On the Practical Stability of Impulsive Differential Equations with Infinite Delay in Terms of Two Measures

Abstract

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 𝑉_𝑗(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: 𝑥̇(𝑡)=𝑓(𝑡,𝑥(𝑠);𝛼≤𝑠≤𝑡),𝑡≥𝑡_∗,𝑡≠𝜏_𝑘,Δ𝑥(𝑡)≜𝑥(𝑡)−𝑥(𝑡⁻)=𝐼_𝑘(𝑥(𝑡⁻)),𝑡=𝜏_𝑘,𝑘=1,2,…, (2.1) where −∞≤𝛼<𝑡_∗,𝛼 could be −∞,𝑡∈𝑅⁺,𝑓∈𝐶[𝑅⁺×𝑃𝐶([𝛼,𝑡],𝑅^𝑛),𝑅^𝑛] is a Volterra-type function. 𝑃𝐶([𝛼,𝑡],𝑅^𝑛) denotes the space of piecewise right continuous functions 𝜑=(𝜑₁,…,𝜑_𝑛)∶[𝛼,𝑡]→𝑅^𝑛 with the sup-norm ||𝜑||=sup_{𝛼≤𝑠≤𝑡}|𝜑(𝑠)|,    |𝜑(𝑠)|=max_{1≤𝑗≤𝑛}|𝜑_𝑗(𝑠)|, 𝑓(𝑡,0)≡0,𝐼_𝑘(0)=0,0=𝜏₀<𝜏₁<𝜏₂<⋯<𝜏_𝑘<⋯,𝜏_𝑘→∞ for 𝑘→∞, and 𝑥(𝑡⁻)=lim_{𝑠→𝑡−}𝑥(𝑠). The functions 𝐼_𝑘∶𝑅^𝑛→𝑅^𝑛,𝑘=1,2,…, are such that if ||𝑥||<𝐻 and 𝐼_𝑘(𝑥)≠0, then ||𝑥+𝐼_𝑘(𝑥)||<𝐻, where 𝐻= const.>0.

The initial condition for system (2.1) is given by 𝑥(𝑡)=𝜑(𝑡),𝑡∈𝛼,𝑡₀, (2.2) where 𝜑∈𝑃𝐶([𝛼,𝑡₀],𝑅^𝑛), 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 𝑃𝐶_𝜌(𝑡)={𝜑∈𝑃𝐶([𝛼,𝑡],𝑅^𝑛)∣||𝜑||<𝜌}. For convenience, we define |𝑥|∶=max_{1≤𝑖≤𝑛}|𝑥_𝑖|,𝑥∈𝑅^𝑛;𝑅_𝛼∶=[𝛼,∞);𝑆(𝜌)={𝑥∈𝑅^𝑛∶||𝑥||<𝜌};𝑆^(𝑗)(𝜌)={𝑥∈𝑅^𝑛𝑗∣||𝑥||<𝜌},𝐾∶={𝑊∈𝐶[𝑅⁺,𝑅⁺],𝑊(0)=0;𝑊(𝑠)>0,𝑠>0},Γ^𝑛∶={ℎ∈𝐶[𝑅⁺×𝑅^𝑛,𝑅⁺𝑅]∣∀𝑡∈₊,inf_𝑥ℎ(𝑡,𝑥)=0},Γ^𝑛_𝛼∶={ℎ∈𝐶[𝑅_𝛼×𝑅^𝑛,𝑅⁺]∣∀𝑡∈𝑅_𝛼,inf_𝑥ℎ(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∈Γ^𝑛_𝛼, 𝑥_𝑡(𝑠)∶=𝑥(𝑠),𝑠∈[𝛼,𝑡] and 𝑥_𝑡∈𝑃𝐶{[𝛼,𝑡],𝑅^𝑛}, for any 𝑡∈𝑅⁺, we define ℎ₀𝑡,𝑥_𝑡=sup_{𝛼≤𝜃≤𝑡}ℎ₀(𝜃,𝑥(𝜃)). (2.3)

