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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 682465, 12 pages
http://dx.doi.org/10.1155/2012/682465
Research Article

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

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: ZhenyuΒ Huang

Copyright Β© 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: 𝑑π‘₯𝑑𝑑=𝑃𝐾(π‘₯βˆ’πœŒπ‘(π‘₯))βˆ’π‘₯,(1.1) where π‘βˆΆβ„π‘›β†’β„π‘› is a single-valued function, 𝜌>0 is a constant, 𝑃𝐾π‘₯ denotes the projection of the point π‘₯ on 𝐾; here πΎβŠ‚β„π‘› 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: 𝑑π‘₯(𝑑)π‘‘π‘‘βˆˆπ‘ƒπΎ[],(𝑔(π‘₯(𝑑))βˆ’πœŒπ‘(π‘₯(𝑑))βˆ’π‘”(π‘₯(𝑑))),fora.a.π‘‘βˆˆ0,𝐽π‘₯(0)=𝑏,(1.2) where π‘βˆΆβ„π‘›β†’2ℝ𝑛 is a set-valued function, π‘”βˆΆβ„π‘›β†’β„π‘› is a single-valued function, 𝜌>0 is a constant, 𝑃𝐾π‘₯ denotes the projection of the point π‘₯ on 𝐾, 𝑏 is a given point in ℝ𝑛.

The concept of generalized 𝑓-projection operator was first introduced by Wu and Huang [11] in 2006. They also proved that the generalized 𝑓-projection operator is an extension of the projection operator 𝑃𝐾 in 𝑅𝑛 and it owns some nice properties as 𝑃𝐾 does; see [12, 13]. Some applications of generalized 𝑓-projection operator are also given in [11–13]. Very recently, Li et al. [14] studied the stability of the generalized 𝑓-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 𝑓-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 𝒳 be a Banach space and let πΎβŠ‚π’³ be a closed convex set, let π‘βˆΆπ’³β†’2𝒳 be a set-valued mapping, and let π‘”βˆΆπ’³β†’π’³ be a single-valued mapping. The normalized duality mapping 𝐽 from 𝒳 to π’³βˆ— is defined by π‘₯𝐽(π‘₯)=βˆ—βˆˆπ’³βˆ—βˆΆβŸ¨π‘₯,π‘₯βˆ—βŸ©=β€–π‘₯β€–2=β€–π‘₯βˆ—β€–2,(2.1) for π‘₯βˆˆπ‘‹. For convenience, we list some properties of 𝐽(β‹…) as follows.𝑋 is a smooth Banach space, 𝐽(β‹…) is single valued and hemicontinuous; that is, 𝐽 is continuous from the strong topology of 𝒳 to the π‘€π‘’π‘Žπ‘˜βˆ— topology of π’³βˆ—.

Let 𝐢(𝒳) denote the family of all nonempty compact subsets of 𝒳 and let β„‹(β‹…,β‹…) denote the Hausdorff metric on 𝐢(𝒳) defined by ξ‚»β„‹(𝐴,𝐡)=maxsupπ‘Žβˆˆπ΄infπ‘βˆˆπ΅π‘‘(π‘Ž,𝑏),supπ‘βˆˆπ΅infπ‘Žβˆˆπ΄ξ‚Όπ‘‘(π‘Ž,𝑏),βˆ€π΄,𝐡∈𝐢(𝒳).(2.2)

In this paper, we consider a new class of generalized set-valued dynamical system, that is, to find those absolutely continuous functions π‘₯(β‹…) from [0,β„Ž]→𝒳 such that 𝑑π‘₯(𝑑)π‘‘π‘‘βˆˆΞ π‘“πΎ[],(𝑔(π‘₯(𝑑))βˆ’πœŒπ‘(π‘₯(𝑑)))βˆ’π‘”(π‘₯(𝑑)),fora.a.π‘‘βˆˆ0,β„Žπ‘₯(0)=𝑏,(2.3) where π‘βˆˆπ’³, 𝜌>0 is a constant and π‘“βˆΆπΎβ†’π‘…βˆͺ{+∞} is proper, convex, and lower semicontinuous and Ξ π‘“πΎβˆΆπ’³β†’2𝐾 is a generalized 𝑓-projection operator denoted by Π𝑓𝐾π‘₯=π‘’βˆˆπΎβˆΆπΊ(𝐽(π‘₯),𝑒)=infπœ‰βˆˆπΎξ‚ΌπΊ(𝐽(π‘₯),πœ‰),βˆ€π‘₯βˆˆπ’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., [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 π‘₯βˆ— is said to be an equilibrium point of global dynamical system (2.3), if π‘₯βˆ— satisfies the following inclusion: 0βˆˆΞ π‘“πΎ(𝑔(π‘₯)βˆ’πœŒπ‘(π‘₯))βˆ’π‘”(π‘₯).(2.5)

