Journal of Applied Mathematics

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Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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Volume 2012 |Article ID 682465 | https://doi.org/10.1155/2012/682465

Yun-zhi Zou, Xi Li, Nan-jing Huang, Chang-yin Sun, "Global Dynamical Systems Involving Generalized š‘“ -Projection Operators and Set-Valued Perturbation in Banach Spaces", Journal of Applied Mathematics, vol. 2012, Article ID 682465, 12 pages, 2012. https://doi.org/10.1155/2012/682465

Global Dynamical Systems Involving Generalized š‘“ -Projection Operators and Set-Valued Perturbation in Banach Spaces

Academic Editor: Zhenyu Huang
Received29 Feb 2012
Accepted16 May 2012
Published25 Jun 2012

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 [1ā€“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: š‘‘š‘„š‘‘š‘”=š‘ƒš¾(š‘„āˆ’šœŒš‘(š‘„))āˆ’š‘„,(1.1) where š‘āˆ¶ā„š‘›ā†’ā„š‘› is a single-valued function, šœŒ>0 is a constant, š‘ƒš¾š‘„ denotes the projection of the point š‘„ on š¾; here š¾āŠ‚ā„š‘› 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: š‘‘š‘„(š‘”)š‘‘š‘”āˆˆš‘ƒš¾[],(š‘”(š‘„(š‘”))āˆ’šœŒš‘(š‘„(š‘”))āˆ’š‘”(š‘„(š‘”))),fora.a.š‘”āˆˆ0,š½š‘„(0)=š‘,(1.2) where š‘āˆ¶ā„š‘›ā†’2ā„š‘› is a set-valued function, š‘”āˆ¶ā„š‘›ā†’ā„š‘› is a single-valued function, šœŒ>0 is a constant, š‘ƒš¾š‘„ denotes the projection of the point š‘„ on š¾, š‘ is a given point in ā„š‘›.

The concept of generalized š‘“-projection operator was first introduced by Wu and Huang [11] in 2006. They also proved that the generalized š‘“-projection operator is an extension of the projection operator š‘ƒš¾ in š‘…š‘› and it owns some nice properties as š‘ƒš¾ does; see [12, 13]. Some applications of generalized š‘“-projection operator are also given in [11ā€“13]. Very recently, Li et al. [14] studied the stability of the generalized š‘“-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 š‘“-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 š’³ be a Banach space and let š¾āŠ‚š’³ be a closed convex set, let š‘āˆ¶š’³ā†’2š’³ be a set-valued mapping, and let š‘”āˆ¶š’³ā†’š’³ be a single-valued mapping. The normalized duality mapping š½ from š’³ to š’³āˆ— is defined by ī‚†š‘„š½(š‘„)=āˆ—āˆˆš’³āˆ—āˆ¶āŸØš‘„,š‘„āˆ—āŸ©=ā€–š‘„ā€–2=ā€–š‘„āˆ—ā€–2ī‚‡,(2.1) for š‘„āˆˆš‘‹. For convenience, we list some properties of š½(ā‹…) as follows.š‘‹ is a smooth Banach space, š½(ā‹…) is single valued and hemicontinuous; that is, š½ is continuous from the strong topology of š’³ to the š‘¤š‘’š‘Žš‘˜āˆ— topology of š’³āˆ—.

Let š¶(š’³) denote the family of all nonempty compact subsets of š’³ and let ā„‹(ā‹…,ā‹…) denote the Hausdorff metric on š¶(š’³) defined by ī‚»ā„‹(š“,šµ)=maxsupš‘Žāˆˆš“infš‘āˆˆšµš‘‘(š‘Ž,š‘),supš‘āˆˆšµinfš‘Žāˆˆš“ī‚¼š‘‘(š‘Ž,š‘),āˆ€š“,šµāˆˆš¶(š’³).(2.2)

In this paper, we consider a new class of generalized set-valued dynamical system, that is, to find those absolutely continuous functions š‘„(ā‹…) from [0,ā„Ž]ā†’š’³ such that š‘‘š‘„(š‘”)š‘‘š‘”āˆˆĪ š‘“š¾[],(š‘”(š‘„(š‘”))āˆ’šœŒš‘(š‘„(š‘”)))āˆ’š‘”(š‘„(š‘”)),fora.a.š‘”āˆˆ0,ā„Žš‘„(0)=š‘,(2.3) where š‘āˆˆš’³, šœŒ>0 is a constant and š‘“āˆ¶š¾ā†’š‘…āˆŖ{+āˆž} is proper, convex, and lower semicontinuous and Ī š‘“š¾āˆ¶š’³ā†’2š¾ is a generalized š‘“-projection operator denoted by Ī š‘“š¾ī‚»š‘„=š‘¢āˆˆš¾āˆ¶šŗ(š½(š‘„),š‘¢)=infšœ‰āˆˆš¾ī‚¼šŗ(š½(š‘„),šœ‰),āˆ€š‘„āˆˆš’³.(2.4)

