Fixed Points and Stability of a Class of Integrodifferential Equations

Abstract

We study a class of integrodifferential functional differential equations 𝑥̈+𝑓(𝑡,𝑥,𝑥̇)𝑥̇+∑^𝑁_𝑗=1∫^𝑡_{𝑡−𝑟𝑗(𝑡)}𝑎_𝑗(𝑡,𝑠)𝑔_𝑗(𝑠,𝑥(𝑠))𝑑𝑠=0 with variable delay. By using the fixed point theory, we establish necessary and sufficient conditions ensuring that the zero solution of this equation is asymptotically stable.

1. Introduction

It is well known that differential equations have many applications in engineering, control theory, neural networks, biology, and so on. The stability of the solution of functional differential equations has drawn a lot of attention for many years. Liapunov’s direct method has been widely applied to study the stability problems for a long time; see, for example, [1–3]. Nowadays, many authors have applied the fixed points theory to study the stability of solution of functional differential equations; some authors even use fixed point theory to study stability conditions for stochastic equations with delay; see, for example, [4–7] and the references therein.

Burton studied stability of a nonconvolution equation 𝑥̇=−^𝑡_𝑡−𝑟𝑎(𝑡,𝑠)𝑔(𝑥(𝑠))𝑑𝑠, (1) where 𝑟 was a positive constant. He obtained sufficient conditions ensuring that the zero solution was asymptotically stable via fixed point theorem; see [8].

Becker and Burton studied the following differential equation: 𝑥̇=−^𝑡_{(𝑡)𝑡−𝑟}𝑎(𝑡,𝑠)𝑔(𝑥(𝑠))𝑑𝑠 (2) with variable delay and its special case 𝑥̇=−^𝑡_𝑡−𝑟𝑎(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠, (3) for 𝑟, is a positive constant, 𝑡≥0, where function 𝑟(𝑡)∶[0,∞)→[0,∞). They found sufficient conditions ensuring that the zero solution was asymptotically stable by changing the supremum metric to an exponentially weighted metric. Moreover, they hoped to reduce condition that the function 𝑡−𝑟(𝑡) was strictly increasing; see [9]. Jin and Luo reduced this condition in their paper. Moreover, they established necessary and sufficient conditions that could ensure that the zero solution of this equation is asymptotically stable; see, for example, [10].

Ardjouni et al. studied the stability of the following neutral integrodifferential equation: 𝑥̇(𝑡)=−^𝑁_𝑗=1^𝑡_{𝑡−𝜏𝑗(𝑡)}𝑎_𝑗(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+^𝑁_𝑗=1𝑐_𝑗(𝑡)𝑥^′𝑡−𝜏_𝑗(𝑡) (4) via fixed point theorem. Also 𝑚_𝑗(0)=inf{𝑡−𝜏_𝑗(𝑡)∶𝑡≥0}. Their main result was as follows.

Theorem A. Suppose that 𝜏_𝑗 is twice differentiable and 𝜏^′_𝑗(𝑡)≠1 for all 𝑡∈𝑅⁺ and there exist continuous functions ℎ_𝑗∶[𝑚_𝑗(0),∞)→𝑅 for 𝑗=1,2,…,𝑁 and a constant 𝛼∈(0,1) such that, for 𝑡≥0,(i)liminf_𝑡→∞∫^𝑡₀𝐻(𝑠)𝑑𝑠>−∞,(ii)^𝑁_𝑗=1𝑐_𝑗(𝑡)1−𝜏^′_𝑗(𝑡)+^𝑁_𝑗=1^𝑡_{𝑡−𝜏𝑗(𝑡)}ℎ_𝑗(𝑠)+𝐵_𝑗(+𝑡,𝑠)𝑑𝑠_{𝑁𝑗=1𝑡0}𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}ℎ_𝑗𝑠−𝜏_𝑗(𝑠)1−𝜏^′_𝑗(𝑠)−𝑟_𝑗(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡0}𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}|𝐻(𝑠×)|_{𝑠𝑠−𝜏𝑗(𝑠)}ℎ_𝑗(𝑢)+𝐵_𝑗(𝑠,𝑢)𝑑𝑢𝑑𝑠≤𝛼, (5) where 𝐻(𝑡)=^𝑁_𝑗=1ℎ_𝑗(𝑡),𝑟_𝑗(𝑡)=𝑐_𝑗(𝑡)𝐻(𝑡)+𝑐^′_𝑗(𝑡)1−𝜏^′_𝑗(𝑡)+𝑐_𝑗(𝑡)𝜏_𝑗^′′(𝑡)1−𝜏^′_𝑗(𝑡)²,𝐵_𝑗(𝑡,𝑠)=^𝑠_𝑡𝑎_𝑗()𝐵𝑢,𝑠𝑑𝑢,_𝑗𝑡,𝑡−𝜏_𝑗(𝑡)=^{𝑡−𝜏𝑗}_𝑡^(𝑡)𝑎_𝑗𝑢,𝑡−𝜏_𝑗(𝑡)𝑑𝑢. (6)Then the zero solution is asymptotically stable if and only if ^𝑡₀𝐻(𝑠)𝑑𝑠⟶∞𝑎𝑠𝑡⟶∞. (7)

They improved the results of [9]; see [11]. Other results on fixed points and stability properties in equations with variable delays can be found in [5].

Recently, Zhao and Yuan investigate the stability of a generalized Volterra-Levin equation: 𝑥̇=−^𝑡_𝑡−𝐿𝑝(𝑠−𝑡)𝑔(𝑠,𝑥(𝑠))𝑑𝑠,𝑡>𝑡₀,𝑥(𝑡)=𝜓(𝑡),𝑡₀−𝐿≤𝑡≤𝑡₀. (8) They give new conditions on the stability of zero solution of this equation; their conditions are different from [5]; see, for example, [12].

The stability of second order equation has always been hot issue of research. Levin and Nohel [13] studied the global asymptotic stability of a class of nonlinear systems: 𝑥̈(𝑡)+ℎ(𝑡,𝑥,𝑥̇)𝑥̇+𝑓(𝑥)=𝑒(𝑡). (9) They obtained asymptotic stability by constructing proper Lyapunov function.

Burton [14] considered the following equation with delay: 𝑥̈+𝑓(𝑡,𝑥,𝑥̇)𝑥̇+𝑏(𝑡)𝑔(𝑥(𝑡−𝐿))=0, (10) where 𝐿 is a positive constant, and obtained some sufficient conditions under which each solution 𝑥(𝑡) satisfies (𝑥(𝑡),𝑥̇(𝑡))→0 via the fixed point theorem.