Definition 2.2. A mapping π‘βˆΆπ’³β†’π’³ is said to be
(i) 𝛼-strongly accretive if there exists some 𝛼>0 such that (𝑁(π‘₯)βˆ’π‘(𝑦),𝐽(π‘₯βˆ’π‘¦))β‰₯𝛼‖π‘₯βˆ’π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπΎ;(2.6)
(ii) πœ‰-Lipschitz continuous if there exists a constant πœ‰β‰₯0 such that ‖𝑁(π‘₯)βˆ’π‘(𝑦)β€–β‰€πœ‰β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΎ.(2.7)

Definition 2.3. A set-valued mapping π‘‡βˆΆπ’³β†’π’³ is said to be πœ‰-Lipschitz continuous if there exists a constant πœ‰>0 such that β„‹(𝑇(π‘₯),𝑇(𝑦))β‰€πœ‰β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΎ,(2.8) where β„‹(β‹…,β‹…) is the Hausdorff metric on 𝐢(𝒳).

Lemma 2.4 (see [14]). Let 𝒳 be a real reflexive and strictly convex Banach space with its dual π’³βˆ— and let 𝐾 be a nonempty closed convex subset of 𝒳. If π‘“βˆΆπΎβ†’π‘…βˆͺ{+∞} is proper, convex, and lower semicontinuous, then Π𝑓𝐾 is single valued. Moreover, if 𝒳 has Kadec-Klee property, then Π𝑓𝐾 is continuous.

Lemma 2.5 (see [18]). Let 𝒳 be a real uniformly smooth Banach space. Then 𝒳 is π‘ž-uniformly smooth if and only if there exists a constant πΆπ‘ž>0 such that, for all π‘₯,π‘¦βˆˆπ’³, β€–π‘₯+π‘¦β€–π‘žβ‰€β€–π‘₯β€–π‘žξ«+π‘žπ‘¦,π½π‘žξ¬(π‘₯)+πΆπ‘žβ€–π‘¦β€–π‘ž.(2.9)

Lemma 2.6 (see [19]). Let (𝒳,𝑑) be a complete metric space and let 𝑇1,𝑇2 be two set-valued contractive mappings with same contractive constants πœƒβˆˆ(0,1). Then ℋ𝐹𝑇1𝑇,𝐹2≀1ξ€Έξ€Έ1βˆ’πœƒsupπ‘₯βˆˆπ’³π»ξ€·π‘‡1(π‘₯),𝑇2ξ€Έ,(π‘₯)(2.10) where 𝐹(𝑇1) and 𝐹(𝑇2) are fixed-point sets of 𝑇1 and 𝑇2, respectively.

Lemma 2.7 (see [19]). Let 𝒳 be a real strictly convex, reflexive, and smooth Banach space. For any π‘₯1,π‘₯2βˆˆπ’³, let Μ‚π‘₯1=Π𝑓𝐾π‘₯1 and Μ‚π‘₯2=Π𝑓𝐾π‘₯2. Then 𝐽π‘₯1ξ€Έξ€·π‘₯βˆ’π½2ξ€Έ,Μ‚π‘₯1βˆ’Μ‚π‘₯2β‰₯2𝑀2𝛿‖‖̂π‘₯1βˆ’Μ‚π‘₯2β€–β€–ξ‚Ά2𝑀,(2.11) where ξƒŽπ‘€=β€–Μ‚π‘₯1β€–2+β€–Μ‚π‘₯2β€–22.(2.12)

We say that 𝒳 is 2-uniformly convex and 2-uniformly smooth Banach space if there exist π‘˜,𝑐>0 such that 𝛿𝑋(πœ–)β‰₯π‘˜πœ–2,πœŒπ‘‹(𝑑)≀𝑐𝑑2,(2.13) where 𝛿𝑋(ξ‚†β€–β€–β€–πœ–)=inf1βˆ’π‘₯+𝑦2‖‖‖,πœŒβˆΆβ€–π‘₯β€–=‖𝑦‖=1,β€–π‘₯βˆ’π‘¦β€–β‰₯πœ–π‘‹ξ‚†1(𝑑)=sup2(.β€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–)βˆ’1βˆΆβ€–π‘₯β€–=1,‖𝑦‖≀𝑑(2.14)

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

Lemma 2.8. Let 𝒳 be 2-uniformly convex and 2-uniformly smooth Banach space. Then ‖‖Π𝑓𝐾π‘₯βˆ’Ξ π‘“πΎπ‘¦β€–β€–π‘β‰€64π‘˜β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπ’³.(2.15)

