# New Results on Impulsive Functional Differential Equations with Infinite Delays

Authors: Jie Yang; Bing Xie
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We investigate the stability for a class of impulsive functional differential equations with infinite delays by using Lyapunov functions and Razumikhin-technique. Some new Razumikhin-type theorems on stability are obtained, which shows that impulses do contribute to the system’s stability behavior. An example is also given to illustrate the importance of our results.


---

## Body

## 1. Introduction

Impulsive differential equations have attracted the interest of many researchers in recent years. It arises naturally from a wide variety of applications such as orbital transfer of satellite, ecosystems management, and threshold theory in biology. There has been a significant development in the theory of impulsive differential equations in the past several years ago, and various interesting results have been reported; see [1–4]. Recently, systems with impulses and time delay have received significant attention [5–16]. In fact, the system stability and convergence properties are strongly affected by time delays, which are often encountered in many industrial and natural processes due to measurement and computational delays, transmission, and transport lags. In [5, 6, 8], the authors considered the stability of impulsive differential equations with finite delay and got some results. In [7], by using Lyapunov functions and Razumikhin technique, some Razumikhin-type theorems on stability are obtained for a class of impulsive functional differential equations with infinite-delay. However, not much has been developed in the direction of the stability theory of impulsive functional differential systems, especially for infinite delays impulsive functional differential systems. As we know, there are a number of difficulties that one must face in developing the corresponding theory of impulsive functional differential systems with infinite-delay; for example, the interval (-∞,σ]  is not compact, and the images of a solution map of closed and bounded sets in C((-∞,0],Rn) space may not be compact. Therefore, it is an interesting and complicated problem to study the stability theory for impulsive functional differential systems with infinite delays.In the present paper, we will consider the stability of impulsive infinite-delay differential equations by using Lyapunov functions and the Razumikhin technique, we get some new results. The effect of delay and impulses which do contribute to the equations’s stability properties will be shown in this paper.The rest of this paper is organized as follows. In Section2, we introduce some notations and definitions. In Section 3, we get some criteria for uniform stability and uniform asymptotic stability for impulsive infinite-delay differential equations, and an example is given to illustrate our results. Finally, concluding remarks are given in Section 4.


## 2. Preliminaries

LetR denote the set of real numbers, R+ the set of nonnegative real numbers, and Rn  the n-dimensional real space equipped with the Euclidean norm ∥·∥. For any t≥t0≥0>α≥-∞, let f(t,x(s)) where s∈[t+α,t] or f(t,x(·)) be a Volterra-type functional. In the case when α=-∞, the interval [t+α,t] is understood to be replaced by (-∞,t].We consider the impulsive functional differential equations(1)x′(t)=f(t,x(·)),t≥t0,t≠tk,Δx|t=tk=x(tk)-x(tk-)=Ik(x(tk-)),k=1,2,…,
where the impulse times tk satisfy 0≤t0<t1<⋯<tk<⋯,limk→+∞tk=+∞ and x′ denotes the right-hand derivative of x. f∈C([tk-1,tk)×C,Rn),f(t,0)=0.C is an open set in PC([α,0],Rn), where PC([α,0],Rn)={ψ:[α,0]→Rn is continuous everywhere except at finite number of points tk, at which ψ(tk+) and  ψ(tk-) exist and ψ(tk+)=ψ(tk)}. For each k=1,2,…,I(t,x)∈C([t0,∞)×Rn,Rn),I(tk,0)=0.For anyρ>0, there exists a ρ1>0(0<ρ1<ρ) such that x∈S(ρ1) implies that x+I(tk,x)∈S(ρ), where S(ρ)={x:∥x∥<ρ,x∈Rn}.DefinePCB(t)={x∈C:xisbounded}. For ψ∈PCB(t), the norm of ψ is defined by ∥ψ∥=supα≤θ≤0|ψ(θ)|. For anyσ≥0, let PCBδ(σ)={ψ∈PCB(σ):∥ψ∥<δ}.For any givenσ≥t0, the initial condition for system (1) is given by
(2)xσ=ϕ,
where ϕ∈PC([α,0],Rn).We assume that the solution for the initial problems, (1)-(2) does exist and is unique which will be written in the form x(t,σ,ϕ); see [4, 10]. Since f(t,0)=0,I(tk,0)=0,k=1,2,…, then x(t)=0 is a solution of (1)-(2), which is called the trivial solution. In this paper, we always assume that the solution x(t,σ,ϕ) of (1)-(2) can be continued to ∞ from the right of σ.For convenience, we also have the following classes in later sections:K1={a∈C(R+,R+)∣a(0)=0 and a(s)>0 for s>0};K2={a∈C(R+,R+)∣a(0)=0 and a(s)>0 for s>0 and a is nondecreasing in s};ΔV(tk,ψ(0))=V(tk,ψ(0)+Ik(tk,ψ))-V(tk-,ψ(0)),k=1,2,…;Δtk=tk-tk-1,k=1,2,….We introduce some definitions as follows.Definition 1 (see [4]).
