# Asymptotic Stability of Neutral Set-Valued Functional Differential Equation by Fixed Point Method

Authors: Junyan Bao; Peiguang Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This paper studies a class of nonlinear neutral set-valued functional differential equations. The globally asymptotic stability theorem with necessary and sufficient conditions is obtained via the fixed point method. Meanwhile, we give an example to illustrate the obtained result.


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## Body

## 1. Introduction

It is well known that Lyapunov’s direct method is an important technique to consider the stability of various differential equations. However, this method is not always valid for stability analysis in the functional differential equation when the delay is unbounded or when the equation has unbounded terms [1–3]. Burton and many researchers found a way to these difficulties by using various fixed point theorems; we can refer to the literature studies [4–14].Recently, the study of qualitative analysis of the set-valued differential equation has attracted much attention. The stability results of various set-valued differential equations were obtained by applying Lyapunov’s direct method. The results can be found in the monograph [15], the papers for the set-valued differential equation [16–24], set-valued functional differential equation [25–29], and other equations [6, 9, 30–32]. However, according to what we know so far, there are few stability results for the set-valued differential equation via the fixed point method. Inspired by the application of the fixed point method mentioned above, in this paper, we study a class of nonlinear neutral set-valued functional differential equations:(1)DHXt=−atXt+btDHXt−τ+Ft,Xt,Xt−τ,t∈I,Xt=Φt,t∈I0,where X∈C1I,Kc∪CI0,Kc, F∈CI×Kc×Kc,Kc, and Ft,θ,θ=θ; Φ∈CI0,Kc, a,b∈CI,ℝ, I=t0,+∞, and I0=t0−τ,t0. Kc denote the collection of all nonempty, compact convex subsets of Banach space ; θ denotes the null set-valued function θ:I⟶Kc, and θt=0 for t∈I, τ>0is a constant.The aim of this paper is to obtain an asymptotic stability theorem with a necessary and sufficient condition via the fixed point method. In addition, an application of the main result is presented.