Definition 2.3 (see[22]). Let ℎ₀∈Γ^𝑛_𝛼,ℎ∈Γ^𝑛. The impulsive functional differential 1 (2.1), (2.2) is said to be(S1)(ℎ₀,ℎ) practically stable, if given (𝑢,𝑣) with 0<𝑢<𝑣, we have ℎ₀(𝑡₀,𝑥_𝑡0)<𝑢 implies ℎ(𝑡,𝑥)<𝑣,𝑡≥𝑡₀ for some 𝑡₀∈𝑅⁺; (S2)(ℎ₀,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 Σ^𝑚_𝑖=1𝑛_𝑖=𝑛 and 𝜑=(𝜑₁⁽¹⁾,…,𝜑_𝑛⁽¹⁾₁,𝜑₁⁽²⁾,…,𝜑_𝑛⁽²⁾₂,…,𝜑₁^(𝑚),…,𝜑_𝑛^(𝑚)_𝑚)^𝑇. For convenience, we define 𝜑^(𝑗)=(𝜑₁^(𝑗),𝜑₂^(𝑗),…,𝜑_𝑛^(𝑗)_𝑗),𝑗=1,2,…,𝑚,and𝜑=(𝜑⁽¹⁾,𝜑⁽²⁾,…,𝜑^(𝑚))^𝑇. For 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛, we adopt notation as for 𝜑. Similarly, let ||𝜑^(𝑗)||=‖𝜑^(𝑗)‖^[𝛼,𝑡]=sup_{𝛼≤𝑠≤𝑡}|𝜑^(𝑗)|,𝑃𝐶^(𝑗)(𝑡)={𝜑^(𝑗)∶[𝛼,𝑡]→𝑅^𝑛𝑗∣𝜑^(𝑗) is piecewise continuous and bounded}, and 𝑆^(𝑗)(𝜌)={𝑥∈𝑅^𝑛𝑗∣||𝑥||<𝜌},𝑃𝐶_𝜌^(𝑗)(𝑡)={𝜑^(𝑗)∈𝑃𝐶^(𝑗)(𝑡)∣||𝜑^(𝑗)||<𝜌}.

3. Main Results

In the sequence, we assume that 𝑓 is defined on 𝑅_𝛼×𝑃𝐶_𝐻(𝑡) for some 𝐻>0. For simplicity, denote 𝑉_𝑖(𝑡,𝑥^(𝑖)),ℎ^(𝑖)(𝑡,𝑥^(𝑖)),ℎ₀^(𝑖)(𝑡,𝑥^(𝑖)) by 𝑉_𝑖(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 Φ_𝑗∶𝑅⁺→𝑅⁺ be continuous, Φ_𝑗∈𝐿¹[0,∞),Φ_𝑗(𝑡)≤𝐾_𝑗 for 𝑡≥0 with some constants 𝐾_𝑗>0, and let 𝑊_𝑖𝑗(𝑖=1,2,3,4) be