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

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: 𝑑𝑥𝑑𝑡=𝑃^𝐾(𝑥−𝜌𝑁(𝑥))−𝑥,(1.1) where 𝑁∶ℝ^𝑛→ℝ^𝑛 is a single-valued function, 𝜌>0 is a constant, 𝑃_𝐾𝑥 denotes the projection of the point 𝑥 on 𝐾; here 𝐾⊂ℝ^𝑛 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: 𝑑𝑥(𝑡)𝑑𝑡∈𝑃^𝐾(𝑔(𝑥(𝑡))−𝜌𝑁(𝑥𝑥((𝑡0)))=−𝑔𝑏,(𝑥(𝑡))),fora.a.𝑡∈[0,𝐽],(1.2) where 𝑁∶ℝ^𝑛→2^ℝ𝑛 is a set-valued function, 𝑔∶ℝ^𝑛→ℝ^𝑛 is a single-valued function, 𝜌>0 is a constant, 𝑃_𝐾𝑥 denotes the projection of the point 𝑥 on 𝐾, 𝑏 is a given point in ℝ^𝑛.

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 𝑃_𝐾 in 𝑅^𝑛 and it owns some nice properties as 𝑃_𝐾 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 𝐾⊂𝒳 be a closed convex set, let 𝑁∶𝒳→2^𝒳 be a set-valued mapping, and let 𝑔∶𝒳→𝒳 be a single-valued mapping. The normalized duality mapping 𝐽 from 𝒳 to 𝒳^∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝒳^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²=‖𝑥^∗‖²,(2.1) for 𝑥∈𝑋. For convenience, we list some properties of 𝐽(⋅) as follows.𝑋 is a smooth Banach space, 𝐽(⋅) is single valued and hemicontinuous; that is, 𝐽 is continuous from the strong topology of 𝒳 to the 𝑤𝑒𝑎𝑘^∗ topology of 𝒳^∗.

Let 𝐶(𝒳) denote the family of all nonempty compact subsets of 𝒳 and let ℋ(⋅,⋅) denote the Hausdorff metric on 𝐶(𝒳) defined by ℋ(𝐴,𝐵)=maxsup_𝑎∈𝐴inf_𝑏∈𝐵𝑑(𝑎,𝑏),sup_𝑏∈𝐵inf_𝑎∈𝐴𝑑(𝑎,𝑏),∀𝐴,𝐵∈𝐶(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 𝑥(⋅) from [0,ℎ]→𝒳 such that 𝑑𝑥(𝑡)𝑑𝑡∈Π^𝑓𝐾(𝑔(𝑥(𝑡))−𝜌𝑁(𝑥𝑥((0𝑡))))=−𝑏𝑔,(𝑥(𝑡)),fora.a.𝑡∈[0,ℎ],(2.3) where 𝑏∈𝒳, 𝜌>0 is a constant and 𝑓∶𝐾→𝑅∪{+∞} is proper, convex, and lower semicontinuous and Π^𝑓𝐾∶𝒳→2^𝐾 is a generalized 𝑓-projection operator denoted by Π^𝑓𝐾𝑥=𝑢∈𝐾∶𝐺(𝐽(𝑥),𝑢)=inf_𝜉∈𝐾𝐺(𝐽(𝑥),𝜉),∀𝑥∈𝒳.(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 𝑥^∗ is said to be an equilibrium point of global dynamical system (2.3), if 𝑥^∗ satisfies the following inclusion: 0∈Π^𝑓𝐾(𝑔(𝑥)−𝜌𝑁(𝑥))−𝑔(𝑥).(2.5)

Definition 2.2. A mapping 𝑁∶𝒳→𝒳 is said to be
(i) 𝛼-strongly accretive if there exists some 𝛼>0 such that (𝑁(𝑥)−𝑁(𝑦),𝐽(𝑥−𝑦))≥𝛼‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐾;(2.6)
(ii) 𝜉-Lipschitz continuous if there exists a constant 𝜉≥0 such that ‖𝑁(𝑥)−𝑁(𝑦)‖≤𝜉‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐾.(2.7)

Definition 2.3. A set-valued mapping 𝑇∶𝒳→𝒳 is said to be 𝜉-Lipschitz continuous if there exists a constant 𝜉>0 such that ℋ(𝑇(𝑥),𝑇(𝑦))≤𝜉‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐾,(2.8) where ℋ(⋅,⋅) is the Hausdorff metric on 𝐶(𝒳).

