We consider the practical stability of impulsive differential equations with infinite delay in terms of two measures. New stability criteria are established by employing Lyapunov functions and Razumikhin technique. Moreover, an example is given to illustrate the advantage of the obtained result.

1. Introduction

One of the trends in the stability theory of the solutions of differential equations is the so-called practical stability, which was introduced by LaSalle and Lefschetz [1]. This is very useful in estimating the worst-case transient and steady-state responses and in verifying pointwise in time constraints imposed on the state trajectories. Fundamental results in this direction were obtained in [2]. In recent years the theory of practical stability and stability has been developed very intensively [3–7].

The theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modelling of many real world phenomena. Impulsive differential equations and impulsive functional differential equations have been intensively researched [8–20].

By employing the Razumikhin technique and Lyapunov functions, several stability criteria are established for general impulsive differential equations with finite delay [5–7, 14, 21]. Systems with infinite delay deserve study because they describe a kind of system present in the real world. For example, it is very useful in a predator-prey system. Therefore, it is an interesting and complicated problem to study the stability of impulsive functional differential systems with infinite delay. Usually, the Lyapunov functions are defined on whole components of system's state x [12–22]. In this paper, we divided the components of x into several groups and correspondingly, we employ several Lyapunov functions Vj(t,x(j))(j=1,2,…,m), where x=(x(1),…,x(m))T for each x(j). In this way, Lyapunov, functions are easier constructed, and the conditions ensuring the required stability are less restrictive. Furthermore, the stability results on impulsive finite delay differential equations considered in [4, 5] are generalized into the results on impulsive infinite delay differential equations in terms of two measures.

The work is organized as follows. In Section 2, we introduce some preliminary definitions which will be employed throughout the paper. In Section 3, based on Lyapunov functions and Razumikhin method, sufficient conditions for the uniformly practical stability in terms of two measures are given; an example is presented to illustrate the effectiveness of the approach.

2. Preliminaries

Consider the following impulsive infinite delay differential equations:
(2.1)x˙(t)=f(t,x(s);α≤s≤t),t≥t*,t≠τk,Δx(t)≜x(t)-x(t-)=Ik(x(t-)),t=τk,k=1,2,…,
where -∞≤α<t*,α could be -∞,t∈R+,f∈C[R+×PC([α,t],Rn),Rn] is a Volterra-type function. PC([α,t],Rn) denotes the space of piecewise right continuous functions φ=(φ1,…,φn):[α,t]→Rn with the sup-norm ||φ||=supα≤s≤t|φ(s)|, |φ(s)|=max1≤j≤n|φj(s)|, f(t,0)≡0,Ik(0)=0,0=τ0<τ1<τ2<⋯<τk<⋯,τk→∞ for k→∞, and x(t-)=lims→t-x(s). The functions Ik:Rn→Rn,k=1,2,…, are such that if ||x||<H and Ik(x)≠0, then ||x+Ik(x)||<H, where H= const.>0.

The initial condition for system (2.1) is given by
(2.2)x(t)=φ(t),t∈[α,t0],
where φ∈PC([α,t0],Rn), for t0≥t*.

We assume that a solution for the initial problem (2.1) and (2.2) does exist and is unique. Since f(t,0)=0, then x(t)=0 is a solution of (2.1), which is called the zero solution. Let PCρ(t)={φ∈PC([α,t],Rn)∣||φ||<ρ}. For convenience, we define |x|:=max1≤i≤n|xi|,x∈Rn;Rα:=[α,∞);S(ρ)={x∈Rn:||x||<ρ};S(j)(ρ)={x∈Rnj∣||x||<ρ},K:={W∈C[R+,R+],W(0)=0;W(s)>0,s>0},Γn:={h∈C[R+×Rn,R+]∣∀t∈R+,infxh(t,x)=0},Γαn:={h∈C[Rα×Rn,R+]∣∀t∈Rα,infxh(t,x)=0}.

Definition 2.1.

