Exponential Stability of Impulsive Delay Differential Equations

Abstract

The main objective of this paper is to further investigate the exponential stability of a class of impulsive delay differential equations. Several new criteria for the exponential stability are analytically established based on Razumikhin techniques. Some sufficient conditions, under which a class of linear impulsive delay differential equations are exponentially stable, are also given. An Euler method is applied to this kind of equations and it is shown that the exponential stability is preserved by the numerical process.

1. Introduction

Impulsive differential equations arise widely in the study of medicine, biology, economics, engineering, and so forth. In recent years, theory of impulsive differential delay equations (IDDEs) has been an object of active research (see [1–18] and references therein). The results about the existence and uniqueness of IDDEs have been studied in [2, 7]. The stability of IDDEs has attracted increasing interest in both theoretical research and practical applications (see [1, 3, 5–18] and references therein). In particular, special attention has been focused on exponential stability of IDDEs (see [1, 3, 8, 9, 15]) because it has played an important role in many areas.

There is a little work done on exponential stability for IDDEs by the Lyapunov-Razumikhin method. Wang and Liu [15] have extended Lyapunov-Razumikhin method to IDDEs and established some exponential stability criteria. In this paper we restrict the length of each impulsive interval instead of some conditions in [15]. As a result, several new criteria on exponential stability are analytically derived.

There are a few papers on numerical methods of impulsive differential equations. In [19], Covachev et al. obtained a convergent difference approximation for a nonlinear impulsive ordinary system in a Banach space. In [20, 21], the authors studied the stability of Runge-Kutta methods for linear impulsive ordinary differential equations. In [4], Ding et al. studied the convergence property of an Euler method for IDDEs. In [18], asymptotic stability of numerical solutions and exact solutions of a class of linear IDDEs was studied by the property of DDEs without impulsive perturbations. The convergence of the numerical methods for this kind of equations was studied. In this paper, we study exponential stability of the numerical solutions of linear IDDEs.

The rest of the paper is organized as follows. In Section 2, we obtained two criteria on exponential stability for IDDEs by the Lyapunov-Razumikhin method. The results obtained are applied to a class of linear IDDEs. In the last section, we prove that the Euler method for the linear IDDEs preserves the analytic exponential stability.

2. Stability of Analytic Solutions

Consider the impulsive delay differential system 𝑥^′(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡∈𝑡_𝑘−1,𝑡_𝑘(𝑡),𝑘=1,2,3,…,Δ𝑥=𝐼_𝑘𝑡,𝑥_𝑡−,𝑡=𝑡_𝑘,𝑘=1,2,3,…, (1) where 𝑓∶𝑅₊×PC([−𝜏,0],𝑅^𝑑)→𝑅^𝑑; 𝐼_𝑘∶PC([−𝜏,0],𝑅^𝑑)→𝑅^𝑑; 0≤𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯, with 𝑡_𝑘→∞ as 𝑘→∞; PC([−𝜏,0],𝑅^𝑑) is a set of piecewise continuous functions 𝑔(𝑡) which have a finite number of points of discontinuity in a finite interval and 𝑔(𝑡)=𝑔(𝑡⁺) for all 𝑡. We assume that 𝑓(𝑡,0)≡0, 𝐼_𝑘(𝑡,0)≡0, so that 𝑥≡0 is a solution of (1) as 𝑥_𝑡0=Φ≡0, which we call the zero solution.

Definition 1 (see [15]). A function 𝑉: 𝑅₊×𝑅^𝑑→𝑅₊ is said to belong to the class 𝑣₀ if(i)𝑉 is continuous in each of the sets [𝑡_𝑘−1,𝑡_𝑘)×𝑅^𝑑 and for each 𝑥∈𝑅^𝑑, 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), lim_{(𝑡,𝑦)→(𝑡}⁻_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑡⁻_𝑘,𝑥) exists, 𝑘=1,2,…;(ii)𝑉(𝑡,𝑥) is locally Lipschitzian in all 𝑥∈𝑅^𝑑, and for all 𝑡≥𝑡₀,𝑉(𝑡,0)≡0.

