Stability for a Class of Differential Equations with Nonconstant Delay

Abstract

Stability is investigated for the following differential equations with nonconstant delay 𝑥^'(𝑡)=𝑞(𝑡)𝐹(𝑥(𝑡))−𝑝(𝑡)𝑓(𝑥(𝑡−𝜏(𝑡))), where 𝑝∶[0,+∞)→[0,+∞), 𝑞∶[0,+∞)→𝑅, 𝜏∶[0,+∞)→[0,𝑟], and 𝐹 and 𝑓∶𝑅→𝑅 with 𝑥𝑓(𝑥)>0for𝑥≠0and|𝑥|≤𝑎 (𝑎 is a positive constant) are continuous functions. A criterion is given for the zero solution of this delay equation being uniformly stable and asymptotically stable.

1. Introduction

Delays are inherent in many physical and technological systems. In particular, pure delays are often used to ideally represent the effects of transmission, transportation, and inertia phenomena. Delay differential equations constitute basic mathematical models of real phenomena, for instance in biology, mechanics, and economics (cf., e.g., [1–17] and references therein). Stability analysis of delay differential equations is particularly relevant in control theory, where one cause of delay is the finite speed of communication. There have been a lot of results on the study of stability of delay differential equations. For example, we can see many earlier results on this issue from Burton's book [2]. Recently, in 2004, Butcher et al. [4] studied the stability properties of delay differential equations with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the system is reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. In 2005, Wahi and Chatterjee [16] used Galerkin-projection to reduce the infinite dimensional dynamics of a delay differential equation to one occurring on a finite number of modes. In 2009, Kalmár-Nagy [7] demonstrated that the method of steps for linear delay differential equation together with the inverse Laplace transform can be used to find a converging sequence of polynomial approximants to the transcendental function determining stability of the delay equation. Most recently, Berezansky and Braverman [3] gave some explicit conditions of asymptotic and exponential stability for the scalar nonautonomous linear delay differential equation with several delays and an arbitrary number of positive and negative coefficients.

This paper is concerned with the following differential equations with nonconstant delay: 𝑥^′(𝑡)=𝑞(𝑡)𝐹(𝑥(𝑡))−𝑝(𝑡)𝑓(𝑥(𝑡−𝜏(𝑡))), (1) where 𝑎∶[0,+∞)→[0,+∞), 𝑞∶[0,+∞)→𝑅, 𝜏∶[0,+∞)→[0,𝑟], and 𝐹 and 𝑓∶𝑅→𝑅 with 𝑥𝑓(𝑥)>0for𝑥≠0,|𝑥|≤𝑎 (2) (𝑎 is a positive constant) are continuous functions. We aim at giving general criterion for the zero solution of this delay equation being uniformly stable and asymptotically stable.

2. Main Result

Denote by 𝐶[𝑡₀−𝑟,𝑡₀] the Banach space of continuous functions from [𝑡₀−𝑟,𝑡₀] to 𝑅 with the sup-norm 𝜑_{𝐶[𝑡0−𝑟,𝑡0]}=max_{𝑡𝑠∈0−𝑟,𝑡0}𝜑(𝑠,𝑡)forevery𝜑∈𝐶₀−𝑟,𝑡₀. (3)

We consider (1) for 𝑡≥𝑡₀ with the initial conditions (for any 𝑡₀≥0) 𝑥(𝑡)=𝜑(𝑡),𝑡₀−𝑟≤𝑡≤𝑡₀, (4) where 𝜑∈𝐶[𝑡₀−𝑟,𝑡₀].

For an initial function 𝜑∈𝐶[𝑡₀−𝑟,𝑡₀], we denote by 𝑥(𝑡;𝑡₀,𝜑) the solution of (1) such that (4) holds.

Definition 1. The zero solution of (1) is said to be stable if for any 𝜀>0 and 𝑡₀≥0, there exists 𝛿(𝑡₀,𝜀)>0 such that if 𝜑_{𝐶[𝑡0−𝑟,𝑡0]}<𝛿, (5) then 𝑥𝑡;𝑡₀,𝜑<𝜀∀𝑡≥𝑡₀. (6) The zero solution of (1) is uniformly stable if the above 𝛿 is independent of 𝑡₀.