We generalized the above equation to an equation with a variable delay [15] as follows: 𝑥̈+𝑓(𝑡,𝑥,𝑥̇)𝑥̇+𝑏(𝑡)𝑔(𝑥(𝑡−𝜏(𝑡)))=0, (11) and obtained some results on asymptotic stability of the zero solution. Before we introduce our new results we recall the main results in [15]. There are basic assumptions on the delay function 𝜏(𝑡)∶(𝒜) 𝑡−𝜏(𝑡) which is strictly increasing and lim_𝑡→∞𝑡−𝜏(𝑡)=∞. 𝑝(𝑡) is the inverse of 𝑡−𝜏(𝑡). Moreover 0≤𝑏(𝑡)≤𝑀 for some constant 𝑀>0.

The main results in [15] can be stated as follows.

Theorem B. Suppose (𝒜) and the following conditions hold.(i)There exists a constant 𝑙>0 such that 𝑔(𝑥) satisfies the Lipschitz condition on [−𝑙,𝑙]. The function 𝑔(𝑥) is odd and is strictly increasing on [−𝑙,𝑙], and 𝑥−𝑔(𝑥) is nondecreasing on [0,𝑙].(ii)There exist an 𝛼∈(0,1) and a continuous function 𝑎(𝑡)∶[0,∞)→[0,∞) such that 𝑓(𝑡,𝑥,𝑦)≥𝑎(𝑡) for 𝑡≥0, 𝑥∈ℝ, 𝑦∈ℝ, ∫^∞₀𝑎(𝑡)𝑑𝑡=∞, and 2sup_𝑡≥0𝑡^{𝑝(𝑡)∞}₀𝑒^{−∫𝑠𝑤+𝑠𝑎(𝑣)𝑑𝑣}𝑏(𝑠)𝑑𝑤𝑑𝑠+2sup_𝑡≥0^𝑡₀^∞_𝑡−𝑠𝑒^{−∫𝑠𝑤+𝑠𝑎(𝑣)𝑑𝑣}𝑏(𝑠)𝑑𝑤𝑑𝑠≤𝛼. (12)(iii)There exist constants 𝑎₀>0 and 𝑄>0 such that, for each 𝑡≥0, if 𝐽≥𝑄, then _𝑡^𝑡+𝐽𝑎(𝑣)𝑑𝑣≥𝑎₀𝐽. (13)(iv)There exist continuous functions 𝐹∶ℝ×ℝ→[0,∞) and 𝑐(𝑡)∶[0,∞)→[0,∞) such that, for all 𝑡≥0, 𝑥∈ℝ, 𝑦∈ℝ, 𝑓(𝑡,𝑥,𝑦)≤𝐹(𝑥,𝑦)𝑐(𝑡). The function 𝑔^′(𝑥) is continuous on [−𝑙,𝑙], 𝑔^′(0)≠0.

If, for each 𝛾>0, ∬^∞₀𝑒^{−∫𝑠𝑤+𝑠𝛾𝑐(𝑣)}𝑏(𝑠)𝑑𝑤𝑑𝑠=∞, then the zero solution of (11) is asymptotically stable.

In [16], we have studied the following equation: 𝑥̈+𝑓(𝑡,𝑥,𝑥̇)𝑥̇+^𝑡_{(𝑡)𝑡−𝑟}𝑎(𝑡,𝑠)𝑔(𝑥(𝑠))𝑑𝑠=0 (14) with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Moreover, we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable.

In this paper we consider 𝑥̈+𝑓(𝑡,𝑥,𝑥̇)𝑥̇+^𝑁_𝑗=1^𝑡_{𝑡−𝑟𝑗(𝑡)}𝑎_𝑗(𝑡,𝑠)𝑔_𝑗(𝑠,𝑥(𝑠))𝑑𝑠=0 (15) for 𝑡≥0, where functions 𝑟_𝑗(𝑡)∶[0,∞)→[0,∞), 𝑎_𝑗(𝑡,𝑠)∶[0,∞)×[−𝑟(0),∞)→𝑅, 𝑔_𝑗∶[−𝑟(0),∞)×𝑅→𝑅, and 𝑓∶𝑅×𝑅×𝑅→𝑅⁺ are all continuous, where 𝑅⁺=[0,+∞). We assume that 𝑡−𝑟_𝑗(𝑡)→∞ as 𝑡→∞. 𝑗=1,2,…,𝑁.

For each 𝑡₀≥0, define 𝑚_𝑗(𝑡₀)=inf{𝑠−𝑟_𝑗(𝑠)∶𝑠≥𝑡₀}, 𝑗=1,2,…,𝑁. Let 𝑚(𝑡₀)=min{𝑚₁(𝑡₀),…,𝑚_𝑁(𝑡₀)}. Also 𝐶(𝑡₀)=𝐶([𝑚(𝑡₀),𝑡₀],𝑅) is a function space endowed with function norm ‖⋅‖, where ‖𝜓‖=sup{|𝜓(𝑠)|∶𝑚(𝑡₀)≤𝑠≤𝑡₀}. We will also use ‖𝜙‖ to express the supremum on [𝑚(𝑡₀),∞) later. It is well known that, in [2], for a given continuous function 𝜙, there exists a solution of (15) on an interval [𝑡₀,𝑇); if the solution remains bounded, then 𝑇=∞. We denote by (𝑥(𝑡),𝑦(𝑡)) the solution (𝑥(𝑡,𝑡₀,𝜙),𝑦(𝑡,𝑡₀,𝜙)). Denote by 𝐴(𝑡)∶=𝑓(𝑡,𝑥(𝑡),𝑦(𝑡)).

We can write (15) in an equivalent form as ̇̇(𝑡)𝑥=𝑦,𝑦=−𝐴𝑦−^𝑁_𝑗=1^𝑡_{𝑡−𝑟𝑗(𝑡)}𝑎_𝑗(𝑡,𝑠)𝑔_𝑗(𝑠,𝑥(𝑠))𝑑𝑠. (16)

We will give a necessary and sufficient condition ensuring that the zero solution of this equation is asymptotically stable. To our knowledge, there are few results about its stability. This paper is organized as follows. In the next section we will state our main results. Their proofs will be given in Sections 3 and 4.

2. Statement of Main Results

First of all, we consider a special case of (15). We consider the following equation:𝑥̈+𝑓(𝑡,𝑥,𝑥̇)𝑥̇+^𝑡_𝑡−𝐿𝑝(𝑠−𝑡)𝑔(𝑠,𝑥(𝑠))𝑑𝑠=0. (17) Rewrite this equation as ̇̇(𝑡)𝑥=𝑦,𝑦=−𝐴𝑦−^𝑡_𝑡−𝐿𝑝(𝑠−𝑡)𝑔(𝑠,𝑥(𝑠))𝑑𝑠. (18) Our main results are as follows.