The functionV:[α,∞)×C→R+ belongs to class v0 if(A1)
V is continuous on each of the sets [tk-1,tk)×C and lim(t,φ)→(tk-,ψ)V(t,φ)=V(tk-,ψ) exists;(A2)
V(t,x) is locally Lipschitzian in x and V(t,0)≡0.Definition 2 (see [4]).
LetV∈v0, for any (t,ψ)∈[tk-1,tk)×C, the upper right-hand Dini derivative of V(t,x) along the solution of (1)-(2) is defined by
(3)D+V(t,ψ(0))=limsuph→0+{V(t+h,ψ(0)+hf(t,ψ))-V(t,ψ(0))}h.
Similarly, we can define D-V(t,ψ(0)),D-V(t,ψ(0),D+V(t,ψ(0)). If V∈C′, then DV(t,ψ(0))=V˙(t,ψ(0)), where D is any of the four Dini derivatives.
ForV∈v0,(t,ψ)∈[tk-1,tk)×C, the upper right-hand Dini derivative of V˙(t,x) along the solution of (1)-(2) is defined by
(4)D+V˙(t,ψ(0))=limsuph→0+{V˙(t+h,ψ(0)+hf(t,ψ))-V˙(t,ψ(0))}h.
Similarly, we can define D-V˙(t,ψ(0)). If V∈C′′, then these are simply the second derivative of V.Definition 3 (see [4]).
Assumex(t)=x(t,σ,ϕ) to be the solution of (1)-(2) through (σ,ϕ). Then, the zero solution of (1)-(2) is said to be(1)
uniformly stable, if for anyε>0 and σ≥t0, there exists a δ=δ(ε)>0 such thatϕ∈PCBδ(σ) implies ∥x(t)∥<ε,t≥σ.(2)
uniformly asymptotically stable, if it is uniformly stable, and there exists aδ>0 such that for anyε>0,σ≥t0, there is a T=T(ε)>0 such that ϕ∈PCBδ(σ) implies ∥x(t)∥<ε,t≥σ+T.


## 3. Main Results

Theorem 4.
Assume that there exist functionswi∈K1,g∈K2,ci,p,q∈C(R+,R+),V(t,x)∈v0,i=1,2, and constants m>1, such that the following conditions hold: (i)
w1(∥x∥)≤V(t,x)≤w2(∥x∥),(t,x)∈[α,∞)×S(ρ);(ii)
for anyσ≥t0 and ψ∈PC([α,0],S(ρ)), if  V(t,ψ(0))≥m-2g(V(t+θ,ψ(θ))), max{α,-q(V(t))}≤θ≤0,t≠tk, then
(5)D+V(t,ψ(0))≤p(t)c1(V(t,ψ(0))),
where s/m≤g(s)<s for any s>0;(iii)
for all(t,ψ(0))∈(tk-1,tk)×PC([α,0],S(ρ1)),
(6)D-V˙(t,ψ(0))≥0.