wedge functions. If there exist Lyapunov functions 𝑉_𝑗∶𝑅_𝛼×𝑆^(𝑗)(𝐻)→𝑅⁺(𝑗=1,2) such that (i)𝑊_1𝑗(ℎ^(𝑗)(𝑡))≤𝑉_𝑗(𝑡)≤𝑊_2𝑗(ℎ₀^(𝑗)(𝑡))+𝑊_3𝑗[∫^𝑡_𝛼Φ_𝑗(𝑡−𝑠)𝑊_4𝑗(ℎ₀^(𝑗)(𝑡))𝑑𝑠], where ℎ₀^(𝑗)∈Γ^𝑛𝑗_𝛼,ℎ^(𝑗)∈Γ^𝑛𝑗; (ii)when 𝑉₁(𝑡)≥𝑉₂(𝑡), there holds 𝑉^′₁(𝑡)≤0 if 𝑉₁(𝑠)<𝑉₁(𝑡) for 𝑠∈[𝛼,𝑡]; when 𝑉₂(𝑡)≥𝑉₁(𝑡), there holds 𝑉^′₂(𝑡)≤0 if 𝑉₂(𝑠)<𝑉₂(𝑡) for 𝑠∈[𝛼,𝑡];(iii)𝑉_𝑗(𝜏_𝑘)≤(1+𝑏_𝑘)𝑉_𝑗(𝜏⁻_𝑘),𝑘=1,2,…,𝑏_𝑘≥0, and ∑^∞_𝑘=1𝑏_𝑘<∞; (iv)0<𝑢<𝑣 are given, 𝜙^(𝑗)(𝑢)<𝑣; when ℎ₀^(𝑗)(𝑡,𝑥_𝑡^(𝑗))<𝑢, there holds ℎ^(𝑗)(𝑡)≤𝜙^(𝑗)(ℎ₀^(𝑗)(𝑡,𝑥_𝑡^(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 (ℎ₀,ℎ) uniformly practically stable with respect to (𝑢,𝑣).

Proof. Since 𝑏_𝑘≥0, and ∑^∞_𝑘=1𝑏_𝑘<∞, it follows that there exists some 𝑀>0, such that ∏^∞_𝑘=1(1+𝑏_𝑘)=𝑀 and 1≤𝑀<∞. Define a function 𝑉(𝑡) for all 𝑡≥𝛼𝑉(𝑡)=𝑉₁(𝑡)if𝑉₁(𝑡)≥𝑉₂(𝑡);𝑉(𝑡)=𝑉₂(𝑡)if𝑉₂(𝑡)≥𝑉₁(𝑡). (3.1)
We claim first that for any 𝑡≥𝛼𝑊₁₁ℎ⁽¹⁾(𝑡)+𝑊₁₂ℎ⁽²⁾(𝑡)2≤𝑉(𝑡)≤𝑊₂₁ℎ₀⁽¹⁾(𝑡)+𝑊₂₂ℎ₀⁽²⁾(𝑡)+𝑊₃₁^𝑡_𝛼Φ₁(𝑡−𝑠)𝑊₄₁ℎ₀⁽¹⁾(𝑡)𝑑𝑠+𝑊₃₂^𝑡_𝛼Φ₂(𝑡−𝑠)𝑊₄₂ℎ₀⁽²⁾(𝑡)𝑑𝑠. (3.2) 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, 𝑉^′(𝑡)≤0,if𝑉(𝑠)≤𝑉(𝑡),𝑠∈[𝛼,𝑡],𝑡≠𝜏_𝑘,𝑉𝜏_𝑘≤1+𝑏_𝑘𝑉𝜏⁻_𝑘,𝑘=1,2,… (3.3) 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=𝜏_𝑗 for some 𝑗∈𝑍⁺, then By (iii) 𝑉(𝜏_𝑗)=𝑉₁(𝜏_𝑗)≤(1+𝑏_𝑗)𝑉₁(𝜏⁻_𝑗)=(1+𝑏_𝑗)𝑉(𝜏⁻_𝑗).