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., [1ā€“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 š‘„āˆ— is said to be an equilibrium point of global dynamical system (2.3), if š‘„āˆ— satisfies the following inclusion: 0āˆˆĪ š‘“š¾(š‘”(š‘„)āˆ’šœŒš‘(š‘„))āˆ’š‘”(š‘„).(2.5)

Definition 2.2. A mapping š‘āˆ¶š’³ā†’š’³ is said to be
(i) š›¼-strongly accretive if there exists some š›¼>0 such that (š‘(š‘„)āˆ’š‘(š‘¦),š½(š‘„āˆ’š‘¦))ā‰„š›¼ā€–š‘„āˆ’š‘¦ā€–2,āˆ€š‘„,š‘¦āˆˆš¾;(2.6)
(ii) šœ‰-Lipschitz continuous if there exists a constant šœ‰ā‰„0 such that ā€–š‘(š‘„)āˆ’š‘(š‘¦)ā€–ā‰¤šœ‰ā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš¾.(2.7)

Definition 2.3. A set-valued mapping š‘‡āˆ¶š’³ā†’š’³ is said to be šœ‰-Lipschitz continuous if there exists a constant šœ‰>0 such that ā„‹(š‘‡(š‘„),š‘‡(š‘¦))ā‰¤šœ‰ā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš¾,(2.8) where ā„‹(ā‹…,ā‹…) is the Hausdorff metric on š¶(š’³).

Lemma 2.4 (see [14]). Let š’³ be a real reflexive and strictly convex Banach space with its dual š’³āˆ— and let š¾ be a nonempty closed convex subset of š’³. If š‘“āˆ¶š¾ā†’š‘…āˆŖ{+āˆž} is proper, convex, and lower semicontinuous, then Ī š‘“š¾ is single valued. Moreover, if š’³ has Kadec-Klee property, then Ī š‘“š¾ is continuous.

Lemma 2.5 (see [18]). Let š’³ be a real uniformly smooth Banach space. Then š’³ is š‘ž-uniformly smooth if and only if there exists a constant š¶š‘ž>0 such that, for all š‘„,š‘¦āˆˆš’³, ā€–š‘„+š‘¦ā€–š‘žā‰¤ā€–š‘„ā€–š‘žī«+š‘žš‘¦,š½š‘žī¬(š‘„)+š¶š‘žā€–š‘¦ā€–š‘ž.(2.9)

Lemma 2.6 (see [19]). Let (š’³,š‘‘) be a complete metric space and let š‘‡1,š‘‡2 be two set-valued contractive mappings with same contractive constants šœƒāˆˆ(0,1). Then ā„‹ī€·š¹ī€·š‘‡1ī€øī€·š‘‡,š¹2ā‰¤1ī€øī€ø1āˆ’šœƒsupš‘„āˆˆš’³š»ī€·š‘‡1(š‘„),š‘‡2ī€ø,(š‘„)(2.10) where š¹(š‘‡1) and š¹(š‘‡2) are fixed-point sets of š‘‡1 and š‘‡2, respectively.

Lemma 2.7 (see [19]). Let š’³ be a real strictly convex, reflexive, and smooth Banach space. For any š‘„1,š‘„2āˆˆš’³, let Ģ‚š‘„1=Ī š‘“š¾š‘„1 and Ģ‚š‘„2=Ī š‘“š¾š‘„2. Then ī«š½ī€·š‘„1ī€øī€·š‘„āˆ’š½2ī€ø,Ģ‚š‘„1āˆ’Ģ‚š‘„2ī¬ā‰„2š‘€2š›æī‚µā€–ā€–Ģ‚š‘„1āˆ’Ģ‚š‘„2ā€–ā€–ī‚¶2š‘€,(2.11) where īƒŽš‘€=ā€–Ģ‚š‘„1ā€–2+ā€–Ģ‚š‘„2ā€–22.(2.12)