## 2. Preliminaries

To get the desired result, we first give some notations, definitions, and propositions briefly; for the details, see the literature [15].LetKc denote the collection of all nonempty, compact convex subsets of Banach space , given A,B∈Kc, defining the Hausdorff metric between A and B as follows:(2)DA,B=maxsupb∈Bdb,A,supa∈Ada,B,where db,A=infdb,a:a∈A and da,B=infda,b:b∈B. The Hausdorff metric satisfies the properties as follows:(3)DA,B≥0withDA,B=0if and only ifA=B,DA,B=DB,A,DλA,λB=λDA,B,DA,B≤DA,C+DC,B,for all A,B,C∈Kc and λ∈ℝ.In the sense of the above metricD, the set Kc is a complete metric space.Definition 1 (see [15]).
The set-valued functionF:a,b⟶Kc is Hukuhara differentiable at t0∈a,b if the limits(4)limh⟶0+Ft0+h−Ft0h,limh⟶0+Ft0−Ft0−hh,exist in Kc and equal to DHFt0. DHFt0∈Kc is called the Hukuhara derivative of Ft at t0∈a,b.
By embeddingKc as a complete cone in a corresponding Banach space and taking into account the differentiation of the Bochner integral, we can find that if(5)Ft=X0+∫atΨsds,X0∈Kc,then DHFt exists and DHFt=Ψt a.e. on a,b holds, where Ψ:a,b⟶Kc is integrable in the sense of Bochner.Proposition 1 (see [15]).
LetX:a,b⟶Kc be Hukuhara differentiable and have continuous Hukuhara derivative DHX on a,b; then, we have(6)Xt=Xt0+∫t0tDHXtdt,a≤t0≤t≤b.Proposition 2 (see [15]).
IfX,Y:a,b⟶Kc is integrable, then(7)D∫t0tXsds,∫t0tYsds≤∫t0tDXs,Ysds.For eacht0,Φ∈I×CI0,Kc, Xt is said to be a solution of (1) through t0,Φ if X∈C1I,Kc∪CI0,Kc satisfies (1) on I and Xt=Φt on I0, and we denote the solution Xt by Xt,t0,Φ.LetX=C1I0∪I,Kc and ℐ=C1I0,Kc with norm(8)X=supt∈IDXt,θ,Φ=maxt∈I0DΦt,θ,respectively, and X0=maxX,DHX. By the properties of Hausdorff metric D and the definition of ⋅0, we can get the following:(i)
Xt0≥0, and Xt0=0 if and only if Xt=θ,(ii)
Xt+Yt0≤Xt0+Yt0,(iii)
λXt=λXt,where Xt,Yt∈X and λ∈ℝ.Definition 2.
The trivial solution of (1) is said to be(i)
Stable inC1 if for any t0∈ℝ+ and ε>0, there exists δ=δε,t0 such that Φ∈ℐ, and Φ0<δ implies that X0<ε and t≥t0.(ii)
Globally asymptotically stable inC1 if it is stable, and for any t0∈I, Φ∈ℐ implies that(9)limt⟶+∞Xt,t0,Φ=limt⟶+∞DHXt,t0,Φ=0.
ForA, B∈Kc and λ∈ℝ, we can define the following addition and scalar multiplication as follows: A+B=a+b:a∈A,b∈B, λA=λa:a∈A. Then, with the algebraic operations of addition and nonnegative scalar multiplication, Kc,D becomes a semilinear metric space.
In order to apply the fixed point method, we first need to prove the following lemma.Lemma 1.
LetX0=Xt∈X:limt⟶+∞Xt=limt⟶+∞DHXt=0. Then, X0,⋅0 is a semilinear complete metric space.Proof.
Firstly, from the previous discussion, we know thatX0,⋅0 is a semilinear metric space. Next, we prove the space is complete. Assume that Xmt is a Cauchy sequence; then, for any ε>0, there exists positive integer N such that, for all m,n>N, we have Xnt−Xmt0<ε, t≥t0−τ. Hence,(10)DXmt,Xnt≤Xmt−Xnt0<ε,t≥t0−τ,DDHXmt,DHXnt≤Xmt−Xnt0<ε,t≥t0−τ.
Then,Xnt and DHXnt are Cauchy sequences in Kc, respectively. Since Kc,D is a complete metric space, there exist set-valued functions Xt and Zt such that(11)limm⟶∞Xmt=Xt,limm⟶∞DHXmt=Zt.
First, we prove thatXt is a continuous function on I0∪I. From (11), there exists a positive integer M such that DXmt,Xt<ε/3 for all m>M and t≥t0−τ. In addition, since Xmt is a continuous set-valued function, there exists δ>0 such that t−s|<δ implies that DXmt,Xms<ε/3 for s,t≥t0−τ. Then, for any t≥t0−τ with t−s<δ and m>M, we have(12)DXt,Xs≤DXt,Xmt+DXmt,Xms+DXms,Xs<ε.
Using the same way, we can also prove thatZt is a continuous function.
Next, we prove thatDHXt=Zt. In fact, for any t0,t∈I and t>t0, we have(13)Xmt=Xmt0+∫t0tDHXmsds.
By (11) and (13), Xt=Xt0+∫t0tZsds holds when m⟶∞. Using Proposition 1, we get DHXt=Zt. Meanwhile, we can prove that limt⟶+∞Xt=limt⟶+∞Zt=0. Therefore, Xt∈X0. This completes the proof of Lemma 1.


## 3. Main Result

In this section, we establish the necessary and sufficient conditions for the global asymptotic stability of trivial solution of equation (1) by using the fixed point method.Theorem 1.
Assume that the following conditions hold:A1  :  the function at is bounded and limt⟶+∞inf∫t0tasds>−∞.A2  :  there exist functions λ1,λ2∈CI,ℝ+ such that(14)Ft,X1,Y1−Ft,X2,Y2≤λ1tX1−X2+λ2tY1−Y2.A3: there exists a constant α∈0,1 such that(15)∫t0te−∫stavdvbs+λ1s+λ2sds≤α,at∫t0te−∫stavdvbs+λ1s+λ2sds+bt+λ1t+λ2t≤α.Then, the trivial solution of equation (1) is globally asymptotically stable in C1 if and only if(16)∫t0+∞asds=+∞.Proof.
We will prove this conclusion in two steps.