A continuous function w:R+→R+ is called a wedge function if w(0)=0 and w(s) is (strictly) increasing.

Definition 2.2.

For h0∈Γαn, xt(s):=x(s),s∈[α,t] and xt∈PC{[α,t],Rn}, for any t∈R+, we define
(2.3)h~0(t,xt)=supα≤θ≤th0(θ,x(θ)).

Let h0∈Γαn,h∈Γn. The impulsive functional differential 1 (2.1), (2.2) is said to be

(h~0,h) practically stable, if given (u,v) with 0<u<v, we have h~0(t0,xt0)<u implies h(t,x)<v,t≥t0 for some t0∈R+;

(h~0,h) uniformly practically stable if (S1) holds for every t0∈R+.

In what follows, we will split φ=∈PC(ρ) into several vectors, such that Σi=1mni=n and φ=(φ1(1),…,φn1(1),φ1(2),…,φn2(2),…,φ1(m),…,φnm(m))T. For convenience, we define φ(j)=(φ1(j),φ2(j),…,φnj(j)),j=1,2,…,m,andφ=(φ(1),φ(2),…,φ(m))T. For x=(x1,x2,…,xn)T∈Rn, we adopt notation as for φ. Similarly, let ||φ(j)||=∥φ(j)∥[α,t]=supα≤s≤t|φ(j)|,PC(j)(t)={φ(j):[α,t]→Rnj∣φ(j) is piecewise continuous and bounded}, and S(j)(ρ)={x∈Rnj∣||x||<ρ},PCρ(j)(t)={φ(j)∈PC(j)(t)∣||φ(j)||<ρ}.

3. Main Results

In the sequence, we assume that f is defined on Rα×PCH(t) for some H>0. For simplicity, denote Vi(t,x(i)),h(i)(t,x(i)),h0(i)(t,x(i)) by Vi(t),h(i)(t),h0(i)(t), respectively, 1≤i≤m. Now we start with the case of m=2.V′(t) be the right-hand derivative of V(t).

Theorem 3.1.

For j=1,2, let Φj:R+→R+ be continuous, Φj∈L1[0,∞),Φj(t)≤Kj for t≥0 with some constants Kj>0, and let Wij(i=1,2,3,4) be wedge functions. If there exist Lyapunov functions Vj:Rα×S(j)(H)→R+(j=1,2) such that

W1j(h(j)(t))≤Vj(t)≤W2j(h0(j)(t))+W3j[∫αtΦj(t-s)W4j(h0(j)(t))ds], where h0(j)∈Γαnj,h(j)∈Γnj;

when V1(t)≥V2(t), there holds V1′(t)≤0 if V1(s)<V1(t) for s∈[α,t]; when V2(t)≥V1(t), there holds V2′(t)≤0 if V2(s)<V2(t) for s∈[α,t];

Vj(τk)≤(1+bk)Vj(τk-),k=1,2,…,bk≥0, and ∑k=1∞bk<∞;

0<u<v are given, ϕ(j)(u)<v; when h~0(j)(t,xt(j))<u, there holds h(j)(t)≤ϕ(j)(h~0(j)(t,xt(j))), where ϕ(j) are wedge functions, and x(t)=(x(1)(t),x(2)(t)) is a solution of (2.1) and (2.2).

Then the zero solution of (2.1) and (2.2) is (h~0,h) uniformly practically stable with respect to (u,v).
Proof.

Since bk≥0, and ∑k=1∞bk<∞, it follows that there exists some M>0, such that ∏k=1∞(1+bk)=M and 1≤M<∞. Define a function V(t) for all t≥α(3.1)V(t)=V1(t)ifV1(t)≥V2(t);V(t)=V2(t)ifV2(t)≥V1(t).