We say that š’³ is 2-uniformly convex and 2-uniformly smooth Banach space if there exist š‘˜,š‘>0 such that š›æš‘‹(šœ–)ā‰„š‘˜šœ–2,šœŒš‘‹(š‘”)ā‰¤š‘š‘”2,(2.13) where š›æš‘‹(ī‚†ā€–ā€–ā€–šœ–)=inf1āˆ’š‘„+š‘¦2ā€–ā€–ā€–ī‚‡,šœŒāˆ¶ā€–š‘„ā€–=ā€–š‘¦ā€–=1,ā€–š‘„āˆ’š‘¦ā€–ā‰„šœ–š‘‹ī‚†1(š‘”)=sup2(ī‚‡.ā€–š‘„+š‘¦ā€–+ā€–š‘„āˆ’š‘¦ā€–)āˆ’1āˆ¶ā€–š‘„ā€–=1,ā€–š‘¦ā€–ā‰¤š‘”(2.14)

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

Lemma 2.8. Let š’³ be 2-uniformly convex and 2-uniformly smooth Banach space. Then ā€–ā€–Ī š‘“š¾š‘„āˆ’Ī š‘“š¾š‘¦ā€–ā€–š‘ā‰¤64š‘˜ā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš’³.(2.15)

Proof. According to Lemma 2.7, we have ī‚¬š½(š‘„)āˆ’š½(š‘¦),Ī š‘“š¾š‘„āˆ’Ī š‘“š¾š‘¦ī‚­ā‰„2š‘€21š›æīƒ©ā€–ā€–Ī š‘“š¾š‘„āˆ’Ī š‘“š¾š‘¦ā€–ā€–2š‘€1īƒŖ,(2.16) where š‘€1=īƒŽā€–Ī š‘“š¾š‘„ā€–2+ā€–Ī š‘“š¾š‘¦ā€–22.(2.17) Since š›æš‘‹(šœ–)ā‰„š‘˜šœ–2, (2.16) yields ā€–ā€–Ī š‘“š¾š‘„āˆ’Ī š‘“š¾š‘¦ā€–ā€–ā‰¤2š‘˜ā€–š½(š‘„)āˆ’š½(š‘¦)ā€–.(2.18) From the property of š½(ā‹…), we have ā€–š½(š‘„)āˆ’š½(š‘¦)ā€–ā‰¤2š‘€22šœŒš‘‹ī€·4ā€–š‘„āˆ’š‘¦ā€–/š‘€2ī€øā€–š‘„āˆ’š‘¦ā€–ā‰¤32š‘ā€–š‘„āˆ’š‘¦ā€–.(2.19) It follows from (2.18) and (2.19) that ā€–ā€–Ī š‘“š¾š‘„āˆ’Ī š‘“š¾š‘¦ā€–ā€–š‘ā‰¤64š‘˜ā€–š‘„āˆ’š‘¦ā€–.(2.20) 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 š’³ be 2-uniformly convex and 2-uniformly smooth Banach space. Let š‘āˆ¶š’³ā†’š¶(š’³) be šœ‡-Lipschitz continuous and let š‘”āˆ¶š’³ā†’š’³ be š›¼-Lipschitz continuous and š›½-strongly accretive. If ī”1+š›¼2āˆ’2š›½š¶2š‘+64š‘˜(š›¼+šœŒšœ‡)<1,(3.1) then the equilibrium points set of the generalized set-valued dynamical system (2.3) is nonempty and closed.

