Journal of Applied Mathematics

Volume 2012 (2012), Article ID 682465, 12 pages

doi:10.1155/2012/682465

Research Article

Global Dynamical Systems Involving Generalized -Projection Operators and Set-Valued Perturbation in Banach Spaces

Yun-zhi Zou,¹ Xi Li,² Nan-jing Huang,² and Chang-yin Sun¹

¹School of Automation, Southeast University, Jiangsu, Nanjing 210096, China

²Department of Mathematics, Sichuan University, Sichuan, Chengdu 610064, China

Received 29 February 2012; Accepted 16 May 2012

Academic Editor: Zhenyu Huang

Copyright © 2012 Yun-zhi Zou 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.

A new class of generalized dynamical systems involving generalized f-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.

1. Introduction

It is well known that dynamics system has long time been an interest of many researchers. This is largely due to its extremely wide applications in a huge variety of scientific fields, for instance, mechanics, optimization and control, economics, transportation, equilibrium, and so on. For details, we refer readers to references [1–10] and the references therein.

In 1994, Friesz et al. [3] introduced a class of dynamics named global projective dynamics based on projection operators. Recently, Xia and Wang [7] analyzed the global asymptotic stability of the dynamical system proposed by Friesz as follows:

where is a single-valued function, is a constant, denotes the projection of the point on ; here is a nonempty, closed, and convex subset.

Later, in 2006, Zou et al. [9] studied a class of global set-valued projected dynamical systems as follows:

where is a set-valued function, is a single-valued function, is a constant, denotes the projection of the point on , is a given point in .

The concept of generalized -projection operator was first introduced by Wu and Huang [11] in 2006. They also proved that the generalized -projection operator is an extension of the projection operator in and it owns some nice properties as does; see [12, 13]. Some applications of generalized -projection operator are also given in [11–13]. Very recently, Li et al. [14] studied the stability of the generalized -projection operator with an application in Banach spaces. We would like to point out that Cojocaru [15] introduced and studied the projected dynamical systems on infinite Hilbert spaces in 2002.

To explore further dynamic systems in infinite dimensional spaces in more general forms has been one of our major motivations and efforts recently, and this paper is a response to those efforts. In this paper, we introduce and study a new class of generalized dynamical systems involving generalized -projection operators. By using the fixed-point theorem due to Nadler [16], we prove that the equilibrium points set of this class of generalized global dynamical systems is nonempty and closed. We also show that the solutions set of the systems with set-valued perturbation is continuous with respect to the initial value. The results presented in this paper generalize many existing results in recent literatures.

2. Preliminaries

Let be a Banach space and let be a closed convex set, let be a set-valued mapping, and let be a single-valued mapping. The normalized duality mapping from to is defined by

for . For convenience, we list some properties of as follows. is a smooth Banach space, is single valued and hemicontinuous; that is, is continuous from the strong topology of to the topology of .

Let denote the family of all nonempty compact subsets of and let denote the Hausdorff metric on defined by

In this paper, we consider a new class of generalized set-valued dynamical system, that is, to find those absolutely continuous functions from such that

where , is a constant and is proper, convex, and lower semicontinuous and is a generalized -projection operator denoted by

It is well known that many problems arising in the economics, physical equilibrium analysis, optimization and control, transportation equilibrium, and linear and nonlinear mathematics programming problems can be formulated as projected dynamical systems (see, e.g., [1–10, 15, 17] and the references therein). We also would like to point out that problem (2.3) includes the problems considered in Friesz et al. [3], Xia and Wang [7], and Zou et al. [9] as special cases. Therefore, it is important and interesting to study the generalized projected dynamical system (2.3).

Definition 2.1. A point is said to be an equilibrium point of global dynamical system (2.3), if satisfies the following inclusion:

Definition 2.2. A mapping is said to be

(i) -strongly accretive if there exists some such that

(ii) -Lipschitz continuous if there exists a constant such that

Definition 2.3. A set-valued mapping is said to be -Lipschitz continuous if there exists a constant such that

where is the Hausdorff metric on .

Lemma 2.4 (see [14]). Let be a real reflexive and strictly convex Banach space with its dual and let be a nonempty closed convex subset of . If is proper, convex, and lower semicontinuous, then is single valued. Moreover, if has Kadec-Klee property, then is continuous.

Lemma 2.5 (see [18]). 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 2.6 (see [19]). Let be a complete metric space and let be two set-valued contractive mappings with same contractive constants . Then

where and are fixed-point sets of and , respectively.

Lemma 2.7 (see [19]). Let be a real strictly convex, reflexive, and smooth Banach space. For any , let and . Then

where

We say that is -uniformly convex and -uniformly smooth Banach space if there exist such that

where

Based on Lemma 2.7, we can obtain the following lemma.