Definition 2 (see [15]). Given a function 𝑉: 𝑅₊×𝑅^𝑑→𝑅₊, the upper right-hand derivative of 𝑉 with respect to system (1) is defined by 𝐷⁺𝑉(𝑡,Ψ(0))=limsup_ℎ→0+1ℎ𝑉𝑡+ℎ,Ψ(0)+ℎ𝑓(𝑡,Ψ)−𝑉(𝑡,Ψ(0)), (2) for (𝑡,Ψ)∈𝑅₊×PC([−𝜏,0],𝑅^𝑑).

Definition 3 (see [15]). The zero solution of (1) is said to be exponentially stable, if there exist constants 𝜆>0 and 𝑀≥1, such that for any initial data 𝑥_𝑡0=Φ∈PC([−𝜏,0],𝑅^𝑑), 𝑥𝑡,𝑡₀,Φ≤𝑀‖Φ‖_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀. (3)

Theorem 4. Assume that there exist a function 𝑉∈𝑣₀; constants 𝑑_𝑘>−1, 𝑘=1,2,…; positive constants 𝐶₁, 𝐶₂, 𝜆, 𝑙₁; and a function 𝑚(𝑡)∈𝑃𝐶([𝑡₀−𝜏,∞),𝑅₊) with inf_{𝑡≥𝑡0−𝜏}𝑚(𝑡)≥𝜆, such that for any Ψ(𝑡)∈𝑃𝐶([−𝜏,0],𝑅^𝑑) with Ψ(0⁻)=Ψ(0) the following conditions hold:(i)𝐶₁‖𝑥‖≤𝑉(𝑡,𝑥)≤𝐶₂‖𝑥‖ for all 𝑡∈𝑅₊, 𝑥∈𝑅^𝑑;(ii)𝐷⁺𝑉(𝑡,Ψ(0))≤−𝑚(𝑡)𝑉(𝑡,Ψ(0)) for all 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), whenever 𝑉(𝑡,Ψ(0))≥𝑉(𝑡+𝑠,Ψ(𝑠))𝑒^{−∫𝑡𝑡−𝜏𝑚(𝑠)𝑑𝑠} for 𝑠∈[−𝜏,0];(iii) 𝑉(𝑡_𝑘,Ψ(0)+𝐼_𝑘(𝑡_𝑘,Ψ(0)))≤(1+𝑑_𝑘)𝑉(𝑡⁻_𝑘,Ψ(0)) for 𝑘=1,2,…;(iv)𝑡_𝑘−𝑡_𝑘−1≥𝑙₁ for 𝑘=1,2,…, and −𝜆𝑙₁+ln𝐻₁<0, where 𝐻₁=sup_𝑘{1+𝑑_𝑘}.
Then the zero solution of (1) is exponentially stable.

Proof. Similar to the proof of Theorem 3.1 in [15], we obtain that 𝑉(𝑡,𝑥)≤𝐶₂^𝑘_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{−∫𝑡𝑡0𝑚(𝑠)𝑑𝑠},𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=1,2,…. (4) Since −𝜆𝑙₁+ln𝐻₁<0, there exists 𝛼 such that 0<𝛼<𝜆 and −(𝜆−𝛼)𝑙₁+ln𝐻₁<0. So 𝐶₁‖𝑥‖≤𝑉(𝑡,𝑥)≤𝐶₂𝐻^𝑘₁‖Φ‖_𝜏𝑒^{−𝜆(𝑡−𝑡0)}≤𝐶₂‖Φ‖_𝜏𝑒^{−𝜆(𝑡−𝑡0)+𝑘ln𝐻1}<𝐶₂‖Φ‖_𝜏𝑒^{−𝜆(𝑡−𝑡0)+(𝜆−𝛼)𝑘𝑙1}≤𝐶₂‖Φ‖_𝜏𝑒^{−𝛼(𝑡−𝑡0)}. (5) Hence the zero solution of (1) is exponentially stable.

