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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 654537, 7 pages
http://dx.doi.org/10.1155/2013/654537
Research Article

A System of Generalized Variational Inclusions Involving a New Monotone Mapping in Banach Spaces

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received 17 April 2013; Accepted 13 July 2013

Academic Editor: Micah Osilike

Copyright © 2013 Jinlin Guan and Changsong Hu. 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 introduce a new monotone mapping in Banach spaces, which is an extension of the -monotone mapping studied by Nazemi (2012), and we generalize the variational inclusion involving the -monotone mapping. Based on the new monotone mapping, we propose a new proximal mapping which combines the proximal mapping studied by Nazemi (2012) with the mapping studied by Lan et al. (2011) and show its Lipschitz continuity. Based on the new proximal mapping, we give an iterative algorithm. Furthermore, we prove the convergence of iterative sequences generated by the algorithm under some appropriate conditions. Our results improve and extend corresponding ones announced by many others.

1. Introduction

Variational inequality theory has emerged as a powerful tool for a wide class of unrelated problems arising in various branches of physical, engineering, pure, and applied sciences in a unified and general framework. As the generalization of variational inequalities, variational inclusions have been widely studied in recent years. One of the most important problems in the theory of variational inclusions is the development of an efficient and implementable iterative algorithm. Therefore, many iterative algorithms and existence results for various variational inclusions have been studied see, for example, [13].

Several years ago, Xia and Huang [4] proposed the concept of general -monotone operators in Banach spaces and studied a class of variational inclusions involving the general -monotone operator in Banach spaces. In 2010, Luo and Huang [5] introduced a new notion of -monotone operators in Banach spaces and gave a new proximal mapping related to these operators. Then, they used it to study a new class of variational inclusions in Banach spaces. Very recently, Nazemi [6] introduced the notion of a new class of -monotone mappings which is an extension of -monotone operators introduced in [5].

Motivated and inspired by the work going on in this direction, in this paper, we propose a new monotone mapping in Banach spaces named --monotone mapping which generalizes the -monotone mapping introduced in [6] from the same -dimensional product space to different -dimensional product space and reduces the mapping from strictly monotone mapping to monotone mapping. Further, we consider a new proximal mapping which associates a mapping introduced in [7] and generalizes the proximal mapping introduced in [6]. Furthermore, in the process of proving the convergence of iterative sequences generated by the algorithm, we change the condition of a uniformly smooth Banach space with to a -uniformly smooth Banach space, which extends the proof of the convergence of iterative sequences in [6]. The results presented in this paper generalize many known and important results in the recent literature and the references therein.

2. Preliminaries

Let be a real Banach space, let be the topological dual space of , and let be the dual pair between and . Let denote the family of all nonempty, closed, and bounded subsets of . Set . Let be the Hausdorff metric on defined by We recall the following definitions and results which are needed in the sequel.

Definition 1 (see [7]). A single-valued mapping is said to be -Lipschitz continuous if there exists a constant such that

Definition 2 (see [8]). A Banach space is called smooth if, for every with , there exists a unique such that . The modulus of smoothness of is the function , defined by

Definition 3 (see [8]). The Banach space is said to be (i)uniformly smooth if (ii)-uniformly smooth, for , if there exists a constant such that

It is well known (see, e.g., [9]) that

Note that if is uniformly smooth, becomes single-valued. In the study of characteristic inequalities in -uniformly smooth Banach space, Xu [8] established the following lemma.

Lemma 4 (see [8]). Let be a real number and let be a smooth Banach space and the normalized duality mapping. Then, is -uniformly smooth if and only if there exists a constant such that for every ,

Definition 5. A single-valued mapping is said to be -relaxed cocoercive if there exist and such that

Definition 6. Let and be a multivalued mapping, , , and single-valued mappings.(i)For each , is said to be -strongly -monotone with respect to (in the th argument) if there exists a constant such that (ii)For each is said to be -relaxed -monotone with respect to (in the th argument) if there exists a constant such that (iii)By assumption that is an even number, is said to be -symmetric -monotone with respect to if, for each is -strongly -monotone with respect to (in the th argument) and for each is -relaxed -monotone with respect to (in the th argument) with (iv)By assumption that is an odd number, is said to be -symmetric -monotone with respect to if, for each , is -strongly -monotone with respect to (in the th argument) and for each , is -relaxed -monotone with respect to (in the th argument) with

Definition 7 (see [10]). Let be a Banach space. A multivalued mapping is said to be -Lipschitz continuous if there exists a constant such that where is the Hausdorff metric on .

Definition 8. Let, for each be a multivalued mapping. A single-valued mapping is said to be -Lipschitz continuous in the th argument with respect to if there exists a constant such that

Definition 9. Let be a Banach space with the dual space single-valued mappings; is said to be -monotone mapping if

3. --Monotone Mapping

First, we define the notion of --monotone mapping.