Also, for all (tk,ψ)∈R+×PC([α,0],S(ρ1)),
(7)ΔtkV˙(tk-,ψ(0))+ΔV(tk,ψ(0))≤-μkc2(V(tk,ψ(0))),
where c2(s)≤sc2′(s),s>0,μk satisfies liminfk→∞μk≥2·sups>0(s/c2(m-1·s));(iv)
there exist constantsM1,M2>0 such that the following inequalities hold:
(8)supt≥0∫tt+τ‍p(s)ds=M1<∞,infs>0∫g(s)s‍dtc1(t)=M2>M1,
where τ=maxk≥1{tk-tk-1}<∞.
Then, the zero solution of (1)-(2) is uniformly asymptotically stable.Proof.
Condition (i) implies that w1(s)≤w2(s) for s∈[0,ρ]. So let W1 and W2 be continuous, strictly increasing functions satisfying W1(s)≤w1(s)≤w2(s)≤W2(s) for all s∈[0,ρ]. Then
(9)W1(∥x∥)≤V(t,x)≤W2(∥x∥),(t,x)∈[α,∞)×S(ρ).
We first show uniform stability.
For anyε>0(<ρ1), one may choose a δ=δ(ε)>0 such that W2(δ)≤g(W1(ε)). Let  x(t)=x(t,σ,ϕ) be a solution of (1)-(2) through (σ,ϕ),σ≥t0. For any ϕ∈PCBδ(σ), we will prove that ∥x(t)∥<ε,t≥σ.
For convenience, letV(t)=V(t,x(t)). Suppose that σ∈[tl-1,tl),l∈Z+. First, for σ+α≤t≤σ, we have
(10)W1(∥x∥)&≤V(t)<W2(δ)≤g(W1(ε))<W1(ε).
So, ∥x(t)∥<ε<ρ1,t∈[σ+α,σ].
Next, we claim that(11)V(t)<W1(ε),t∈[σ,tl).
Suppose on the contrary that there exists some t∈[σ,tl) such that V(t)≥W1(ε). Since V(σ)<W1(ε), we can define t^=inf{t∈[σ,tl)∣V(t)≥W1(ε)}. Thus, t^∈(σ,tl),V(t^)=W1(ε), and V(t)<W1(ε),t∈[σ,t^). Also, from (10) we obtain
(12)V(t)<W1(ε),t∈[σ+α,t^).
On the other hand, note that V(t^)=W1(ε)>g(W1(ε)) and V(σ)<g(W1(ε)) in view of (10), we can define t*=sup{t∈[σ,t^]V(t)≤g(W1(ε))}; it is obvious that t*∈[σ,t^),V(t*)=g(W1(ε)) and V(t)>g(W1(ε)) for t∈(t*,t^]. Therefore, combining (12), we have for t∈(t*,t^)(13)V(t)>g(W1(ε))>g(V(t+θ)),α≤θ≤0;
that is,
(14)V(t,ψ(0))>m-2g(V(t+θ,ψ(θ))),max{α,-q(V(t))}≤θ≤0.
By assumption (ii), (iv), we have
(15)∫V(t*)V(t^)‍dsc1(s)=∫g(W1(ε))W1(ε)‍dsc1(s)≥M2>M1.
However, we also have
(16)∫V(t*)V(t^)‍dsc1(s)≤∫t*t^‍p(s)ds<∫t*t*+τ‍p(s)ds≤M1,
which is a contradiction. So, (11) holds.
Hence,W1(∥x∥)≤V(t)<W1(ε),t∈[σ,tl) implies that ∥x(tl-)∥<ε<ρ1. Thus, x(tl)∈S(ρ).