Remark 5. Theorem 3.1 in [15] requires that 𝑑_𝑖≥0, and ∑^∞_𝑖=0𝑑_𝑖<∞, which implies lim_𝑘→∞𝑑_𝑘=0. In our Theorem 4, we require 𝑡_𝑘−𝑡_𝑘−1≥𝑙₁ for 𝑘=1,2,…, and sup_𝑘{1+𝑑_𝑘}<𝑒^𝜆𝑙1 instead. This means that the impulsive effects are bounded instead of tending to zero (see Example 13).

Theorem 6. Assume that there exist a function 𝑉∈𝑣₀; constants 𝑑_𝑘∈(−1,0), 𝑘=1,2,…; positive constants 𝑙₁, 𝑙₂, 𝐶₁, 𝐶₂, 𝜆; and a function 𝑚(𝑡)∈𝑃𝐶([𝑡₀−𝜏,∞),𝑅₊) with sup_{𝑡≥𝑡0−𝜏}𝑚(𝑡)≤𝜆, such that for any Ψ(𝑡)∈𝑃𝐶([−𝜏,0],𝑅^𝑑) with Ψ(0⁻)=Ψ(0), the following conditions hold:(i)𝐶₁‖𝑥‖≤𝑉(𝑡,𝑥)≤𝐶₂‖𝑥‖ for all 𝑡∈𝑅₊, 𝑥∈𝑅^𝑑;(ii)𝐷⁺𝑉(𝑡,Ψ(0))≤𝑚(𝑡)𝑉(𝑡,Ψ(0)) for all 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘=1,2,…, whenever 𝑉(𝑡,Ψ(0))≥𝛾𝑉(𝑡+𝑠,Ψ(𝑠)) for 𝑠∈[−𝜏,0], where 𝛾 is a constant and 0<𝛾<𝐻^𝑞₂,  𝐻₂=inf_𝑘{1+𝑑_𝑘} and 𝑞 is the smallest integer larger than or equal to 𝜏/𝑙₁; (iii)𝑉(𝑡_𝑘,Ψ(0)+𝐼_𝑘(𝑡_𝑘,Ψ))≤(1+𝑑_𝑘)𝑉(𝑡⁻_𝑘,Ψ(0)),𝑘=1,2,…;(iv)𝑙₁≤𝑡_𝑘−𝑡_𝑘−1≤𝑙₂ for 𝑘=1,2,…, and 0<𝐻₂≤𝐻₁<𝑒^−𝜆𝑙2, where 𝐻₁=sup_𝑘{1+𝑑_𝑘}, 𝐻₂=inf_𝑘{1+𝑑_𝑘}.
Then the zero solution of (1) is exponentially stable.