Proof. Let š‘‡(š‘„)=š‘„āˆ’š‘”(š‘„)+Ī š‘“š¾(š‘”(š‘„)āˆ’šœŒš‘(š‘„)),āˆ€š‘„āˆˆš¾.(3.2) Since š‘āˆ¶š’³ā†’š¶(š’³) and Ī š‘“š¾ are continuous, we know that š‘‡āˆ¶š’³ā†’š¶(š’³). From Definition 2.1, it is easy to see that š‘„āˆ— is an equilibrium point of the generalized set-valued dynamical system (2.3) if and only if š‘„āˆ— is a fixed-point of š‘‡ in š’³, that is: š‘„āˆ—ī€·š‘„āˆˆš‘‡āˆ—ī€ø=š‘„āˆ—ī€·š‘„āˆ’š‘”āˆ—ī€ø+Ī š‘“š¾ī€·š‘”ī€·š‘„āˆ—ī€øī€·š‘„āˆ’šœŒš‘āˆ—.ī€øī€ø(3.3) Thus, the equilibrium points set of (2.3) is the same as the fixed-points set of š‘‡. We first prove that š¹(š‘‡) is nonempty. In fact, for any š‘„,š‘¦āˆˆš’³ and š‘Ž1āˆˆš‘‡(š‘„), there exists š‘¢āˆˆš‘(š‘„) such that š‘Ž1=š‘„āˆ’š‘”(š‘„)+Ī š‘“š¾(š‘”(š‘„)āˆ’šœŒš‘¢).(3.4) Since š‘¢āˆˆš‘(š‘„), and š‘āˆ¶š’³ā†’š¶(š’³), it follows from Nadler [16] that there exists š‘£āˆˆš‘(š‘¦) such that ā€–š‘¢āˆ’š‘£ā€–ā‰¤ā„‹(š‘(š‘„),š‘(š‘¦)).(3.5) Let š‘Ž2=š‘¦āˆ’š‘”(š‘¦)+Ī š‘“š¾(š‘”(š‘¦)āˆ’šœŒš‘£).(3.6) Then š‘Ž2āˆˆš‘‡(š‘¦). From (3.4) to (3.6), we have ā€–ā€–š‘Ž1āˆ’š‘Ž2ā€–ā€–=ā€–ā€–š‘„āˆ’š‘¦āˆ’(š‘”(š‘„)āˆ’š‘”(š‘¦))+Ī š‘“š¾(š‘”(š‘„)āˆ’šœŒš‘¢)āˆ’Ī š‘“š¾ā€–ā€–ā€–ā€–Ī (š‘”(š‘¦)āˆ’šœŒ(š‘£))ā‰¤ā€–š‘„āˆ’š‘¦āˆ’(š‘”(š‘„)āˆ’š‘”(š‘¦))ā€–+š‘“š¾(š‘”(š‘„)āˆ’šœŒš‘¢)āˆ’Ī š‘“š¾ā€–ā€–.(š‘”(š‘¦)āˆ’šœŒ(š‘£))(3.7) Since š‘” is š›¼-Lipschitz continuous and š›½-strongly accretive, ā€–ā€–š‘„āˆ’š‘¦āˆ’(š‘”(š‘„)āˆ’š‘”(š‘¦)2ā‰¤ā€–š‘„āˆ’š‘¦ā€–2āˆ’2āŸØš‘”(š‘„)āˆ’š‘”(š‘¦),š½(š‘„āˆ’š‘¦)āŸ©+š¶2ā€–š‘”(š‘„)āˆ’š‘”(š‘¦)ā€–2ā‰¤ī€·1+š›¼2āˆ’2š›½š¶2ī€øā€–š‘„āˆ’š‘¦ā€–2.(3.8) From Lemma 2.8, where Ī š‘“š¾ is Lipchitz continuous, we have ā€–ā€–Ī š‘“š¾(š‘”(š‘„)āˆ’šœŒš‘¢)āˆ’Ī š‘“š¾ā€–ā€–š‘(š‘”(š‘¦)āˆ’(šœŒš‘£))ā‰¤64š‘˜()š‘ā€–š‘”(š‘„)āˆ’š‘”(š‘¦)ā€–+šœŒā€–š‘¢āˆ’š‘£ā€–ā‰¤64š‘˜(š›¼ā€–š‘„āˆ’š‘¦ā€–+šœŒā€–š‘¢āˆ’š‘£ā€–).(3.9) From the selection of š‘£ and the Lipschitz continuity of š‘, ā€–š‘¢āˆ’š‘£ā€–ā‰¤ā„‹(š‘(š‘„),š‘(š‘¦))ā‰¤šœ‡ā€–š‘„āˆ’š‘¦ā€–.(3.10) In light of (3.7)ā€“(3.10), we have ā€–ā€–š‘Ž1āˆ’š‘Ž2ā€–ā€–ā‰¤ī‚µī”1+š›¼2āˆ’2š›½š¶2š‘+64š‘˜ī‚¶(š›¼+šœŒšœ‡)ā€–š‘„āˆ’š‘¦ā€–=šæā€–š‘„āˆ’š‘¦ā€–,(3.11) where ī”šæ=1+š›¼2āˆ’2š›½š¶2š‘+64š‘˜(š›¼+šœŒšœ‡).(3.12) Now (3.11) implies that š‘‘ī€·š‘Ž1ī€ø,š‘‡(š‘¦)=infš‘Ž2āˆˆš‘‡(š‘¦)ā€–ā€–š‘Ž1āˆ’š‘Ž2ā€–ā€–ā‰¤šæā€–š‘„āˆ’š‘¦ā€–.(3.13) Since š‘Ž1āˆˆš‘‡(š‘„) is arbitrary, we have supš‘Ž1āˆˆš‘‡(š‘„)š‘‘ī€·š‘Ž1ī€ø,š‘‡(š‘¦)ā‰¤šæā€–š‘„āˆ’š‘¦ā€–.(3.14) Similarly, we can prove supš‘Ž2āˆˆš‘‡(š‘¦)š‘‘ī€·š‘‡(š‘„),š‘Ž2ī€øā‰¤šæā€–š‘„āˆ’š‘¦ā€–.(3.15) From (3.14), (3.15), and the definition of the Hausdorff metric ā„‹ on š¶(š’³), we have ā„‹(š‘‡(š‘„),š‘‡(š‘¦))ā‰¤šæā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš¾.(3.16) Now the assumption of the theorem implies that šæ<1 and so š‘‡(š‘„) is a set-valued contractive mapping. By the fixed-point theorem of Nadler [16], there exists š‘„āˆ— such that š‘„āˆ—āˆˆš‘‡(š‘„āˆ—), and thus š‘„āˆ— is the equilibrium point of (2.3). This means that š¹(š‘‡) is nonempty.
Now we prove that š¹(š‘‡) is closed. Let {š‘„š‘›}āŠ‚š¹(š‘‡) with š‘„š‘›ā†’š‘„0(š‘›ā†’āˆž). Then š‘„š‘›āˆˆš‘‡(š‘„š‘›) and (3.16) imply that ā„‹ī€·š‘‡ī€·š‘„š‘›ī€øī€·š‘„,š‘‡0ā€–ā€–š‘„ī€øī€øā‰¤šæš‘›āˆ’š‘„0ā€–ā€–.(3.17) Thus, š‘‘ī€·š‘„0ī€·š‘„,š‘‡0ā‰¤ā€–ā€–š‘„ī€øī€ø0āˆ’š‘„š‘›ā€–ā€–ī€·š‘„+š‘‘š‘›ī€·š‘„,š‘‡š‘›ī€·š‘‡ī€·š‘„ī€øī€ø+ā„‹š‘›ī€øī€·š‘„,š‘‡0ā€–ā€–š‘„ī€øī€øā‰¤(1+šæ)š‘›āˆ’š‘„0ā€–ā€–āŸ¶0asš‘›ā†’āˆž.(3.18) It follows that š‘„0āˆˆš¹(š‘‡) and so š¹(š‘‡) are closed. This completes the proof.

