative numbers that satisfy the following conditions:

(i) for , and ;(ii) for .

Then, converges to 0 as .

3. Main Result

Let be -uniformly smooth real Banach space and a nonempty closed convex set. Let , be single-valued operators, for all . For any fix , we let , -accretive set-valued operator and a set-valued mapping which nonempty values. The system of nonlinear set-valued variational inclusions is to find , such that

If , then system of nonlinear set-valued variational inclusions (3.1) becomes following system of variational inclusions: finding , and such that

If , then system of nonlinear set-valued variational inclusions (3.1) becomes the following class of nonlinear set-valued variational inclusions see [15]: finding , such that

For solving the system of nonlinear set-valued variational inclusions involving a finite family of -accretive operators in Banach spaces, let us give the following assumptions.

For any , we suppose that

(A1) is -strongly accretive with respect to , -relaxed accretive with respect to and ,(A2) is an -accretive single-valued mapping,(A3) is a contraction set-valued mapping with and nonempty values,(A4) is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to ,(A5) is -Lipschitz continuous with respect to its first argument and -Lipschitz continuous with respect to its second argument,(A6) is -strongly accretive with respect to .

Theorem 3.1. For given , , it is a solution of problem (3.1) if and only if

where are constants.

Proof. We note from the Definition 2.6 that , is a solution of (3.1) if and only if, for each , we have

Algorithm 3.2. For given , , we let

for all , where . By Nadler theorem [20], there exists such that

where is the Hausdorff pseudo metric on . Continuing the above process inductively, we can obtain the sequences and such that

for all , where with . Therefore, by Nadler theorem [20], there exists such that

The idea of the proof of the next theorem is contained in the paper of Verma [15] and Zou and Huang [17].

Theorem 3.3. Let be -uniformly smooth real Banach space. Let be single-valued operators, a single-valued operator satisfy (A1) and , , , , satisfy conditions (A2)–(A6), respectively. If there exists a constant such that

for all , then problem (3.1) has a solution , .

Proof. For any and , we define by

for all . Let . For any , we note by (3.11) and Lemma 2.7 that

By Lemma 2.4, we have

Moreover, by (A4), we obtain

From (A6), we have

Moreover, from (A5), we obtain

From (3.13)–(3.16), we have

It follows from (3.12), (3.17), and (3.18) that

Put

Define on by for all . It is easy to see that is a Banach space. For any given , we choose a finite sequence . Define by . Set , where are contraction constants of , respectively. We note that , for all , and so . Let , and , . By (A3), we get

and so is a contraction on . Hence there exists such that . From Theorem 3.1, , is the solution of the problem (3.1).

Theorem 3.4. Let be -uniformly smooth real Banach space. For . Let be single-valued operators, single-valued operator satisfy (A1) and suppose that , , , , satisfy conditions (A2)–(A6), respectively. Then, for any , the sequences and generated by Algorithm 3.2 converge strongly to , , respectively.

Proof. By Theorem 3.3, the problem (3.1) has a solution , . From Theorem 3.1, we note that

for all . Hence, by (3.8) and (3.22), we have

By Lemma 2.4, we obtain

From (A4), we note that

From (3.24) and (A6), it follows that

By (3.23), (3.24), and (A5), we have

From (3.23)–(3.28), we obtain

Hence, by (3.23), (3.28) and (3.29), we have

Put , where

It follows from (3.30) that

Set and . From (3.32), we obtain

Since , we have . Thus, it follows from Lemma 2.8 that and hence . Therefore, is a Cauchy sequence and hence there exists such that as , for all . Next, we will show that as . Hence, it follows from (3.9) that is also a Cauchy sequence. Thus there exists such that as . Consider

as . Since is a closed set and , we have . By continuing the above process, there exist such that as . Hence, by (3.8), we obtain

Therefore, it follows from Theorem 3.1 that is a solution of problem (3.1).

Setting in Theorem 3.3, we have the following result.

Corollary 3.5. Let be -uniformly smooth real Banach spaces. Let be singled valued operators, a single-valued operator such that is -strongly accretive with respect to , -relaxed accretive with respect to and and suppose that is an -accretive set-valued mapping and contraction set-valued mapping with and nonempty values, for all . Assume that is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to , is -Lipschitz continuous with respect to its first argument and -Lipschitz continuous with respect to its second argument, is -strongly accretive with respect to , and is -strongly accretive with respect to , for all . If

for all , then problem (3.2) has a solution , .

Setting in Theorem 3.3, we have the following result.

Corollary 3.6. Let be -uniformly smooth real Banach spaces. Let be two singled valued operators, a single-valued operator such that is -strongly accretive with respect to , -relaxed accretive with respect to , and and suppose that is an -accretive set-valued mapping, is contraction set-valued mapping with and nonempty values. Assume that is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to , is -Lipschitz continuous with respect to its first argument and -Lipschitz continuous with respect to its second argument, is -strongly accretive with respect to . If

then problem (3.3) has a solution and .

Acknowledgments

The first author would like to thank the Office of the Higher Education Commission, Thailand, financial support under Grant CHE-Ph.D-THA-SUP/191/2551, Thailand. Moreover, the second author would like to thank the Thailand Research Fund for financial support under Grant BRG5280016.

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