Proof. Let 𝑥(𝑡)=𝑥(𝑡,𝑡₀,Φ) be the solution of system (1) and 𝑉(𝑡)=𝑉(𝑡,𝑥(𝑡)). We will prove 𝑉(𝑡)≤𝐶₂^𝑘_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡𝑡0𝑚(𝑠)𝑑𝑠},𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=1,2,…. (6)
Let 𝑄(𝑡)=𝑉(𝑡)−𝐶∏₂^𝑘_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡𝑡0𝑚(𝑠)𝑑𝑠},𝑡∈𝑡_𝑘,𝑡_𝑘+1𝑉(𝑡),𝑘=1,2,…,−𝐶₂‖Φ‖_𝜏𝑒^{∫𝑡𝑡0𝑚(𝑠)𝑑𝑠},𝑡∈𝑡₀,𝑡₁,𝑉(𝑡)−𝐶₂‖Φ‖_𝜏,𝑡∈𝑡₀−𝜏,𝑡₀. (7) We need to show that 𝑄(𝑡)≤0 for all 𝑡≥𝑡₀. It is clear that 𝑄(𝑡)≤0 for 𝑡∈[𝑡₀−𝜏,𝑡₀], since 𝑄(𝑡)=𝑉(𝑡)−𝐶₂‖Φ‖_𝜏≤0 by condition (i).
Next we shall show 𝑄(𝑡)≤0, for 𝑡∈[𝑡₀,𝑡₁). Suppose this is not true. Then there is a 𝑡^∗ such that 𝑡^∗≤inf{𝑡∈[𝑡₀,𝑡₁),𝑄(𝑡)>0}, 𝑄(𝑡^∗)≤0, 𝑄(𝑡^∗)≥(𝛾−1)𝐶₂‖Φ‖_𝜏, and 𝐷⁺𝑄𝑡^∗>0. (8) Note that 𝑉(𝑡^∗)=𝑄(𝑡^∗)+𝐶₂‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝜎)𝑑𝜎}. Then 𝑉(𝑡^∗+𝑠)≤𝐶₂‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝜎)𝑑𝜎}≤𝛾⁻¹(𝑄(𝑡^∗)+𝐶₂‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝜎)𝑑𝜎})=𝛾⁻¹𝑉(𝑡^∗) for 𝑠∈[−𝜏,0]. By condition (ii), 𝐷⁺𝑉(𝑡^∗)≤𝑚(𝑡^∗)𝑉(𝑡^∗). So 𝐷⁺𝑄𝑡^∗=𝐷⁺𝑉𝑡^∗−𝑚𝑡^∗𝐶₂‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝑠)𝑑𝑠}≤𝑚𝑡^∗𝑉𝑡^∗−𝐶₂‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝑠)𝑑𝑠}=𝑚𝑡^∗𝑄𝑡^∗=0, (9) which contradicts (8). Hence 𝑄(𝑡)≤0, for all 𝑡∈[𝑡₀,𝑡₁).
Assume that 𝑄(𝑡)≤0, for 𝑡∈[𝑡₀,𝑡_𝑚),𝑚≥1. We shall show that 𝑄(𝑡)≤0, for 𝑡∈[𝑡₀,𝑡_𝑚+1). Obviously, by condition (iii) 𝑄𝑡_𝑚=𝑉𝑡_𝑚−𝐶₂^𝑚_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡𝑚𝑡0𝑚(𝑠)𝑑𝑠}≤1+𝑑_𝑚𝑉𝑡⁻_𝑚−𝐶₂^𝑚_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡𝑚𝑡0𝑚(𝑠)𝑑𝑠}=1+𝑑_𝑚𝑄𝑡⁻_𝑚≤0. (10) Suppose