### 3.1. Proof of Sufficiency

LetD=X∈X0:Xt=Φt,t∈I0. Then, D is a nonempty, closed subset of X0. At the same time, we can get equation (1) with initial condition Xt0=Φt which is equivalent to the following integral equation:(17)Xt=Φt0e−∫t0tavdv+∫t0te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds,t∈I,Xt=Φt,t∈I0.We define the mappingQ:D⟶C1I0∪I,Kc as follows:(18)QXt=Φt0e−∫t0tavdv+∫t0te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds,t∈I,Φt,t∈I0.Firstly, we prove thatQ is a mapping of D⟶D. In fact, we just need to prove that QXt⟶0 and DHQXt⟶0 as t⟶+∞. Since limt⟶+∞Xt=limt⟶+∞DHXt=0, for any ε>0, there exists T>0 such that t>T implies(19)maxXt,Xt−τ,DHXt−τ<ε.Thus, from (18), (19), A2, and (A3), we have(20)QXt≤Φt0e−∫t0tavdv+∫t0Te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds+∫Tte−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τ−Fs,θ,θds≤e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds+∫Tte−∫stavdvbsDHXs−τ+λ1sXs+λ2sXs−τds≤e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds+ε∫Tte−∫stavdvbs+λ1s+λ2sds≤e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds+αε,t>T.Then, by condition (16), there exists T1>T such that, for t>T1 implies(21)e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds<ε.Thus, we haveQXt<1+αε for t>T1, and we can obtain QXt⟶0 as t⟶+∞. In addition, we can also get(22)DHQXt=−atΦt0e−∫t0tavdv−at∫t0te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds+btDHXt−τ+Ft,Xt,Xt−τ=−atQXt+btDHXt−τ+Ft,Xt,Xt−τ=−atQXt+DHXt+atXt.Furthermore, from (22), we can get(23)DHQXt≤atQXt+DHXt+atXt.From (A1) and limt⟶+∞QXt=limt⟶+∞Xt=0, we have limt⟶+∞DHQXt=0 for X∈D. Therefore, QXt∈D for X∈D, that is, Q:D⟶D.Secondly, we prove that the mappingQ has a unique fixed point.LetX,Y∈D, and by (18), (A2), and (A3), we have(24)QXt−QYt≤∫t0te−∫stavdvbsDHXs−DHYs+Fs,Xs,Xs−τ−Fs,Ys,Ys−τds≤Xt−Yt0∫t0te−∫stavdvbs+λ1s+λ2sds≤αXt−Yt0,α<1,t≥t0.Similarly, from (18), (24), (A2), and (A3), we can also get(25)DHQXt−DHQYt≤atQXt−QYt+btDHXt−DHYt+Ft,Xt,Xt−τ−Ft,Yt,Yt−τ≤Xt−Yt0at∫t0te−∫stavdvbs+λ1s+λ2sds+bt+λ1t+λ2t≤αXt−Yt0,t≥t0.By (24) and (25), we know that Q is a contraction mapping. Therefore, using the principle of contraction mapping, Q has a unique fixed point X∈D which is a unique solution of equation (1) and satisfies(26)limt⟶+∞Xt=limt⟶+∞DHXt=0.Finally, we prove the global asymptotic stability of equation (1). To do this, we first prove that the trivial solution of (1) is stable. Let(27)M1=supt∈Ie−∫t0tavdv,M2=supt∈Iat.For anyε>0, we choose δ>0 satisfying maxδ,M1δ+αε,M1M2δ+αε<ε. If Xt=Xt,t0,Φ is a solution of equation (1) with Φt0<δ, we can claim that Xt0<ε on t∈I. In fact, if this is not true and we notice that Xt0<ε on I0, then there exists t¯>t0 such that(28)maxXt¯,DHXt¯=ε,maxXt,DHXt<ε,fort∈t0−τ,t¯.IfXt¯=ε, then it follows from (A2) and (A3) that(29)Xt¯≤Φt0e−∫t0t¯avdv+∫t0t¯e−∫st¯avdvbsDHXs−τ+Fs,Xs,Xs−τds≤M1δ+ε∫t0t¯e−∫st¯avdvbs+λ1s+λ2sds≤M1δ+αε<ε,which contradicts the definition of t¯.IfDHXt¯=ε, then it follows from (A2) and (A3) that(30)DHXt¯≤at¯Φt0e−∫t0t¯avdv+bt¯DHXt¯−τ+Ft¯,Xt¯,Xt¯−τ+at¯∫t0t¯e−∫st¯avdvbsDHXs−τ+Fs,Xs,Xs−τds≤M1M2δ+εat∫t0te−∫stavdvbs+λ1s+λ2sds≤M1M2δ+αε<ε,which also contradicts the definition t¯. Hence, Xt0<ε for all t≥t0, and the trivial solution of (1) is stable. This, together with (26), claims that the trivial solution of (1) is globally asymptotically stable.