Definition 2. The zero solution of (1) is said to be asymptotically stable if it is stable and if for any 𝑡₀≥0, there exists 𝛿(𝑡₀)>0 such that if 𝜑_{𝐶[−𝑟,0]}<𝛿, (7) then 𝑥𝑡;𝑡₀,𝜑⟶0,as𝑡⟶+∞. (8)

Theorem 3. Assume that (1)the zero solution to (1) is unique;(2)if 𝑞 is nontrivial function and 𝐹(⋅) is nontrivial in any interval [−𝑏,𝑏](𝑏>0), then lim_𝑡→+∞𝑞(𝑡)=0,lim_𝑡→+∞^𝑡_{(𝑡)𝑡−𝜏}𝑞(𝑠𝑝(𝑡))𝑑𝑠=0,≥𝜇>0,𝑡≥0, (9) for a constant 𝜇; (3)lim_𝑡→+∞∫^𝑡_{𝑡−𝜏(𝑡)}𝑝(𝑠)𝑑𝑠=𝐴; (4)if 𝐴≠0, then 𝑓(𝑥≤𝜆|𝑥|)2𝐴,for𝑥∈𝑅, (10) where 0<𝜆<1.
Then the zero solution of (1) is uniformly stable.

Proof. For each 𝜀>0, we set 𝑆𝑓,𝜀∶=sup𝑓(𝑥);|𝑥|≤𝜀, (11) and when 𝑞 is a nontrivial function and 𝐹(⋅) is nontrivial in any interval [−𝑏,𝑏] (𝑏>0), we set 𝑆()𝐹(𝑥;|𝑥|},𝐼(𝜀)𝑦𝐹,𝜀∶=sup{|)|≤𝜀∶=inf𝑥𝑓;𝑥𝑦>0,1−𝜆2𝜀≤|𝑥|≤𝜀,1−𝜆2𝜀≤𝑦≤𝜀. (12)(13) From (3) and (2), it follows that for every 𝜀>0, there exists 𝑡(𝜀)>0 such that ^𝑡_{(𝑡)𝑡−𝜏}𝑝(𝑠)<𝑑𝑠1−𝜆4𝑆𝑓,𝜀+1𝜀,if𝐴=0,1−𝜆4𝑆{}(𝜀),𝑓,𝜀+1min𝐴,1𝜀+𝐴,if𝐴≠0,∀𝑡>𝑡 (14) and when 𝑞 is a nontrivial function and 𝐹(⋅) is nontrivial in any interval [−𝑏,𝑏] (𝑏>0), such that ^𝑡_{(𝑡)𝑡−𝜏}𝑞(𝑠)𝑑𝑠<1−𝜆4(𝑆())(𝜀),𝑞(𝑡𝐼(𝜀)𝐹,𝜀+1𝜀,∀𝑡>𝑡)≤𝜇2(𝑆())(𝜀).𝐹,𝜀+1)(𝜀+1,∀𝑡>𝑡 (15)(16) We claim that for any 𝜀>0 and 𝑡₀≥𝑡(𝜀), if 𝜑_{𝐶[𝑡0−𝑟,𝑡0]}<1−𝜆2𝜀, (17) then 𝑥𝑡;𝑡₀,𝜑<𝜀∀𝑡≥𝑡₀, (18) which means that the zero solution of (1) is eventually uniformly stable. Actually, if this is not true, then there exist 𝜀₀≤min{𝑎,1} (19) and a solution 𝑥(𝑡)∶=𝑥𝑡;𝑡₀,𝜑 (20) to (1) with ‖𝜑‖_{𝐶[𝑡0−𝑟,𝑡0]}<((1−𝜆)/2)𝜀 and 𝑡₀>𝑡𝜀₀ (21) such that there is a 𝑡>𝑡₀, 𝑥𝑡≥𝜀₀. (22) Define 𝑡₂∶=inf𝑡≥𝑡₀;|𝑥(𝑡)|=𝜀₀,𝑡₁∶=sup𝑡₀≤𝑡<𝑡₂;|𝑥(𝑡)|=1−𝜆2𝜀₀,𝑉(𝑥)=𝑥²,𝑥∈𝑅. (23)(24) Then, together with (21) and (22), we obtain 𝑡𝜀₀<𝑡₁<𝑡₂,𝑉𝑥𝑡₁=(1−𝜆)²4𝜀²₀,𝑉𝑥𝑡₂>𝜀²₀, (25) and, for 𝑡∈(𝑡₁,𝑡₂), (1−𝜆)²4𝜀²₀<𝑉(𝑥(𝑡))<𝜀²₀, (26) and for arbitrary 𝜂>0, there exists 𝜉∈[𝑡₂−𝜂,𝑡₂] such that 𝑉^′(𝑥(𝜉))>0. (27) Therefore, 𝑉^′𝑥𝑡₂≥0. (28) This implies that 𝑡₁≥𝑡₂−𝜏𝑡₂. (29) In fact, if 𝑡₁<𝑡₂−𝜏𝑡₂, (30) then by (23)–(25), we have 1−𝜆2𝜀₀≤𝑥𝑡₂−𝜏𝑡₂≤𝜀₀,𝑡₂−𝜏𝑡₂>𝑡𝜀₀. (31) It is not hard to see that we can choose 𝑡₁ and 𝑡₂ above to make 𝑥(𝑡) have constant sign in [𝑡₁,𝑡₂].
Case I. When 𝑞(𝑡)≡0 or 𝐹(𝑥)≡0for|𝑥|≤𝑏, (32) where 𝑏 is a positive real number.