On the other hand, from condition (iii), we note for k=1,2,…,
(17)V(tk)-V(tk-1)=V(tk)-V(tk-)+V(tk-)-V(tk-1)=ΔV(tk)+∫tk-1tk‍V˙(t)dt≤ΔV(tk)+ΔtkV˙(tk-)≤-μkc2(V(tk))≤0.
Hence, we obtain V(tk)≤V(tk-1),k=1,2,…. particularly, V(tl)≤V(tl-1). In view of (10), we get
(18)V(tl)≤V(tl-1)<g(W1(ε))<W1(ε).
Next, we claim that
(19)V(t)<W1(ε),t∈[tl,tl+1).
Suppose on the contrary that there exists some t∈[tl,tl+1) such that V(t)≥W1(ε). Then applying exactly the same argument as in the proof of (11) yields our desired contradiction.
By induction hypothesis, we may prove, in general, that fort∈[tl+k,tl+k+1),k>0,
(20)V(t)<W1(ε).
In view of condition (i), we obtain that ∥x(t)∥<ε,t≥σ. Therefore, we have proved that the solutions of (1)-(2) are uniformly stable.
Next, we claim that they are uniformly asymptotically stable. Since the zero solution of (1)-(2) is uniformly stable, for any given constant H>0(<ρ1), then there exists δ>0 such that ϕ∈PCBδ(σ) implies that V(t)<W1(H),∥x(t)∥<ρ1,t≥σ.
For anyε∈(0,H), let
(21)d<min{d^,W1(ε)},d^=inf{s-g(s)∣m-1W1(ε)≤s≤W1(H)},h=sup{q(s)∣m-1W1(ε)≤s≤W1(H)},n0=W1(H)2·sups>0(s/c2(m-1s))c2(m-1W1(ε))+1.
From condition (iii), we get that there exists a n1>0 such that for k>n1,
(22)μk≥2·sups>0sc2(m-1·s).
Choose a positive integer N satisfying
(23)W1(ε)+(N-1)d<W1(H)≤W1(ε)+Nd,
and define T=N(h+n0τ)+n1, we will prove that ϕ∈PCBδ(σ) implies ∥x(t)∥<ε,t≥σ+T.
First, we prove that there existst^∈[σ+h+n1,σ+h+n1+n0τ] such that
(24)V(t^)<m-1[W1(ε)+(N-1)d].
Suppose on the contrary that for all t∈[σ+h+n1,σ+h+n1+n0τ],
(25)V(t)≥m-1[W1(ε)+(N-1)d]≥m-1W1(ε).
Let tk1=min{tk:tk≥σ+h+n1}, from (17), we get
(26)V(tk1)-V(tk1-1)≤-μk1c2(V(tk1))≤-μk1c2(m-1W1(ε)),V(tk1+1)-V(tk1)≤-μk1+1c2(m-1W1(ε)),⋮V(tk1+n0)-V(tk1+n0-1)≤-μk1+n0c2(m-1W1(ε)),
In general, combining (22), we deduce that
(27)V(tk1+n0)≤V(tk1-1)-∑s=0n0‍μk1+sc2(m-1W1(ε))≤W1(H)-2(n0+1)·sups>0sc2(m-1s)c2(m-1W1(ε))=-4·sups>0sc2(m-1s)c2(m-1W1(ε))<0,
which is a contradiction. So, (24) holds.
Supposet^∈[tl-1,tl),l>1. Furthermore, we can prove that for t∈[t^,tl)(28)V(t)<W1(ε)+(N-1)d.