### 3.2. Proof of Necessity

Assume that (16) is not valid. Then, by condition (A1), there exists a sequence tn such that limn⟶+∞∫t0tnavdv=l for some l∈R, where tn⟶+∞ as n⟶∞. Furthermore, we also choose a positive constant p satisfying(31)−p≤∫t0tnavdv≤p,forn≥1.By (A3), we have(32)In=∫t0tne∫t0savdvbs+λ1s+λ2sds≤αe∫t0tnavdv,that is, the sequence of In is bounded. Thus, there exists a convergent subsequence. For convenience, we may assume that limn⟶+∞In=λ. Let(33)K=supt∈Ie−∫t0tavdv,A=supt∈Iat.Then, there exists a sufficient large positive integerk such that, for all n>k,(34)In−Ik=∫tktne∫t0savdvbs+λ1s+λ2sds<1−α4Me2p,where M=maxKep,KAep.For anyδ>0, we consider the solution Xt,tk,Φ of (1) with Φtk>1/2δ and Φt<δ, t∈tk−τ,tk. It follows from (17), (22), (31), (A2), and (A3) that(35)Xt≤Φtke−∫tktavdv+∫tkte−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds≤Φtke−∫t0tavdve∫t0tkavdv+Xt0k∫tkte−∫stavdvbs+λ1s+λ2sds≤Kepδ+αXt0k,t∈tk,+∞,DHXt≤atΦtke−∫tktavdv+btDHXt−τ+Ft,Xt,Xt−τ+at∫tkte−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds≤atΦtke−∫t0tavdve∫t0tkavdv+Xt0kbt+λ1t+λ2t+Xt0k∫tkte−∫stavdvbs+λ1s+λ2sds≤KAepδ+Xt0kat∫t0te−∫stavdvbs+λ1s+λ2sds+bt+λ1t+λ2t≤KAepδ+αXt0k,t∈tk,+∞.Hence, we haveXt0k≤Mδ+αXt0k, that is, Xt0k≤M/1−αδ, where Xt0k=maxXt,DHXt for t∈tk,+∞. Furthermore, it follows from (17), (34), and (A2) that, for n≥k,(36)Xtn≥Φtke−∫tktnavdv−e−∫t0tnavdv∫tktne∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds≥Φtke−∫tktnavdv−Xt0ke−∫t0tnavdv∫tktne∫t0savdvbs+λ1s+λ2sds>12δe−p−M1−αδep1−α4Me2p=e−p4δ,which contradicts with (26). Hence, condition (16) is necessary for globally asymptotic stability of the trivial solution of (1). This completes the proof of Theorem 1.To illustrate our result, we give an example.Example 1.
In equation (1), let at=1/1+t, bt=1/81+t, and(37)Ft,Xt,Xt−τ=181+tXt+Xt−τ.
By a straightforward computation, we can obtain(38)at≤1,∫t0+∞asds=+∞,fort∈I.
In addition,(39)Ft,X1,Y1−Ft,X2,Y2=181+tX1+Y1−X2−Y2≤181+tX1−X2+Y1−Y2.
Letλ1t=λ2t=1/81+t; then, condition (A2) of Theorem 1 holds. By direct calculations, we have(40)∫t0te−∫stavdvbs+λ1s+λ2sds≤38e−21+t∫t0te21+s11+sds≤381−e2−21+t<38,(41)at∫t0te−∫stavdvbs+λ1s+λ2sds+bt+λ1t+λ2t≤381+t1−e2−21+t+381+t<34.
Letα=3/4, and by (38)–(41), we find that conditions (A1)–(A3) and (16) hold. Thus, (1) is globally asymptotically stable.