Theorem 1. Assume that ℎ(𝑡)∈𝐶(𝑅⁺,𝑅⁺) and the following conditions hold.(i)Consider that ∫⁰_−𝐿𝑝(𝑠)𝑑𝑠=1. There exist a constant 𝜆>0 and a function 𝑄(𝑥)∈𝐶(𝑅,𝑅⁺) such that, for ∀𝑥,𝑦₁,𝑦₂∈𝑅, |𝑔(𝑥,𝑦₁)−𝑔(𝑥,𝑦₂)|≤𝑄(𝑥)|𝑦₁−𝑦₂| and the inequality sup_𝑡≥0∫⁰_−𝐿|𝑝(𝑠)|∫^𝑡+𝑠_𝑡−𝐿𝑄(𝑢)𝑑𝑢𝑑𝑠≤𝜆 hold. There exists a constant 𝐾>0 such that 𝑄(𝑡)≤𝐾.(ii)There exist 𝛼∈(0,1) and a continuous function 𝐴₁(𝑡)∶[0,∞)→[0,∞) such that 𝑓(𝑡,𝑥,𝑦)≥𝐴₁(𝑡) for 𝑡≥0. For 𝑡≥0, 𝑥∈𝑅, 𝑦∈𝑅^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴1(𝑣)𝑑𝑣}𝑄(𝑠−𝐿)𝑑𝑠𝑑𝑢+2𝜆^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝑑𝑢≤𝛼. (19)(iii)There exist constants 𝑎₀>0 and 𝑄₀>0 such that, for each 𝑡≥0, if 𝐽≥𝑄₀, then _𝑡^𝑡+𝐽𝐴₁(𝑠)𝑑𝑠≥𝑎₀𝐽. (20) Then the zero solution of (17) is stable.

In addition, we have the following.

Theorem 2. Assume that lim_𝑡→∞𝑡−𝑟_𝑗(𝑡)=∞, 𝑟_𝑗(𝑡) is differentiable, 𝑗=1,2,…,𝑁, and there exist some functions ℎ_𝑗(𝑡)∈𝐶(𝑅⁺,𝑅⁺) such that for 𝑡≥0 the following conditions hold.(i)There exist a constant Γ>0 and some functions 𝑄_𝑗(𝑥)∈𝐶(𝑅,𝑅⁺) such that, for ∀𝑥,𝑦₁,𝑦₂∈𝑅, 𝑔_𝑗𝑥,𝑦₁−𝑔_𝑗𝑥,𝑦₂≤𝑄_𝑗(𝑥)𝑦₁−𝑦₂,𝑗=1,2,…,𝑁,sup^𝑁_{𝑡≥0𝑗=1}^𝑡_{𝑡−𝑟𝑗(𝑡)}^𝑢_𝑡𝑎_𝑗(𝑣,𝑡)𝑑𝑣𝑄_𝑗(𝑢)𝑑𝑢≤Γ. (21) There exists a constant 𝐾^∗>0 such that 𝑄_𝑗(𝑡)≤𝐾^∗, 𝑗=1,2,…,𝑁.(ii)Let 𝐻(𝑡)=∑^𝑁_𝑗=1ℎ_𝑗(𝑡). There exist a constant 𝛼∈(0,1) and a continuous function 𝐴₁(𝑡)∶𝑅⁺→𝑅⁺ such that 𝑓(𝑡,𝑥,𝑦)≥𝐴₁(𝑡). For ∀𝑡≥0, 𝑥∈𝑅, 𝑦∈𝑅, 2^𝑁_𝑗=1^𝑡_{𝑡−𝑟𝑗(𝑡)}ℎ_𝑗(𝑣)𝑑𝑣+2Γ^𝑡₀𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣}+𝑑𝑢_{𝑁𝑗=1𝑡0}𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}ℎ_𝑗𝑠−𝑟_𝑗(𝑠)1−𝑟^′_𝑗(𝑠+)𝑑𝑠_{𝑁𝑗=1𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣𝑢}₀𝑒^{−∫𝑢𝑠𝐴1(𝑣)𝑑𝑣}𝐾^∗𝑊_𝑗(𝑠)𝑑𝑠𝑑𝑢≤𝛼, (22) where 𝑊_𝑗(𝑠)=(1−𝑟^′_𝑗(𝑠))∫^{𝑠−𝑟𝑗}_𝑠^(𝑠)𝑎_𝑗(𝑣,𝑠−𝑟_𝑗(𝑠))𝑑𝑣.(iii)There exist constants 𝑎₀>0 and 𝑄₀>0 such that, for each 𝑡≥0, if 𝐽≥𝑄₀, then _𝑡^𝑡+𝐽𝐴₁(𝑠)𝑑𝑠≥𝑎₀𝐽. (23) Then the zero solution of (15) is asymptotically stable if and only if ^𝑡₀𝐻(𝑠)𝑑𝑠⟶∞𝑎𝑠𝑡⟶∞. (24)

3. Proof of Theorem 1

In this section, we will prove Theorem 1 by applying the fixed point theory. We will give the expression of the solution of the related equation. The following result can be found in [12].

Lemma 3. Let the function 𝑝∶[−𝐿,0]→𝑅, 𝑔∈𝐶(𝑅×𝑅,𝑅), ∫⁰_−𝐿𝑝(𝑠)𝑑𝑠=1. Then 𝑥̇(𝑡)=−^𝑡_𝑡−𝐿𝑝(𝑠−𝑡)𝑔(𝑠,𝑥(𝑠))𝑑𝑠 (25) is equivalent to 𝑥̇(𝑡)((+𝑑=−𝑔𝑡−𝐿,𝑥𝑡−𝐿))𝑑𝑡⁰_−𝐿𝑝(𝑠)^𝑡−𝐿_𝑡+𝑠𝑔(𝑢,𝑥(𝑢))𝑑𝑢𝑑𝑠. (26)

Lemma 4. Let 𝜓∶[−𝐿,0]→𝑅 be a given continuous function; if (𝑥(𝑡),𝑦(𝑡)) is the solution of (17) satisfying 𝑥(𝑡)=𝜓(𝑡), 𝑡∈[−𝐿,0], and 𝑦(0)=𝑥̇(0), then 𝑥(𝑡) is the solution of the following integral equation: 𝑥(𝑡)=𝜓(0)𝑒^{−∫𝑡0ℎ(𝑠)𝑑𝑠}+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝐵(𝑢)−𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑔(𝑠−𝐿,𝑥(𝑠−𝐿))𝑑𝑠𝑑𝑢+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝑒^{−∫𝑢0𝐴(𝑣)𝑑𝑣}×⁰_−𝐿𝑝(𝑣)_𝑣^−𝐿𝑔(𝑤)−𝑤,𝜓𝑑𝑤𝑑𝑣𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠0}_−𝐿𝑝(𝑣)^𝑢−𝐿_𝑢+𝑣𝑔((𝑤−𝑤,𝑥))𝑑𝑤𝑑𝑣𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝐴(𝑠)×⁰_−𝐿𝑝(𝑣)×^𝑠−𝐿_𝑠+𝑣𝑔(𝑤,𝑥(𝑤))𝑑𝑤𝑑𝑣𝑑𝑠𝑑𝑢, (27) where 𝐵(𝑡)=𝑥̇(0)𝑒^{−∫𝑡0𝐴(𝑠)𝑑𝑠}.