Proof. According to Lemma 2.7, we have 𝐽(π‘₯)βˆ’π½(𝑦),Π𝑓𝐾π‘₯βˆ’Ξ π‘“πΎπ‘¦ξ‚­β‰₯2𝑀21𝛿‖‖Π𝑓𝐾π‘₯βˆ’Ξ π‘“πΎπ‘¦β€–β€–2𝑀1ξƒͺ,(2.16) where 𝑀1=ξƒŽβ€–Ξ π‘“πΎπ‘₯β€–2+‖Π𝑓𝐾𝑦‖22.(2.17) Since 𝛿𝑋(πœ–)β‰₯π‘˜πœ–2, (2.16) yields ‖‖Π𝑓𝐾π‘₯βˆ’Ξ π‘“πΎπ‘¦β€–β€–β‰€2π‘˜β€–π½(π‘₯)βˆ’π½(𝑦)β€–.(2.18) From the property of 𝐽(β‹…), we have ‖𝐽(π‘₯)βˆ’π½(𝑦)‖≀2𝑀22πœŒπ‘‹ξ€·4β€–π‘₯βˆ’π‘¦β€–/𝑀2ξ€Έβ€–π‘₯βˆ’π‘¦β€–β‰€32𝑐‖π‘₯βˆ’π‘¦β€–.(2.19) It follows from (2.18) and (2.19) that ‖‖Π𝑓𝐾π‘₯βˆ’Ξ π‘“πΎπ‘¦β€–β€–π‘β‰€64π‘˜β€–π‘₯βˆ’π‘¦β€–.(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 𝒳 be 2-uniformly convex and 2-uniformly smooth Banach space. Let π‘βˆΆπ’³β†’πΆ(𝒳) be πœ‡-Lipschitz continuous and let π‘”βˆΆπ’³β†’π’³ be 𝛼-Lipschitz continuous and 𝛽-strongly accretive. If 1+𝛼2βˆ’2𝛽𝐢2𝑐+64π‘˜(𝛼+πœŒπœ‡)<1,(3.1) then the equilibrium points set of the generalized set-valued dynamical system (2.3) is nonempty and closed.