## 3.1. Proof of Sufficiency

LetD=X∈X0:Xt=Φt,t∈I0. Then, D is a nonempty, closed subset of X0. At the same time, we can get equation (1) with initial condition Xt0=Φt which is equivalent to the following integral equation:(17)Xt=Φt0e−∫t0tavdv+∫t0te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds,t∈I,Xt=Φt,t∈I0.We define the mappingQ:D⟶C1I0∪I,Kc as follows:(18)QXt=Φt0e−∫t0tavdv+∫t0te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds,t∈I,Φt,t∈I0.Firstly, we prove thatQ is a mapping of D⟶D. In fact, we just need to prove that QXt⟶0 and DHQXt⟶0 as t⟶+∞. Since limt⟶+∞Xt=limt⟶+∞DHXt=0, for any ε>0, there exists T>0 such that t>T implies(19)maxXt,Xt−τ,DHXt−τ<ε.Thus, from (18), (19), A2, and (A3), we have(20)QXt≤Φt0e−∫t0tavdv+∫t0Te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds+∫Tte−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τ−Fs,θ,θds≤e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds+∫Tte−∫stavdvbsDHXs−τ+λ1sXs+λ2sXs−τds≤e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds+ε∫Tte−∫stavdvbs+λ1s+λ2sds≤e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds+αε,t>T.Then, by condition (16), there exists T1>T such that, for t>T1 implies(21)e−∫t0tavdvΦt0+∫t0Te∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds<ε.Thus, we haveQXt<1+αε for t>T1, and we can obtain QXt⟶0 as t⟶+∞. In addition, we can also get(22)DHQXt=−atΦt0e−∫t0tavdv−at∫t0te−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds+btDHXt−τ+Ft,Xt,Xt−τ=−atQXt+btDHXt−τ+Ft,Xt,Xt−τ=−atQXt+DHXt+atXt.Furthermore, from (22), we can get(23)DHQXt≤atQXt+DHXt+atXt.From (A1) and limt⟶+∞QXt=limt⟶+∞Xt=0, we have limt⟶+∞DHQXt=0 for X∈D. Therefore, QXt∈D for X∈D, that is, Q:D⟶D.Secondly, we prove that the mappingQ has a unique fixed point.LetX,Y∈D, and by (18), (A2), and (A3), we have(24)QXt−QYt≤∫t0te−∫stavdvbsDHXs−DHYs+Fs,Xs,Xs−τ−Fs,Ys,Ys−τds≤Xt−Yt0∫t0te−∫stavdvbs+λ1s+λ2sds≤αXt−Yt0,α<1,t≥t0.Similarly, from (18), (24), (A2), and (A3), we can also get(25)DHQXt−DHQYt≤atQXt−QYt+btDHXt−DHYt+Ft,Xt,Xt−τ−Ft,Yt,Yt−τ≤Xt−Yt0at∫t0te−∫stavdvbs+λ1s+λ2sds+bt+λ1t+λ2t≤αXt−Yt0,t≥t0.By (24) and (25), we know that Q is a contraction mapping. Therefore, using the principle of contraction mapping, Q has a unique fixed point X∈D which is a unique solution of equation (1) and satisfies(26)limt⟶+∞Xt=limt⟶+∞DHXt=0.Finally, we prove the global asymptotic stability of equation (1). To do this, we first prove that the trivial solution of (1) is stable. Let(27)M1=supt∈Ie−∫t0tavdv,M2=supt∈Iat.For anyε>0, we choose δ>0 satisfying maxδ,M1δ+αε,M1M2δ+αε<ε. If Xt=Xt,t0,Φ is a solution of equation (1) with Φt0<δ, we can claim that Xt0<ε on t∈I. In fact, if this is not true and we notice that Xt0<ε on I0, then there exists t¯>t0 such that(28)maxXt¯,DHXt¯=ε,maxXt,DHXt<ε,fort∈t0−τ,t¯.IfXt¯=ε, then it follows from (A2) and (A3) that(29)Xt¯≤Φt0e−∫t0t¯avdv+∫t0t¯e−∫st¯avdvbsDHXs−τ+Fs,Xs,Xs−τds≤M1δ+ε∫t0t¯e−∫st¯avdvbs+λ1s+λ2sds≤M1δ+αε<ε,which contradicts the definition of t¯.IfDHXt¯=ε, then it follows from (A2) and (A3) that(30)DHXt¯≤at¯Φt0e−∫t0t¯avdv+bt¯DHXt¯−τ+Ft¯,Xt¯,Xt¯−τ+at¯∫t0t¯e−∫st¯avdvbsDHXs−τ+Fs,Xs,Xs−τds≤M1M2δ+εat∫t0te−∫stavdvbs+λ1s+λ2sds≤M1M2δ+αε<ε,which also contradicts the definition t¯. Hence, Xt0<ε for all t≥t0, and the trivial solution of (1) is stable. This, together with (26), claims that the trivial solution of (1) is globally asymptotically stable.


