New results on almost periodic solutions for a Nicholson’s blowflies model with a

New results on almost periodic solutions for a Nicholson’s blowflies model with a

linear harvesting term

Aiping Zhang

School of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, P. R. China Received 4 February 2014, appeared 31 July 2014

Communicated by Leonid Berezansky

Abstract. In this paper, we study the global dynamic behavior of a non-autonomous delayed Nicholson’s blowflies model with a linear harvesting term. Under proper con- ditions, we employ a novel argument to establish a criterion on the global exponential stability of positive almost periodic solutions for the model. Moreover, we also provide a numerical example to support the theoretical results.

Keywords: Nicholson’s blowflies model, linear harvesting term, positive almost peri- odic solution, global exponential stability.

2010 Mathematics Subject Classification: 34C25, 34K13.

1 Introduction

Since the exploitation of biological resources and the harvest of population species are com- monly practiced in fishery, forestry, and wildlife management, the study of population dy- namics with harvesting is an important subject in mathematical bioeconomics, which is re- lated to the optimal management of renewable resources (see [3,7,13]). Recently, Berezansky et al. [2] presented the following Nicholson’s blowflies model

x⁰(_t) =−δx(_t) +_px(_t−τ)_e^{−ax(t−τ)}−Hx(_t−σ)_{, δ,p,τ,a,H,}σ∈ (_0, +_∞)_{, (1.1)} whereHx(t−σ)is a linear harvesting term,x(t)is the size of the population at timet, pis the maximum per capita daily egg production, ¹_a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time.

Furthermore, the authors in [1,12,15,16] extended (1.1) to non-autonomous equations with periodic time-varying coefficient and delays, and they also established some criteria to guar- antee the existence of positive periodic solutions for these generalized models by applying the method of coincidence degree and the fixed-point theorem in cones. L. Berezansky et al. [2]

formulated an open problem: what can be said about the dynamic behavior of (1.1).

BEmail: aipingzhang2012@aliyun.com

It is well known that the focus in theoretical models of population and community dy- namics must not be only on how populations depend on their own population densities or the population densities of other organisms, but also on how populations change in re- sponse to the physical environment. To consider almost periodic environmental factors, it is reasonable to study the non-autonomous Nicholson’s blowflies model with almost periodic coefficients and delays. Most recently, some sufficient conditions were obtained in [10,11,14]

to ensure the local existence and exponential stability of positive almost periodic solution for the non-autonomous Nicholson’s blowflies model with a linear harvesting term. However, as pointed out by Liu [8], it is difficult to study the global dynamic behavior of the Nicholson’s blowflies model with a linear harvesting term. So far, there is no literature considering the global exponential stability of positive almost periodic solutions for (1.1) and its generalized equations. Thus, it is worthwhile to continue to investigate the global dynamic behavior of positive almost periodic solutions for non-autonomous Nicholson’s blowflies model with the linear harvesting term.

Motivated by the above discussions, the main purpose of this paper is to establish some criteria for the global dynamic behavior of positive almost periodic solutions for a general Nicholson’s blowflies model with the linear harvesting term given by

x⁰(t) = −a(t)x(t) +

∑

β_j(t)x(t−τ_j(t))e^{−γj(t)x(t−τj(t))} +β₁(t)x(t−τ₁(t))e^{−γ1(t)x(t)}−H(t)x(t−σ(t))_,

(1.2)

where a,H,σ,γ_j: R → (0,+_∞) and β_j,τ_j: R → [0,+_∞) are almost periodic functions for j=1, 2, . . . ,m. Obviously, (1.1) is a special case of (1.2) with constant coefficients and delays.

For convenience, we introduce some notations. In the following part of this paper, given a bounded continuous functiongdefined on R, letg⁺andg⁻be defined as

g⁺=sup

t∈R

|g(t)|, g⁻= inf

t∈R|g(t)|. It will be assumed that

γ⁻_j ≥1(j=1, 2, . . . ,m), r :=max{max

1≤j≤mτ_j⁺, σ⁺}. (1.3) LetC = C([−r, 0],R)be the continuous functions space equipped with the usual supremum normk · k, and letC+= C([−r, 0],(0,+_∞)). Ifx(t)is continuous and defined on[−r+t₀,σ) witht₀,σ∈ R, then we define x_t∈C, wherex_t(θ) =x(t+θ)for all θ∈ [−r, 0].