In this case, if 𝑞(𝑡)≡0, then 𝑉^′𝑥𝑡₂=−2𝑝𝑡₂𝑥𝑡₂𝑓𝑥𝑡₂−𝜏𝑡₂<0, (33) which contradicts with (28). Moreover, if 𝐹(𝑥)≡0for|𝑥|≤𝑏, (34) for a positive real number 𝑏, then it is clear that we can require 𝜀₀<𝑏. Hence, 𝑉^′𝑥𝑡₂=−2𝑝𝑡₂𝑥𝑡₂𝑓𝑥𝑡₂−𝜏𝑡₂<0, (35) which contradicts with (28) too.
Consequently, in this case we have the following observation.
Case I-1. If 𝐴=0, then we deduce by (23), (24), (1), and (11) that 𝜀₀2+𝜆2𝜀₀=𝑥𝑡₂−𝑥𝑡₁≤𝑥𝑡₂−𝑥𝑡₁≤^𝑡2_𝑡1𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_𝑡2𝑡1𝑞(𝑠)|𝐹(𝑥(𝑠))|𝑑𝑠≤𝑆𝑓,𝜀₀^𝑡2_𝑡1𝑝(𝑠)𝑑𝑠≤𝑆𝑓,𝜀₀^𝑡2_{𝑡2𝑡−𝜏2}𝑝(𝑠)<𝜀𝑑𝑠₀2. (36) This is clearly impossible.
Case I-2. If 𝐴≠0, then we deduce by (23), (24), (1), (11), and (14) that 𝜀₀2+𝜆2𝜀₀=𝑥𝑡₂−𝑥𝑡₁≤𝑥𝑡₂−𝑥𝑡₁≤^𝑡2_𝑡1𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_𝑡2𝑡1𝑞(𝑠𝐹(𝑥(𝑠≤)|))|𝑑𝑠𝜆𝜀₀2𝐴^𝑡2_𝑡1𝑝(𝑠)≤𝑑𝑠𝜆𝜀₀2𝐴^𝑡2_{𝑡2𝑡−𝜏2}𝑝(𝑠)≤𝑑𝑠𝜆𝜀₀2𝐴1−𝜆4𝑆𝑓,𝜀₀{}𝜀+1min𝐴,1₀<+𝐴1−𝜆4𝜆𝜀+2𝜀<𝜀2. (37) This is clearly impossible too.
Therefore, in this case, the zero solution of (1) is eventually uniformly stable. This, together with assumption (1), implies that the zero solution of (1) is uniformly stable.
Case II. 𝑞 is a nontrivial function and 𝐹(⋅) is nontrivial in any interval [−𝑏,𝑏] (𝑏>0).
In this case, by virtue of (1), and assumption (2), (12), (13), and (16), we get 𝑉^′𝑥𝑡₂=−2𝑝𝑡₂𝑥𝑡₂𝑓𝑥𝑡₂−𝜏𝑡₂+2𝑥𝑡₂𝑞𝑡₂𝐹𝑥𝑡₂≤−2𝜇𝐼𝜀₀+2𝜀₀𝜇𝐼𝜀₀2𝑆𝐹,𝜀₀+1𝜀₀𝑆()(𝜀)+1𝐹,𝜀≤−𝜇𝐼<0, (38) which contradicts with (28).
Consequently, in this case we have the following observation,
Case II-1. If 𝐴=0, then we deduce by (23), (24), (1), (11), (12), (14), and (15) that 𝜀₀2+𝜆2𝜀₀=𝑥𝑡₂−𝑥𝑡₁≤𝑥𝑡₂−𝑥𝑡₁≤^𝑡2_𝑡1𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_𝑡2𝑡1𝑞(𝑠)|𝐹(𝑥(𝑠))|𝑑𝑠≤𝑆𝑓,𝜀^𝑡2_𝑡1𝑝(𝑠)()𝑑𝑠+𝑆𝐹,𝜀^𝑡2_𝑡1𝑞(𝑠)𝑑𝑠≤𝑆𝑓,𝜀^𝑡2_{𝑡2𝑡−𝜏2}𝑝(𝑠)()𝑑𝑠+𝑆𝐹,𝜀^𝑡2_{𝑡2𝑡−𝜏2}𝑞(𝑠<)𝑑𝑠1−𝜆4𝜀+1−𝜆4𝜀<𝜀2. (39)
This is a contradiction.
Case II-2. If 𝐴≠0, then we deduce by (23), (24), (1), (11), (12), (14), and (15) that 𝜀2+𝜆2𝜀=𝑥𝑡₂−𝑥𝑡₁≤𝑥𝑡₂−𝑥𝑡₁≤^𝑡2_𝑡1𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_𝑡2𝑡1𝑞(𝑠𝐹(𝑥(𝑠≤)|))|𝑑𝑠𝜆𝜀₀2𝐴^𝑡2_𝑡1𝑝(𝑠)𝑑𝑠+𝑆(𝐹,𝜀)^𝑡2_𝑡1𝑞(𝑠≤)𝑑𝑠𝜆𝜀₀2𝐴^𝑡2_{𝑡2𝑡−𝜏2}𝑝(𝑠)𝑑𝑠+𝑆(𝐹,𝜀)^𝑡2_{𝑡2𝑡−𝜏2}𝑞(𝑠≤)𝑑𝑠𝜆𝜀₀2𝐴1−𝜆4𝑆𝑓,𝜀₀{}𝜀+1min𝐴,1₀++𝐴1−𝜆4𝜀<1−𝜆4𝜆𝜀+2𝜀+1−𝜆4𝜀=𝜀2. (40) This is a contradiction too.
Therefore, in this case, the zero solution of (1) is eventually uniformly stable. This, together with assumption (1), implies that the zero solution of (1) is uniformly stable.