Remark 3.2. Theorem 3.1 is a generalization of Theorem 1 in Zou et al. [9] from ā„š‘› to Banach space š’³.

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: š‘‘š‘„(š‘”)š‘‘š‘”āˆˆĪ š‘“š¾[],(š‘”(š‘„(š‘”))āˆ’šœŒš‘(š‘„(š‘”)))āˆ’š‘”(š‘„(š‘”))+š¹(š‘„(š‘”)),fora.a.š‘”āˆˆ0,ā„Žš‘„(0)=š‘,(4.1) where š‘” and š‘ are the same as in (2.3), š¹āˆ¶š’³ā†’2š’³ is a set-valued mapping, and š‘āˆ¶š’³ā†’š’³ is a single-valued mapping. Let š‘†(š‘) denote the set of all solutions of (4.1) on [0,ā„Ž] with š‘„(0)=š‘.

Now we prove the following result.

Theorem 4.1. Let š’³ be 2-uniformly convex and 2-uniformly smooth Banach space. Let š‘”āˆ¶š’³ā†’š’³ be š›¼-Lipschitz continuous, let š‘āˆ¶š’³ā†’š’³ be šœ‡-Lipschitz continuous, and let š¹āˆ¶š’³ā†’š¶(š’³) be a šœ”-Lipschitz continuous set-valued mapping with compact convex values. If š‘64š‘˜ī‚€š‘(š›¼+šœŒšœ‡)+š›¼+šœ”<1,ā„Ž64š‘˜ī‚(š›¼+šœŒšœ‡)+š›¼+šœ”<1,(4.2) then š‘†(š‘) is nonempty and continuous.