Proof. Let 𝑇(π‘₯)=π‘₯βˆ’π‘”(π‘₯)+Π𝑓𝐾(𝑔(π‘₯)βˆ’πœŒπ‘(π‘₯)),βˆ€π‘₯∈𝐾.(3.2) Since π‘βˆΆπ’³β†’πΆ(𝒳) and Π𝑓𝐾 are continuous, we know that π‘‡βˆΆπ’³β†’πΆ(𝒳). From Definition 2.1, it is easy to see that π‘₯βˆ— is an equilibrium point of the generalized set-valued dynamical system (2.3) if and only if π‘₯βˆ— is a fixed-point of 𝑇 in 𝒳, that is: π‘₯βˆ—ξ€·π‘₯βˆˆπ‘‡βˆ—ξ€Έ=π‘₯βˆ—ξ€·π‘₯βˆ’π‘”βˆ—ξ€Έ+Π𝑓𝐾𝑔π‘₯βˆ—ξ€Έξ€·π‘₯βˆ’πœŒπ‘βˆ—.ξ€Έξ€Έ(3.3) Thus, the equilibrium points set of (2.3) is the same as the fixed-points set of 𝑇. We first prove that 𝐹(𝑇) is nonempty. In fact, for any π‘₯,π‘¦βˆˆπ’³ and π‘Ž1βˆˆπ‘‡(π‘₯), there exists π‘’βˆˆπ‘(π‘₯) such that π‘Ž1=π‘₯βˆ’π‘”(π‘₯)+Π𝑓𝐾(𝑔(π‘₯)βˆ’πœŒπ‘’).(3.4) Since π‘’βˆˆπ‘(π‘₯), and π‘βˆΆπ’³β†’πΆ(𝒳), it follows from Nadler [16] that there exists π‘£βˆˆπ‘(𝑦) such that β€–π‘’βˆ’π‘£β€–β‰€β„‹(𝑁(π‘₯),𝑁(𝑦)).(3.5) Let π‘Ž2=π‘¦βˆ’π‘”(𝑦)+Π𝑓𝐾(𝑔(𝑦)βˆ’πœŒπ‘£).(3.6) Then π‘Ž2βˆˆπ‘‡(𝑦). From (3.4) to (3.6), we have β€–β€–π‘Ž1βˆ’π‘Ž2β€–β€–=β€–β€–π‘₯βˆ’π‘¦βˆ’(𝑔(π‘₯)βˆ’π‘”(𝑦))+Π𝑓𝐾(𝑔(π‘₯)βˆ’πœŒπ‘’)βˆ’Ξ π‘“πΎβ€–β€–β€–β€–Ξ (𝑔(𝑦)βˆ’πœŒ(𝑣))≀‖π‘₯βˆ’π‘¦βˆ’(𝑔(π‘₯)βˆ’π‘”(𝑦))β€–+𝑓𝐾(𝑔(π‘₯)βˆ’πœŒπ‘’)βˆ’Ξ π‘“πΎβ€–β€–.(𝑔(𝑦)βˆ’πœŒ(𝑣))(3.7) Since 𝑔 is 𝛼-Lipschitz continuous and 𝛽-strongly accretive, β€–β€–π‘₯βˆ’π‘¦βˆ’(𝑔(π‘₯)βˆ’π‘”(𝑦)2≀‖π‘₯βˆ’π‘¦β€–2βˆ’2βŸ¨π‘”(π‘₯)βˆ’π‘”(𝑦),𝐽(π‘₯βˆ’π‘¦)⟩+𝐢2‖𝑔(π‘₯)βˆ’π‘”(𝑦)β€–2≀1+𝛼2βˆ’2𝛽𝐢2ξ€Έβ€–π‘₯βˆ’π‘¦β€–2.(3.8) From Lemma 2.8, where Π𝑓𝐾 is Lipchitz continuous, we have ‖‖Π𝑓𝐾(𝑔(π‘₯)βˆ’πœŒπ‘’)βˆ’Ξ π‘“πΎβ€–β€–π‘(𝑔(𝑦)βˆ’(πœŒπ‘£))≀64π‘˜()𝑐‖𝑔(π‘₯)βˆ’π‘”(𝑦)β€–+πœŒβ€–π‘’βˆ’π‘£β€–β‰€64π‘˜(𝛼‖π‘₯βˆ’π‘¦β€–+πœŒβ€–π‘’βˆ’π‘£β€–).(3.9) From the selection of 𝑣 and the Lipschitz continuity of 𝑁, β€–π‘’βˆ’π‘£β€–β‰€β„‹(𝑁(π‘₯),𝑁(𝑦))β‰€πœ‡β€–π‘₯βˆ’π‘¦β€–.(3.10) In light of (3.7)–(3.10), we have β€–β€–π‘Ž1βˆ’π‘Ž2‖‖≀1+𝛼2βˆ’2𝛽𝐢2𝑐+64π‘˜ξ‚Ά(𝛼+πœŒπœ‡)β€–π‘₯βˆ’π‘¦β€–=𝐿‖π‘₯βˆ’π‘¦β€–,(3.11) where 𝐿=1+𝛼2βˆ’2𝛽𝐢2𝑐+64π‘˜(𝛼+πœŒπœ‡).(3.12) Now (3.11) implies that π‘‘ξ€·π‘Ž1ξ€Έ,𝑇(𝑦)=infπ‘Ž2βˆˆπ‘‡(𝑦)β€–β€–π‘Ž1βˆ’π‘Ž2‖‖≀𝐿‖π‘₯βˆ’π‘¦β€–.(3.13) Since π‘Ž1βˆˆπ‘‡(π‘₯) is arbitrary, we have supπ‘Ž1βˆˆπ‘‡(π‘₯)π‘‘ξ€·π‘Ž1ξ€Έ,𝑇(𝑦)≀𝐿‖π‘₯βˆ’π‘¦β€–.(3.14) Similarly, we can prove supπ‘Ž2βˆˆπ‘‡(𝑦)𝑑𝑇(π‘₯),π‘Ž2≀𝐿‖π‘₯βˆ’π‘¦β€–.(3.15) From (3.14), (3.15), and the definition of the Hausdorff metric β„‹ on 𝐢(𝒳), we have β„‹(𝑇(π‘₯),𝑇(𝑦))≀𝐿‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΎ.(3.16) Now the assumption of the theorem implies that 𝐿<1 and so 𝑇(π‘₯) is a set-valued contractive mapping. By the fixed-point theorem of Nadler [16], there exists π‘₯βˆ— such that π‘₯βˆ—βˆˆπ‘‡(π‘₯βˆ—), and thus π‘₯βˆ— is the equilibrium point of (2.3). This means that 𝐹(𝑇) is nonempty.
Now we prove that 𝐹(𝑇) is closed. Let {π‘₯𝑛}βŠ‚πΉ(𝑇) with π‘₯𝑛→π‘₯0(π‘›β†’βˆž). Then π‘₯π‘›βˆˆπ‘‡(π‘₯𝑛) and (3.16) imply that ℋ𝑇π‘₯𝑛π‘₯,𝑇0β€–β€–π‘₯ξ€Έξ€Έβ‰€πΏπ‘›βˆ’π‘₯0β€–β€–.(3.17) Thus, 𝑑π‘₯0ξ€·π‘₯,𝑇0≀‖‖π‘₯ξ€Έξ€Έ0βˆ’π‘₯𝑛‖‖π‘₯+𝑑𝑛π‘₯,𝑇𝑛𝑇π‘₯ξ€Έξ€Έ+ℋ𝑛π‘₯,𝑇0β€–β€–π‘₯≀(1+𝐿)π‘›βˆ’π‘₯0β€–β€–βŸΆ0asπ‘›β†’βˆž.(3.18) It follows that π‘₯0∈𝐹(𝑇) and so 𝐹(𝑇) are closed. This completes the proof.

