New Results on Impulsive Functional Differential Equations with Infinite Delays

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.

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],𝑅_𝑛) 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 Section 2, 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

Let 𝑅 denote the set of real numbers, 𝑅₊ the set of nonnegative real numbers, and 𝑅^𝑛 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 𝑥^′(𝑡)=𝑓(𝑡,𝑥(⋅)),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,Δ𝑥|_{𝑡=𝑡𝑘}=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,𝑘=1,2,…, (1) where the impulse times 𝑡_𝑘 satisfy 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯,lim_𝑘→+∞𝑡_𝑘=+∞ and 𝑥^′ denotes the right-hand derivative of 𝑥. 𝑓∈𝐶([𝑡_𝑘−1,𝑡_𝑘)×𝐶,𝑅^𝑛),𝑓(𝑡,0)=0.𝐶 is an open set in PC([𝛼,0],𝑅^𝑛), where PC([𝛼,0],𝑅^𝑛)={𝜓∶[𝛼,0]→𝑅^𝑛 is continuous everywhere except at finite number of points 𝑡_𝑘, at which 𝜓(𝑡⁺_𝑘) and 𝜓(𝑡⁻_𝑘) exist and 𝜓(𝑡⁺_𝑘)=𝜓(𝑡_𝑘)}. For each 𝑘=1,2,…,𝐼(𝑡,𝑥)∈𝐶([𝑡₀,∞)×𝑅^𝑛,𝑅^𝑛),𝐼(𝑡_𝑘,0)=0.

For any 𝜌>0, there exists a 𝜌₁>0(0<𝜌₁<𝜌) such that 𝑥∈𝑆(𝜌₁) implies that 𝑥+𝐼(𝑡_𝑘,𝑥)∈𝑆(𝜌), where 𝑆(𝜌)={𝑥∶‖𝑥‖<𝜌,𝑥∈𝑅^𝑛}.

Define PCB(𝑡)={𝑥∈𝐶∶𝑥isbounded}. For 𝜓∈PCB(𝑡), the norm of 𝜓 is defined by ‖𝜓‖=sup_{𝛼≤𝜃≤0}|𝜓(𝜃)|. For any𝜎≥0, let PCB_𝛿(𝜎)={𝜓∈PCB(𝜎)∶‖𝜓‖<𝛿}.

For any given 𝜎≥𝑡₀, the initial condition for system (1) is given by 𝑥_𝜎=𝜙, (2) where 𝜙∈PC([𝛼,0],𝑅^𝑛).

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(𝑡_𝑘,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(𝑡_𝑘,𝜓(0))=𝑉(𝑡_𝑘,𝜓(0)+𝐼_𝑘(𝑡_𝑘,𝜓))−𝑉(𝑡⁻_𝑘,𝜓(0)),𝑘=1,2,…;  Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1,𝑘=1,2,….

We introduce some definitions as follows.

Definition 1 (see [4]). The function 𝑉∶[𝛼,∞)×𝐶→𝑅₊ belongs to class 𝑣₀ if(𝐴₁)𝑉 is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×𝐶 and lim_{(𝑡,𝜑)→(𝑡}⁻_𝑘,𝜓)𝑉(𝑡,𝜑)=𝑉(𝑡⁻_𝑘,𝜓) exists;(𝐴₂)𝑉(𝑡,𝑥) is locally Lipschitzian in 𝑥 and 𝑉(𝑡,0)≡0.

Definition 2 (see [4]). Let 𝑉∈𝑣₀, for any (𝑡,𝜓)∈[𝑡_𝑘−1,𝑡_𝑘)×𝐶, the upper right-hand Dini derivative of 𝑉(𝑡,𝑥) along the solution of (1)-(2) is defined by 𝐷⁺𝑉(0=𝑡,𝜓)limsup_ℎ→0+𝑉𝑡+ℎ,𝜓(0)+ℎ𝑓𝑡,𝜓−𝑉𝑡,𝜓(0)ℎ. (3) 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))=𝑉̇(𝑡,𝜓(0)), where 𝐷 is any of the four Dini derivatives.
For 𝑉∈𝑣₀,(𝑡,𝜓)∈[𝑡_𝑘−1,𝑡_𝑘)×𝐶, the upper right-hand Dini derivative of 𝑉̇(𝑡,𝑥) along the solution of (1)-(2) is defined by 𝐷⁺𝑉̇(0)=𝑡,𝜓limsup_ℎ→0+𝑉̇𝑡+ℎ,𝜓(0)+ℎ𝑓𝑡,𝜓−𝑉̇𝑡,𝜓(0)ℎ. (4) Similarly, we can define 𝐷⁻𝑉̇(𝑡,𝜓(0)). If 𝑉∈𝐶^′′, then these are simply the second derivative of 𝑉.