Proof. We apply the variation of parameters formula to the second equation of (18); we obtain 𝑥̇(𝑡)=𝑥̇(0)𝑒^{−∫𝑡0𝐴(𝑠)𝑑𝑠}−^𝑡₀𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣𝑠}_𝑠−𝐿𝑝(𝑣−𝑠)𝑔(𝑣,𝑥(𝑣))𝑑𝑣𝑑𝑠. (28) By Lemma 3, (28) can be written as 𝑥̇(𝑡)=𝐵(𝑡)−^𝑡₀𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣}𝑔((+𝑠−𝐿,𝑥𝑠−𝐿))𝑑𝑠_𝑡0𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣}×𝑑𝑑𝑠⁰_−𝐿𝑝(𝑠)^𝑠−𝐿_𝑠+𝑣𝑔(𝑢,𝑥(𝑢))𝑑𝑢𝑑𝑣𝑑𝑠; (29) multiplying both sides of the above equation by 𝑒^{∫𝑡0ℎ(𝑠)𝑑𝑠}, we see that 𝑥(𝑡)=𝜓(0)𝑒^{−∫𝑡0ℎ(𝑠)𝑑𝑠}+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝐵(𝑢)−𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑔((+𝑠−𝐿,𝑥𝑠−𝐿))𝑑𝑠𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}×^𝑢₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}×𝑑𝑑𝑠⁰_−𝐿𝑝(𝑠)×^𝑠−𝐿_𝑠+𝑣𝑔(𝑤,𝑥(𝑤))𝑑𝑤𝑑𝑣𝑑𝑠𝑑𝑢. (30)
If we integrate the last term by parts, we have 𝑥(𝑡)=𝜓(0)𝑒^{−∫𝑡0ℎ(𝑠)𝑑𝑠}+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝐵(𝑢)−𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑔(𝑠−𝐿,𝑥(𝑠−𝐿))𝑑𝑠𝑑𝑢+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝑒^{−∫𝑢0𝐴(𝑣)𝑑𝑣}×⁰_−𝐿𝑝(𝑣)_𝑣^−𝐿𝑔(𝑤)−𝑤,𝜓𝑑𝑤𝑑𝑣𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠0}_−𝐿𝑝(𝑣)^𝑢−𝐿_𝑢+𝑣𝑔(𝑤,𝑥(𝑤))𝑑𝑤𝑑𝑣𝑑𝑢−^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}×^𝑢₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝐴(𝑠)×⁰_−𝐿𝑝(𝑣)×^𝑠−𝐿_𝑠+𝑣𝑔(𝑤,𝑥(𝑤))𝑑𝑤𝑑𝑣𝑑𝑠𝑑𝑢. (31)

Let (𝐶,‖⋅‖) be the Banach space of bounded continuous functions on [−𝐿,∞) with the supremum norm. For a given continuous initial function 𝜓∶[−𝐿,0]→𝑅, define the set 𝐶_𝜓⊂𝐶 by 𝐶_𝜓=𝜙∶[−𝐿,∞)⟶𝑅∣𝜙∈𝐶,𝜙(𝑡)=𝜓(𝑡),𝑡∈[−𝐿,0] (32) and its subset 𝐶^𝑙_𝜓=[)(𝑡)(𝑡),[],𝜙(𝑡,𝜙∶−𝐿,∞⟶𝑅∣𝜙∈𝐶,𝜙=𝜓𝑡∈−𝐿,0)≤𝑙,𝑡>0 (33) where 𝜓∶[−𝐿,0]→[−𝑙,𝑙] is a given initial function and 𝑙 is a positive constant. Let 𝑃₁ be a mapping defined on 𝐶^𝑙_𝜓 as follows: for 𝜙∈𝐶^𝑙_𝜓, if 𝑡∈[−𝐿,0], (𝑃₁𝜙)(𝑡)=𝜓(𝑡). If 𝑡>0, 𝑃₁𝜙(𝑡)=𝜓(0)𝑒^{−∫𝑡0ℎ(𝑠)𝑑𝑠}+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝐵(𝑢)−𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑔()+𝑠−𝐿,𝜙𝑠−𝐿𝑑𝑠𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}𝑒^{−∫𝑢0𝐴(𝑣)𝑑𝑣}×⁰_−𝐿𝑝(𝑣)_𝑣^−𝐿𝑔(𝑤)−𝑤,𝜓𝑑𝑤𝑑𝑣𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠0}_−𝐿𝑝(𝑣)^𝑢−𝐿_𝑢+𝑣𝑔(𝑤)−𝑤,𝜙𝑑𝑤𝑑𝑣𝑑𝑢_𝑡0𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝐴(𝑠)×⁰_−𝐿𝑝(𝑣)×^𝑠−𝐿_𝑠+𝑣𝑔𝑤,𝜙(𝑤)𝑑𝑤𝑑𝑣𝑑𝑠𝑑𝑢. (34) Easy calculation shows that 𝑃₁𝜙(𝑡)−𝑃₁𝜂(𝑡≤_𝑡0)𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠𝑢}₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑔()()𝑠−𝐿,𝜙𝑠−𝐿−𝑔𝑠−𝐿,𝜂𝑠−𝐿𝑑𝑠𝑑𝑢+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠0}_−𝐿𝑝(𝑣)^𝑢+𝑣_𝑢−𝐿𝑔(𝑤)(𝑤)𝑤,𝜙−𝑔𝑤,𝜂𝑑𝑤𝑑𝑣𝑑𝑢+^𝑡₀𝑒^{−∫𝑡𝑢ℎ(𝑠)𝑑𝑠}×^𝑢₀𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝐴(𝑠)×⁰_−𝐿𝑝(𝑣)×^𝑠+𝑣_𝑠−𝐿𝑔(𝑤(𝑤)𝑤,𝜙)−𝑔𝑤,𝜂𝑑𝑤𝑑𝑣𝑑𝑠𝑑𝑢. (35)

We can easily obtain that (𝑃₁𝜙)(𝑡) is a contraction mapping by using condition (ii). We continue to prove Theorem 1. Choose 𝜓∶[−𝐿,0]→𝑅 and 𝑥̇(0) such that 𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀|𝑥̇(0)|+𝛿+𝜆𝛿𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀≤(1−𝛼)𝑙. (36) Condition (i) implies that 𝑔(𝑡,0)=0, since 𝑔(𝑡,𝑥) satisfies the Lipschitz condition with respect to 𝑥, such that 𝛿 exists and 𝛿<𝑙. By the expression of (𝑃₁𝜙)(𝑡), condition (ii), and condition (iii) of Theorem 1, we have ^𝑡₀𝑒^{−∫𝑢0𝐴(𝑠)𝑑𝑠}𝑑𝑢=^𝑄0₀𝑒^{−∫𝑢0𝐴(𝑠)𝑑𝑠}𝑑𝑢+^𝑡_𝑄0𝑒^{−∫𝑢0𝐴(𝑠)𝑑𝑠}≤𝑑𝑢_𝑄00𝑑𝑢+^𝑡_𝑄0𝑒^−𝑎0𝑢𝑑𝑢≤𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀. (37)

According to condition (ii) and (36), we have 𝑃₁𝜙(𝑡)≤(1−𝛼)𝑙+𝛼𝑙=𝑙. (38) Note that if 𝑡∈[−𝐿,0], then (𝑃₁𝜙)(𝑡)=𝜓(𝑡). We obtain that |(𝑃₁𝜙)(𝑡)|≤𝑙, 𝑡∈[−𝐿,∞). Thus 𝑃₁∶𝐶^𝑙_𝜓→𝐶^𝑙_𝜓. Since we have proved that 𝑃₁ is a contraction mapping, hence 𝑃₁ has a unique fixed point 𝑥(𝑡) and |𝑥(𝑡)|≤𝑙.