Proof. Let š‘€(š‘„)=Ī š‘“š¾(š‘”(š‘„)āˆ’šœŒš‘(š‘„))āˆ’š‘”(š‘„)+š¹(š‘„).(4.3) Then š‘€āˆ¶š’³ā†’š¶(š’³) is a set-valued mapping with compact convex values since š¹āˆ¶š’³ā†’š¶(š’³) is a set-valued mapping with compact convex values. For any š‘„1,š‘„2āˆˆš’³ and š‘Ž1āˆˆš‘€(š‘„1), there exists š‘¢āˆˆš¹(š‘„1) such that š‘Ž1=Ī š‘“š¾ī€·š‘”ī€·š‘„1ī€øī€·š‘„āˆ’šœŒš‘1ī€·š‘„ī€øī€øāˆ’š‘”1ī€ø+š‘¢.(4.4) Since š‘¢āˆˆš¹(š‘„1), and š¹āˆ¶š’³ā†’š¶(š’³), it follows from Nadler [16] that there exists š‘£āˆˆš¹(š‘„2) such that ī€·š¹ī€·š‘„ā€–š‘¢āˆ’š‘£ā€–ā‰¤ā„‹1ī€øī€·š‘„,š¹2ī€øī€ø.(4.5) Let š‘Ž2=Ī š‘“š¾ī€·š‘”ī€·š‘„2ī€øī€·š‘„āˆ’šœŒš‘2ī€·š‘„ī€øī€øāˆ’š‘”2ī€ø+š‘£.(4.6) Then š‘Ž2āˆˆš‘€(š‘„2). From (4.4) and (4.6), we have ā€–ā€–š‘Ž1āˆ’š‘Ž2ā€–ā€–=ā€–ā€–Ī š‘“š¾ī€·š‘”ī€·š‘„1ī€øī€·š‘„āˆ’šœŒš‘1ī€øī€øāˆ’Ī š‘“š¾ī€·š‘”ī€·š‘„2ī€øī€·š‘„āˆ’šœŒš‘2āˆ’ī€·š‘”ī€·š‘„ī€øī€ø2ī€øī€·š‘„āˆ’š‘”2ā€–ā€–ā‰¤ā€–ā€–š‘”ī€·š‘„ī€øī€ø+š‘¢āˆ’š‘£1ī€øī€·š‘„āˆ’š‘”2ī€øā€–ā€–+ā€–ā€–Ī š‘“š¾ī€·ī€·š‘„š‘”(š‘„)āˆ’šœŒš‘1ī€øī€øāˆ’Ī š‘“š¾ī€·ī€·š‘„š‘”(š‘¦)āˆ’šœŒš‘2ā€–ā€–ī€øī€ø+ā€–š‘¢āˆ’š‘£ā€–.(4.7) Since š‘” is š›¼-Lipschitz continuous, ā€–ā€–š‘”ī€·š‘„1ī€øī€·š‘„āˆ’š‘”2ī€øā€–ā€–ā€–ā€–š‘„ā‰¤š›¼2āˆ’š‘„2ā€–ā€–.(4.8) From Lemma 2.8, Ī š‘“š¾ is Lipschitz continuous. It follows from the continuity of š‘ and š‘” that ā€–ā€–Ī š‘“š¾ī€·š‘”ī€·š‘„1ī€øī€·š‘„āˆ’šœŒš‘1ī€øī€øāˆ’Ī š‘“š¾ī€·š‘”ī€·š‘„2ī€øī€·š‘„āˆ’šœŒš‘2ā€–ā€–š‘ī€øī€øā‰¤64š‘˜ī€·ā€–ā€–š‘”ī€·š‘„1ī€øī€·š‘„āˆ’š‘”2ī€øā€–ā€–ā€–ā€–š‘ī€·š‘„+šœŒ1ī€øī€·š‘„āˆ’š‘2ī€øā€–ā€–ī€øš‘ā‰¤64š‘˜ā€–ā€–š‘„(š›¼+šœŒšœ‡)1āˆ’š‘„2ā€–ā€–.(4.9) From the selection of š‘£ and the Lipschitz continuity of š¹, we know ī€·š¹ī€·š‘„ā€–š‘¢āˆ’š‘£ā€–ā‰¤ā„‹1ī€øī€·š‘„,š¹2ā€–ā€–š‘„ī€øī€øā‰¤šœ”1āˆ’š‘„2ā€–ā€–.(4.10) In light of (4.7)ā€“(4.10), we have ā€–ā€–š‘Ž1āˆ’š‘Ž2ā€–ā€–ā‰¤ī‚€š‘š›¼+64š‘˜ī‚ā€–ā€–š‘„(š›¼+šœŒšœ‡)+šœ”1āˆ’š‘„2ā€–ā€–=šœƒā€–š‘„āˆ’š‘¦ā€–,(4.11) where š‘šœƒ=š›¼+64š‘˜(š›¼+šœŒšœ‡)+šœ”.