Definition 3 (see [4]). Assume 𝑥(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 any 𝜀>0 and 𝜎≥𝑡₀, there exists a 𝛿=𝛿(𝜀)>0 such that𝜙∈PCB_𝛿(𝜎) implies ‖𝑥(𝑡)‖<𝜀,𝑡≥𝜎.(2)uniformly asymptotically stable, if it is uniformly stable, and there exists a 𝛿>0 such that for any𝜀>0,𝜎≥𝑡₀, there is a 𝑇=𝑇(𝜀)>0 such that 𝜙∈PCB_𝛿(𝜎) implies ‖𝑥(𝑡)‖<𝜀,𝑡≥𝜎+𝑇.

3. Main Results

Theorem 4. Assume that there exist functions 𝑤_𝑖∈𝐾₁,𝑔∈𝐾₂,𝑐_𝑖,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 any 𝜎≥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≠𝑡_𝑘, then 𝐷⁺𝑉𝑡,𝜓(0)≤𝑝(𝑡)𝑐₁𝑉𝑡,𝜓(0), (5) where 𝑠/𝑚≤𝑔(𝑠)<𝑠 for any 𝑠>0;(iii)for all (𝑡,𝜓(0))∈(𝑡_𝑘−1,𝑡_𝑘)×PC([𝛼,0],𝑆(𝜌₁)), 𝐷⁻𝑉̇𝑡,𝜓(0)≥0. (6) Also, for all (𝑡_𝑘,𝜓)∈𝑅₊×PC([𝛼,0],𝑆(𝜌₁)), Δ𝑡_𝑘𝑉̇𝑡⁻_𝑘,𝜓(0)+Δ𝑉𝑡_𝑘,𝜓(0)≤−𝜇_𝑘𝑐₂𝑉𝑡_𝑘,𝜓(0), (7) where 𝑐₂(𝑠)≤𝑠𝑐^′₂(𝑠),𝑠>0,𝜇_𝑘 satisfies liminf_𝑘→∞𝜇_𝑘≥2⋅sup_𝑠>0(𝑠/𝑐₂(𝑚⁻¹⋅𝑠));(iv)there exist constants 𝑀₁,𝑀₂>0 such that the following inequalities hold: sup_𝑡≥0𝑡^𝑡+𝜏𝑝(𝑠)𝑑𝑠=𝑀₁<∞,inf_𝑠>0^𝑠_𝑔(𝑠)𝑑𝑡𝑐₁(𝑡)=𝑀₂>𝑀₁, (8) where 𝜏=max_𝑘≥1{𝑡_𝑘−𝑡_𝑘−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 𝑊₁(‖𝑥‖)≤𝑉(𝑡,𝑥)≤𝑊₂(‖𝑥‖),(𝑡,𝑥)∈[𝛼,∞)×𝑆𝜌. (9)
We first show uniform stability.
For any 𝜀>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∈PCB_𝛿(𝜎), we will prove that ‖𝑥(𝑡)‖<𝜀,𝑡≥𝜎.
For convenience, let 𝑉(𝑡)=𝑉(𝑡,𝑥(𝑡)). Suppose that 𝜎∈[𝑡_𝑙−1,𝑡_𝑙),𝑙∈𝑍₊. First, for 𝜎+𝛼≤𝑡≤𝜎, we have 𝑊₁(‖𝑥‖)&≤𝑉(𝑡)<𝑊₂(𝛿)≤𝑔𝑊₁(𝜀)<𝑊₁(𝜀). (10) So, ‖𝑥(𝑡)‖<𝜀<𝜌₁,𝑡∈[𝜎+𝛼,𝜎].
Next, we claim that 𝑉(𝑡)<𝑊₁(𝜀),𝑡∈𝜎,𝑡_𝑙. (11) Suppose on the contrary that there exists some 𝑡∈[𝜎,𝑡_𝑙) such that 𝑉(𝑡)≥𝑊₁(𝜀). Since 𝑉(𝜎)<𝑊₁(𝜀), we can define 𝑡̂=inf{𝑡∈[𝜎,𝑡_𝑙)∣𝑉(𝑡)≥𝑊₁(𝜀)}. Thus, 𝑡̂∈(𝜎,𝑡_𝑙),𝑉(𝑡̂)=𝑊₁(𝜀), and 𝑉(𝑡)<𝑊₁(𝜀),𝑡∈[𝜎,𝑡)̂. Also, from (10) we obtain 𝑉(𝑡)<𝑊₁(𝜀),𝑡∈𝜎+𝛼,𝑡̂. (12) 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)̂𝑉(𝑡)>𝑔𝑊₁(𝜀)>𝑔(𝑉(𝑡+𝜃)),𝛼≤𝜃≤0; (13) that is, 𝑉𝑡,𝜓(0)>𝑚⁻²𝑔𝑉(𝜃,(𝑉(𝑡𝑡+𝜃,𝜓)max𝛼,−𝑞))≤𝜃≤0. (14) By assumption (ii), (iv), we have _)𝑉(𝑡_𝑡∗)̂_𝑉(𝑑𝑠𝑐₁(𝑠)=^𝑊1_𝑔𝑊^(𝜀)_1(𝜀)𝑑𝑠𝑐₁(𝑠)≥𝑀₂>𝑀₁. (15) However, we also have _)𝑉(𝑡_𝑡∗)̂_𝑉(𝑑𝑠𝑐₁(𝑠)≤_𝑡𝑡_∗̂𝑝(𝑠)𝑑𝑠<^𝑡∗_𝑡^+𝜏_∗𝑝(𝑠)𝑑𝑠≤𝑀₁, (16) which is a contradiction. So, (11) holds.