We claim first that for any t≥α(3.2)[W11(h(1)(t))+W12(h(2)(t))]2≤V(t)≤W21(h0(1)(t))+W22(h0(2)(t))+W31∫αtΦ1(t-s)W41(h0(1)(t))ds+W32∫αtΦ2(t-s)W42(h0(2)(t))ds.
In fact, if V1(t)≥V2(t), then by (3.1) and condition (i), V(t)=V1(t)≥[V1(t)+V2(t)]/2≥[W11(h(1)(t))+W12(h(2)(t))]/2; whereas, if V2(t)≥V1(t), it also holds. On the other hand, the right-hand inequality in (3.2) is trivially valid.

Step 1. We aim to show that for each t≥t0,
(3.3)V′(t)≤0,ifV(s)≤V(t),s∈[α,t],t≠τk,V(τk)≤(1+bk)V(τk-),k=1,2,…
Indeed, suppose V1(t0)≥V2(t0) and there exists some t1>t0 such that for t∈[t0,t1],V1(t)≥V2(t). Then by (3.1), V(t)=V1(t),t∈[t0,t1].

Case 1 ..

If t=τj for some j∈Z+, then By (iii) V(τj)=V1(τj)≤(1+bj)V1(τj-)=(1+bj)V(τj-).

Case 2 ..

t≠τj for any j∈Z+, and V(s)≤V(t),s∈[α,t]. Then if V1(s)≤V2(s) we have V(s)=V2(s). Clearly, V(s)≤V(t) implies V1(s)≤V2(s)=V(s)≤V(t)=V1(t). If V1(s)≥V2(s) we have V(s)=V1(s). Obviously, V(s)≤V(t) implies V1(s)=V(s)≤V(t)=V1(t). In conclusion, V(s)≤V(t),s∈[α,t],t≠τk, implies V1(s)≤V1(t),s∈[α,t],t≠τk. So by (ii) we have V′(t)=V1′(t)≤0.

If t1=∞ we arrive at the assertion that (3.3) is true for all t≥t0. Otherwise, there exists a t2>t1 such that V1(t)≤V2(t),t∈[t1,t2]. When t1=τi for some i∈Z+ we have V1(τi-)≥V2(τi-) and V1(τi)≤V2(τi). In this case, by (iii) we have V(τi)=V2(τi)≤(1+bi)V2(τi-)≤(1+bi)V(τi-). When t1≠τi for any i∈Z+, we set V(t)=V2(t) for t∈[t1,t2].

By the similar analysis to Cases 1 and 2, we also have (3.3) when t,τk∈[t1,t2].

If t2=∞ then (3.3) holds for all t≥t0. Otherwise, repeat the above argument to arrive at the assertion that (3.3) is valid for all t≥t0. As for the case of V1(t)≤V2(t) for t∈[t0,t1], the process is similar and thus omitted.

For any t0∈R+, we assume there is a unique solution of (2.1), (2.2) through (t0,φ). Furthermore, we denote
(3.4)h(t,x(t)):=max{h(j)(t),j=1,2};h~0(t):=max{h~0(j)(t,xt(j)),j=1,2}.

If (t0,xt0)∈R+×PC([α,t0],Rn), such that h~0(t0,xt0)<u. By condition (iv),
(3.5)h(j)(t0)≤ϕ(j)(h~0(j)(t0))<ϕ(j)(u)<v.
From the definition of h(t,x(t)), we have h(t0,x(t0))<v.

Let v*=(1/M)min{W11(v),W12(v)}, we assume W2j(u)<v*/8 and W3j(Jj×W4j(u))<v*/8, where Jj=∫0∞Φj(s)ds,j=1,2.

Step 2. We aim to prove that V(t)≤Mv*/2,forallt≥t0.

First, for any t∈[α,t0], from Definition 2.2 and condition (iv), we know h0(j)(t,x(j)(t))≤h~0(j)(t0,xt0(j))<u. Then by (3.2), Vj(t)≤W21(u)+W22(u)+W31(J1W41(u))+W32(J2W42(u))<v*/2 for t∈[α,t0]. Hence, V(t)≤v*/2,t∈[α,t0].