Lemma 2.4 (see [14]). Let 𝒳 be a real reflexive and strictly convex Banach space with its dual 𝒳^∗ and let 𝐾 be a nonempty closed convex subset of 𝒳. If 𝑓∶𝐾→𝑅∪{+∞} is proper, convex, and lower semicontinuous, then Π^𝑓𝐾 is single valued. Moreover, if 𝒳 has Kadec-Klee property, then Π^𝑓𝐾 is continuous.

Lemma 2.5 (see [18]). Let 𝒳 be a real uniformly smooth Banach space. Then 𝒳 is 𝑞-uniformly smooth if and only if there exists a constant 𝐶_𝑞>0 such that, for all 𝑥,𝑦∈𝒳, ‖𝑥+𝑦‖^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝐽_𝑞(𝑥)+𝐶_𝑞‖𝑦‖^𝑞.(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 𝜃∈(0,1). Then ℋ𝐹𝑇₁,𝐹𝑇₂≤11−𝜃sup_𝑥∈𝒳𝐻𝑇₁(𝑥),𝑇₂(𝑥),(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 𝑥₁,𝑥₂∈𝒳, let 𝑥̂₁=Π^𝑓𝐾𝑥₁ and 𝑥̂₂=Π^𝑓𝐾𝑥₂. Then 𝐽𝑥₁−𝐽𝑥₂,𝑥̂₁−𝑥̂₂≥2𝑀²𝛿‖‖𝑥̂¹−𝑥̂₂‖‖2𝑀,(2.11) where 𝑀=‖𝑥̂₁‖²+‖𝑥̂₂‖²2.(2.12)

We say that 𝒳 is 2-uniformly convex and 2-uniformly smooth Banach space if there exist 𝑘,𝑐>0 such that 𝛿𝜌_𝑋𝑋(𝜖)≥𝑘𝜖²,(𝑡)≤𝑐𝑡²,(2.13) where 𝛿_𝑋(𝜖)=inf1−‖‖‖𝑥+𝑦𝜌2‖‖‖∶‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜖_𝑋,(𝑡)=sup12(‖𝑥+𝑦‖+‖𝑥−𝑦‖)−1∶‖𝑥‖=1,‖𝑦‖≤𝑡.(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 ‖‖Π^𝑓𝐾𝑥−Π^𝑓𝐾𝑦‖‖≤64𝑐𝑘‖𝑥−𝑦‖,∀𝑥,𝑦∈𝒳.(2.15)

Proof. According to Lemma 2.7, we have 𝐽(𝑥)−𝐽(𝑦),Π^𝑓𝐾𝑥−Π^𝑓𝐾𝑦≥2𝑀²¹𝛿‖‖Π^𝑓𝐾𝑥−Π^𝑓𝐾𝑦‖‖2𝑀₁,(2.16) where 𝑀₁=‖Π^𝑓𝐾𝑥‖²+‖Π^𝑓𝐾𝑦‖²2.(2.17) Since 𝛿_𝑋(𝜖)≥𝑘𝜖², (2.16) yields ‖‖Π^𝑓𝐾𝑥−Π^𝑓𝐾𝑦‖‖≤2𝑘‖𝐽(𝑥)−𝐽(𝑦)‖.(2.18) From the property of 𝐽(⋅), we have ‖𝐽(𝑥)−𝐽(𝑦)‖≤2𝑀²²𝜌_𝑋4‖𝑥−𝑦‖/𝑀₂≤32𝑐‖𝑥−‖𝑦𝑥‖−.𝑦‖(2.19) It follows from (2.18) and (2.19) that ‖‖Π^𝑓𝐾𝑥−Π^𝑓𝐾𝑦‖‖≤64𝑐𝑘‖𝑥−𝑦‖.(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 𝑁∶𝒳→𝐶(𝒳) be 𝜇-Lipschitz continuous and let 𝑔∶𝒳→𝒳 be 𝛼-Lipschitz continuous and 𝛽-strongly accretive. If 1+𝛼²−2𝛽𝐶₂+64𝑘𝑐(𝛼+𝜌𝜇)<1,(3.1) then the equilibrium points set of the generalized set-valued dynamical system (2.3) is nonempty and closed.