Hence, 𝑊₁(‖𝑥‖)≤𝑉(𝑡)<𝑊₁(𝜀),𝑡∈[𝜎,𝑡_𝑙) implies that ‖𝑥(𝑡⁻_𝑙)‖<𝜀<𝜌₁. Thus, 𝑥(𝑡_𝑙)∈𝑆(𝜌).
On the other hand, from condition (iii), we note for 𝑘=1,2,…, 𝑉𝑡_𝑘−𝑉𝑡_𝑘−1=𝑉𝑡_𝑘−𝑉𝑡⁻_𝑘+𝑉𝑡⁻_𝑘−𝑉𝑡_𝑘−1=Δ𝑉𝑡_𝑘+^𝑡𝑘_𝑡𝑘−1𝑉̇(𝑡)𝑡𝑑𝑡≤Δ𝑉_𝑘+Δ𝑡_𝑘𝑉̇𝑡⁻_𝑘≤−𝜇_𝑘𝑐₂𝑉𝑡_𝑘≤0. (17) Hence, we obtain 𝑉(𝑡_𝑘)≤𝑉(𝑡_𝑘−1),𝑘=1,2,…. particularly, 𝑉(𝑡_𝑙)≤𝑉(𝑡_𝑙−1). In view of (10), we get 𝑉𝑡_𝑙≤𝑉𝑡_𝑙−1<𝑔𝑊₁(𝜀)<𝑊₁(𝜀). (18) Next, we claim that 𝑉(𝑡)<𝑊₁(𝜀),𝑡∈𝑡_𝑙,𝑡_𝑙+1. (19) Suppose on the contrary that there exists some 𝑡∈[𝑡_𝑙,𝑡_𝑙+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 for 𝑡∈[𝑡_𝑙+𝑘,𝑡_𝑙+𝑘+1),𝑘>0, 𝑉(𝑡)<𝑊₁(𝜀). (20) 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∈PCB_𝛿(𝜎) implies that 𝑉(𝑡)<𝑊₁(𝐻),‖𝑥(𝑡)‖<𝜌₁,𝑡≥𝜎.
For any 𝜀∈(0,𝐻), let 𝑑<min𝑑,𝑊₁(𝜀,(𝑠))𝑑=inf𝑠−𝑔∣𝑚⁻¹𝑊₁(𝜀)≤𝑠≤𝑊₁(𝐻,𝑞(𝑠))ℎ=sup∣𝑚⁻¹𝑊₁(𝜀)≤𝑠≤𝑊₁(𝐻,𝑛₀)=𝑊₁(𝐻)2⋅sup_𝑠>0𝑠/𝑐₂𝑚⁻¹𝑠𝑐₂𝑚⁻¹𝑊₁(𝜀)+1. (21) From condition (iii), we get that there exists a 𝑛₁>0 such that for 𝑘>𝑛₁, 𝜇_𝑘≥2⋅sup_𝑠>0𝑠𝑐₂𝑚⁻¹.⋅𝑠 (22) Choose a positive integer 𝑁 satisfying 𝑊₁(𝜀)+(𝑁−1)𝑑<𝑊₁(𝐻)≤𝑊₁(𝜀)+𝑁𝑑, (23) and define 𝑇=𝑁(ℎ+𝑛₀𝜏)+𝑛₁, we will prove that 𝜙∈PCB_𝛿(𝜎) implies ‖𝑥(𝑡)‖<𝜀,𝑡≥𝜎+𝑇.
First, we prove that there exists 𝑡̂∈[𝜎+ℎ+𝑛₁,𝜎+ℎ+𝑛₁+𝑛₀𝜏] such that 𝑉𝑡̂<𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑. (24) Suppose on the contrary that for all 𝑡∈[𝜎+ℎ+𝑛₁,𝜎+ℎ+𝑛₁+𝑛₀𝜏], 𝑉(𝑡)≥𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑≥𝑚⁻¹𝑊₁(𝜀). (25) Let 𝑡_𝑘1=min{𝑡_𝑘∶𝑡_𝑘≥𝜎+ℎ+𝑛₁}, from (17), we get 𝑉𝑡_𝑘1−𝑉𝑡_𝑘1−1≤−𝜇_𝑘1𝑐₂𝑉𝑡_𝑘1≤−𝜇_𝑘1𝑐₂𝑚⁻¹𝑊₁(𝜀),𝑉𝑡_𝑘1+1−𝑉𝑡_𝑘1≤−𝜇_𝑘1+1𝑐₂𝑚⁻¹𝑊₁(𝜀),⋮𝑉𝑡_𝑘1+𝑛0−𝑉𝑡_{𝑘1+𝑛0−1}≤−𝜇_𝑘1+𝑛0𝑐₂𝑚⁻¹𝑊₁(𝜀), (26) In general, combining (22), we deduce that 𝑉𝑡_𝑘1+𝑛0≤𝑉𝑡_𝑘1−1−^𝑛0_𝑠=0𝜇_𝑘1+𝑠𝑐₂𝑚⁻¹𝑊₁(𝜀)≤𝑊₁(𝐻)−2𝑛₀+1⋅sup_𝑠>0𝑠𝑐₂𝑚⁻¹𝑠𝑐₂𝑚⁻¹𝑊₁(𝜀)=−4⋅sup_𝑠>0𝑠𝑐₂𝑚⁻¹𝑠𝑐₂𝑚⁻¹𝑊₁(𝜀)<0, (27) which is a contradiction. So, (24) holds.
Suppose 𝑡̂∈[𝑡_𝑙−1,𝑡_𝑙),𝑙>1. Furthermore, we can prove that for 𝑡∈[𝑡,̂𝑡_𝑙)𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−1)𝑑. (28) Suppose this assertion is false, then there exists some 𝑡∈[𝑡,̂𝑡_𝑙) such that 𝑉(𝑡)≥𝑊₁(𝜀)+(𝑁−1)𝑑. Since 𝑉(𝑡̂)<𝑚⁻¹[𝑊₁(𝜀)+(𝑁−1)𝑑]<𝑊₁(𝜀)+(𝑁−1)𝑑, so define 𝑡^∗=inf𝑡∈𝑡,̂𝑡_𝑙∣𝑉(𝑡)≥𝑊₁(𝜀)+(𝑁−1)𝑑; (29) then 𝑡^∗∈(𝑡,̂𝑡_𝑙),𝑉(𝑡^∗)=𝑊₁(𝜀)+(𝑁−1)𝑑 