We also consider admissible initial conditions

xt₀ = ϕ, ϕ∈C+. (1.4)

We denote by x_t(t₀, ϕ)(x(t;t₀,ϕ)) an admissible solution of the admissible initial value problem (1.2) and (1.4). Also, let[t0,η(ϕ)) be the maximal right-interval of the existence of x_t(t₀,ϕ).

Since the function ¹⁻_ex^x is decreasing on the interval [0, 1] with the range [0, 1], it follows easily that there exists a uniqueκ∈(0, 1)such that

1−κ e^κ = ¹

e². (1.5)

Obviously,

sup

x≥κ

1−x e^x

= ¹

e². (1.6)

Moreover, since xe^−x increases on[0, 1]and decreases on[1,+_∞), leteκ be the unique number in (1,+_∞)such that

κe^−κ =_eκe^−eκ. (1.7)

2 Preliminary results

In this section, we shall first recall some basic definitions, lemmas which are used in what follows.

Definition 2.1 ([4,5]). A continuous function u: R → R is said to be almost periodic on R if, for any e> 0, the set T(u,e) = {δ : |u(t+δ)−u(t)| < efor allt∈ R}is relatively dense, i.e., for any e > 0, it is possible to find a real number l = l(e) > 0 with the property that, for any interval with length l(e), there exists a number δ = δ(e) in this interval such that

|u(t+δ)−u(t)|<efor allt ∈R.

From the theory of almost periodic functions in [4,5], it follows that for any e > 0, it is possible to find a real number l = l(e) > 0, for any interval with length l(e), there exists a numberδ =δ(e)in this interval such that

(|a(t+δ)−a(t)|<e, |H(t+δ)−H(t)|< e, |β_j(t+δ)−β_j(t)|< e,

|τ_j(t+δ)−τ_j(t)|< e, |γ_j(t+δ)−γ_j(t)|< e, |σ(t+δ)−σ(t)|<e, (2.1) for all t∈ Randj=1, 2, . . . ,m.

Lemma 2.2([8, Theorem 2.1]). Assume that

tinf∈R

β₁(t)e^−eκ−H(t) >_0, and τ₁(t)≡ σ(t) for all t∈ R. (2.2) Then, the solution x_t(t₀,ϕ) ∈ C+ for all t ∈ [t₀,η(ϕ)), the set of {x_t(t₀,ϕ) : t ∈ [t₀, η(ϕ))} is bounded, andη(ϕ) = +_∞.

Lemma 2.3([8, Theorem 3.1]). Suppose that all conditions in Lemma2.2are satisfied. Let lim inf

t→+_∞

_m

j

∑

β_j(t) a(t) +

β₁(t)

a(t) − ^H(t)

a(t) >1. (2.3)

Then, there exist two positive constants K₁ and K₂such that K₁≤lim inf

t→+_∞ x(t;t₀,ϕ)≤lim sup

t→+_∞

x(t;t₀,ϕ)≤K₂.

Lemma 2.4 ([6, Lemma 2.3]). Let(2.2) hold. Suppose that there exists a positive constant M such that

1max≤j≤mγ⁺_j ≤ ^eκ

M, (2.4)

and

sup

t∈R

(

−a(t) + ¹ eM

∑

β_j(t) γ_j(_t)

)

<0, inf

t∈R

(

−a(t) +e^−κ

∑

β_j(t) γ_j(_t)

)

>0. (2.5)

Then, the set of {x_t(t₀,ϕ) : t ∈ [t₀, η(ϕ))} is bounded, andη(ϕ) = +∞. Moreover, there exists tϕ >t0such that

κ<x(t;t₀,ϕ)< M for all t≥ tϕ. (2.6) Lemma 2.5. Suppose(2.2),(2.4)and(2.5)hold, and

sup

t∈R

−a(t) +

∑

β_j(t)¹

e² +β₁(t)e^−κ(M+1) +H(t)

<0. (2.7)

Moreover, assume that x(t) =x(t;t₀,ϕ)is a solution of equation(1.2)with initial condition(1.4)and ϕ⁰ is bounded continuous on [−r, 0]. Then for any e > 0,there exists l = l(e) > 0, such that every interval[α,α+l]contains at least one numberδfor which there exists N>0satisfying

|x(t+δ)−x(t)| ≤e, for all t> N. (2.8) Proof. Define a continuous functionΓ(µ)by setting