Proof. Let 𝑇(𝑥)=𝑥−𝑔(𝑥)+Π^𝑓𝐾(𝑔(𝑥)−𝜌𝑁(𝑥)),∀𝑥∈𝐾.(3.2) Since 𝑁∶𝒳→𝐶(𝒳) and Π^𝑓𝐾 are continuous, we know that 𝑇∶𝒳→𝐶(𝒳). From Definition 2.1, it is easy to see that 𝑥^∗ is an equilibrium point of the generalized set-valued dynamical system (2.3) if and only if 𝑥^∗ is a fixed-point of 𝑇 in 𝒳, that is: 𝑥^∗∈𝑇𝑥^∗=𝑥^∗−𝑔𝑥^∗+Π^𝑓𝐾𝑔𝑥^∗−𝜌𝑁𝑥^∗.(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 𝑥,𝑦∈𝒳 and 𝑎₁∈𝑇(𝑥), there exists 𝑢∈𝑁(𝑥) such that 𝑎₁=𝑥−𝑔(𝑥)+Π^𝑓𝐾(𝑔(𝑥)−𝜌𝑢).(3.4) Since 𝑢∈𝑁(𝑥), and 𝑁∶𝒳→𝐶(𝒳), it follows from Nadler [16] that there exists 𝑣∈𝑁(𝑦) such that ‖𝑢−𝑣‖≤ℋ(𝑁(𝑥),𝑁(𝑦)).(3.5) Let 𝑎₂=𝑦−𝑔(𝑦)+Π^𝑓𝐾(𝑔(𝑦)−𝜌𝑣).(3.6) Then 𝑎₂∈𝑇(𝑦). From (3.4) to (3.6), we have ‖‖𝑎₁−𝑎₂‖‖≤=‖‖‖𝑥𝑥−−𝑦𝑦−−((𝑔𝑔((𝑥𝑥))−−𝑔𝑔((𝑦𝑦))))‖++Π‖‖^𝑓𝐾Π(𝑔_𝑓𝐾(𝑥)−𝜌𝑢)−Π^𝑓𝐾(𝑔(𝑦)−𝜌(𝑣))‖‖(𝑔(𝑥)−𝜌𝑢)−Π^𝑓𝐾(𝑔(𝑦)−𝜌(𝑣))‖‖.(3.7) Since 𝑔 is 𝛼-Lipschitz continuous and 𝛽-strongly accretive, ‖𝑥−𝑦−(𝑔(𝑥)−𝑔(𝑦)‖²≤‖𝑥−𝑦‖²−2⟨𝑔(𝑥)−𝑔(𝑦),𝐽(𝑥−𝑦)⟩+𝐶₂‖𝑔(𝑥)−𝑔(𝑦)‖²≤1+𝛼²−2𝛽𝐶₂‖𝑥−𝑦‖².(3.8) From Lemma 2.8, where Π^𝑓𝐾 is Lipchitz continuous, we have ‖‖Π^𝑓𝐾(𝑔(𝑥)−𝜌𝑢)−Π^𝑓𝐾(𝑔(𝑦)−(𝜌𝑣))‖‖≤64𝑐≤64𝑘𝑐(‖𝑔(𝑥)−𝑔(𝑦)‖+𝜌‖𝑢−𝑣‖)𝑘(𝛼‖𝑥−𝑦‖+𝜌‖𝑢−𝑣‖).(3.9) From the selection of 𝑣 and the Lipschitz continuity of 𝑁, ‖𝑢−𝑣‖≤ℋ(𝑁(𝑥),𝑁(𝑦))≤𝜇‖𝑥−𝑦‖.(3.10) In light of (3.7)–(3.10), we have ‖‖𝑎₁−𝑎₂‖‖≤1+𝛼²−2𝛽𝐶₂+64𝑐𝑘(𝛼+𝜌𝜇)‖𝑥−𝑦‖=𝐿‖𝑥−𝑦‖,(3.11) where 𝐿=1+𝛼²−2𝛽𝐶₂+64𝑐𝑘(𝛼+𝜌𝜇).