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

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: 𝑑π‘₯(𝑑)π‘‘π‘‘βˆˆΞ π‘“πΎ[],(𝑔(π‘₯(𝑑))βˆ’πœŒπ‘(π‘₯(𝑑)))βˆ’π‘”(π‘₯(𝑑))+𝐹(π‘₯(𝑑)),fora.a.π‘‘βˆˆ0,β„Žπ‘₯(0)=𝑏,(4.1) where 𝑔 and 𝑏 are the same as in (2.3), πΉβˆΆπ’³β†’2𝒳 is a set-valued mapping, and π‘βˆΆπ’³β†’π’³ is a single-valued mapping. Let 𝑆(𝑏) denote the set of all solutions of (4.1) on [0,β„Ž] with π‘₯(0)=𝑏.

Now we prove the following result.

Theorem 4.1. Let 𝒳 be 2-uniformly convex and 2-uniformly smooth Banach space. Let π‘”βˆΆπ’³β†’π’³ be 𝛼-Lipschitz continuous, let π‘βˆΆπ’³β†’π’³ be πœ‡-Lipschitz continuous, and let πΉβˆΆπ’³β†’πΆ(𝒳) be a πœ”-Lipschitz continuous set-valued mapping with compact convex values. If 𝑐64π‘˜ξ‚€π‘(𝛼+πœŒπœ‡)+𝛼+πœ”<1,β„Ž64π‘˜ξ‚(𝛼+πœŒπœ‡)+𝛼+πœ”<1,(4.2) then 𝑆(𝑏) is nonempty and continuous.