Case 2 .. 𝑡≠𝜏_𝑗 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≠𝜏_𝑘, implies 𝑉₁(𝑠)≤𝑉₁(𝑡),𝑠∈[𝛼,𝑡],𝑡≠𝜏_𝑘. So by (ii) we have 𝑉^′(𝑡)=𝑉^′₁(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=𝜏_𝑖 for some 𝑖∈𝑍⁺ we have 𝑉₁(𝜏⁻_𝑖)≥𝑉₂(𝜏⁻_𝑖) and 𝑉₁(𝜏_𝑖)≤𝑉₂(𝜏_𝑖). In this case, by (iii) we have 𝑉(𝜏_𝑖)=𝑉₂(𝜏_𝑖)≤(1+𝑏_𝑖)𝑉₂(𝜏⁻_𝑖)≤(1+𝑏_𝑖)𝑉(𝜏⁻_𝑖). When 𝑡₁≠𝜏_𝑖 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,𝜏_𝑘∈[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 ℎ(𝑡,𝑥(𝑡))∶=maxℎ^(𝑗)(𝑡),𝑗=1,2;ℎ₀(𝑡)∶=maxℎ₀^(𝑗)𝑡,𝑥_𝑡^(𝑗),𝑗=1,2. (3.4)
If (𝑡₀,𝑥_𝑡0)∈𝑅⁺×𝑃𝐶([𝛼,𝑡₀],𝑅^𝑛), such that ℎ₀(𝑡₀,𝑥_𝑡0)<𝑢. By condition (iv), ℎ^(𝑗)𝑡₀≤𝜙^(𝑗)ℎ₀^(𝑗)𝑡₀<𝜙^(𝑗)(𝑢)<𝑣. (3.5) 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 𝑊_2𝑗(𝑢)<𝑣^∗/8 and 𝑊_3𝑗(𝐽_𝑗×𝑊_4𝑗(𝑢))<𝑣^∗/8, where 𝐽_𝑗=∫^∞₀Φ_𝑗(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))≤ℎ₀^(𝑗)(𝑡₀,𝑥_𝑡^(𝑗)₀)<𝑢. Then by (3.2), 𝑉_𝑗(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 𝜏_𝑙 is the first impulse of all 𝜏_𝑖,𝑖∈𝑍⁺ such that 𝑡₀<𝜏_𝑖. Now we claim that 𝑉(𝑡)≤𝑣^∗2for𝑡₀≤𝑡<𝜏_𝑙. (3.6)
If it does not hold, then there is a 𝑡̂∈(𝑡₀,𝜏_𝑙) 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(𝜏_𝑙)≤𝑉₂(𝜏_𝑙), then 𝑉(𝜏_𝑙)=𝑉₂(𝜏_𝑙); from inequality (3.6) and condition (iii) we have 𝑉(𝜏_𝑙)=𝑉₂(𝜏_𝑙)≤(1+𝑏_𝑙)𝑉₂(𝜏⁻_𝑙)≤(1+𝑏_𝑙)𝑣^∗/2. Thus, 𝑉𝜏_𝑙≤1+𝑏_𝑙𝑣^∗2. (3.7)
Similarly, with the process in proving (3.6) and (3.7), we have 𝑉(𝑡)≤1+𝑏_𝑙𝑣^∗2for𝜏_𝑙≤𝑡<𝜏_𝑙+1;𝑉𝜏_𝑙+1≤1+𝑏_𝑙+11+𝑏_𝑙𝑣^∗2. (3.8) By simple induction, we can prove that, in general 𝑉(𝑡)≤1+𝑏_𝑙+𝑖+1⋯1+𝑏_𝑙𝑣^∗2for𝜏_𝑙+𝑖≤𝑡≤𝜏_𝑙+𝑖+1. (3.9) Taking this together with (3.2) and ∏^∞_𝑘=1(1+𝑏_𝑘)=𝑀, we have 𝑤₁₁ℎ⁽¹⁾(𝑡)+𝑤₁₂ℎ⁽²⁾(𝑡)2(𝑡)𝑣≤𝑉≤𝑀^∗2,∀𝑡≥𝑡₀. (3.10) Since 