(3.12) Now (3.11) implies that 𝑑𝑎₁,𝑇(𝑦)=inf_{𝑎2∈𝑇(𝑦)}‖‖𝑎₁−𝑎₂‖‖≤𝐿‖𝑥−𝑦‖.(3.13) Since 𝑎₁∈𝑇(𝑥) is arbitrary, we have sup_{𝑎1∈𝑇(𝑥)}𝑑𝑎₁,𝑇(𝑦)≤𝐿‖𝑥−𝑦‖.(3.14) Similarly, we can prove sup_{𝑎2∈𝑇(𝑦)}𝑑𝑇(𝑥),𝑎₂≤𝐿‖𝑥−𝑦‖.(3.15) From (3.14), (3.15), and the definition of the Hausdorff metric ℋ on 𝐶(𝒳), we have ℋ(𝑇(𝑥),𝑇(𝑦))≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐾.(3.16) Now the assumption of the theorem implies that 𝐿<1 and so 𝑇(𝑥) is a set-valued contractive mapping. By the fixed-point theorem of Nadler [16], there exists 𝑥^∗ such that 𝑥^∗∈𝑇(𝑥^∗), and thus 𝑥^∗ is the equilibrium point of (2.3). This means that 𝐹(𝑇) is nonempty.
Now we prove that 𝐹(𝑇) is closed. Let {𝑥_𝑛}⊂𝐹(𝑇) with 𝑥_𝑛→𝑥₀(𝑛→∞). Then 𝑥_𝑛∈𝑇(𝑥_𝑛) and (3.16) imply that ℋ𝑇𝑥_𝑛,𝑇𝑥₀≤𝐿‖‖𝑥_𝑛−𝑥₀‖‖.(3.17) Thus, 𝑑𝑥₀,𝑇𝑥₀≤≤(‖‖1𝑥+₀−𝐿𝑥)_𝑛‖‖‖‖𝑥_𝑛+𝑑𝑥_𝑛,𝑇𝑥_𝑛+ℋ𝑇𝑥_𝑛,𝑇𝑥₀−𝑥₀‖‖⟶0as𝑛→∞.(3.18) It follows that 𝑥₀∈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 ℝ^𝑛 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: 𝑑𝑥(𝑡)𝑑𝑡∈Π^𝑓𝐾(𝑔(𝑥(𝑡))−𝜌𝑁(𝑥(𝑡))𝑥)−(0𝑔)=(𝑥𝑏(𝑡,))+𝐹(𝑥(𝑡)),fora.a.[𝑡∈0,ℎ],(4.1) where 𝑔 and 𝑏 are the same as in (2.3), 𝐹∶𝒳→2^𝒳 is a set-valued mapping, and 𝑁∶𝒳→𝒳 is a single-valued mapping. Let 𝑆(𝑏) denote the set of all solutions of (4.1) on [0,ℎ] with 𝑥(0)=𝑏.