and 𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−1)𝑑,𝑡∈(𝑡,̂𝑡^∗). Note that 𝑉𝑡^∗=𝑊₁(𝜀)+(𝑁−1)𝑑>𝑔𝑊₁(𝜀)+()𝑑,𝑉𝑡̂𝑁−1<𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑<𝑔𝑊₁(𝜀)+(𝑁−1)𝑑; (30) thus, we can define 𝑡=sup𝑡∈𝑡,̂𝑡^∗∣𝑉(𝑡)≤𝑔𝑊₁(𝜀)+(𝑁−1)𝑑, (31) 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 for 𝑡∈(𝑡,𝑡^∗]𝑉(𝑡)>𝑔𝑊₁(𝜀)+(𝑁−1)𝑑≥𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑≥𝑚⁻¹𝑊₁(𝜀), (32) which implies that for 𝑡∈(𝑡,𝑡^∗]𝑉(𝑡)≥𝑔(𝑉(𝑡))+𝑑≥𝑚⁻¹𝑉(𝑡)>(𝑡)+𝑑𝑚𝑉𝑚²+𝑑𝑚²≥𝑊₁(𝜀)+𝑁𝑑𝑚²≥𝑊₁(𝐻)𝑚²≥𝑉(𝑠)𝑚²>𝑔(𝑉(𝑠))𝑚²,𝑡+𝛼<𝑠≤𝑡. (33) 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^∗), ^{𝑉(𝑡∗)}_𝑉𝑡𝑑𝑠𝑐₁(𝑠)=^𝑊1_𝑔𝑊^{(𝜀)+(𝑁−1)𝑑}_{1(𝜀)+()𝑑𝑁−1}𝑑𝑠𝑐₁(𝑠)≥𝑀₂>𝑀₁. (34) However, we also have ^{𝑉(𝑡∗)}_𝑉𝑡𝑑𝑠𝑐₁(𝑠)<^𝑡∗_𝑡𝑝(𝑠)𝑑𝑠<^𝑡+𝜏_𝑡𝑝(𝑠)𝑑𝑠<𝑀₁, (35) which is a contradiction. So, (28) holds.
On the other hand, it is easy to prove that the functions 𝑠/𝑐₂(𝑚⁻¹𝑠) are nonincreasing for 𝑠∈(0,+∞) in view of condition 𝑐₂(𝑠)≤𝑠𝑐^′₂(𝑠) for any 𝑠>0.
Hence, the following inequalities hold: for 𝑘>𝑛₁, 𝑊₁(𝜀)+(𝑁−𝑖)𝑑𝑐₂𝑚⁻¹𝑊₁(𝜀)+()𝑑≤𝑊𝑁−𝑖−1¹(𝜀)+𝑑𝑐₂𝑚⁻¹𝑊₁(𝜀)<2𝑊₁(𝜀)𝑐₂𝑚⁻¹𝑊₁(𝜀)≤𝜇_𝑘,𝑖=1,2,…,𝑁−1. (36) Next, we claim that 𝑉𝑡_𝑙<𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑. (37) Or else, then 𝑉(𝑡_𝑙)≥𝑚⁻¹[𝑊₁(𝜀)+(𝑁−1)𝑑]; from (17), we get 𝑉𝑡_𝑙−𝑉𝑡_𝑙−1≤−𝜇_𝑙𝑐₂𝑉𝑡_𝑙≤−𝜇_𝑙𝑐₂𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑. (38) Considering (36), it holds that 𝑉𝑡_𝑙≤𝑉𝑡_𝑙−1−𝜇_𝑙𝑐₂𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑≤𝑊₁(𝐻)−𝜇_𝑙𝑐₂𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑≤𝑊₁(𝜀)+𝑁𝑑−𝜇_𝑙𝑐₂𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑≤𝑐₂𝑚⁻¹𝑊₁(𝜀)+()𝑑×𝑊𝑁−1₁(𝜀)+𝑁𝑑𝑐₂𝑚⁻¹𝑊₁(𝜀)+(𝑁−1)𝑑−𝜇_𝑙<0, (39) which is a contradiction and (37) holds.
Next, we can prove that for 𝑡∈[𝑡_𝑙,𝑡_𝑙+1)𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−1)𝑑. (40) Suppose that this assertion is false, then there exists some 𝑡∈[𝑡,̂𝑡_𝑙) 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, for 𝑡∈[𝑡_𝑙+𝑘,𝑡_𝑙+𝑘+1),𝑘=1,2,…, 𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−1)𝑑; (41) that is, 𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−1)𝑑,𝑡≥𝑡.