Suppose this assertion is false, then there exists some t∈[t^,tl) such that V(t)≥W1(ε)+(N-1)d. Since V(t^)<m-1[W1(ε)+(N-1)d]<W1(ε)+(N-1)d, so define
(29)t*=inf{t∈[t^,tl)∣V(t)≥W1(ε)+(N-1)d};
then t*∈(t^,tl),V(t*)=W1(ε)+(N-1)d and V(t)<W1(ε)+(N-1)d,t∈(t^,t*). Note that
(30)V(t*)=W1(ε)+(N-1)d>g(W1(ε)+(N-1)d),V(t^)<m-1[W1(ε)+(N-1)d]<g(W1(ε)+(N-1)d);
thus, we can define
(31)t-=sup{t∈[t^,t*]∣V(t)≤g(W1(ε)+(N-1)d)},
then t-∈[t^,t*),V(t-)=g(W1(ε)+(N-1)d) and V(t)>g(W1(ε)+(N-1)d) for t∈(t-,t*].
Hence, we get fort∈(t-,t*](32)V(t)>g(W1(ε)+(N-1)d)≥m-1[W1(ε)+(N-1)d]≥m-1W1(ε),
which implies that for t∈(t-,t*](33)V(t)≥g(V(t))+d≥m-1V(t)+d>mV(t)m2+dm2≥W1(ε)+Ndm2≥W1(H)m2≥V(s)m2>g(V(s))m2,t+α<s≤t.
Thus, V(t)≥(1/m2)g(V(t+θ,ψ(θ))),max{α,-q(V(t))}≤θ≤0.
By assumption, (ii),(iv), we have for t∈(t-,t*),
(34)∫V(t-)V(t*)‍dsc1(s)=∫g(W1(ε)+(N-1)d)W1(ε)+(N-1)d‍dsc1(s)≥M2>M1.
However, we also have
(35)∫V(t-)V(t*)‍dsc1(s)<∫t-t*‍p(s)ds<∫t-t-+τ‍p(s)ds<M1,
which is a contradiction. So, (28) holds.
On the other hand, it is easy to prove that the functionss/c2(m-1s) are nonincreasing for s∈(0,+∞) in view of condition c2(s)≤sc2′(s) for any s>0.
Hence, the following inequalities hold: fork>n1,
(36)W1(ε)+(N-i)dc2(m-1(W1(ε)+(N-i-1)d))≤W1(ε)+dc2(m-1W1(ε))<2W1(ε)c2(m-1W1(ε))≤μk,i=1,2,…,N-1.
Next, we claim that
(37)V(tl)<m-1[W1(ε)+(N-1)d].
Or else, then V(tl)≥m-1[W1(ε)+(N-1)d]; from (17), we get
(38)V(tl)-V(tl-1)≤-μlc2(V(tl))≤-μlc2(m-1[W1(ε)+(N-1)d]).
Considering (36), it holds that
(39)V(tl)≤V(tl-1)-μlc2(m-1[W1(ε)+(N-1)d])≤W1(H)-μlc2(m-1[W1(ε)+(N-1)d])≤W1(ε)+Nd-μlc2(m-1[W1(ε)+(N-1)d])≤c2(m-1[W1(ε)+(N-1)d])×{W1(ε)+Ndc2(m-1[W1(ε)+(N-1)d])-μl}<0,
which is a contradiction and (37) holds.
Next, we can prove that fort∈[tl,tl+1)(40)V(t)<W1(ε)+(N-1)d.
Suppose that this assertion is false, then there exists some t∈[t^,tl) such that V(t)≥W1(ε)+(N-1)d. Then applying exactly the same argument as in the proof of (24) and (28) yields our desired contradiction. Here, we omit it.
By induction hypothesis, we may prove, fort∈[tl+k,tl+k+1),k=1,2,…,
(41)V(t)<W1(ε)+(N-1)d;
that is,
(42)V(t)<W1(ε)+(N-1)d,t≥t^.
Hence, we obtain
(43)V(t)<W1(ε)+(N-1)d,t≥σ+h+n1+n0τ.
Next, we prove that there exists t^2∈[σ+2h+n1+n0τ,σ+2h+n1+2n0τ] such that
(44)V(t^2)<m-1[W1(ε)+(N-2)d].