Assume τl is the first impulse of all τi,i∈Z+ such that t0<τi. Now we claim that
(3.6)V(t)≤v*2fort0≤t<τl.

If it does not hold, then there is a t^∈(t0,τl) such that V(t^)>v*/2 and V′(t^)>0,V(t)≤V(t^) for t∈[α,t^]. From (3.3) we have V′(t^)≤0. It is a contradiction, so (3.6) holds.

Without loss of generality, we assume V1(τl)≤V2(τl), then V(τl)=V2(τl); from inequality (3.6) and condition (iii) we have V(τl)=V2(τl)≤(1+bl)V2(τl-)≤(1+bl)v*/2. Thus,
(3.7)V(τl)≤(1+bl)v*2.

Similarly, with the process in proving (3.6) and (3.7), we have
(3.8)V(t)≤(1+bl)v*2forτl≤t<τl+1;V(τl+1)≤(1+bl+1)(1+bl)v*2.
By simple induction, we can prove that, in general
(3.9)V(t)≤(1+bl+i+1)⋯(1+bl)v*2forτl+i≤t≤τl+i+1.
Taking this together with (3.2) and ∏k=1∞(1+bk)=M, we have
(3.10)[w11(h(1)(t))+w12(h(2)(t))]2≤V(t)≤Mv*2,∀t≥t0.
Since Mv*=min{w11(v),w12(v)}, we have
(3.11)w1j(h(j)(t))≤w1j(v),thatis,h(j)(t)≤v,j=1,2,∀t≥t0.
Therefore, by the definition of h(t,x), we have h(t,x)≤v. Thus the zero solution of (2.1), (2.2) with respect to (u,v) is (h~0,h)-uniformly practically stable.

Remark 3.2.

Since in our result α may be -∞ and the upper bound of the Lyapunov functions in our paper is improved by w3j,j=1,2, the result we have obtained is more general than that in [4–7, 14] with or without finite delay; furthermore, we have divided the components of x into several groups, correspondingly, several Lyapunov functions Vj(t,x(j))(j=1,2,…,m) are employed, where x=(x(1),…,x(m))T for each x(j). In this way, construction of the suitable Lyapunov functions is much easier than for x as [4, 6, 7, 10]. In additional, compared with [9, 12] where the infinite delay was considered in the Lyapunov stability of differential equations, we obtain the uniformly practical stability in terms of two measures.

Now, we may develop the ideas behind Theorem 3.1 to obtain the following more general results.

Theorem 3.3.

For j=1,2,…,m, let Φj:R+→R+ be continuous, Φj∈L1[0,∞),Φj(t)≤Kj for t≥0 with some constants Kj>0, and let Wij(i=1,2,3,4) be wedge functions. If there also exist Lyapunov functions Vj:Rα×S(j)(H)→R+ such that

W1j(h(j)(t))≤Vj(t)≤W2j(h0(j)(t))+W3j[∫αtΦj(t-s)W4j(h0(j)(t))ds], where h0(j)∈Γαnj,h(j)∈Γnj;

when Vl(t)=max{Vj(t)∣j=1,2,…,m}, there holds Vl′(t)≤0 if Vl(s)<Vl(t) for s∈[α,t];l=1,2,…,m;

Vj(τk)≤(1+bk)Vj(τk-),k=1,2,…,bk≥0, and ∑k=1∞bk<∞;

0<u<v are given, ϕ(j)(u)<v; when h~0(j)(t,xt(j))<u,h(j)(t)≤ϕ(j)(h~0(j)(t,xt(j))) where ϕ(j) are wedge functions, and x(t)=(x(1)(t),…,x(m)(t)) is a solution of (2.1) and (2.2).

Then the zero solution of (2.1) and (2.2) is (h~0,h)-uniformly practically stable.

It suffices to mention a few points in the proofs of Theorem 3.3, the rest are the same as in the proofing of Theorem 3.1, thus, are omitted.