̂ (42) Hence, we obtain 𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−1)𝑑,𝑡≥𝜎+ℎ+𝑛₁+𝑛₀𝜏. (43) Next, we prove that there exists 𝑡₂̂∈[𝜎+2ℎ+𝑛₁+𝑛₀𝜏,𝜎+2ℎ+𝑛₁+2𝑛₀𝜏] such that 𝑉𝑡₂̂<𝑚⁻¹𝑊₁(𝜀)+(𝑁−2)𝑑. (44) Suppose that for all 𝑡∈[𝜎+2ℎ+𝑛₁+𝑛₀𝜏,𝜎+2ℎ+𝑛₁+2𝑛₀𝜏], 𝑉(𝑡)≥𝑚⁻¹𝑊₁(𝜀)+(𝑁−2)𝑑≥𝑚⁻¹𝑊₁(𝜀). (45) Using the same argument as in the proof of (24), we get 𝑉𝑡_𝑘2+𝑛0≤𝑉𝑡_𝑘2−1−^𝑛0_𝑠=0𝜇_𝑘2+𝑠𝑐₂𝑚⁻¹𝑊₁(𝜀)≤𝑊₁(𝐻)−2𝑛₀+1⋅sup_𝑠>0𝑠𝑐₂𝑚⁻¹𝑠𝑐₂𝑚⁻¹𝑊₁(𝜀)=−4⋅sup_𝑠>0𝑠𝑐₂𝑚⁻¹𝑠𝑐₂𝑚⁻¹𝑊₁(𝜀)<0, (46) where 𝑡_𝑘2=min{𝑡_𝑘∶𝑘≥𝜎+2ℎ+𝑛₁+𝑛₀𝜏}.
This is a contradiction. So, (44) holds.
Suppose 𝑡₂̂∈[𝑡_𝑘−1,𝑡_𝑘),𝑘>𝑙. Furthermore, we claim that for 𝑡∈[𝑡₂̂,𝑡_𝑘) 𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−2)𝑑. (47) Suppose on the contrary, that there exists some 𝑡∈[𝑡₂̂,𝑡_𝑘) such that 𝑉(𝑡)≥𝑊₁(𝜀)+(𝑁−2)𝑑. We define 𝑡^⋆=inf𝑡∈𝑡₂̂,𝑡_𝑘∣𝑉(𝑡)≥𝑊₁(𝜀)+(𝑁−2)𝑑, (48) since 𝑉(𝑡₂̂)<𝑚⁻¹[𝑊₁(𝜀)+(𝑁−2)𝑑]<𝑊₁(𝜀)+(𝑁−2)𝑑 in view of (44). Thus, 𝑡^⋆∈(𝑡₂̂,𝑡_𝑘),𝑉(𝑡^⋆)=𝑊₁(𝜀)+(𝑁−2)𝑑 and 𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−2)𝑑,𝑡∈(𝑡₂̂,𝑡^⋆). Note that 𝑉𝑡^⋆=𝑊₁(𝜀)+(𝑁−2)𝑑>𝑔𝑊₁(𝜀)+()𝑑,𝑉𝑡̂𝑁−2₂<𝑚⁻¹𝑊₁(𝜀)+(𝑁−2)𝑑<𝑔𝑊₁(𝜀)+(𝑁−2)𝑑; (49) furthermore, we can define ^∼𝑡=sup𝑡∈𝑡₂̂,𝑡^⋆∣𝑉(𝑡)≤𝑔𝑊₁(𝜀)+(𝑁−2)𝑑, (50) then ^∼𝑡∈[𝑡₂̂,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 for 𝑡∈(^∼𝑡,𝑡^⋆]𝑉(𝑡)>𝑔𝑊₁(𝜀)+(𝑁−2)𝑑≥𝑚⁻¹𝑊₁(𝜀)+(𝑁−2)𝑑≥𝑚⁻¹𝑊₁(𝜀); (51) considering the definition of 𝑑 and (43), we get for 𝑡∈(^∼𝑡,𝑡^⋆]𝑉(𝑡)≥𝑔(𝑉(𝑡))+𝑑≥𝑚⁻¹𝑉(𝑡)>(𝑡)+𝑑𝑚𝑉𝑚²+𝑑𝑚²≥𝑊₁(𝜀)+(𝑁−1)𝑑𝑚²≥𝑉(𝑠)𝑚²>𝑔(𝑉(𝑠))𝑚²,𝑡−ℎ<𝑠≤𝑡. (52) Thus, 𝑉(𝑡)≥(1/𝑚²)𝑔(𝑉(𝑡+𝜃,𝜓(𝜃))),max{𝛼,−𝑞(𝑉(𝑡))}≤𝜃≤0.
Using assumptions (ii),(iv), we have ^{𝑉(𝑡⋆)}_𝑉∼𝑡𝑑𝑠𝑐₁(𝑠)=^{𝑊1(𝜀)+(𝑁−2)𝑑}_{𝑔(𝑊1(𝜀)+(𝑁−2)𝑑)}𝑑𝑠𝑐₁(𝑠)≥𝑀₂>𝑀₁. (53) However, ^{𝑉(𝑡⋆)}_{𝑉(∼𝑡)}𝑑𝑠𝑐₁(𝑠)<^𝑡⋆_∼𝑡𝑝(𝑠)𝑑𝑠<^∼𝑡+𝜏_∼𝑡𝑝(𝑠)𝑑𝑠<𝑀_1, (54) giving us a contradiction. So, (47) holds.
Next, we claim that 𝑉𝑡_𝑙<𝑚⁻¹𝑊₁(𝜀)+()𝑑,𝑉(𝑡)𝑁−1<𝑊₁(𝜀)+(𝑁−1)𝑑,𝑡∈𝑡_𝑙,𝑡_𝑙+1, (55) 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 𝑉(𝑡)<𝑊₁(𝜀)+(𝑁−2)𝑑,𝑡≥𝜎+2ℎ+𝑛₁+2𝑛₀𝜏. (56) By induction hypothesis, we have 𝑉(𝑡)≤𝑊₁(𝜀)+(𝑁−𝑖)𝑑,𝑡≥𝜎+𝑖ℎ+𝑛₁+𝑖𝑛₀𝜏. (57) Let 𝑖=𝑁, then for 𝑡≥𝜎+𝑁(ℎ+𝑛₀𝜏)+𝑛₁, 𝑉(𝑡)<𝑊₁(𝜀). (58) Therefore, we arrive at ‖𝑥(𝑡)‖<𝜀,𝑡≥𝑇. The proof of Theorem 4 is complete.