Now we prove the following result.

Theorem 4.1. Let 𝒳 be 2-uniformly convex and 2-uniformly smooth Banach space. Let 𝑔∶𝒳→𝒳 be 𝛼-Lipschitz continuous, let 𝑁∶𝒳→𝒳 be 𝜇-Lipschitz continuous, and let 𝐹∶𝒳→𝐶(𝒳) be a 𝜔-Lipschitz continuous set-valued mapping with compact convex values. If 64𝑐𝑘(𝛼+𝜌𝜇)+𝛼+𝜔<1,ℎ64𝑘𝑐(𝛼+𝜌𝜇)+𝛼+𝜔<1,(4.2) then 𝑆(𝑏) is nonempty and continuous.

Proof. Let 𝑀(𝑥)=Π^𝑓𝐾(𝑔(𝑥)−𝜌𝑁(𝑥))−𝑔(𝑥)+𝐹(𝑥).(4.3) Then 𝑀∶𝒳→𝐶(𝒳) is a set-valued mapping with compact convex values since 𝐹∶𝒳→𝐶(𝒳) is a set-valued mapping with compact convex values. For any 𝑥₁,𝑥₂∈𝒳 and 𝑎₁∈𝑀(𝑥₁), there exists 𝑢∈𝐹(𝑥₁) such that 𝑎₁=Π^𝑓𝐾𝑔𝑥₁−𝜌𝑁𝑥₁−𝑔𝑥₁+𝑢.(4.4) Since 𝑢∈𝐹(𝑥₁), and 𝐹∶𝒳→𝐶(𝒳), it follows from Nadler [16] that there exists 𝑣∈𝐹(𝑥₂) such that ‖𝑢−𝑣‖≤ℋ𝐹𝑥₁,𝐹𝑥₂.(4.5) Let 𝑎₂=Π^𝑓𝐾𝑔𝑥₂−𝜌𝑁𝑥₂−𝑔𝑥₂+𝑣.(4.6) Then 𝑎₂∈𝑀(𝑥₂). From (4.4) and (4.6), we have ‖‖𝑎₁−𝑎₂‖‖≤=‖‖‖‖𝑔Π^𝑓𝐾𝑥𝑔𝑥₁−𝜌𝑁𝑥₁−Π^𝑓𝐾𝑔𝑥₂−𝜌𝑁𝑥₂−𝑔𝑥₂−𝑔𝑥₂+𝑢−𝑣₁‖‖−𝑔𝑥₂‖‖+‖‖Π^𝑓𝐾𝑔(𝑥)−𝜌𝑁𝑥₁−Π^𝑓𝐾𝑔(𝑦)−𝜌𝑁𝑥₂‖‖+‖𝑢−𝑣‖.(4.7) Since 𝑔 is 𝛼-Lipschitz continuous, ‖‖𝑔𝑥₁−𝑔𝑥₂‖‖≤𝛼‖‖𝑥₂−𝑥₂‖‖.(4.8) From Lemma 2.8, Π^𝑓𝐾 is Lipschitz continuous. It follows from the continuity of 𝑁 and 𝑔 that ‖‖Π^𝑓𝐾𝑔𝑥₁−𝜌𝑁𝑥₁−Π^𝑓𝐾𝑔𝑥₂−𝜌𝑁𝑥₂‖‖≤64𝑐≤64𝑘𝑐‖‖𝑔𝑥¹−𝑔𝑥₂‖‖+𝜌‖‖𝑁𝑥₁−𝑁𝑥₂‖‖𝑘(𝛼+𝜌𝜇)‖‖𝑥¹−𝑥₂‖‖.(4.9) From the selection of 𝑣 and the Lipschitz continuity of 𝐹, we know ‖𝑢−𝑣‖≤ℋ𝐹𝑥₁,𝐹𝑥₂≤𝜔‖‖𝑥₁−𝑥₂‖‖.(4.10) In light of (4.7)–(4.10), we have ‖‖𝑎₁−𝑎₂‖‖≤𝛼+64𝑐𝑘(𝛼+𝜌𝜇)+𝜔‖‖𝑥¹−𝑥₂‖‖=𝜃‖𝑥−𝑦‖,(4.11) where 𝜃=𝛼+64𝑐𝑘(𝛼+𝜌𝜇)+𝜔.(4.12) Now (4.11) implies that 𝑑𝑎₁,𝑀𝑥₂=inf_{𝑎2∈𝑀(𝑥2)}‖‖𝑎₁−𝑎₂‖‖≤𝜃‖‖𝑥₁−𝑥₂‖‖.(4.13) Since 𝑎₁∈𝑀(𝑥₁) is arbitrary, we obtain sup_{𝑎1𝑥∈𝑀1}𝑑𝑎₁,𝑀𝑥₂≤𝜃‖‖𝑥₁−𝑥₂‖‖.(4.14) Similarly, we can prove sup_{𝑎2𝑥∈𝑀2}𝑑𝑀𝑥₁,𝑎₂≤𝜃‖‖𝑥₁−𝑥₂‖‖.(4.15) From (4.13), to (4.15), and the definition of the Hausdorff metric ℋ on 𝐶(𝒳), we have ℋ𝑀𝑥₁,𝑀𝑥₂≤𝜃‖‖𝑥₁−𝑥₂‖‖,∀𝑥₁,𝑥₂∈𝒳.(4.16) Now (4.2) implies that 0<𝜃<1, and so 𝑀(𝑥) is a set-valued contractive mapping. Let 𝑄(𝑥,𝑏)=𝑦∈𝐶([0,ℎ],𝒳)∣𝑦(𝑡)=𝑏+^𝑡0𝑧(𝑠)𝑑𝑠,𝑧(𝑠)∈𝑀(𝑥(𝑠)),(4.17) where 𝐶([0,ℎ],𝒳)={𝑓∶[0,ℎ]⟶𝒳∣𝑓iscontinuous}.