Γ(µ) =sup

t∈R

−[a(t)−µ] +

∑

β_j(t)¹

e²e^µr+β₁(t)e^−κ(M+e^µr) +H(t)e^µr

, µ∈[0, 1]. Then, we have

Γ(0) =sup

t∈R

−a(t) +

∑

β_j(t)¹

e² +β₁(t)e^−κ(M+1) +H(t)

<0, which implies that there exist two constantsη>0 andλ∈ (0, 1]such that

Γ(λ) =sup

t∈R

−[a(t)−λ] +

∑

β_j(t)¹

e²e^λr+β₁(t)e^−κ(M+e^λr) +H(t)e^λr

<−η<0. (2.9) Fort∈ (−_∞,t0−r], we add the definition ofx(t)withx(t)≡x(t0−r). Set

e(δ,t) = −[a(t+δ)−a(t)]x(t+δ) +

∑

[β_j(t+δ)−β_j(t)]x(t+δ−τ_j(t+δ))e^{−γj(t+δ)x(t+δ−τj(t+δ))} +

∑

β_j(t)^hx(t+δ−τ_j(t+δ))e^{−γj(t+δ)x(t+δ−τj(t+δ))}

−x(t−τ_j(t) +δ)e^{−γj(t+δ)x(t−τj(t)+δ)}i +

∑

β_j(t)^hx(t−τ_j(t) +δ)e^{−γj(t+δ)x(t−τj(t)+δ)}

−x(t−τ_j(t) +δ)e^{−γj(t)x(t−τj(t)+δ)}i

+ [β₁(t+δ)−β₁(t)]x(t+δ−τ₁(t+δ))e^{−γ1(t+δ)x(t+δ)}

+β₁(t)^hx(t+δ−τ₁(t+δ))e^{−γ1(t+δ)x(t+δ)}−x(t−τ₁(t) +δ)e^{−γ1(t)x(t+δ)}i

−[H(t+δ)−H(t)]x(t+δ−σ(t+δ))

−H(t)[x(t+δ−σ(t+δ))−x(t−σ(t) +δ)], t ∈R.

(2.10)

By Lemma2.2, the solutionx(t)is bounded and

κ< _x(_t)< _{M for allt}≥ tϕ, (2.11) which implies that the right side of (1.2) is also bounded, andx⁰(t)is a bounded function on [t₀−r,+_∞). Thus, in view of the fact thatx(t)≡ x(t₀−r)fort∈ (−_∞,t₀−r], we obtain that x(t) is uniformly continuous onR. From (2.1), for anye > 0, there exists l = l(e)> _{0, such} that every interval[α,α+l],α∈ R, contains aδfor which

|e(_δ,t)| ≤ ¹

2ηe for allt ∈R. (2.12)

Let N₀ ≥ max{t₀, t₀−δ, t_ϕ+r, t_ϕ+r−δ}. For t ∈ R, denote u(t) = x(t+δ)−x(t). Then, for all t≥ N₀, we get

du(t)

dt = −a(t)[x(t+δ)−x(t)]

+

∑

β_j(t)^hx(t−τ_j(t) +δ)e^{−γj(t)x(t−τj(t)+δ)}−x(t−τ_j(t))e^{−γj(t)x(t−τj(t))}i +β₁(t)^hx(t−τ₁(t) +δ)e^{−γ1(t)x(t+δ)}−x(t−τ₁(t) +δ)e^{−γ1(t)x(t)}i

+β₁(t)^hx(t−τ₁(t) +δ)e^{−γ1(t)x(t)}−x(t−τ₁(t))e^{−γ1(t)x(t)}i

−H(t)[x(t−σ(t) +δ)−x(t−σ(t))] +e(δ,t).

(2.13)

From (1.6), (1.7), (2.10), (2.13) and the inequalities

|e^−s−e^−t|=e^{−(s+θ(t−s))}|s−t| ≤e^−κ|s−t|, where s,t ∈[κ,κe], 0<θ <1, (2.14) and

|se^−s−te^−t|=

1−(s+θ(t−s)) e^s+θ(t−s)

|s−t| ≤ ¹

e²|s−t|, where s,t∈[κ,+_∞), 0<θ <1, (2.15) we obtain

D⁻(e^λs|u(s)|)|_s=t

≤λe^λt|u(t)|+e^λt

−a(t)|x(t+δ)−x(t)|

+

∑

β_j(t)^hx(t−τ_j(t) +δ)e^{−γj(t)x(t−τj(t)+δ)}−x(t−τ_j(t))e^{−γj(t)x(t−τj(t))}i +β₁(t)^hx(t−τ₁(t) +δ)e^{−γ1(t)x(t+δ)}−x(t−τ₁(t) +δ)e^{−γ1(t)x(t)}i