Proof. Let 𝑀(π‘₯)=Π𝑓𝐾(𝑔(π‘₯)βˆ’πœŒπ‘(π‘₯))βˆ’π‘”(π‘₯)+𝐹(π‘₯).(4.3) Then π‘€βˆΆπ’³β†’πΆ(𝒳) is a set-valued mapping with compact convex values since πΉβˆΆπ’³β†’πΆ(𝒳) is a set-valued mapping with compact convex values. For any π‘₯1,π‘₯2βˆˆπ’³ and π‘Ž1βˆˆπ‘€(π‘₯1), there exists π‘’βˆˆπΉ(π‘₯1) such that π‘Ž1=Π𝑓𝐾𝑔π‘₯1ξ€Έξ€·π‘₯βˆ’πœŒπ‘1ξ€·π‘₯ξ€Έξ€Έβˆ’π‘”1ξ€Έ+𝑒.(4.4) Since π‘’βˆˆπΉ(π‘₯1), and πΉβˆΆπ’³β†’πΆ(𝒳), it follows from Nadler [16] that there exists π‘£βˆˆπΉ(π‘₯2) such that 𝐹π‘₯β€–π‘’βˆ’π‘£β€–β‰€β„‹1ξ€Έξ€·π‘₯,𝐹2ξ€Έξ€Έ.(4.5) Let π‘Ž2=Π𝑓𝐾𝑔π‘₯2ξ€Έξ€·π‘₯βˆ’πœŒπ‘2ξ€·π‘₯ξ€Έξ€Έβˆ’π‘”2ξ€Έ+𝑣.(4.6) Then π‘Ž2βˆˆπ‘€(π‘₯2). From (4.4) and (4.6), we have β€–β€–π‘Ž1βˆ’π‘Ž2β€–β€–=‖‖Π𝑓𝐾𝑔π‘₯1ξ€Έξ€·π‘₯βˆ’πœŒπ‘1ξ€Έξ€Έβˆ’Ξ π‘“πΎξ€·π‘”ξ€·π‘₯2ξ€Έξ€·π‘₯βˆ’πœŒπ‘2βˆ’ξ€·π‘”ξ€·π‘₯ξ€Έξ€Έ2ξ€Έξ€·π‘₯βˆ’π‘”2‖‖≀‖‖𝑔π‘₯ξ€Έξ€Έ+π‘’βˆ’π‘£1ξ€Έξ€·π‘₯βˆ’π‘”2ξ€Έβ€–β€–+‖‖Π𝑓𝐾π‘₯𝑔(π‘₯)βˆ’πœŒπ‘1ξ€Έξ€Έβˆ’Ξ π‘“πΎξ€·ξ€·π‘₯𝑔(𝑦)βˆ’πœŒπ‘2β€–β€–ξ€Έξ€Έ+β€–π‘’βˆ’π‘£β€–.(4.7) Since 𝑔 is 𝛼-Lipschitz continuous, ‖‖𝑔π‘₯1ξ€Έξ€·π‘₯βˆ’π‘”2ξ€Έβ€–β€–β€–β€–π‘₯≀𝛼2βˆ’π‘₯2β€–β€–.(4.8) From Lemma 2.8, Π𝑓𝐾 is Lipschitz continuous. It follows from the continuity of 𝑁 and 𝑔 that ‖‖Π𝑓𝐾𝑔π‘₯1ξ€Έξ€·π‘₯βˆ’πœŒπ‘1ξ€Έξ€Έβˆ’Ξ π‘“πΎξ€·π‘”ξ€·π‘₯2ξ€Έξ€·π‘₯βˆ’πœŒπ‘2‖‖𝑐≀64π‘˜ξ€·β€–β€–π‘”ξ€·π‘₯1ξ€Έξ€·π‘₯βˆ’π‘”2‖‖‖‖𝑁π‘₯+𝜌1ξ€Έξ€·π‘₯βˆ’π‘2‖‖𝑐≀64π‘˜β€–β€–π‘₯(𝛼+πœŒπœ‡)1βˆ’π‘₯2β€–β€–.(4.9) From the selection of 𝑣 and the Lipschitz continuity of 𝐹, we know 𝐹π‘₯β€–π‘’βˆ’π‘£β€–β‰€β„‹1ξ€Έξ€·π‘₯,𝐹2β€–β€–π‘₯ξ€Έξ€Έβ‰€πœ”1βˆ’π‘₯2β€–β€–.(4.10) In light of (4.7)–(4.10), we have β€–β€–π‘Ž1βˆ’π‘Ž2‖‖≀𝑐𝛼+64π‘˜ξ‚β€–β€–π‘₯(𝛼+πœŒπœ‡)+πœ”1βˆ’π‘₯2β€–β€–=πœƒβ€–π‘₯βˆ’π‘¦β€–,(4.11) where π‘πœƒ=𝛼+64π‘˜(𝛼+πœŒπœ‡)+πœ”.(4.12) Now (4.11) implies that π‘‘ξ€·π‘Ž1ξ€·π‘₯,𝑀2ξ€Έξ€Έ=infπ‘Ž2βˆˆπ‘€(π‘₯2)β€–β€–π‘Ž1βˆ’π‘Ž2β€–β€–β€–β€–π‘₯β‰€πœƒ1βˆ’π‘₯2β€–β€–.(4.13) Since π‘Ž1βˆˆπ‘€(π‘₯1) is arbitrary, we obtain supπ‘Ž1ξ€·π‘₯βˆˆπ‘€1ξ€Έπ‘‘ξ€·π‘Ž1ξ€·π‘₯,𝑀2β€–β€–π‘₯ξ€Έξ€Έβ‰€πœƒ1βˆ’π‘₯2β€–β€–.(4.14) Similarly, we can prove supπ‘Ž2ξ€·π‘₯βˆˆπ‘€2𝑑𝑀π‘₯1ξ€Έ,π‘Ž2ξ€Έβ€–β€–π‘₯β‰€πœƒ1βˆ’π‘₯2β€–β€–.(4.15) From (4.13), to (4.15), and the definition of the Hausdorff metric β„‹ on 𝐢(𝒳), we have ℋ𝑀π‘₯1ξ€Έξ€·π‘₯,𝑀2β€–β€–π‘₯ξ€Έξ€Έβ‰€πœƒ1βˆ’π‘₯2β€–β€–,βˆ€π‘₯1,π‘₯2βˆˆπ’³.(4.16) Now (4.2) implies that 0<πœƒ<1, and so 𝑀(π‘₯) is a set-valued contractive mapping. Let ξ‚»[]ξ€œπ‘„(π‘₯,𝑏)=π‘¦βˆˆπΆ(0,β„Ž,𝒳)βˆ£π‘¦(𝑑)=𝑏+𝑑0ξ‚Ό,𝑧(𝑠)𝑑𝑠,𝑧(𝑠)βˆˆπ‘€(π‘₯(𝑠))(4.17) where [][]𝐢(0,β„Ž,𝒳)={π‘“βˆΆ0,β„ŽβŸΆπ’³βˆ£π‘“iscontinuous}.