Suppose that for all t∈[σ+2h+n1+n0τ,σ+2h+n1+2n0τ],
(45)V(t)≥m-1[W1(ε)+(N-2)d]≥m-1W1(ε).
Using the same argument as in the proof of (24), we get
(46)V(tk2+n0)≤V(tk2-1)-∑s=0n0‍μk2+sc2(m-1W1(ε))≤W1(H)-2(n0+1)W1·sups>0sc2(m-1s)c2(m-1W1(ε))=-4·sups>0sc2(m-1s)c2(m-1W1(ε))<0,
where tk2=min{tk:k≥σ+2h+n1+n0τ}.
This is a contradiction. So, (44) holds.
Supposet^2∈[tk-1,tk),k>l. Furthermore, we claim that for t∈[t^2,tk)
(47)V(t)<W1(ε)+(N-2)d.
Suppose on the contrary, that there exists some t∈[t^2,tk) such that V(t)≥W1(ε)+(N-2)d. We define
(48)t⋆=inf{t∈[t^2,tk)∣V(t)≥W1(ε)+(N-2)d},
since V(t^2)<m-1[W1(ε)+(N-2)d]<W1(ε)+(N-2)d in view of (44). Thus, t⋆∈(t^2,tk),V(t⋆)=W1(ε)+(N-2)d and V(t)<W1(ε)+(N-2)d,t∈(t^2,t⋆). Note that
(49)V(t⋆)=W1(ε)+(N-2)d>g(W1(ε)+(N-2)d),V(t^2)<m-1[W1(ε)+(N-2)d]<g(W1(ε)+(N-2)d);
furthermore, we can define
(50)t~=sup{t∈[t^2,t⋆]∣V(t)≤g(W1(ε)+(N-2)d)},
then t~∈[t^2,t⋆),V(t~)=g(W1(ε)+(N-2)d) and V(t)>g(W1(ε)+(N-2)d) for t∈(t~,t⋆].
Hence, we get fort∈(t~,t⋆](51)V(t)>g(W1(ε)+(N-2)d)≥m-1[W1(ε)+(N-2)d]≥m-1W1(ε);
considering the definition of d and (43), we get for t∈(t~,t⋆](52)V(t)≥g(V(t))+d≥m-1V(t)+d>mV(t)m2+dm2≥W1(ε)+(N-1)dm2≥V(s)m2>g(V(s))m2,t-h<s≤t.
Thus, V(t)≥(1/m2)g(V(t+θ,ψ(θ))),max{α,-q(V(t))}≤θ≤0.
Using assumptions(ii),(iv), we have
(53)∫V(t~)V(t⋆)‍dsc1(s)=∫g(W1(ε)+(N-2)d)W1(ε)+(N-2)d‍dsc1(s)≥M2>M1.
However,
(54)∫V(t~)V(t⋆)‍dsc1(s)<∫t~t⋆‍p(s)ds<∫t~t~+τ‍p(s)ds<M1,
giving us a contradiction. So, (47) holds.
Next, we claim that(55)V(tl)<m-1[W1(ε)+(N-1)d],V(t)<W1(ε)+(N-1)d,t∈[tl,tl+1),
whose arguments are the same as was employed in the proof of (36), (37). there we omit it.
Repeating this process, it is easy to check that(56)V(t)<W1(ε)+(N-2)d,t≥σ+2h+n1+2n0τ.
By induction hypothesis, we have
(57)V(t)≤W1(ε)+(N-i)d,t≥σ+ih+n1+in0τ.
Let i=N, then for t≥σ+N(h+n0τ)+n1,
(58)V(t)<W1(ε).
Therefore, we arrive at ∥x(t)∥<ε,t≥T. The proof of Theorem 4 is complete.Corollary 5.