that there exists a 𝑡 such that 𝑡∈[𝑡_𝑚,𝑡_𝑚+1) and 𝑄(𝑡)>0. There is a 𝑡^∗ such that 𝑡^∗≤inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1),𝑄(𝑡)>0}, 𝑄(𝑡^∗)≤0, 𝑄(𝑡^∗)≥(𝛾𝐻₂^−𝑞−1)𝐶₂∏^𝑚_𝑖=1(1+𝑑_𝑖)‖Φ‖_𝜏, and 𝐷⁺𝑄𝑡^∗>0. (11)
Since 𝑉(𝑡^∗)=𝑄(𝑡^∗)+𝐶₂∏^𝑚_𝑖=1(1+𝑑_𝑖)‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝑠)𝑑𝑠}, then for any 𝑠∈[−𝜏,0], we have 𝑉𝑡^∗+𝑠≤𝑄𝑡^∗+𝑠+𝐶₂^𝑚−𝑞_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0+𝑠𝑚(𝜎)𝑑𝜎}≤𝐻₂^−𝑞𝐶₂^𝑚_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝜎)𝑑𝜎}≤𝛾⁻¹𝑄𝑡^∗+𝐶₂^𝑚_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝜎)𝑑𝜎}≤𝛾⁻¹𝑉𝑡^∗. (12) Thus by condition (ii), 𝐷⁺𝑉(𝑡^∗)≤𝑚(𝑡^∗)𝑉(𝑡^∗), then 𝐷⁺𝑄𝑡^∗=𝐷⁺𝑉𝑡^∗+𝑚𝑡^∗𝐶₂^𝑚_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝑠)𝑑𝑠}≤𝑚𝑡^∗𝑉𝑡^∗−𝐶₂^𝑚_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡∗𝑡0𝑚(𝑠)𝑑𝑠}=𝑚𝑡^∗𝑄𝑡^∗=0, (13) which contradicts (11). Hence 𝑄(𝑡)≤0 for all 𝑡∈[𝑡_𝑚,𝑡_𝑚+1). By induction, 𝑄(𝑡)≤0 for all 𝑡≥𝑡₀. In view of 𝑚(𝑡)≤𝜆 for all 𝑡≥𝑡₀−𝜏, we obtain 𝑉(𝑡)≤𝐶₂^𝑘_𝑖=11+𝑑_𝑖‖Φ‖_𝜏𝑒^{∫𝑡𝑡0𝑚(𝑠)𝑑𝑠}≤𝐶₂‖Φ‖_𝜏𝐻^𝑘₁𝑒^{∫𝑡𝑡0𝑚(𝑠)𝑑𝑠}≤𝐶₂𝐻^𝑘₁‖Φ‖_𝜏𝑒^{𝜆(𝑡−𝑡0)}, (14) for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘=1,2,…. Since 𝐻₁<𝑒^−𝜆𝑙2, there exists 𝛼 such that 0<𝛼<𝜆 and 𝜆𝑙₂+ln𝐻₂<−𝛼𝑙₂. By condition (i) 𝐶₁‖𝑥‖≤𝑉(𝑡,𝑥)≤𝐶₂𝐻^𝑘₁‖Φ‖_𝜏𝑒^{𝜆(𝑡−𝑡0)}≤𝐶₂‖Φ‖_𝜏𝑒^{𝜆(𝑡−𝑡0)+𝑘ln𝐻1}≤𝐶₂‖Φ‖_𝜏𝑒^{𝜆(𝑡−𝑡0)−(𝜆+𝛼)𝑘𝑙2}≤𝑀‖Φ‖_𝜏𝑒^{−𝛼(𝑡−𝑡0)}, (15) where 𝑀=𝐶₂𝑒^{(𝜆+𝛼)𝑙2}. Hence the zero solution of (1) is exponentially stable.