Recalling (28), we have 𝑦(𝑡)≤|𝑥̇(0)|𝑒^{−∫𝑡0𝐴(𝑠)𝑑𝑠}+^𝑡₀𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣𝑠}_𝑠−𝐿𝑝(𝑠−𝑣)𝑔(𝑣,𝑥(𝑣))𝑑𝑣𝑑𝑠; (39) then 𝑦(𝑡)≤|𝑥̇(0)|+𝐾𝑙⋅^𝑡₀𝑒^{−∫𝑡𝑢𝐴(𝑠)𝑑𝑠}𝑄𝑑𝑢<𝑙1+𝐾₀+𝑒^{−𝑎0𝑄0}𝑎₀. (40) It follows that |𝑥(𝑡)|+𝑦(𝑡)<𝑙2+𝐾𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀. (41) For any given 𝜖>0; we only need to replace 𝜖 by 𝑙. Then we obtain the stability of zero solution. This completes the proof of Theorem 1.

4. Proof of Theorem 2

In this section, we will prove Theorem 2. First of all, we will obtain a useful expression for the solution of (15) in the following proof.

We apply the variation of parameters formula to the second equation of (15); then 𝑥̇(𝑡)=𝐵₁(𝑡)−^𝑁_𝑗=1^𝑡_𝑡0𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣}×^𝑠_{𝑠−𝑟𝑗(𝑠)}𝑎_𝑗(𝑠,𝑣)𝑔_𝑗(𝑣,𝑥(𝑣))𝑑𝑣𝑑𝑠, (42) where 𝐵₁(𝑡)=𝑥̇(𝑡₀)𝑒^{−∫𝑡𝑡0𝐴(𝑠)𝑑𝑠}.

Lemma 5. The equation 𝑥̇(𝑡)=−^𝑁_𝑗=1^𝑡_{𝑡−𝑟𝑗(𝑡)}𝑎_𝑗(𝑡,𝑠)𝑔_𝑗(𝑠,𝑥(𝑠))𝑑𝑠 (43) is equivalent to 𝑥̇(𝑡)=^𝑁_𝑗=1𝐶_𝑗𝑡,𝑡−𝑟_𝑗(𝑡)1−𝑟^′_𝑗(𝑡)𝑔_𝑗𝑡−𝑟_𝑗(𝑡),𝑥𝑡−𝑟_𝑗(𝑡)+^𝑁_𝑗=1𝑑𝑑𝑡^𝑡_{𝑡−𝑟𝑗(𝑡)}𝐶_𝑗(𝑡,𝑠)𝑔_𝑗(𝑠,𝑥(𝑠))𝑑𝑠, (44) where 𝐶_𝑗(𝑡,𝑠)=^𝑠_𝑡𝑎_𝑗()𝐶𝑢,𝑠𝑑𝑢,_𝑗𝑡,𝑡−𝑟_𝑗(𝑡)=^{𝑡−𝑟𝑗}_𝑡^(𝑡)𝑎_𝑗𝑢,𝑡−𝑟_𝑗(𝑡)𝑑𝑢. (45)

Proof. Differentiating the integral term in (44), we have 𝑑𝑑𝑡^𝑡_{𝑡−𝑟𝑗(𝑡)}𝐶_𝑗(𝑡,𝑠)𝑔_𝑗((𝑠=𝑠,𝑥))𝑑𝑠_{𝑡𝑡−𝑟𝑗(𝑡)}𝜕𝐶𝜕𝑡^𝑗(𝑡,𝑠)𝑔_𝑗(𝑠,𝑥(𝑠))𝑑𝑠+𝐶_𝑗(𝑡,𝑡)𝑔_𝑗(𝑡,𝑥(𝑡))−𝐶_𝑗𝑡,𝑡−𝑟_𝑗(𝑡)𝑔_𝑗𝑡−𝑟_𝑗(𝑡),𝑥𝑡−𝑟_𝑗(𝑡)×1−𝑟^′_𝑗(𝑡). (46) It follows that if 𝐶_𝑗(𝑡,𝑡)=0, (𝜕/𝜕𝑡)𝐶_𝑗(𝑡,𝑠)=−𝑎_𝑗(𝑡,𝑠), then (43) is equivalent to (44). Easy calculation shows that 𝐶_𝑗(𝑡,𝑠)=^𝑠_𝑡𝑎_𝑗()𝐶𝑢,𝑠𝑑𝑢,_𝑗𝑡,𝑡−𝑟_𝑗(𝑡)=^{𝑡−𝑟𝑗}_𝑡^(𝑡)𝑎_𝑗𝑢,𝑡−𝑟_𝑗(𝑡)𝑑𝑢. (47)