## 3.2. Proof of Necessity

Assume that (16) is not valid. Then, by condition (A1), there exists a sequence tn such that limn⟶+∞∫t0tnavdv=l for some l∈R, where tn⟶+∞ as n⟶∞. Furthermore, we also choose a positive constant p satisfying(31)−p≤∫t0tnavdv≤p,forn≥1.By (A3), we have(32)In=∫t0tne∫t0savdvbs+λ1s+λ2sds≤αe∫t0tnavdv,that is, the sequence of In is bounded. Thus, there exists a convergent subsequence. For convenience, we may assume that limn⟶+∞In=λ. Let(33)K=supt∈Ie−∫t0tavdv,A=supt∈Iat.Then, there exists a sufficient large positive integerk such that, for all n>k,(34)In−Ik=∫tktne∫t0savdvbs+λ1s+λ2sds<1−α4Me2p,where M=maxKep,KAep.For anyδ>0, we consider the solution Xt,tk,Φ of (1) with Φtk>1/2δ and Φt<δ, t∈tk−τ,tk. It follows from (17), (22), (31), (A2), and (A3) that(35)Xt≤Φtke−∫tktavdv+∫tkte−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds≤Φtke−∫t0tavdve∫t0tkavdv+Xt0k∫tkte−∫stavdvbs+λ1s+λ2sds≤Kepδ+αXt0k,t∈tk,+∞,DHXt≤atΦtke−∫tktavdv+btDHXt−τ+Ft,Xt,Xt−τ+at∫tkte−∫stavdvbsDHXs−τ+Fs,Xs,Xs−τds≤atΦtke−∫t0tavdve∫t0tkavdv+Xt0kbt+λ1t+λ2t+Xt0k∫tkte−∫stavdvbs+λ1s+λ2sds≤KAepδ+Xt0kat∫t0te−∫stavdvbs+λ1s+λ2sds+bt+λ1t+λ2t≤KAepδ+αXt0k,t∈tk,+∞.Hence, we haveXt0k≤Mδ+αXt0k, that is, Xt0k≤M/1−αδ, where Xt0k=maxXt,DHXt for t∈tk,+∞. Furthermore, it follows from (17), (34), and (A2) that, for n≥k,(36)Xtn≥Φtke−∫tktnavdv−e−∫t0tnavdv∫tktne∫t0savdvbsDHXs−τ+Fs,Xs,Xs−τds≥Φtke−∫tktnavdv−Xt0ke−∫t0tnavdv∫tktne∫t0savdvbs+λ1s+λ2sds>12δe−p−M1−αδep1−α4Me2p=e−p4δ,which contradicts with (26). Hence, condition (16) is necessary for globally asymptotic stability of the trivial solution of (1). This completes the proof of Theorem 1.To illustrate our result, we give an example.Example 1.
In equation (1), let at=1/1+t, bt=1/81+t, and(37)Ft,Xt,Xt−τ=181+tXt+Xt−τ.
By a straightforward computation, we can obtain(38)at≤1,∫t0+∞asds=+∞,fort∈I.
In addition,(39)Ft,X1,Y1−Ft,X2,Y2=181+tX1+Y1−X2−Y2≤181+tX1−X2+Y1−Y2.
Letλ1t=λ2t=1/81+t; then, condition (A2) of Theorem 1 holds. By direct calculations, we have(40)∫t0te−∫stavdvbs+λ1s+λ2sds≤38e−21+t∫t0te21+s11+sds≤381−e2−21+t<38,(41)at∫t0te−∫stavdvbs+λ1s+λ2sds+bt+λ1t+λ2t≤381+t1−e2−21+t+381+t<34.
Letα=3/4, and by (38)–(41), we find that conditions (A1)–(A3) and (16) hold. Thus, (1) is globally asymptotically stable.


## 4. Conclusion

Stability is one of the main problems encountered in applications and has recently attracted considerable attention. The fixed point method is an effective method to discuss the stability for the differential equation with unbounded delay or the differential equation with unbounded terms. In this paper, we investigate a class of neutral set-valued functional differential equations and obtain a criterion for the globally asymptotic stability theorem with necessary and sufficient conditions by the fixed point method. Finally, we verify the validity of the result by an example.



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*Source: 6569308-2020-04-27.xml*