𝑀𝑣^∗=min{𝑤₁₁(𝑣),𝑤₁₂(𝑣)}, we have 𝑤_1𝑗ℎ^(𝑗)(𝑡)≤𝑤_1𝑗(𝑣),thatis,ℎ^(𝑗)(𝑡)≤𝑣,𝑗=1,2,∀𝑡≥𝑡₀. (3.11) 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 (ℎ₀,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 𝑤_3𝑗,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 𝑉_𝑗(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 Φ_𝑗∶𝑅⁺→𝑅⁺ be continuous, Φ_𝑗∈𝐿¹[0,∞),Φ_𝑗(𝑡)≤𝐾_𝑗 for 𝑡≥0 with some constants 𝐾_𝑗>0, and let 𝑊_𝑖𝑗(𝑖=1,2,3,4) be wedge functions. If there also exist Lyapunov functions 𝑉_𝑗∶𝑅_𝛼×𝑆^(𝑗)(𝐻)→𝑅⁺ such that (i)𝑊_1𝑗(ℎ^(𝑗)(𝑡))≤𝑉_𝑗(𝑡)≤𝑊_2𝑗(ℎ₀^(𝑗)(𝑡))+𝑊_3𝑗[∫^𝑡_𝛼Φ_𝑗(𝑡−𝑠)𝑊_4𝑗(ℎ₀^(𝑗)(𝑡))𝑑𝑠], where ℎ₀^(𝑗)∈Γ^𝑛𝑗_𝛼,ℎ^(𝑗)∈Γ^𝑛𝑗;(ii)when 𝑉_𝑙(𝑡)=max{𝑉_𝑗(𝑡)∣𝑗=1,2,…,𝑚}, there holds 𝑉^′_𝑙(𝑡)≤0 if 𝑉_𝑙(𝑠)<𝑉_𝑙(𝑡) for 𝑠∈[𝛼,𝑡];𝑙=1,2,…,𝑚;(iii)𝑉_𝑗(𝜏_𝑘)≤(1+𝑏_𝑘)𝑉_𝑗(𝜏⁻_𝑘),𝑘=1,2,…,𝑏_𝑘≥0, and ∑^∞_𝑘=1𝑏_𝑘<∞;(iv)0<𝑢<𝑣 are given, 𝜙^(𝑗)(𝑢)<𝑣; when ℎ₀^(𝑗)(𝑡,𝑥_𝑡^(𝑗))<𝑢,ℎ^(𝑗)(𝑡)≤𝜙^(𝑗)(ℎ₀^(𝑗)(𝑡,𝑥_𝑡^(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 (ℎ₀,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 𝑉(𝑡)=𝑉_𝑙(𝑡),𝑉_𝑙(𝑡)=max𝑉_𝑗(𝑡)∣𝑗=1,2,…,𝑚; (3.12) Second, instead of (3.2) we can claim that for any 𝑡≥𝛼∑^𝑚_𝑗=1𝑊_1𝑗ℎ^(𝑗)(𝑡)𝑚≤𝑉(𝑡)≤^𝑚_𝑗=1𝑊_2𝑗ℎ₀^(𝑗)(𝑡)+^𝑚_𝑗=1𝑊_3𝑗^𝑡_𝛼Φ_𝑗(𝑡−𝑠)𝑊_4𝑗ℎ₀^(𝑗)(𝑡)𝑑𝑠. (3.13)

Example 3.4. Consider the equation 𝑥^′₁(𝑡)=−𝑎₁(𝑡)𝑥₁(𝑡)+𝑎₂(𝑡)𝑥₂(𝑡)+𝑏₁(𝑡)𝑥₁𝑡−𝑟₁(𝑡)+⁰_−∞𝑔₁𝑡,𝑢,𝑥₁(𝑡+𝑢)𝑑𝑢,𝑡≠𝑡_𝑘,𝑥^′₂(𝑡)=𝑐₁(𝑡)𝑥₁(𝑡)−𝑐₂(𝑡)𝑥₂(𝑡)+𝑏₂(𝑡)𝑥₂𝑡−𝑟₂(𝑡)+⁰_−∞𝑔₂𝑡,𝑢,𝑥₂(𝑡+𝑢)𝑑𝑢,𝑡≠𝑡_𝑘,𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝐼_𝑘𝑥_𝑖𝑡⁻_𝑘,𝑘∈𝑍⁺,𝑖=1,2, (3.14) where |𝑥+𝐼_𝑘(𝑥)|²≤(1+𝑏_𝑘)²𝑥², with 𝑏_𝑘≥0,∑^∞_𝑘=1𝑏_𝑘<∞. Let 𝑀=∏^∞_𝑘=1(1+𝑏_𝑘)<∞. 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖,𝑟_𝑖 and 𝑔_𝑖(𝑖=1,2) are all continuous functions.