By Lemma 5, (42) can be written as 𝑥̇(𝑡)=𝐵₁(𝑡)+^𝑁_𝑗=1^𝑡_𝑡0𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣}×𝑑𝑑𝑠^𝑠_{𝑠−𝑟𝑗(𝑠)}𝐶_𝑗(𝑠,𝑣)×𝑔_𝑗((𝑣+𝑣,𝑥))𝑑𝑣𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣}𝐶_𝑗𝑠,𝑠−𝑟_𝑗(𝑠)1−𝑟^′_𝑗(𝑠)×𝑔_𝑗𝑠−𝑟_𝑗(𝑠),𝑥𝑠−𝑟_𝑗(𝑠)𝑑𝑠. (48) We define 𝐶⁰_𝜓=𝜙∶𝑚𝑡₀(𝑡)(𝑡),𝑚𝑡,∞⟶𝑅∣𝜙∈𝐶,𝜙=𝜓𝑡∈₀,𝑡₀,𝜙(𝑡)⟶0as𝑡⟶∞. (49) Multiplying both sides of the above equation by 𝑒^{∫𝑡𝑡0𝐻(𝑢)𝑑𝑢}, easy calculation shows that 𝑥(𝑡)=𝜓𝑡₀𝑒^{−∫𝑡𝑡0𝐻(𝑢)𝑑𝑢}+^𝑡_𝑡0𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}𝐵₁(𝑠)+𝑑𝑠_𝑡𝑡0𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}𝐻(𝑠)𝑥(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣}×^𝑢_𝑡0𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑑𝑑𝑠×^𝑠_{𝑠−𝑟𝑗(𝑠)}𝐶_𝑗(𝑠,𝑣)𝑔_𝑗((𝑣+𝑣,𝑥))𝑑𝑣𝑑𝑠𝑑𝑢_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣𝑢}_𝑡0𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑊_𝑗(𝑠)×𝑔_𝑗𝑠−𝑟_𝑗(𝑠),𝑥𝑠−𝑟_𝑗(𝑠)𝑑𝑠𝑑𝑢, (50) where 𝑊_𝑗(𝑠)=𝐶_𝑗(𝑠,𝑠−𝑟_𝑗(𝑠))(1−𝑟^′_𝑗(𝑠)). Performing an integration by parts, then 𝑥(𝑡)=𝜓𝑡₀𝑒^{−∫𝑡𝑡0𝐻(𝑠)𝑑𝑠}+^𝑡_𝑡0𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}𝐵₁(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡𝑡−𝑟𝑗(𝑡)}ℎ_𝑗(𝑣)𝑥(𝑣)−𝑑𝑣_{𝑁𝑗=1𝑡0𝑡0−𝑟𝑗(𝑡0)}ℎ_𝑗(𝑢)𝜓(𝑢)𝑑𝑢⋅𝑒^{−∫𝑡𝑡0𝐻(𝑢)𝑑𝑢}−^𝑁_𝑗=1^𝑡_𝑡0^𝑠_{𝑠−𝑟𝑗(𝑠)}ℎ_𝑗(𝑣)𝑥(𝑣)𝑑𝑣𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}𝐻(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}ℎ_𝑗𝑠−𝑟_𝑗(𝑠)𝑥𝑠−𝑟_𝑗(𝑠)×1−𝑟^′_𝑗(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣𝑢}_{𝑢−𝑟𝑗(𝑢)}𝐶_𝑗(𝑢,𝑣)𝑔_𝑗((𝑣−𝑣,𝑥))𝑑𝑣𝑑𝑢_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣𝑢}_𝑡0𝑒^{−∫𝑢𝑡0𝐴(𝑣)𝑑𝑣}×^𝑡0_{𝑡0−𝑟𝑗𝑡0}𝐶_𝑗𝑡₀,𝑣×𝑔_𝑗(𝑣)−𝑣,𝜓𝑑𝑣𝑑𝑠𝑑𝑢_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢𝑠}_𝑡0𝑒^{−∫𝑠𝑣𝐴(𝑟)𝑑𝑟}𝐴(𝑣)×^𝑣_{𝑣−𝑟𝑗(𝑣)}𝐶_𝑗(𝑣,𝑢)×𝑔_𝑗((𝑢+𝑢,𝑥))𝑑𝑢𝑑𝑣𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣𝑢}_𝑡0𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}𝑊_𝑗(𝑠)×𝑔_𝑗𝑠−𝑟_𝑗(𝑠),𝑥𝑠−𝑟_𝑗(𝑠)𝑑𝑠𝑑𝑢. (51) Let 𝑃₂ be a mapping defined on 𝐶⁰_𝜓 as follows: for 𝜙∈𝐶_𝜓, if 𝑡∈[𝑚(𝑡₀),𝑡₀], (𝑃₂𝜙)(𝑡)=𝜓(𝑡). If 𝑡>𝑡₀, we define 𝑃₂𝜙(𝑡)=𝜓𝑡₀𝑒^{−∫𝑡𝑡0𝐻(𝑠)𝑑𝑠}+^𝑡_𝑡0𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}𝐵₁(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡𝑡−𝑟𝑗(𝑡)}ℎ_𝑗(𝑣)𝜙(𝑣)−𝑑𝑣_{𝑁𝑗=1𝑡0𝑡0−𝑟𝑗(𝑡0)}ℎ_𝑗(𝑢)𝜓(𝑢)𝑑𝑢⋅𝑒^{−∫𝑡𝑡0𝐻(𝑢)𝑑𝑢}−^𝑁_𝑗=1^𝑡_𝑡0^𝑠_{𝑠−𝑟𝑗(𝑠)}ℎ_𝑗(𝑣)𝜙(𝑣)𝑑𝑣𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}𝐻(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}ℎ_𝑗(𝑠−𝑟(𝑠))𝜙𝑠−𝑟_𝑗(𝑠)×1−𝑟^′_𝑗(𝑠)+𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣𝑢}_{𝑢−𝑟𝑗(𝑢)}𝐶_𝑗(𝑢,𝑣)𝑔_𝑗(𝑣)−𝑣,𝜙𝑑𝑣𝑑𝑢_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣}×^𝑢_𝑡0𝑒^{−∫𝑢𝑡0𝐴(𝑣)𝑑𝑣}×^𝑡0_{𝑡0−𝑟𝑗𝑡0}𝐶_𝑗𝑡₀,𝑣×𝑔_𝑗(𝑣)−𝑣,𝜓𝑑𝑣𝑑𝑠𝑑𝑢_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑠𝐻(𝑢)𝑑𝑢}×^𝑠_𝑡0𝑒^{−∫𝑠𝑣𝐴(𝑟)𝑑𝑟}𝐴(𝑣)×^𝑣_{𝑣−𝑟𝑗(𝑣)}𝐶_𝑗(𝑣,𝑢)𝑔_𝑗(𝑢+𝑢,𝜙)𝑑𝑢𝑑𝑣𝑑𝑠_{𝑁𝑗=1𝑡𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣}×^𝑢_𝑡0𝑒^{−∫𝑢𝑠𝐴(𝑣)𝑑𝑣}×𝑊_𝑗(𝑠)𝑔_𝑗𝑠−𝑟_𝑗(𝑠),𝜙𝑠−𝑟_𝑗(𝑠)𝑑𝑠𝑑𝑢. (52)

If 𝜙∈𝐶⁰_𝜓, since ∫^𝑡₀𝐻(𝑠)𝑑𝑠→∞ as 𝑡→∞, 𝑡₀≥0, hence ∫^𝑡_𝑡0𝐻(𝑠)𝑑𝑠→∞ as 𝑡→∞. The first term and fourth term of (𝑃₂𝜙)(𝑡)→0, as 𝑡→∞.