(4.18) Since 𝑀∶𝒳→𝐶(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(𝑥,𝑏) is nonempty for each 𝑥 and 𝑏∈𝒳. Moreover, it is easy to see that the set of fixed-points of 𝑄(𝑥,𝑏) coincides with 𝑆(𝑏). It follows from [21] or [8] that 𝑄(𝑥,𝑏) is compact and convex for each 𝑥 and 𝑏∈𝒳. Suppose that 𝑏_𝑚 is the initial value of (4.1); that is, 𝑥(0)=𝑏_𝑚(𝑚=0,1,2,…) and 𝑏_𝑚→𝑏₀(𝑚→∞). Since 𝑄𝑥,𝑏₀=𝑄𝑥,𝑏_𝑚−𝑏_𝑚+𝑏₀,(4.19) it is obvious that 𝑄(𝑥,𝑏_𝑚) converges uniformly to 𝑄(𝑥,𝑏₀).
Next we prove that 𝑄(𝑥,𝑏_𝑚) is a set-valued contractive mapping. For any given 𝑥₁,𝑥₂∈𝐶([0,ℎ],𝒳), since 𝑀∶𝒳→𝐶(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(𝑥₁(𝑠)) has a continuous selection 𝑟₁(𝑠)∈𝑀(𝑥₁(𝑠)). Let 𝑐₁(𝑡)=𝑏_𝑚+^𝑡0𝑟₁(𝑠)𝑑𝑠.(4.20) Then 𝑐₁∈𝑄(𝑥₁,𝑏_𝑚). Since 𝑟₁(𝑠)∈𝑀(𝑥₁(𝑠)) is measurable and 𝑀(𝑥₂(𝑠)) is a measurable mapping with compact values, we know that there exists a measurable selection 𝑟₂(𝑠)∈𝑀(𝑥₂(𝑠)) such that ‖‖𝑟₁(𝑠)−𝑟₂(𝑠)‖‖≤ℋ𝑀𝑥₁(𝑠),𝑀𝑥₂(𝑠).(4.21) Thus, it follows from (4.16) that ‖‖𝑟₁(𝑠)−𝑟₂(𝑠)‖‖≤ℋ𝑀𝑥₁(𝑠),𝑀𝑥₂(𝑠)≤𝜃‖‖𝑥₁−𝑥₂‖‖.(4.22) Let 𝑐₂(𝑡)=𝑏_𝑚+^𝑡0𝑟₂(𝑠)𝑑𝑠.(4.23) Then 𝑐₂∈𝑄(𝑥₂,𝑏_𝑚) and ‖‖𝑐₁−𝑐₂‖‖≤≤ℎ𝜃^ℎ0‖‖‖‖𝑥𝑟₁₁(𝑠)−𝑟₂(𝑠)‖‖𝑑𝑠≤ℎℋ𝑀𝑥₁(𝑠),𝑀𝑥₂(𝑠)−𝑥₂‖‖.(4.24) Hence, we have 𝑑𝑐₁,𝑄𝑥₂,𝑏_𝑚=inf_{𝑐2∈𝑄(𝑥2,𝑏𝑚)}‖‖𝑐₁−𝑐₂‖‖≤ℎ𝜃‖‖𝑥₁−𝑥₂‖‖.(4.25) Since 𝑐₁∈𝑄(𝑥₁,𝑏_𝑚) is arbitrary, we obtain sup_{𝑐1𝑥∈𝑄1,𝑏𝑚}𝑑𝑐₁,𝑄𝑥₂,𝑏_𝑚≤ℎ𝜃‖‖𝑥₁−𝑥₂‖‖.(4.26) Similarly, we can prove that sup_{𝑐2𝑥∈𝑄2,𝑏𝑚}𝑑𝑄𝑥₁,𝑏_𝑚,𝑐₂≤ℎ𝜃‖‖𝑥₁−𝑥₂‖‖.(4.27) From the definition of the Hausdorff metric ℋ on 𝐶(𝒳), (4.26) and (4.27) imply that ℋ𝑄𝑥₁,𝑏_𝑚,𝑄𝑥₂,𝑏_𝑚≤ℎ𝜃‖‖𝑥₁−𝑥₂‖‖,∀𝑥₁,𝑥₂∈𝒳,𝑚=0,1,2,….(4.28) Since ℎ𝜃<1, it is easy to see that 𝑄(𝑥,𝑏) has a fixed-point for each given 𝑏∈𝒳, and so 𝑆(𝑏) is nonempty for each given 𝑏∈𝒳. Setting 𝑊_𝑚(𝑥)=𝑄𝑥,𝑏_𝑚,𝑚=0,1,2,…,(4.29) we know that 𝑊_𝑚(𝑥) are contractive mappings with the same contractive constant ℎ𝜃. By Lemma 2.6 and (4.28), we have ℋ𝐹𝑊_𝑚,𝐹𝑊₀≤11−ℎ𝜃sup_𝑥∈𝑋ℋ𝑊_𝑚(𝑥),𝑊₀(𝑥)⟶0.(4.30) Thus, 𝐹(𝑊_𝑚)→𝐹(𝑊₀), which implies that 𝑆(𝑏_𝑚)→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 ℝ^𝑛 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).