Lemma 2.8. Let be 2-uniformly convex and 2-uniformly smooth Banach space. Then

Proof. According to Lemma 2.7, we have

where

Since , (2.16) yields

From the property of , we have

It follows from (2.18) and (2.19) that

This completes the proof.

3. Equilibrium Points Set

In this section, we prove that the equilibrium points set of the generalized set-valued dynamical system (2.3) is nonempty and closed.

Theorem 3.1. Let be 2-uniformly convex and 2-uniformly smooth Banach space. Let be -Lipschitz continuous and let be -Lipschitz continuous and -strongly accretive. If

then the equilibrium points set of the generalized set-valued dynamical system (2.3) is nonempty and closed.

Proof. Let

Since and are continuous, we know that . From Definition 2.1, it is easy to see that is an equilibrium point of the generalized set-valued dynamical system (2.3) if and only if is a fixed-point of in , that is:

Thus, the equilibrium points set of (2.3) is the same as the fixed-points set of . We first prove that is nonempty. In fact, for any and , there exists such that

Since , and , it follows from Nadler [16] that there exists such that

Let

Then . From (3.4) to (3.6), we have

Since is -Lipschitz continuous and -strongly accretive,

From Lemma 2.8, where is Lipchitz continuous, we have

From the selection of and the Lipschitz continuity of ,

In light of (3.7)–(3.10), we have

where

Now (3.11) implies that

Since is arbitrary, we have

Similarly, we can prove

From (3.14), (3.15), and the definition of the Hausdorff metric on , we have

Now the assumption of the theorem implies that and so is a set-valued contractive mapping. By the fixed-point theorem of Nadler [16], there exists such that , and thus is the equilibrium point of (2.3). This means that is nonempty.

Now we prove that is closed. Let with . Then and (3.16) imply that

Thus,

It follows that and so are closed. This completes the proof.

Remark 3.2. Theorem 3.1 is a generalization of Theorem 1 in Zou et al. [9] from to Banach space .

4. Sensitivity of the Solutions Set

In this section, we study the sensitivity of the solutions set of the generalized dynamical system with set-valued perturbation for (2.3) as follows:

where and are the same as in (2.3), is a set-valued mapping, and is a single-valued mapping. Let denote the set of all solutions of (4.1) on with .

Now we prove the following result.

Theorem 4.1. Let be 2-uniformly convex and 2-uniformly smooth Banach space. Let be -Lipschitz continuous, let be -Lipschitz continuous, and let be a -Lipschitz continuous set-valued mapping with compact convex values. If

then is nonempty and continuous.

Proof. Let

Then is a set-valued mapping with compact convex values since is a set-valued mapping with compact convex values. For any and , there exists such that

Since , and , it follows from Nadler [16] that there exists such that

Let

Then . From (4.4) and (4.6), we have

Since is -Lipschitz continuous,

From Lemma 2.8, is Lipschitz continuous. It follows from the continuity of and that

From the selection of and the Lipschitz continuity of , we know

In light of (4.7)–(4.10), we have

where

Now (4.11) implies that

Since is arbitrary, we obtain

Similarly, we can prove

From (4.13), to (4.15), and the definition of the Hausdorff metric on , we have

Now (4.2) implies that , and so is a set-valued contractive mapping. Let

where

Since is a continuous set-valued mapping with compact convex values, by the Michael's selection theorem (see, e.g., Theorem 16.1 in [20]), we know that is nonempty for each and . Moreover, it is easy to see that the set of fixed-points of coincides with . It follows from [21] or [8] that is compact and convex for each and . Suppose that is the initial value of (4.1); that is, and . Since

it is obvious that converges uniformly to .

Next we prove that is a set-valued contractive mapping. For any given , since is a continuous set-valued mapping with compact convex values, by the Michael's selection theorem (see, e.g., Theorem 16.1 in [20]), we know that has a continuous selection . Let

Then . Since is measurable and is a measurable mapping with compact values, we know that there exists a measurable selection such that

Thus, it follows from (4.16) that

Let

Then and

Hence, we have

Since is arbitrary, we obtain

Similarly, we can prove that

From the definition of the Hausdorff metric on , (4.26) and (4.27) imply that

Since , it is easy to see that has a fixed-point for each given , and so is nonempty for each given . Setting

we know that are contractive mappings with the same contractive constant . By Lemma 2.6 and (4.28), we have

Thus, , which implies that ; that is, the solution of (4.1) is continuous with respect to the initial value of (4.1). This completes the proof.

Remark 4.2. Theorem 4.1 is a generalization of Theorem 2 in Zou et al. [9] from to Banach space .

Acknowledgments

The authors appreciate greatly the editor and two anonymous referees for their useful comments and suggestions, which have helped to improve an early version of the paper. This work was supported by the National Natural Science Foundation of China (11171237) and the Key Program of NSFC (Grant no. 70831005).

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