(4.18) Since π‘€βˆΆπ’³β†’πΆ(𝒳) 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 𝑄(π‘₯,𝑏) is nonempty for each π‘₯ and π‘βˆˆπ’³. Moreover, it is easy to see that the set of fixed-points of 𝑄(π‘₯,𝑏) coincides with 𝑆(𝑏). It follows from [21] or [8] that 𝑄(π‘₯,𝑏) is compact and convex for each π‘₯ and π‘βˆˆπ’³. Suppose that π‘π‘š is the initial value of (4.1); that is, π‘₯(0)=π‘π‘š(π‘š=0,1,2,…) and π‘π‘šβ†’π‘0(π‘šβ†’βˆž). Since 𝑄π‘₯,𝑏0ξ€Έξ€·=𝑄π‘₯,π‘π‘šξ€Έβˆ’π‘π‘š+𝑏0,(4.19) it is obvious that 𝑄(π‘₯,π‘π‘š) converges uniformly to 𝑄(π‘₯,𝑏0).
Next we prove that 𝑄(π‘₯,π‘π‘š) is a set-valued contractive mapping. For any given π‘₯1,π‘₯2∈𝐢([0,β„Ž],𝒳), since π‘€βˆΆπ’³β†’πΆ(𝒳) 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 𝑀(π‘₯1(𝑠)) has a continuous selection π‘Ÿ1(𝑠)βˆˆπ‘€(π‘₯1(𝑠)). Let 𝑐1(𝑑)=π‘π‘š+ξ€œπ‘‘0π‘Ÿ1(𝑠)𝑑𝑠.(4.20) Then 𝑐1βˆˆπ‘„(π‘₯1,π‘π‘š). Since π‘Ÿ1(𝑠)βˆˆπ‘€(π‘₯1(𝑠)) is measurable and 𝑀(π‘₯2(𝑠)) is a measurable mapping with compact values, we know that there exists a measurable selection π‘Ÿ2(𝑠)βˆˆπ‘€(π‘₯2(𝑠)) such that β€–β€–π‘Ÿ1(𝑠)βˆ’π‘Ÿ2‖‖𝑀π‘₯(𝑠)≀ℋ1ξ€Έξ€·π‘₯(𝑠),𝑀2(𝑠)ξ€Έξ€Έ.(4.21) Thus, it follows from (4.16) that β€–β€–π‘Ÿ1(𝑠)βˆ’π‘Ÿ2‖‖𝑀π‘₯(𝑠)≀ℋ1ξ€Έξ€·π‘₯(𝑠),𝑀2β€–β€–π‘₯(𝑠)ξ€Έξ€Έβ‰€πœƒ1βˆ’π‘₯2β€–β€–.(4.22) Let 𝑐2(𝑑)=π‘π‘š+ξ€œπ‘‘0π‘Ÿ2(𝑠)𝑑𝑠.(4.23) Then 𝑐2βˆˆπ‘„(π‘₯2,π‘π‘š) and ‖‖𝑐1βˆ’π‘2β€–β€–β‰€ξ€œβ„Ž0β€–β€–π‘Ÿ1(𝑠)βˆ’π‘Ÿ2‖‖𝑀π‘₯(𝑠)π‘‘π‘ β‰€β„Žβ„‹1ξ€Έξ€·π‘₯(𝑠),𝑀2β€–β€–π‘₯(𝑠)ξ€Έξ€Έβ‰€β„Žπœƒ1βˆ’π‘₯2β€–β€–.(4.24) Hence, we have 𝑑𝑐1ξ€·π‘₯,𝑄2,π‘π‘šξ€Έξ€Έ=inf𝑐2βˆˆπ‘„(π‘₯2,π‘π‘š)‖‖𝑐1βˆ’π‘2β€–β€–β€–β€–π‘₯β‰€β„Žπœƒ1βˆ’π‘₯2β€–β€–.(4.25) Since 𝑐1βˆˆπ‘„(π‘₯1,π‘π‘š) is arbitrary, we obtain sup𝑐1ξ€·π‘₯βˆˆπ‘„1,π‘π‘šξ€Έπ‘‘ξ€·π‘1ξ€·π‘₯,𝑄2,π‘π‘šβ€–β€–π‘₯ξ€Έξ€Έβ‰€β„Žπœƒ1βˆ’π‘₯2β€–β€–.(4.26) Similarly, we can prove that sup𝑐2ξ€·π‘₯βˆˆπ‘„2,π‘π‘šξ€Έπ‘‘ξ€·π‘„ξ€·π‘₯1,π‘π‘šξ€Έ,𝑐2ξ€Έβ€–β€–π‘₯β‰€β„Žπœƒ1βˆ’π‘₯2β€–β€–.(4.27) From the definition of the Hausdorff metric β„‹ on 𝐢(𝒳), (4.26) and (4.27) imply that ℋ𝑄π‘₯1,π‘π‘šξ€Έξ€·π‘₯,𝑄2,π‘π‘šβ€–β€–π‘₯ξ€Έξ€Έβ‰€β„Žπœƒ1βˆ’π‘₯2β€–β€–,βˆ€π‘₯1,π‘₯2βˆˆπ’³,π‘š=0,1,2,….(4.28) Since β„Žπœƒ<1, it is easy to see that 𝑄(π‘₯,𝑏) has a fixed-point for each given π‘βˆˆπ’³, and so 𝑆(𝑏) is nonempty for each given π‘βˆˆπ’³. Setting π‘Šπ‘šξ€·(π‘₯)=𝑄π‘₯,π‘π‘šξ€Έ,π‘š=0,1,2,…,(4.29) we know that π‘Šπ‘š(π‘₯) are contractive mappings with the same contractive constant β„Žπœƒ. By Lemma 2.6 and (4.28), we have β„‹ξ€·πΉξ€·π‘Šπ‘šξ€Έξ€·π‘Š,𝐹0≀1ξ€Έξ€Έ1βˆ’β„Žπœƒsupπ‘₯βˆˆπ‘‹β„‹ξ€·π‘Šπ‘š(π‘₯),π‘Š0ξ€Έ(π‘₯)⟢0.(4.30) Thus, 𝐹(π‘Šπ‘š)→𝐹(π‘Š0), which implies that 𝑆(π‘π‘š)→𝑆(𝑏); 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 ℝ𝑛 to Banach space 𝒳.