Definition 10. Let be a Banach space with the dual space . Let and be single-valued mappings and a multivalued mapping.(i)In case that is an even number, is said to be a --monotone mapping if is -symmetric -monotone with respect to and , for every .(ii)In case that is an odd number, is said to be a --monotone mapping if is -symmetric -monotone with respect to and , for every .

Remark 11. (i) If , , and is monotone, then the --monotone mapping reduces to the general -monotone mapping considered in [4].
(ii) If , , then the --monotone mapping reduces to the -monotone mapping considered in [5].
(iii) If , , and are -relaxed monotone, then the --monotone mapping reduces to the -monotone mapping considered in [11].
(iv) If reduces to ,  , and reduce to , then the --monotone mapping reduces to the -monotone mapping considered in [6].

Example 12. Let and then ; and assume is an even number; let , where is the equivalent norm on space, , for ; let , , , , , , where , are constants such that Let , where , ; let , , . Then is a --monotone mapping.

With no loss of generality, we may assume that is an even number in the next text.

Lemma 13. Let , , be single-valued mappings; a -symmetric -monotone with respect to . Then for one has where .

Proof. Setting  , , . From Definition 10, we have where .
This completes the proof.

Theorem 14. Let be a Banach space with the dual space . Let and , , single-valued mappings, a -monotone mapping, and a --monotone mapping. Then,   is a single-valued mapping.

Proof. Suppose, on the contrary, that there exists , such that then Now, by using Lemma 13 and since is a -monotone mapping, we have Thus, we have , which implies that is single valued. This completes the proof.

By Theorem 14, we can define the proximal mapping as follows.

Definition 15. Let be a Banach space with the dual space . Let and , , be single-valued mappings, a -monotone mapping, and a --monotone mapping. A proximal mapping is defined by

Theorem 16. Let be a Banach space with the dual space . Let be a -Lipschitz continuous mapping. Let and , , be single-valued mappings, a -monotone mapping, and a --monotone mapping. Then, the proximal mapping is -Lipschitz continuous, where .

Proof. Let be any given points. It follows from Definition 15 that Setting This implies that By using Lemma 13, we have since is -Lipschitz continuous, we have thus that is, where .
This completes the proof.

4. System of Variational Inclusions: Iterative Algorithm

Let and , , , , be single-valued mappings and , , be multivalued mappings. We will study the following variational inclusion problem: for any given , find , , , such that

We remark that problem (30) includes as special cases many kinds of variational inclusion and variational inequality of [4, 5, 10, 12, 13].

Theorem 17. Let and , , , , be single-valued mappings and let , , be multivalued mappings. Let be a -monotone mapping and a --monotone mapping with respect to . Then, is a solution of problem (30) if and only if where , , and is a constant.

Proof. Let be a solution of problem (30); then we have then thus Setting , from the definition of , we have Conversely, let ; then thus we have
This completes the proof.

Based on Theorem 17, we construct the following iterative algorithm for solving problem (30).

Iterative Algorithm  1
For any given , we choose and compute by iterative schemes for all .

Now, we give some sufficient conditions which guarantee the convergence of iterative sequences generated by Algorithm 4.1.

Theorem 18. Let be a -uniformly smooth Banach space with and the dual space of . Let   -Lipschitz continuous. Let and , , be single-valued mappings, a -monotone and -Lipschitz continuous mapping, a -relaxed cocoercive and -Lipschitz continuous mapping, and a --monotone mapping. Let be a -Lipschitz continuous mapping and, for each , let be -Lipschitz continuous with constant . Suppose that is -Lipschitz continuous in the th argument with respect to and the following condition is satisfied: where

Then, the iterative sequences generated by Algorithm 4.1 converge strongly to , respectively, and is a solution of problem (30).

Proof. By using Algorithm 4.1 and Theorem 16, we have From the Lipschitz continuity of , and -relaxed cocoercivity of and Lemma 4, we have where is the normalized duality mapping.
Since are Lipschitz continuous, we have It follows from (41)–(45) that where Letting , we obtain , where From condition (39), we know that , and hence is a Cauchy sequence in . Thus, there exists such that , as . Now, we prove that . In fact, it follows from the Lipschitz continuity of and Algorithm 4.1 that From (49), we know that is also a Cauchy sequence. In a similar way, are Cauchy sequences. Thus, there exist , such that , , as . Furthermore, Since is closed, we have . In a similar way, we can show that , . By continuity of and Algorithm 4.1, we have By Theorem 17, is a solution of problem (30). This completes the proof.

5. Conclusions

The purpose of this paper is to study a new monotone mapping in Banach spaces, which generalizes the -monotone mapping in [6], and generalizes the concepts of many monotone mappings. Moreover, the result of Theorem 18 improves and generalizes the corresponding results of [46, 10, 12, 13].

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