Theorem 4. Assume that (1)the zero solution to (1) is unique;(2)if 𝑞(𝑡)≡0 or 𝐹(𝑥)≡0for|𝑥|≤𝑏, (41) for a positive real number 𝑏, then ₀^+∞𝑝(𝑠)𝑑𝑠=+∞; (42)(3)if 𝑞 is nontrivial function and 𝐹(⋅) is nontrivial in any interval [−𝑏,𝑏]  (𝑏>0), then lim_𝑡→+∞𝑞(𝑡)=0,lim_𝑡→+∞^𝑡_{(𝑡)𝑡−𝜏}𝑞(𝑠)𝑑𝑠=0,_0+∞𝑞(𝑠)𝑝(𝑡)𝑑𝑠<+∞,≥𝜇>0,𝑡≥0, (43) for a constant 𝜇;(4)lim_𝑡→+∞∫^𝑡_{𝑡−𝜏(𝑡)}𝑝(𝑠)𝑑𝑠=𝐴; (5)if 𝐴≠0, then 𝑓(𝑥≤𝜆|𝑥|)2𝐴,for𝑥∈𝑅, (44) where 0<𝜆<1. Then the zero solution of (1) is asymptotically stable.

Proof. It follows from Theorem 3 that the zero solution of (1) is uniformly stable; that is, for arbitrarily given 𝜀>0 and 𝑡₀≥0, there exists 𝛿=𝛿(𝜀)>0 such that if 𝜑_{𝐶[𝑡0−𝑟,𝑡0]}<𝛿, (45) then 𝑥𝑡;𝑡₀,𝜑<𝜀∀𝑡≥𝑡₀. (46) Next, we will prove that 𝑥𝑡;𝑡₀,𝜑⟶0,as𝑡⟶+∞. (47) First, we show that liminf_𝑡→+∞𝑥𝑡;𝑡₀,𝜑=0. (48) Suppose that this is not true. Then liminf_𝑡→+∞𝑥𝑡;𝑡₀,𝜑>0. (49) Hence, for the arbitrarily given 0<𝜀<min{𝑎,𝑏}, (50) there exist 0<𝜀₀<𝜀 and 𝑇>𝑡₀ such that 𝑥𝑡;𝑡₀,𝜑>𝜀₀∀𝑡≥𝑇, (51) or 𝑥𝑡;𝑡₀,𝜑<−𝜀₀∀𝑡≥𝑇. (52) Let us now consider 𝑥𝑡;𝑡₀,𝜑>𝜀₀∀𝑡≥𝑇. (53)
Case I. When 𝑞(𝑡)≡0 or 𝐹(𝑥)≡0for|𝑥|≤𝑏, (54) for a positive real number 𝑏, we obtain by assumption (2), (46), (50), and (53) 𝑥(𝑡)=𝑥(𝑇+𝑟)−^𝑡_𝑇+𝑟𝑝(𝑠)𝑓(𝑥((𝑠+𝑠−𝜏)))𝑑𝑠_{𝑡𝑇+𝑟}𝑞(𝑠)𝐹(𝑥(𝑠()𝑓(𝑥)𝜀))𝑑𝑠≤𝑥𝑇+𝑟−inf;𝑥∈₀,𝜀^𝑡_𝑇+𝑟𝑝(𝑠)𝑑𝑠. (55) This implies that 𝑥(𝑡)⟶−∞as𝑡⟶+∞, (56) which contradicts with (53).