(4.12) Now (4.11) implies that š‘‘ī€·š‘Ž1ī€·š‘„,š‘€2ī€øī€ø=infš‘Ž2āˆˆš‘€(š‘„2)ā€–ā€–š‘Ž1āˆ’š‘Ž2ā€–ā€–ā€–ā€–š‘„ā‰¤šœƒ1āˆ’š‘„2ā€–ā€–.(4.13) Since š‘Ž1āˆˆš‘€(š‘„1) is arbitrary, we obtain supš‘Ž1ī€·š‘„āˆˆš‘€1ī€øš‘‘ī€·š‘Ž1ī€·š‘„,š‘€2ā€–ā€–š‘„ī€øī€øā‰¤šœƒ1āˆ’š‘„2ā€–ā€–.(4.14) Similarly, we can prove supš‘Ž2ī€·š‘„āˆˆš‘€2ī€øš‘‘ī€·š‘€ī€·š‘„1ī€ø,š‘Ž2ī€øā€–ā€–š‘„ā‰¤šœƒ1āˆ’š‘„2ā€–ā€–.(4.15) From (4.13), to (4.15), and the definition of the Hausdorff metric ā„‹ on š¶(š’³), we have ā„‹ī€·š‘€ī€·š‘„1ī€øī€·š‘„,š‘€2ā€–ā€–š‘„ī€øī€øā‰¤šœƒ1āˆ’š‘„2ā€–ā€–,āˆ€š‘„1,š‘„2āˆˆš’³.(4.16) Now (4.2) implies that 0<šœƒ<1, and so š‘€(š‘„) is a set-valued contractive mapping. Let ī‚»[]ī€œš‘„(š‘„,š‘)=š‘¦āˆˆš¶(0,ā„Ž,š’³)āˆ£š‘¦(š‘”)=š‘+š‘”0ī‚¼,š‘§(š‘ )š‘‘š‘ ,š‘§(š‘ )āˆˆš‘€(š‘„(š‘ ))(4.17) where [][]š¶(0,ā„Ž,š’³)={š‘“āˆ¶0,ā„ŽāŸ¶š’³āˆ£š‘“iscontinuous}.(4.18) Since š‘€āˆ¶š’³ā†’š¶(š’³) 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 š‘„(š‘„,š‘) is nonempty for each š‘„ and š‘āˆˆš’³. Moreover, it is easy to see that the set of fixed-points of š‘„(š‘„,š‘) coincides with š‘†(š‘). It follows from [21] or [8] that š‘„(š‘„,š‘) is compact and convex for each š‘„ and š‘āˆˆš’³. Suppose that š‘š‘š is the initial value of (4.1); that is, š‘„(0)=š‘š‘š(š‘š=0,1,2,ā€¦) and š‘š‘šā†’š‘0(š‘šā†’āˆž). Since š‘„ī€·š‘„,š‘0ī€øī€·=š‘„š‘„,š‘š‘šī€øāˆ’š‘š‘š+š‘0,(4.19) it is obvious that š‘„(š‘„,š‘š‘š) converges uniformly to š‘„(š‘„,š‘0).
Next we prove that š‘„(š‘„,š‘š‘š) is a set-valued contractive mapping. For any given š‘„1,š‘„2āˆˆš¶([0,ā„Ž],š’³), since š‘€āˆ¶š’³ā†’š¶(š’³) 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 š‘€(š‘„1(š‘ )) has a continuous selection š‘Ÿ1(š‘ )āˆˆš‘€(š‘„1(š‘ )). Let š‘1(š‘”)=š‘š‘š+ī€œš‘”0š‘Ÿ1(š‘ )š‘‘š‘ .(4.20) Then š‘1āˆˆš‘„(š‘„1,š‘š‘š). Since š‘Ÿ1(š‘ )āˆˆš‘€(š‘„1(š‘ )) is measurable and š‘€(š‘„2(š‘ )) is a measurable mapping with compact values, we know that there exists a measurable selection š‘Ÿ2(š‘ )āˆˆš‘€(š‘„2(š‘ )) such that ā€–ā€–š‘Ÿ1(š‘ )āˆ’š‘Ÿ2ā€–ā€–ī€·š‘€ī€·š‘„(š‘ )ā‰¤ā„‹1ī€øī€·š‘„(š‘ ),š‘€2(š‘ )ī€øī€ø.