Assume that there exist functionswi∈K1,g∈K2,c,p∈C(R+,R+),V(t,x)∈v0,i=1,2, and constants m>1, such that the following conditions hold: (i)
w1(∥x∥)≤V(t,x)≤w2(∥x∥),(t,x)∈[α,∞)×S(ρ);(ii)
for anyσ≥t0 and ψ∈PC([α,0],S(ρ)), if  V(t,ψ(0))≥g(V(t+θ,ψ(θ))),α≤θ≤0,t≠tk, then
(59)D+V(t,ψ(0))≤p(t)c(V(t,ψ(0))),
where (s/m)≤g(s)<s for any s>0;(iii)
for all(t,ψ(0))∈(tk-1,tk)×PC([α,0],S(ρ1)),
(60)D-V˙(t,ψ(0))≥0.
Also, for all (tk,ψ)∈R+×PC([α,0],S(ρ1)),
(61)ΔtkV˙(tk-,ψ(0))+ΔV(tk,ψ(0))≤0;(iv)
there exist constantsM1,M2>0 such that the following inequalities hold:
(62)supt≥0∫tt+τ‍p(s)ds=M1<∞,infs>0∫g(s)s‍dtc(t)=M2>M1,
where τ=maxk≥1{tk-tk-1}<∞.
Then the zero solution of (1)-(2) is uniformly stable.Theorem4 has a dual result when V˙ is nonincreasing on (tk-1,tk). Here, we only give the results whose proof is very similar to the proof of Theorem 4.Theorem 6.
Assume that there exist functionswi∈K1,g∈K2,ci,p,q∈C(R+,R+),V(t,x)∈v0,i=1,2, and constants m>1, such that the following conditions hold: (i)
w1(∥x∥)≤V(t,x)≤w2(∥x∥),(t,x)∈[α,∞)×S(ρ);(ii)
for anyσ≥t0 and ψ∈PC([α,0],S(ρ)), if  V(t,ψ(0))≥m-2g(V(t+θ,ψ(θ))),max{α,-q(V(t))}≤θ≤0,t≠tk, then
(63)D+V(t,ψ(0))≤p(t)c1(V(t,ψ(0))),
where (s/m)≤g(s)<s for any s>0;(iii)
for all(t,ψ(0))∈(tk-1,tk)×PC([α,0],S(ρ1)),
(64)D-V˙(t,ψ(0))≤0.
Also, for all (tk,ψ)∈R+×PC([α,0],S(ρ1)),
(65)ΔtkV˙(tk-1-,ψ(0))+ΔV(tk,ψ(0))≤-μkc2(V(tk,ψ(0))),
where c2(s)≤sc2′(s),s>0,μk satisfies liminfk→∞μk≥2·sups>0(s/c2(m-1·s));(iv)
there exist constantsM1,M2>0 such that the following inequalities hold:
(66)supt≥0∫tt+τ‍p(s)ds=M1<∞,(67)infs>0∫g(s)s‍dtc1(t)=M2>M1,
where τ=maxk≥1{tk-tk-1}<∞.
Then the zero solution of (1)-(2) is uniformly asymptotically stable.Example 7.
Consider the impulsive delay differential equations:(68)x′(t)=ax(t)-b∫-∞0‍esx(t+s)ds,t≥0,t≠tk,Δx|t=tk=Ik(x),k=1,2,…,x0=ϕ>0,
where a∈(0,3],b∈(0,2],|x+Ik(x)|≤λ·|x|,k=1,2,…,λ∈(0,1). For any given ϕ>0, we always suppose that (68) has and only has positive solutions, and assume without loss of generality that x(t)=x(t,0,ϕ) is a solutions of (68) through (0,ϕ). Suppose that there exists m>1 such that the following inequalities hold:
(69)τ<min{lnm2(a-bm),1-λ-2λm2a},a>2b-1,
where τ=maxk≥1{tk-tk-1}<∞. Then, the zero solution of (68) is uniformly asymptotically stable.In fact, letV(t,x)=x2/2,g(s)=m-1s(m>1),andc1(s)=s then V(t,x(t))>g(V(s,x(s))),-∞≤s≤t implies that m·|x(t)|>|x(s)|,-∞≤s≤t. Thus, for t≠tk(70)D+V(t,x(·))=x(t)x′(t)=x(t){ax(t)-b∫-∞0‍esx(t+s)ds}≤x2(t){a-bm∫-∞0‍esds}=x2(t){a-bm}=p(t)V(t,x(t)),
where p(t)=2(a-bm).In view of condition (69), we note
(71)supt≥0∫tt+τ‍p(s)ds=2τ(a-bm)<lnm=infs>0∫g(s)s‍dtc1(t).