Corollary 5. Assume that there exist functions 𝑤_𝑖∈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 any 𝜎≥𝑡₀ and 𝜓∈PC([𝛼,0],𝑆(𝜌)), if  𝑉(𝑡,𝜓(0))≥𝑔(𝑉(𝑡+𝜃,𝜓(𝜃))),𝛼≤𝜃≤0,𝑡≠𝑡_𝑘, then 𝐷⁺𝑉𝑡,𝜓(0)≤𝑝(𝑡)𝑐𝑉𝑡,𝜓(0), (59) where (𝑠/𝑚)≤𝑔(𝑠)<𝑠 for any 𝑠>0;(iii)for all (𝑡,𝜓(0))∈(𝑡_𝑘−1,𝑡_𝑘)×PC([𝛼,0],𝑆(𝜌₁)), 𝐷⁻𝑉̇𝑡,𝜓(0)≥0. (60) Also, for all (𝑡_𝑘,𝜓)∈𝑅₊×PC([𝛼,0],𝑆(𝜌₁)), Δ𝑡_𝑘𝑉̇𝑡⁻_𝑘,𝜓(0)+Δ𝑉𝑡_𝑘,𝜓(0)≤0; (61)(iv)there exist constants 𝑀₁,𝑀₂>0 such that the following inequalities hold: sup_𝑡≥0𝑡^𝑡+𝜏𝑝(𝑠)𝑑𝑠=𝑀₁<∞,inf_𝑠>0^𝑠_𝑔(𝑠)𝑑𝑡𝑐(𝑡)=𝑀₂>𝑀₁, (62) where 𝜏=max_𝑘≥1{𝑡_𝑘−𝑡_𝑘−1}<∞.
Then the zero solution of (1)-(2) is uniformly stable.