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).

References

  1. J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1984. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. P. Dupuis and A. Nagurney, β€œDynamical systems and variational inequalities,” Annals of Operations Research, vol. 44, no. 1–4, pp. 9–42, 1993, Advances in equilibrium modeling, analysis and computation. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. T. L. Friesz, D. Bernstein, N. J. Mehta, R. L. Tobin, and S. Ganjalizadeh, β€œDay-to-day dynamic network disequilibria and idealized traveler information systems,” Operations Research, vol. 42, no. 6, pp. 1120–1136, 1994. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1992.
  5. G. Isac, β€œSome solvability theorems for nonlinear equations with applications to projected dynamical systems,” Applicable Analysis and Discrete Mathematics, vol. 3, no. 1, pp. 3–13, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. Y. S. Xia, β€œFurther results on global convergence and stability of globally projected dynamical systems,” Journal of Optimization Theory and Applications, vol. 122, no. 3, pp. 627–649, 2004. View at Publisher Β· View at Google Scholar
  7. Y. S. Xia and J. Wang, β€œOn the stability of globally projected dynamical systems,” Journal of Optimization Theory and Applications, vol. 106, no. 1, pp. 129–150, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. D. Zhang and A. Nagurney, β€œOn the stability of projected dynamical systems,” Journal of Optimization Theory and Applications, vol. 85, no. 1, pp. 97–124, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. Y. Z. Zou, K. Ding, and N. J. Huang, β€œNew global set-valued projected dynamical systems,” Impulsive Dynamical Systems and Applications, vol. 4, pp. 233–237, 2006. View at Google Scholar
  10. Y.-z. Zou, N.-j. Huang, and B.-S. Lee, β€œA new class of generalized global set-valued dynamical systems involving (H,η)-monotone operators in Hilbert spaces,” Nonlinear Analysis Forum, vol. 12, no. 2, pp. 183–191, 2007. View at Google Scholar
  11. K.-q. Wu and N.-j. Huang, β€œThe generalised f-projection operator with an application,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006. View at Publisher Β· View at Google Scholar
  12. K.-q. Wu and N.-j. Huang, β€œProperties of the generalized f-projection operator and its applications in Banach spaces,” Computers & Mathematics with Applications, vol. 54, no. 3, pp. 399–406, 2007. View at Publisher Β· View at Google Scholar
  13. K.-Q. Wu and N.-J. Huang, β€œThe generalized f-projection operator and set-valued variational inequalities in Banach spaces,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2481–2490, 2009. View at Publisher Β· View at Google Scholar
  14. X. Li, N. J. Huang, and Y. Z. Zou, β€œOn the stability of generalized f-projection operators with an application,” Acta Mathematica Sinica, vol. 54, pp. 1–12, 2011. View at Google Scholar
  15. M. G. Cojocaru, in Projected dynamical systems on Hilbert spaces [Ph.D. thesis], Queen’s University, Kingston, Canada, 2002. View at Zentralblatt MATH
  16. S. B. Nadler, Jr., β€œMulti-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969. View at Google Scholar Β· View at Zentralblatt MATH
  17. M.-G. Cojocaru, β€œMonotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 3, pp. 793–804, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. H. K. Xu, β€œInequalities in Banach spaces with applications,” Nonlinear Analysis, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. T.-C. Lim, β€œOn fixed point stability for set-valued contractive mappings with applications to generalized differential equations,” Journal of Mathematical Analysis and Applications, vol. 110, no. 2, pp. 436–441, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, vol. 495 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1999.
  21. Y. Xia and J. Wang, β€œA general projection neural network for solving monotone variational inequalities and related optimization problems,” IEEE Transactions on Neural Networks, vol. 15, no. 2, pp. 318–328, 2004. View at Publisher Β· View at Google Scholar Β· View at Scopus