Note that ^𝑡_𝑡0𝑒^{−∫𝑡𝑢𝐻(𝑠)𝑑𝑠}𝐵₁(𝑢)𝑑𝑢=𝑥̇𝑡₀^𝑡_𝑡0𝑒^{−∫𝑡𝑢𝐻(𝑠)𝑑𝑠}𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}𝑑𝑢. (53) Since 𝐴(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑦(𝑡))≥𝐴₁(𝑡)≥0, then ^𝑡_𝑡0𝑒^{−∫𝑡𝑢𝐻(𝑠)𝑑𝑠}𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}𝑑𝑢≤^𝑡_𝑡0𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}𝑑𝑢. (54)

For a given 𝜖>0, there exists 𝑇₁>𝑄₀+𝑡₀ such that 𝑒^{𝑎0𝑡0−𝑎0𝑇}<𝑎₀𝜖. For 𝑇₁<𝑡 and 𝑡₀<𝑄₀<𝑇₁≤𝑢, we have ^𝑡_𝑇1𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}𝑑𝑢≤^𝑡_𝑇1𝑒^{−𝑎0(𝑢−𝑡0)}𝑒𝑑𝑢=^𝑎0𝑡0𝑎₀𝑒^{−𝑎0𝑇1}−𝑒^−𝑎0𝑡<𝑒^{𝑎0𝑡0−𝑎0𝑇1}𝑎₀<𝜖. (55) For 𝑡>𝑇₁, we have ^𝑇1_𝑡0𝑒^{−∫𝑡𝑢𝐻(𝑠)𝑑𝑠}𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}𝑑𝑢=𝑒^{−∫𝑡𝑇1𝐻(𝑠)𝑑𝑠}⋅^𝑇1_𝑡0𝑒^{−∫𝑇1𝑢𝐻(𝑠)𝑑𝑠}𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}𝑑𝑢. (56) The second term of (𝑃₂𝜙)(𝑡)→0, as 𝑡→∞.

Note that ∫^𝑡0_{𝑡0−𝑟𝑗(𝑡0)}𝐶_𝑗(𝑡₀,𝑣)𝑔_𝑗(𝑣,𝜓(𝑣))𝑑𝑣 is bounded; we can check the eighth term of (𝑃₂𝜙)(𝑡)→0, as 𝑡→∞ similarly.

Then we can easily check that 𝑃₂ is a contraction mapping on 𝐶⁰_𝜓 by using condition (ii). By the contraction mapping principle, 𝑃₂ has a unique fixed point 𝑥(𝑡) in 𝐶⁰_𝜓. Also 𝑥(𝑡)→0 as 𝑡→+∞.

In order to obtain the asymptotic stability, we still need to show that the zero solution is stable. Let 𝜖>0 be given, we choose 𝜓∶[𝑚(𝑡₀),𝑡₀]→𝑅 and 𝑥̇(𝑡₀) such that 𝛿+𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀𝑥̇𝑡₀+^𝑁_𝑗=1^𝑡0_{𝑡0−𝑟𝑗𝑡0}ℎ_𝑗(𝑠)𝑒𝑑𝑠⋅𝛿+𝛼Γ𝑄+^−𝑎0𝑄𝑎₀𝛿≤(1−𝛼)𝜖. (57) Then we have |𝑥(𝑡)|≤𝛿+^𝑡_𝑡0𝑒^{−∫𝑡𝑢𝐻(𝑠)𝑑𝑠}𝑥̇𝑡₀𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}+𝑑𝑢_{𝑁𝑗=1𝑡0𝑡0−𝑟𝑗𝑡0}ℎ_𝑗(𝑠)𝑒𝑑𝑠⋅𝛿+𝛼Γ𝑄+^−𝑎0𝑄𝑎₀𝛿+𝛼𝜖. (58) Easy calculation shows that ^𝑡_𝑡0𝑒^{−∫𝑢𝑡0𝐴(𝑠)𝑑𝑠}𝑑𝑢≤^𝑡0+𝑄0_𝑡0𝑑𝑢+^𝑡_𝑡0+𝑄0𝑒^{−𝑎0(𝑢−𝑡0)}𝑑𝑢≤𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀. (59) Hence |𝑥(𝑡)|≤𝜖. (60) Therefore |𝑥̇(𝑡≤𝑥̇𝑡)|₀+^𝑁_𝑗=1^𝑡_𝑡0𝑒^{−∫𝑡𝑠𝐴(𝑣)𝑑𝑣}×^𝑠_{𝑠−𝑟𝑗(𝑠)}𝑎_𝑗(𝑠,𝑣)𝑔_𝑗(𝑣,𝑥(𝑣))𝑑𝑣𝑑𝑠. (61) It follows that |𝑥(𝑡)|+𝑦(𝑡)<𝜖2+𝑁𝐾^∗𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀. (62) Therefore the zero solution is stable; since we have obtained that 𝑥(𝑡)→0 as 𝑡→∞, it follows that the zero solution is asymptotically stable.

A necessary condition is as follows. For each 𝑡₀≥0, we denote that 𝑀=sup_𝑡≥0{𝑒^{−∫𝑡0𝐻(𝑠)𝑑𝑠}}. We will prove that ∫^∞₀𝐻(𝑠)𝑑𝑠=∞ by way of contradiction. If ∫^∞₀𝐻(𝑠)𝑑𝑠<∞, since 𝐻(𝑡)≥0, ∃ being a sequence {𝑡_𝑛}, 𝑡_𝑛→∞ as 𝑛→∞ such that lim_𝑛→∞∫^𝑡𝑛₀𝐻(𝑠)𝑑𝑠=𝜎, where 𝜎 is a finite number. Choose 𝜂 such that −𝜂≤∫^𝑡𝑛₀𝐻(𝑠)𝑑𝑠≤𝜂 holds, for all 𝑛≥1.

Denote that 𝜔(𝑢)≜𝐻(𝑢)^𝑁_𝑗=1^𝑢_{𝑢−𝑟𝑗(𝑢)}ℎ_𝑗(𝑣+)𝑑𝑣_𝑁𝑗=1ℎ_𝑗𝑠−𝑟_𝑗(𝑠)1−𝑟^′_𝑗(𝑠)+^𝑁_𝑗=1^𝑢_𝑡0𝑒^{−∫𝑢𝑣𝐴(𝑟)𝑑𝑟}𝐴(𝑣)×^𝑣_{𝑣−𝑟𝑗(𝑣)}𝐶_𝑗(𝑣,𝑤)𝑄_𝑗(𝑤)+𝑑𝑤𝑑𝑣_{𝑁𝑗=1𝑡0}𝑒^{−∫𝑡𝑢𝐻(𝑣)𝑑𝑣𝑢}₀𝑒^{−∫𝑢𝑠𝐴1(𝑣)𝑑𝑣}𝐾^∗×𝑊_𝑗(𝑠+)𝑑𝑠𝑑𝑢_{𝑁𝑗=1𝑢𝑢−𝑟𝑗(𝑢)}𝐶_𝑗(𝑢,𝑣)𝑄_𝑗(𝑣)𝑑𝑣𝑑𝑢, (63) where 𝑊_𝑗(𝑠)=(1−𝑟^′_𝑗(𝑠))∫^{𝑠−𝑟𝑗}_𝑠^(𝑠)𝑎_𝑗(𝑣,𝑠−𝑟_𝑗(𝑠))𝑑𝑣.