We first assume that 𝑟_𝑖(𝑡)≥0 and |𝑔_𝑖(𝑡,𝑢,𝑥)|≤𝑚_𝑖(𝑢)|𝑥|,𝑡≥0,𝑖=1,2, with ∫⁰_−∞𝑚₁(𝑢)𝑑𝑢≤𝑎₁(𝑡)−|𝑎₂(𝑡)|−|𝑏₁(𝑡)|, and ∫⁰_−∞𝑚₂(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 𝑉_𝑗(𝑡,𝑥_𝑗(𝑡))=𝑥²_𝑗(𝑡),ℎ₀^(𝑗)(𝑡,𝑥_𝑗)=𝑥²_𝑗(𝑡),𝑤_1𝑗(𝑠)=(1/2)𝑠,𝑤_2𝑗(𝑠)=2𝑠, then from the definition ℎ₀^(𝑗)(𝑡,𝑥_𝑗𝑡)=sup_{−∞<𝜃≤𝑡}𝑥²_𝑗(𝜃)=||𝑥²_𝑗𝑡||,𝑗=1,2. For given 0<𝑢<𝑣, we assume ||𝑥²_𝑗𝑡||<𝑢 implies that there exists a 𝐾∈𝑅⁺ such that 𝑥²_𝑗(𝑡)<𝐾𝑥²_𝑗(𝜃) for any 𝜃∈(−∞,𝑡]. Let ℎ^(𝑗)(𝑡,𝑥_𝑗)=𝑥²_𝑗(𝑡)/(𝐾+1),𝜙^(𝑖)(𝑡)=(𝐾/(𝐾+1))𝑡,   then 𝜙^(𝑖)(𝑢)<𝑢<𝑣; furthermore, when ℎ₀^(𝑗)(𝑡,𝑥_𝑗𝑡)=||𝑥²_𝑗𝑡||<𝑢, then for 𝜃∈(−∞,𝑡], ℎ^(𝑗)𝑡,𝑥_𝑗=𝑥²_𝑗(𝑡)≤𝐾𝐾+1𝑥𝐾+1^2𝑗(𝜃)≤𝐾𝑥𝐾+1^2𝑗𝑡=𝜙^(𝑗)ℎ₀^(𝑗)𝑡,𝑥_𝑗𝑡, (3.15) 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 𝑉^′₁(𝑡)=−2𝑎₁(𝑡)𝑥²₁(𝑡)+2𝑎₂(𝑡)𝑥₁(𝑡)𝑥₂(𝑡)+2𝑏₁(𝑡)𝑥₁(𝑡)𝑥₁𝑡−𝑟₁(𝑡)+2𝑥₁(𝑡)⁰_−∞𝑔₁𝑡,𝑢,𝑥₁()𝑎𝑡+𝑢𝑑𝑢≤−2₁(𝑡)−𝑎₂(𝑡)−𝑏₁(𝑡)−⁰_−∞𝑚₁(𝑢)𝑑𝑢𝑥²₁(𝑡)≤0, (3.16) similarly, when 𝑉₁(𝑡)≤𝑉₂(𝑡) and for 𝑠∈(−∞,𝑡],𝑉₂(𝑠)≤𝑉₂(𝑡), we also have 𝑉^′₂(𝑡)≤0. Thus, condition (ii) in Theorem 3.1 is satisfied and the zero solution of system (3.14) is (ℎ₀,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).