Remark 7. Theorem 6 says that the delay differential equation is unstable and the suitable impulse effects are given, then it will become stable (see Example 14). Compared with the Theorem 3.1 in [16], we do not require that 𝜏≤𝑡_𝑘−𝑡_𝑘−1. For example, by Theorem 6 we know that the zero solution of the following system is exponentially stable: 𝑥^′(𝑡)=12𝑥(𝑡)+18𝑥(𝑡−1),𝑡≥0,𝑡≠𝑡_𝑘,𝑡_𝑘=𝑘2𝑥𝑡,𝑘=1,2,…,_𝑘=13𝑥𝑡⁻_𝑘,𝑡_𝑘=𝑘2,𝑘=1,2,…. (16)
In the following we consider 𝑥^′(𝑡)(𝑡)(),))𝑥𝑡=𝑎𝑥+𝑏𝑥𝑡−𝜏𝑡≥0,𝑡∈[(𝑘−1𝜏,𝑘𝜏𝑘=1,2,…,_𝑘=1+𝑐_𝑘𝑥𝑡⁻_𝑘,𝑡_𝑘=𝑘𝜏,𝑘=1,2,…, (17) where 𝜏>0, and 𝑑_𝑘,𝑎,𝑏∈𝑅 are constants.

Theorem 8. Assume that 𝑐_𝑘≠−1,𝑘=1,2,…, and there is a constant 𝜆>0 such that 𝑎+|𝑏|𝑒^𝜆𝜏≤−𝜆 and 0<𝐻₁<𝑒^𝜆𝜏, where 𝐻₁=sup_𝑘{|1+𝑐_𝑘|}. Then the zero solution of (17) is exponentially stable.

Proof. Assume that 𝑉(𝑥)=𝑉(𝑡,𝑥)=|𝑥|.(i)Obviously, there exist 𝐶₁=𝐶₂=1, such that 𝐶₁|𝑥|≤𝑉(𝑥)≤𝐶₂|𝑥|.(ii) Assume that 𝑚(𝑡)=𝜆 for all 𝑡≥−𝜏. For any Ψ∈PC([−𝜏,0],𝑅), if 𝑉(𝑡,Ψ(0))≥𝑉(𝑡+𝑠,Ψ(𝑠))𝑒^{−∫𝑡𝑡−𝜏𝑚(𝑠)𝑑𝑠}=𝑉(𝑡+𝑠,Ψ(𝑠))𝑒^−𝜆𝜏, (18)we have |Ψ(−𝜏)|≤𝑒^𝜆𝜏|Ψ(0)|. For 𝑠∈[−𝜏,0], we have 𝐷⁺𝑉((0|Ψ(0+|𝑏Ψ(|Ψ(0+|𝑏|𝑒𝑡,Ψ))≤𝑎)|||−𝜏)|≤𝑎)|^𝜆𝜏|Ψ(0=|𝑏|𝑒)|𝑎+^𝜆𝜏|Ψ(0|Ψ(0(𝑡)𝑉((0.)|≤−𝜆)|=−𝑚𝑡,Ψ)) (19)(iii) Suppose that 1+𝑑_𝑘=|(1+𝑐_𝑘)|. Hence 𝑉𝑥𝑡_𝑘=𝑥𝑡_𝑘=1+𝑐_𝑘𝑥𝑡⁻_𝑘=1+𝑑_𝑘𝑥𝑡⁻_𝑘=1+𝑑_𝑘𝑉𝑥𝑡⁻_𝑘. (20)(iv) Obviously, 𝑙₁=𝜏=𝑡_𝑘−𝑡_𝑘−1 and −𝜆𝑙₁+ln𝐻₁<0.
By Theorem 4, the zero solution of (11) is exponentially stable.

Similarly, by Theorem 6 we have the following theorem.

Theorem 9. Assume that 0<|1+𝑐_𝑘|<1, 𝑘=1,2,…, and there are constants 𝜆 and 𝛾 such that 𝜆>0, 0<𝛾<𝐻₂≤𝐻₁<𝑒^−𝜆𝜏, and 𝑎+|𝑏|𝛾⁻¹≤𝜆, where 𝐻₁=sup_𝑘{|1+𝑐_𝑘|}, 𝐻₂=inf_𝑘{|1+𝑐_𝑘|}. Then the zero solution of (17) is exponentially stable.

3. The Euler Method for Linear IDDEs

In this section, we consider the exponential stability of the Euler method for (17). The convergence property can be proved similarly to [4]. The Euler method for (17) with initial function Φ∈PC([−𝜏,0],𝑅) is given by 𝑥_𝑘,𝑙+1=𝑥_𝑘,𝑙+ℎ𝑎𝑥_𝑘,𝑙+ℎ𝑏𝑥_{𝑘−1,𝑙},𝑥𝑙=0,1,…,𝑚−1,𝑘=0,1,…,_(𝑘+1),0=1+𝑐_𝑘+1𝑥_𝑘,𝑚𝑥,𝑘=0,1,2,…,_−1,𝑙=Φ(−𝜏+𝑙ℎ), (21) where ℎ=𝜏/𝑚. Let 𝑛=𝑘𝑚+𝑙, then 𝑥_𝑛=𝑥_{𝑘𝑚+𝑙}=𝑥_𝑘,𝑙 is an approximation for the exact solution 𝑥((𝑘𝑚+𝑙)ℎ) for 𝑘=0,1,2,…,𝑙=0,1,2,…,𝑚−1, and 𝑥_𝑘,𝑚 is an approximation for 𝑥((𝑘+1)𝜏⁻).