First, for x(t)=(x(1)(t),…,x(m)(t)), we define
(3.12)V(t)=Vl(t),Vl(t)=max{Vj(t)∣j=1,2,…,m};
Second, instead of (3.2) we can claim that for any t≥α(3.13)∑j=1mW1j(h(j)(t))m≤V(t)≤∑j=1mW2j(h0(j)(t))+∑j=1mW3j∫αtΦj(t-s)W4j(h0(j)(t))ds.

Example 3.4.

Consider the equation
(3.14)x1′(t)=-a1(t)x1(t)+a2(t)x2(t)+b1(t)x1(t-r1(t))+∫-∞0g1(t,u,x1(t+u))du,t≠tk,x2′(t)=c1(t)x1(t)-c2(t)x2(t)+b2(t)x2(t-r2(t))+∫-∞0g2(t,u,x2(t+u))du,t≠tk,xi(tk)-xi(tk-)=Ik(xi(tk-)),k∈Z+,i=1,2,
where |x+Ik(x)|2≤(1+bk)2x2, with bk≥0,∑k=1∞bk<∞. Let M=∏k=1∞(1+bk)<∞. ai,bi,ci,ri and gi(i=1,2) are all continuous functions.

We first assume that ri(t)≥0 and |gi(t,u,x)|≤mi(u)|x|,t≥0,i=1,2, with ∫-∞0m1(u)du≤a1(t)-|a2(t)|-|b1(t)|, and ∫-∞0m2(u)du≤c2(t)-|c1(t)|-|b2(t)|. Without loss of generality, we may assume that the right-hand sides of (3.14) are defined on R×PC1(t), then set α=-∞ and t*=0.

Let Vj(t,xj(t))=xj2(t),h0(j)(t,xj)=xj2(t),w1j(s)=(1/2)s,w2j(s)=2s, then from the definition h~0(j)(t,xjt)=sup-∞<θ≤txj2(θ)=||xjt2||,j=1,2. For given 0<u<v, we assume ||xjt2||<u implies that there exists a K∈R+ such that xj2(t)<Kxj2(θ) for any θ∈(-∞,t]. Let h(j)(t,xj)=xj2(t)/(K+1),ϕ(i)(t)=(K/(K+1))t, then ϕ(i)(u)<u<v; furthermore, when h~0(j)(t,xjt)=||xjt2||<u, then for θ∈(-∞,t],
(3.15)h(j)(t,xj)=xj2(t)K+1≤KK+1xj2(θ)≤KK+1||xjt2||=ϕ(j)(h~0(j)(t,xjt)),
so conditions (i) and (iv) in Theorem 3.1 are verified.

Moreover, when V1(t)≥V2(t), that is, |x1(t)|≥|x2(t)|, and for s∈(-∞,t],V1(s)≤V1(t), we have
(3.16)V1′(t)=-2a1(t)x12(t)+2a2(t)x1(t)x2(t)+2b1(t)x1(t)x1(t-r1(t))+2x1(t)∫-∞0g1(t,u,x1(t+u))du≤-2[a1(t)-|a2(t)|-|b1(t)|-∫-∞0m1(u)du]x12(t)≤0,
similarly, when V1(t)≤V2(t) and for s∈(-∞,t],V2(s)≤V2(t), we also have V2′(t)≤0. Thus, condition (ii) in Theorem 3.1 is satisfied and the zero solution of system (3.14) is (h~0,h)-uniformly practically stable.

It is easy to see that if we put two variables x1,x2 in one Lyapunov function, then the arguments to get the desired stability conclusions would be much more complicated and the imposed conditions would be more restrictive. Furthermore, we extend the uniformly practically stable results to the infinite delay systems, and it is easy to see that the criteria in [3–10] are limited to judge the practical stability of Example 3.4.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11101373 and 61074011), the Natural Science Foundation of Zhejiang Province of China (no. Y6100007), and Zhejiang Innovation Project (no. T200905).

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