Theorem 4 has a dual result when 𝑉̇ is nonincreasing on (𝑡_𝑘−1,𝑡_𝑘). Here, we only give the results whose proof is very similar to the proof of Theorem 4.

Theorem 6. Assume that there exist functions 𝑤_𝑖∈𝐾₁,𝑔∈𝐾₂,𝑐_𝑖,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 any 𝜎≥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≠𝑡_𝑘, then 𝐷⁺𝑉𝑡,𝜓(0)≤𝑝(𝑡)𝑐₁𝑉𝑡,𝜓(0), (63) where (𝑠/𝑚)≤𝑔(𝑠)<𝑠 for any 𝑠>0;(iii)for all (𝑡,𝜓(0))∈(𝑡_𝑘−1,𝑡_𝑘)×PC([𝛼,0],𝑆(𝜌₁)), 𝐷⁻𝑉̇𝑡,𝜓(0)≤0. (64) Also, for all (𝑡_𝑘,𝜓)∈𝑅₊×PC([𝛼,0],𝑆(𝜌₁)), Δ𝑡_𝑘𝑉̇𝑡⁻_𝑘−1,𝜓(0)+Δ𝑉𝑡_𝑘,𝜓(0)≤−𝜇_𝑘𝑐₂𝑉𝑡_𝑘,𝜓(0), (65) where 𝑐₂(𝑠)≤𝑠𝑐^′₂(𝑠),𝑠>0,𝜇_𝑘 satisfies liminf_𝑘→∞𝜇_𝑘≥2⋅sup_𝑠>0(𝑠/𝑐₂(𝑚⁻¹⋅𝑠));(iv)there exist constants 𝑀₁,𝑀₂>0 such that the following inequalities hold: sup_𝑡≥0𝑡^𝑡+𝜏𝑝(𝑠)𝑑𝑠=𝑀₁<∞,inf_𝑠>0^𝑠_𝑔(𝑠)𝑑𝑡𝑐₁(𝑡)=𝑀₂>𝑀₁, (66)(67) where 𝜏=max_𝑘≥1{𝑡_𝑘−𝑡_𝑘−1}<∞.
Then the zero solution of (1)-(2) is uniformly asymptotically stable.

Example 7. Consider the impulsive delay differential equations: 𝑥^′(𝑡)=𝑎𝑥(𝑡)−𝑏⁰_−∞𝑒^𝑠𝑥(𝑡+𝑠)𝑑𝑠,𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑥)𝑥,𝑘=1,2,…,₀=𝜙>0, (68) where 𝑎∈(0,3],𝑏∈(0,2],|𝑥+𝐼_𝑘(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: 𝜏<minln𝑚2𝑎−𝑏√𝑚,1−𝜆−2𝜆𝑚2𝑎,𝑎>2𝑏−1, (69) where 𝜏=max_𝑘≥1{𝑡_𝑘−𝑡_𝑘−1}<∞. Then, the zero solution of (68) is uniformly asymptotically stable.

In fact, let 𝑉(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≠𝑡_𝑘𝐷⁺𝑉(𝑡,𝑥(⋅))=𝑥(𝑡)𝑥^′(𝑡)=𝑥(𝑡)𝑎𝑥(𝑡)−𝑏⁰_−∞𝑒^𝑠𝑥(𝑡+𝑠)𝑑𝑠≤𝑥²(𝑡)𝑎−𝑏√𝑚⁰_−∞𝑒^𝑠𝑑𝑠=𝑥²(𝑡)𝑎−𝑏√𝑚=𝑝(𝑡)𝑉(𝑡,𝑥(𝑡)), (70) where 𝑝(𝑡)=2(𝑎−𝑏√𝑚).