Definition 10. The Euler method for (11) is said to be exponentially stable if there exist positive constants 𝜆,𝑀, and 𝑀₁, for any Φ∈PC([−𝜏,0],𝑅), such that ‖𝑥_𝑛‖≤𝑀‖Φ‖_𝜏𝑒^{−𝑛𝜆ℎ} for ℎ=𝜏/𝑚, 𝑚≥𝑀₁, and 𝑛=1,2,….

The following theorem indicates that the Euler method preserves the property of exact solutions which was obtained above.

Theorem 11. Under the conditions of Theorem 8, the Euler method for (17) is exponentially stable.

Proof. If 𝑎<0, then 𝑀₁=−𝑎𝜏. (i) If 𝐻₁>1, we want to prove that 𝑥_𝑘,𝑙≤‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−(𝑘𝑚+𝑙)𝜆ℎ}, (22) for 𝑘=0,1,2,…, 𝑙=0,1,2,…,𝑚. Obviously, |𝑥_0,0|=|Φ(0)|≤‖Φ‖_𝜏. Firstly, we consider the case 𝑘=0 and 𝑙=1. Because ℎ=𝜏/𝑚 and 𝑚≥𝑀₁, so 1+ℎ𝑎≥0. Hence we have 𝑥_0,1=𝑥_0,0+ℎ𝑎𝑥_0,0+ℎ𝑏𝑥_−1,0≤(1+ℎ𝑎)𝑥_0,0+ℎ|𝑏|𝑥_−1,0≤(1+ℎ𝑎)‖Φ‖_𝜏+ℎ|𝑏|‖Φ‖_𝜏≤1+ℎ𝑎+ℎ|𝑏|𝑒^𝜆𝜏‖Φ‖_𝜏. (23) Because 𝑎+|𝑏|𝑒^𝜆𝜏≤−𝜆, we have |𝑥_0,1|≤(1−ℎ𝜆)‖Φ‖_𝜏. By the inequality 𝑒^−𝑥≥1−𝑥 holding for all 𝑥∈𝑅, we get |𝑥_0,1|≤‖Φ‖_𝜏𝑒^−𝜆ℎ.
Assume that |𝑥_0,𝑝|≤‖Φ‖_𝜏𝑒^{−𝜆𝑝ℎ} for 𝑝<𝑙≤𝑚. Then 𝑥_0,𝑙=𝑥_0,𝑙−1+ℎ𝑎𝑥_0,𝑙−1+ℎ𝑏𝑥_{−1,𝑙−1}≤(1+ℎ𝑎)𝑥_0,𝑙−1+ℎ|𝑏|𝑥_{−1,𝑙−1}≤(1+ℎ𝑎)‖Φ‖_𝜏𝑒^{−𝜆(𝑙−1)ℎ}+ℎ|𝑏|‖Φ‖_𝜏≤1+ℎ𝑎+ℎ|𝑏|𝑒^𝜆𝜏‖Φ‖_𝜏𝑒^{−𝜆(𝑙−1)ℎ}≤(1−ℎ𝜆)‖Φ‖_𝜏𝑒^{−𝜆(𝑙−1)ℎ}≤‖Φ‖_𝜏𝑒^{−𝜆𝑙ℎ}. (24) So (22) holds for 𝑘=0, 𝑙=0,1,2,…,𝑚. Suppose that (22) holds for 𝑛<𝑘, 𝑙=0,1,2,…,𝑚. Next, we shall prove (22) holds, when 𝑛=𝑘, 𝑙=0,1,2,…,𝑚. Hence 𝑥_𝑘,0=1+𝑐_𝑘𝑥_{𝑘−1,𝑚}≤‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−𝑘𝑚𝜆ℎ}. (25) Assume that |𝑥_𝑘,𝑢|≤‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−(𝑘𝑚+𝑢)𝜆ℎ} for 𝑢<𝑙≤𝑚. Then 𝑥_𝑘,𝑙=𝑥_{𝑘,𝑙−1}+ℎ𝑎𝑥_{𝑘,𝑙−1}+ℎ𝑏𝑥_{𝑘−1,𝑙−1}≤(1+ℎ𝑎)𝑥_{𝑘,𝑙−1}+ℎ|𝑏|𝑥_{𝑘−1,𝑙−1}≤(1+ℎ𝑎)‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−(𝑘𝑚+𝑙−1)𝜆ℎ}+ℎ|𝑏|⋅‖Φ‖_𝜏𝐻₁^𝑘−11+𝑑_𝑖𝑒^{−(𝑘𝑚−𝑚+𝑙−1)𝜆ℎ}≤1+ℎ𝑎+ℎ|𝑏|𝑒^𝜆𝜏‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−(𝑘𝑚+𝑙−1)𝜆ℎ}≤(1−ℎ𝜆)‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−(𝑘𝑚+𝑙−1)𝜆ℎ}≤‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−(𝑘𝑚+𝑙)𝜆ℎ}. (26) Hence (22) holds. Since −𝜆𝑙₁+ln𝐻₁<0, there exists 𝛼 such that 0<𝛼<𝜆 and −(𝜆−𝛼)𝑙₁+ln𝐻₁<0. Hence 𝑥_𝑘,𝑙≤‖Φ‖_𝜏𝐻^𝑘₁𝑒^{−(𝑘𝑚+𝑙)𝜆ℎ}≤‖Φ‖_𝜏𝑒^{ln𝐻1−𝜆𝑚ℎ𝑘}𝑒^{−𝜆𝑙ℎ}≤‖Φ‖_𝜏𝑒^{−𝛼(𝑚𝑘+𝑙)ℎ}. (27)(ii)If 𝐻₁≤1, we can prove that 𝑥_𝑘,𝑙≤‖Φ‖_𝜏𝑒^{−𝜆(𝑘𝑚+𝑙)ℎ}. (28)Consequently, the theorem holds.