Case II. When 𝑞 is a nontrivial function and 𝐹(⋅) is nontrivial in any interval [−𝑏,𝑏] (𝑏>0), we obtain by assumptions (3), (46), (50), and (53) 𝑥(𝑡)=𝑥(𝑇+𝑟)−^𝑡_𝑇+𝑟𝑝(𝑠)𝑓(𝑥((𝑠+𝑠−𝜏)))𝑑𝑠_{𝑡𝑇+𝑟}𝑞(𝑠)𝐹(𝑥(𝑠()𝑓(𝑥)𝜀))𝑑𝑠≤𝑥𝑇+𝑟−𝜇inf;𝑥∈₀)𝐹(𝑥𝜀,𝜀(𝑡−𝑇−𝑟+sup|)|;𝑥∈₀,𝜀^𝑡_𝑇+𝑟𝑞(𝑠)𝑑𝑠. (57) This, together with assumption (2), implies that 𝑥(𝑡)⟶−∞as𝑡⟶+∞, (58) which contradicts with (53).
Moreover, in a similar way, we can prove that 𝑥𝑡;𝑡₀,𝜑<−𝜀∀𝑡≥𝑇 (59) is impossible.
Therefore, (48) is true.
Based on (48), we will show that limsup_𝑡→+∞𝑥𝑡;𝑡₀,𝜑=0. (60) Actually, if this is not true, that is, limsup_𝑡→+∞𝑥𝑡;𝑡₀,𝜑>0, (61) then by (48) we see that there are 𝜀₀ with 0<𝜀₀<min{𝑎,𝑏,1}, (62) and two sequences {𝜃_𝑛} and {𝑡_𝑛} such that 𝜃_𝑛<𝑡_𝑛𝜃,𝑛=1,2,…,_𝑛⟶+∞𝑡_𝑛𝑉𝑥𝜃⟶+∞as𝑛⟶+∞,_𝑛=(1−𝜆)²4𝜀²₀,𝑉𝑥𝑡_𝑛>𝜀²₀,𝑉^′𝑥𝑡_𝑛>0, (63) and for 𝑡∈(𝜃_𝑛,𝑡_𝑛), (1−𝜆)²4𝜀²₀<𝑉(𝑥(𝑡))<𝜀²₀. (64) By the same reason as that in the proof of Theorem 3, we know that 𝑡_𝑛−𝜏𝑡_𝑛≤𝜃_𝑛≤𝑡_𝑛. (65) Define 𝑆(𝑓,𝜀), 𝑆(𝐹,𝜀), 𝐼(𝜀), and 𝑡(𝜀) as those in the proof of Theorem 3. Then when 𝑛 is large enough, we have 𝑡_𝑛>𝑡(𝜀). (66)
Case I. When 𝑞(𝑡)≡0 or 𝐹(𝑥)≡0for|𝑥|≤𝑏, (67) where 𝑏 is a positive real number.
Case I-1. If 𝐴=0, then we deduce that 𝜀₀2+𝜆2𝜀₀=𝑥𝑡_𝑛−𝑥𝜃_𝑛≤𝑥𝑡_𝑛−𝑥𝜃_𝑛≤^𝑡𝑛_𝜃𝑛𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_{𝑡𝑛𝜃𝑛}𝑞(𝑠)|𝐹(𝑥(𝑠))|𝑑𝑠≤𝑆𝑓,𝜀₀^𝑡𝑛_𝜃𝑛𝑝(𝑠)𝑑𝑠≤𝑆𝑓,𝜀₀^𝑡𝑛_{𝑡𝑛𝑡−𝜏𝑛}𝑝(𝑠)<𝜀𝑑𝑠₀2. (68) This is impossible.