In view of condition (69), we note sup_𝑡≥0𝑡^𝑡+𝜏𝑝(𝑠)𝑑𝑠=2𝜏𝑎−𝑏√𝑚<ln𝑚=inf_𝑠>0^𝑠_𝑔(𝑠)𝑑𝑡𝑐₁(𝑡). (71) So, condition (iv) in Corollary 5 holds.

On the other hand, we have for 𝑡≠𝑡_𝑘𝐷⁻𝑉̇(𝑡,𝑥(⋅))=𝑥(𝑡)𝑥^′(𝑡)^′=𝑥(𝑡)𝑥^′′(𝑡)+𝑥^′(𝑡)²=𝑥^′(𝑡)²+𝑥(𝑡)𝑎𝑥(𝑡)−𝑏⁰_−∞𝑒^𝑠𝑥(𝑡+𝑠)𝑑𝑠^′=𝑥^′(𝑡)²+𝑎𝑥(𝑡)𝑥^′(𝑡)−𝑏𝑥(𝑡)⁰_−∞𝑒^𝑠𝑥(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()+()𝑥𝑎−1𝑎−𝑏²(𝑡)=3−𝑎2𝑥^′(𝑡)²+𝑎+12−𝑏𝑥²(𝑡)≥0, (72) in view of condition 𝑎>2𝑏−1. Also, considering 𝑥(𝑡) to be a positive solution of (68), we get Δ𝑡_𝑘𝑉̇𝑡⁻_𝑘,𝜓(0)+Δ𝑉𝑡_𝑘,𝜓(0)≤𝑎𝜏𝑥²𝑡⁻_𝑘+()𝑥𝜆−1²𝑡⁻_𝑘2=(2𝑎𝜏+𝜆−1)𝑥²𝑡⁻_𝑘𝑥²𝑡⁻_𝑘2=(2𝑎𝜏+𝜆−1)𝑉𝑡⁻_𝑘≤−1−2𝑎𝜏−𝜆𝜆𝑉𝑡_𝑘=−𝜇_𝑘𝑐₂𝑉𝑡_𝑘,𝜓(0), (73) where 𝑐₂=𝑠,𝜇_𝑘=(1−2𝑎𝜏−𝜆)/𝜆.

Note that sup_𝑠>02𝑠𝑐₂𝑚⁻¹⋅𝑠=2𝑚<1−2𝑎𝜏−𝜆𝜆=𝜇_𝑘, (74) in view of (69). So, the zero solution of (68) is uniformly stable by Corollary 5.

Furthermore, choose 𝑞(𝑠)=−ln(1−1/𝑚) (positive constants), which implies that ∫⁰_{−𝑞(𝑠)}𝑒^𝑠𝑑𝑠=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 𝑚^3/2|𝑥(𝑡)|>|𝑥(𝑠)|,max{𝛼,𝑡−𝑞(𝑉(𝑡))}≤𝑠≤𝑡, then 𝐷⁺𝑉₍₆₈₎(𝑡,𝑥(⋅))≤𝑎𝑥²(𝑡)−𝑏𝑥(𝑡)⁰_−∞𝑒^𝑠|𝑥(𝑡+𝑠)|𝑑𝑠≤𝑎𝑥²(𝑡)−𝑏𝑥(𝑡)^𝑡_−∞𝑒^𝑠−𝑡|𝑥(𝑠)|𝑑𝑠≤𝑎𝑥²(𝑡)−𝑏𝑥(𝑡)^𝑡_{(𝑉((⋅𝑡−𝑞𝑡,𝑥)))}𝑒^𝑠−𝑡|𝑥(𝑠(𝑡))|𝑑𝑠−𝑥^{𝑡−𝑞(𝑉(𝑡,𝑥(⋅)))}_−∞𝑒^𝑠−𝑡|𝑥(𝑠)|𝑑𝑠≤𝑎𝑥²(𝑡)−𝑏𝑥(𝑡)^𝑡_{(𝑉((⋅𝑡−𝑞𝑡,𝑥)))}𝑒^𝑠−𝑡|𝑥(𝑠)|𝑑𝑠≤𝑥²(𝑡)𝑎−𝑏𝑚^3/20_{(𝑉((⋅−𝑞𝑡,𝑥)))}𝑒^𝑠𝑑𝑠≤𝑥²(𝑡)𝑎−𝑏√𝑚=𝑐(𝑉(𝑡,𝑥(𝑡)))𝑝(𝑡). (75) 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)𝑒^𝑡 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.