Theorem 12. Under the conditions of Theorem 9, the Euler method for (17) is exponentially stable.

Example 13. Consider the system 𝑥^′(𝑡)(𝑡)()𝑥(𝑘)(𝑘=−4𝑥+𝑥𝑡−1,𝑡≥0,𝑡≠𝑘,𝑘=1,2,…,=2𝑥⁻),𝑘=1,2,…. (29) Obviously, 𝜆=1 satisfies the Theorem 8. Therefore the zero solution of (29) is exponentially stable. By Theorem 11, the Euler method for (29) is also exponentially stable (see Figure 1).

Figure 1 

The solutions of (29), as Φ≡1, ℎ=1/10.

Example 14. Obviously, the zero solution of the system 𝑥^′(𝑡)=12𝑥(𝑡)+18𝑥(𝑡−1),𝑡≥0, (30) is unstable (see Figure 2) while the zero solution of the following system 𝑥^′(𝑡)=12𝑥(𝑡)+18𝑥()𝑥(𝑘)=1𝑡−1,𝑡≥0,𝑡≠𝑘,𝑘=1,2,…,3𝑥(𝑘⁻),𝑘=1,2,…, (31) is exponentially stable by Theorem 9 with 𝜆=1. By Theorem 12, the Euler method for (31) is also exponentially stable (see Figure 3).

Figure 2 

The solution of (30) as Φ(𝑡)≡1, 𝑡∈[−𝜏,0].

Figure 3 

The solutions of (31), as Φ≡1, ℎ=1/10.

Acknowledgments

The authors wish to thank referees for valuable comments. The research was supported by the NSF of China no. 11071050.