So, condition (iv) in Corollary 5 holds.On the other hand, we have fort≠tk(72)D-V˙(t,x(·))=(x(t)x′(t))′=x(t)x′′(t)+(x′(t))2=(x′(t))2+x(t)(ax(t)-b∫-∞0‍esx(t+s)ds)′=(x′(t))2+ax(t)x′(t)-bx(t)∫-∞0‍esx(t+s)ds-bx2(t)=(x′(t))2+ax(t)x′(t)+ax2(t)-x(t)x′(t)-bx2(t)=(x′(t))2+(a-1)x(t)x′(t)+(a-b)x2(t)≥(x′(t))2-(x′(t))2+x2(t)2(a-1)+(a-b)x2(t)=3-a2(x′(t))2+(a+12-b)x2(t)≥0,
in view of condition a>2b-1. Also, considering x(t) to be a positive solution of (68), we get
(73)ΔtkV˙(tk-,ψ(0))+ΔV(tk,ψ(0))≤aτx2(tk-)+(λ-1)x2(tk-)2=(2aτ+λ-1)x2(tk-)x2(tk-)2=(2aτ+λ-1)V(tk-)≤-1-2aτ-λλV(tk)=-μkc2(V(tk,ψ(0))),
where c2=s,μk=(1-2aτ-λ)/λ.Note that(74)sups>02sc2(m-1·s)=2m<1-2aτ-λλ=μk,
in view of (69). So, the zero solution of (68) is uniformly stable by Corollary 5.Furthermore, chooseq(s)=-ln(1-1/m) (positive constants), which implies that ∫-q(s)0‍esds=m-1. On the other hand, since V(t,x(t))>m-2g(V(s,x(s))),max{α,t-q(V(t))}≤s≤t, implying that m3/2|x(t)|>|x(s)|,max{α,t-q(V(t))}≤s≤t, then
(75)D+V|(68)(t,x(·))D+V|≤ax2(t)-bx(t)∫-∞0‍es|x(t+s)|dsD+V|≤ax2(t)-bx(t)∫-∞t‍es-t|x(s)|dsD+V|≤ax2(t)-bx(t)∫t-q(V(t,x(·)))t‍es-t|x(s)|ds6666666-x(t)∫-∞t-q(V(t,x(·)))‍es-t|x(s)|dsD+V|≤ax2(t)-bx(t)∫t-q(V(t,x(·)))t‍es-t|x(s)|dsD+V|≤x2(t){a-bm3/2∫-q(V(t,x(·)))0‍esds}D+V|≤x2(t){a-bm}D+V|=c(V(t,x(t)))p(t).
By Theorem 4, we obtain that if (69) holds, then the zero solution of (68) is uniformly asymptotically stable.Remark 8.
In fact,x(t)=ϕ(0)et is a positive solution of (68) through (0,ϕ) in the absence of impulses. It is obvious that the solution is unstable. However, the solution is uniformly asymptotically stable under proper impulses effect, which shows that impulses do contribute to the system’s stability behavior.


## 4. Conclusion

In this work, we have considered the stability of impulsive infinite-delay differential systems. By using Lyapunov functions and the Razumikhin technique, we have obtained some new results. We can see that impulses and delay do contribute to the system’s stability behavior.



---

*Source: 132619-2013-04-27.xml*