(4.21) Thus, it follows from (4.16) that ā€–ā€–š‘Ÿ1(š‘ )āˆ’š‘Ÿ2ā€–ā€–ī€·š‘€ī€·š‘„(š‘ )ā‰¤ā„‹1ī€øī€·š‘„(š‘ ),š‘€2ā€–ā€–š‘„(š‘ )ī€øī€øā‰¤šœƒ1āˆ’š‘„2ā€–ā€–.(4.22) Let š‘2(š‘”)=š‘š‘š+ī€œš‘”0š‘Ÿ2(š‘ )š‘‘š‘ .(4.23) Then š‘2āˆˆš‘„(š‘„2,š‘š‘š) and ā€–ā€–š‘1āˆ’š‘2ā€–ā€–ā‰¤ī€œā„Ž0ā€–ā€–š‘Ÿ1(š‘ )āˆ’š‘Ÿ2ā€–ā€–ī€·š‘€ī€·š‘„(š‘ )š‘‘š‘ ā‰¤ā„Žā„‹1ī€øī€·š‘„(š‘ ),š‘€2ā€–ā€–š‘„(š‘ )ī€øī€øā‰¤ā„Žšœƒ1āˆ’š‘„2ā€–ā€–.(4.24) Hence, we have š‘‘ī€·š‘1ī€·š‘„,š‘„2,š‘š‘šī€øī€ø=infš‘2āˆˆš‘„(š‘„2,š‘š‘š)ā€–ā€–š‘1āˆ’š‘2ā€–ā€–ā€–ā€–š‘„ā‰¤ā„Žšœƒ1āˆ’š‘„2ā€–ā€–.(4.25) Since š‘1āˆˆš‘„(š‘„1,š‘š‘š) is arbitrary, we obtain supš‘1ī€·š‘„āˆˆš‘„1,š‘š‘šī€øš‘‘ī€·š‘1ī€·š‘„,š‘„2,š‘š‘šā€–ā€–š‘„ī€øī€øā‰¤ā„Žšœƒ1āˆ’š‘„2ā€–ā€–.(4.26) Similarly, we can prove that supš‘2ī€·š‘„āˆˆš‘„2,š‘š‘šī€øš‘‘ī€·š‘„ī€·š‘„1,š‘š‘šī€ø,š‘2ī€øā€–ā€–š‘„ā‰¤ā„Žšœƒ1āˆ’š‘„2ā€–ā€–.(4.27) From the definition of the Hausdorff metric ā„‹ on š¶(š’³), (4.26) and (4.27) imply that ā„‹ī€·š‘„ī€·š‘„1,š‘š‘šī€øī€·š‘„,š‘„2,š‘š‘šā€–ā€–š‘„ī€øī€øā‰¤ā„Žšœƒ1āˆ’š‘„2ā€–ā€–,āˆ€š‘„1,š‘„2āˆˆš’³,š‘š=0,1,2,ā€¦.(4.28) Since ā„Žšœƒ<1, it is easy to see that š‘„(š‘„,š‘) has a fixed-point for each given š‘āˆˆš’³, and so š‘†(š‘) is nonempty for each given š‘āˆˆš’³. Setting š‘Šš‘šī€·(š‘„)=š‘„š‘„,š‘š‘šī€ø,š‘š=0,1,2,ā€¦,(4.29) we know that š‘Šš‘š(š‘„) are contractive mappings with the same contractive constant ā„Žšœƒ. By Lemma 2.6 and (4.28), we have ā„‹ī€·š¹ī€·š‘Šš‘šī€øī€·š‘Š,š¹0ā‰¤1ī€øī€ø1āˆ’ā„Žšœƒsupš‘„āˆˆš‘‹ā„‹ī€·š‘Šš‘š(š‘„),š‘Š0ī€ø(š‘„)āŸ¶0.(4.30) Thus, š¹(š‘Šš‘š)ā†’š¹(š‘Š0), which implies that š‘†(š‘š‘š)ā†’š‘†(š‘); 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 ā„š‘› to Banach space š’³.

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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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.


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