Case I-2. If 𝐴≠0, then we obtain 𝜀₀2+𝜆2𝜀₀=𝑥𝑡_𝑛−𝑥𝜃_𝑛≤𝑥𝑡_𝑛−𝑥𝜃_𝑛≤^𝑡𝑛_𝜃𝑛𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_{𝑡𝑛𝜃𝑛}𝑞(𝑠𝐹(𝑥(𝑠≤)|))|𝑑𝑠𝜆𝜀₀2𝐴^𝑡𝑛_𝜃𝑛𝑝(𝑠)≤𝑑𝑠𝜆𝜀₀2𝐴^𝑡𝑛_{𝑡𝑛𝑡−𝜏𝑛}𝑝(𝑠)≤𝑑𝑠𝜆𝜀₀2𝐴1−𝜆4𝑆𝑓,𝜀₀{}𝜀+1min𝐴,1₀<+𝐴1−𝜆4𝜆𝜀+2𝜀<𝜀2. (69) This is clearly impossible too.
Consequently, (60) is true in this case.
Case II. When 𝑞 is nontrivial function and 𝐹(⋅) is nontrivial in any interval [−𝑏,𝑏] (𝑏>0).
Case II-1. If 𝐴=0, then we deduce that 𝜀₀2+𝜆2𝜀₀=𝑥𝑡_𝑛−𝑥𝜃_𝑛≤𝑥𝑡_𝑛−𝑥𝜃_𝑛≤^𝑡𝑛_𝜃𝑛𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_{𝑡𝑛𝜃𝑛}𝑞(𝑠)|𝐹(𝑥(𝑠))|𝑑𝑠≤𝑆𝑓,𝜀^𝑡𝑛_𝜃𝑛𝑝(𝑠)()𝑑𝑠+𝑆𝐹,𝜀^𝑡𝑛_𝜃𝑛𝑞(𝑠)𝑑𝑠≤𝑆𝑓,𝜀^𝑡𝑛_{𝑡𝑛𝑡−𝜏𝑛}𝑝(𝑠)()𝑑𝑠+𝑆𝐹,𝜀^𝑡𝑛_{𝑡𝑛𝑡−𝜏𝑛}𝑞(𝑠<)𝑑𝑠1−𝜆4𝜀+1−𝜆4𝜀<𝜀2. (70) This is a contradiction.
Case II-2. If 𝐴≠0, then we obtain 𝜀₀2+𝜆2𝜀₀=𝑥𝑡_𝑛−𝑥𝜃_𝑛≤𝑥𝑡_𝑛−𝑥𝜃_𝑛≤^𝑡𝑛_𝜃𝑛𝑝(𝑠𝑓(𝑥((𝑠+)𝑠−𝜏)))𝑑𝑠_{𝑡𝑛𝜃𝑛}𝑞(𝑠𝐹(𝑥(𝑠≤)|))|𝑑𝑠𝜆𝜀₀2𝐴^𝑡𝑛_𝜃𝑛𝑝(𝑠)𝑑𝑠+𝑆(𝐹,𝜀)^𝑡𝑛_𝜃𝑛𝑞(𝑠≤)𝑑𝑠𝜆𝜀₀2𝐴^𝑡𝑛_{𝑡𝑛𝑡−𝜏𝑛}𝑝(𝑠)()𝑑𝑠+𝑆𝐹,𝜀^𝑡𝑛_{𝑡𝑛𝑡−𝜏𝑛}𝑞(𝑠≤)𝑑𝑠𝜆𝜀₀2𝐴1−𝜆4𝑆𝑓,𝜀₀{}𝜀+1min𝐴,1₀++𝐴1−𝜆4𝜀<1−𝜆4𝜆𝜀+2𝜀+1−𝜆4𝜀=𝜀2. (71) This is a contradiction too.
Therefore, (60) is true in this case. So, (60) holds truly. This means that the zero solution of (4) is asymptotically stable.

Remark 5. Our results are new comparing with the results in [2, 3] since 𝜏(𝑡) could go to 0 or a big number as 𝑡→+∞ and in this case 𝑝(𝑡) also could be very large in our theorems. Moreover, for the case of 𝐴=0, the condition on 𝑓 in our results is very weak.

Acknowledgment

This work was supported by the NSF of China (11171210).