By conditions of Theorem 2, we have 0≤^𝑡𝑛₀𝜔(𝑢)𝑒^{−∫𝑡𝑛𝑢𝐻(𝑠)𝑑𝑠}𝑑𝑢≤𝛼. (64) Then ^𝑡𝑛₀𝜔(𝑢)𝑒^{∫𝑢0𝐻(𝑠)𝑑𝑠}𝑑𝑢≤𝛼𝑒^{∫𝑡𝑛0𝐻(𝑠)𝑑𝑠}≤𝛼𝑒^𝜂. (65) Thus the sequence {∫^𝑡𝑛₀𝜔(𝑢)𝑒^{∫𝑢0𝐻(𝑠)𝑑𝑠}𝑑𝑢} is bounded; there exists a convergent subsequence; we assume that lim_𝑛→∞∫^𝑡𝑛₀𝜔(𝑢)𝑒^{∫𝑢0𝐻(𝑠)𝑑𝑠}𝑑𝑢=𝐼, 𝐼≥0. We can choose a positive integer 𝑘 large enough such that ^𝑡𝑛_𝑡𝑘𝜔(𝑢)𝑒^{∫𝑢0𝐻(𝑠)𝑑𝑠}𝛿𝑑𝑢<⁰8𝑀 (66) for ∀𝑛≥𝑘, where 0<𝛿₀(𝛿₀<𝜖) is satisfying 𝛿₀+max𝑄₀+𝑒^{−𝑎0𝑄0}𝑎₀,1𝛿₀+𝛿₀^𝑁_𝑗=1^𝑡𝑘_{𝑡𝑘−𝑟𝑗𝑡𝑘}ℎ_𝑗(𝑠)𝑄𝑑𝑠+𝛼Γ₀+𝑒^{−𝑎0𝑄0}𝑎₀𝛿₀≤(1−𝛼). (67)

Now we consider the solution 𝑥(𝑡)=𝑥(𝑡,𝜓,𝑥̇(𝑡_𝑘)) of (11) which satisfies 𝜓𝑡_𝑘=3𝛿₀4,𝑥̇𝑡_𝑘=𝛿₀4,𝜓(𝑠)+|𝑥̇(𝑠)|≤𝛿₀,𝑠≤𝑡_𝑘. (68) We can obtain that |𝑥(𝑡)|≤1 by a similar argument with (60) if we replace 𝜖 by 1. Consider 𝜓𝑡_𝑘−^𝑁_𝑗=1^𝑡𝑘_{𝑡𝑘−𝑟𝑗(𝑡𝑘)}ℎ_𝑗(𝑠)𝜓(𝑠)𝑡𝑑𝑠≥𝜓_𝑘−^𝑁_𝑗=1^𝑡𝑘_{𝑡𝑘−𝑟𝑗𝑡𝑘}ℎ_𝑗(𝑠𝜓(𝑠≥))𝑑𝑠3𝛿₀4−𝛿₀2=𝛿₀4. (69)

By (27) and |𝑥(𝑡)|≤1, if 𝑡_𝑛≥𝑡_𝑘, we have 𝑥𝑡_𝑛−^𝑁_𝑗=1^𝑡𝑛_{𝑡𝑛−𝑟𝑗𝑡𝑛}ℎ_𝑗(𝑠)𝑥(𝑠)≥𝜓𝑡𝑑𝑠_𝑘𝑒^{−∫𝑡𝑛𝑡𝑘𝐻(𝑠)𝑑𝑠}+^𝑡𝑛_𝑡𝑘𝑒^{−∫𝑡𝑛𝑢𝐻(𝑠)𝑑𝑠}𝐵(𝑢)−𝑑𝑢_{𝑁𝑗=1𝑡𝑘𝑡𝑗𝑘−𝑟(𝑡𝑘)}ℎ_𝑗(𝑠)𝜓(𝑠)𝑑𝑠⋅𝑒^{−∫𝑡𝑛𝑡𝑘𝐻(𝑢)𝑑𝑢}−^𝑡𝑛_𝑡𝑘𝑒^{−∫𝑡𝑛𝑢𝐻(𝑠)𝑑𝑠}𝜔(𝑢)𝑑𝑢. (70) This implies that 𝑥𝑡_𝑛−^𝑁_𝑗=1^𝑡𝑛_{𝑡𝑛−𝑟𝑗𝑡𝑛}ℎ_𝑗(𝑠)𝑥(𝑠)𝑑𝑠≥𝑒^{−∫𝑡𝑛𝑡𝑘𝐻(𝑠)𝑑𝑠}𝛿₀4+^𝑡𝑛_𝑡𝑘𝑒^{∫𝑢𝑡𝑘𝐻(𝑠)𝑑𝑠}𝑥̇𝑡_𝑘𝑒^{∫𝑢𝑡𝑘𝐴(𝑠)𝑑𝑠}−𝑑𝑢_{𝑡𝑛𝑡𝑘}𝜔(𝑢)𝑒^{∫𝑢𝑡𝑘𝐻(𝑠)𝑑𝑠}𝑑𝑢. (71) Note that 𝑥̇(𝑡_𝑘)=𝛿₀/4>0; it follows that 𝑥𝑡_𝑛−^𝑁_𝑗=1^𝑡𝑛_{𝑡𝑛−𝑟𝑗𝑡𝑛}ℎ_𝑗(𝑠)𝑥(𝑠)𝑑𝑠≥𝑒^{−∫𝑡𝑛𝑡𝑘ℎ(𝑠)𝑑𝑠}𝛿₀4−𝑒^{−∫𝑡𝑘0𝐻(𝑠)𝑑𝑠}⋅^𝑡𝑛_𝑡𝑘𝜔(𝑢)𝑒^{∫𝑢0𝐻(𝑠)𝑑𝑠}≥𝛿𝑑𝑢₀8𝑒^{−∫𝑡𝑛𝑡𝑘𝐻(𝑠)𝑑𝑠}≥𝛿₀8𝑒^−2𝜂>0. (72)

If the zero solution 𝑥(𝑡) is asymptotically stable, then 𝑥(𝑡)→0 as 𝑡→∞. By the mean value theorem and condition (ii), we have ^𝑡𝑛_{𝑡𝑛−𝑟𝑗𝑡𝑛}ℎ_𝑗(𝑠)𝑥(𝑠)𝑑𝑠=(𝑥(𝜛))^𝑡𝑛_{𝑡𝑛−𝑟𝑗𝑡𝑛}ℎ_𝑗(𝑠)≤|𝑥(𝜛.𝑑𝑠)| (73) Thus ^𝑁_𝑗=1^𝑡𝑛_{𝑡𝑛−𝑟𝑗𝑡𝑛}ℎ_𝑗(𝑠)𝑥(𝑠)𝑑𝑠⟶0as𝑡_𝑛⟶∞. (74) Consequently, lim_{𝑡𝑛→∞}𝑥𝑡_𝑛−^𝑁_𝑗=1^𝑡𝑛_{𝑡𝑛−𝑟𝑗𝑡𝑛}ℎ_𝑗(𝑠)𝑥(𝑠)𝑑𝑠=0. (75) This is a contradiction to (72).

Hence ^∞₀𝐻(𝑠)𝑑𝑠=∞. (76)

This completes the proof of Theorem 2.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by NNSF of China, Grants 11226145 and 11271046, and a Research Foundation of Huaqiao University 12BS112.