+β₁(t)^hx(t−τ₁(t) +δ)e^{−γ1(t)x(t)}−x(t−τ₁(t))e^{−γ1(t)x(t)}i

−H(t)x(t−σ(t) +δ)−x(t−σ(t))+e(δ,t)

=λe^λt|u(t)|+e^λt

−a(t)|u(t)|

+

∑

β_j(t) γ_j(_t) h

γ_j(t)x(t−τ_j(t) +δ)e^{−γj(t)x(t−τj(t)+δ)}−γ_j(t)x(t−τ_j(t))e^{−γj(t)x(t−τj(t))}i

+β₁(t)^hx(t−τ₁(t) +δ)e^{−γ1(t)x(t+δ)}−x(t−τ₁(t) +δ)e^{−γ1(t)x(t)}i +β₁(t)^hx(t−τ₁(t) +δ)e^{−γ1(t)x(t)}−x(t−τ₁(t))e^{−γ1(t)x(t)}i

−H(t)x(t−σ(t) +δ)−x(t−σ(t))+e(δ,t)

≤ λe^λt|u(t)|+e^λt

−a(t)|u(t)|+

∑

β_j(t)¹

e²|u(t−τ_j(t))|+β₁(t)Me^−κ|u(t)|

+β₁(t)e^−κ|u(t−τ₁(t))|+H(t)e^λσ(t)e^{λ(t−σ(t))}|u(t−σ(t))|+|e(δ,t)|

= −[a(t)−λ]e^λt|u(t)|+

∑

β_j(t)¹

e²e^λτj(t)e^{λ(t−τj(t))}|u(t−τ_j(t))|+β₁(t)Me^−κe^λt|u(t)|

+β₁(t)e^−κe^λτ1(t)e^{λ(t−τ1(t))}|u(t−τ₁(t))|

+H(t)e^λσ(t)e^{λ(t−σ(t))}|u(t−σ(t))|+e^λt|e(δ,t)|, for allt≥ N0. (2.16)

Let

U(t) = sup

−_∞<s≤t

{e^λs|u(s)|}. (2.17)

It is obvious thate^λt|u(t)| ≤U(t), andU(t)is non-decreasing.

Now, we distinguish two cases to finish the proof.

Case one.

U(t)>e^λt|u(t)| for all t≥ N0. (2.18) We claim that

U(_t)≡U(_N0) is a constant for allt ≥N0. (2.19) Assume, by way of contradiction, that (2.19) does not hold. Then, there existst₁ > N₀ such thatU(t₁)>U(N₀)_{. Since}

e^λt|u(t)| ≤U(N₀) for allt ≤ N₀. There must existβ∈(N₀, t₁)such that

e^λβ|u(β)|=U(t₁)≥U(β),

which contradicts (2.18). This contradiction implies that (2.19) holds. It follows that there existst₂ > N₀such that

|u(t)| ≤e^−λtU(t) =e^−λtU(N₀)< e for allt≥ t₂. (2.20) Case two. There is a t^∗₀ ≥ N₀ that U(t^∗₀) =e^λt∗0|u(t^∗₀)|. Then, in view of (2.13) and (2.20),

we get

0≤ D⁻(e^λs|u(s)|)|_s=t^∗

0

≤ −[a(t^∗₀)−λ]e^λt∗0|u(t^∗₀)|

+

∑

β_j(t₀^∗)¹

e²e^λτj(t∗0)e^{λ(t∗0−τj(t0∗))}|u(t^∗₀−τ_j(t^∗₀))|+β₁(t^∗₀)Me^−κe^λt|u(t^∗₀)|

+β₁(t^∗₀)e^−κe^λτ1(t∗0)e^{λ(t0∗−τ1(t∗0))}|u(t₀^∗−τ₁(t^∗₀))|

+H(t^∗₀)e^λσ(t∗0)e^{λ(t0∗−σ(t∗0))}|u(t^∗₀−σ(t^∗₀))|

+e^λt∗0|e(δ,t^∗₀)|

≤

−[a(t^∗₀)−λ] +

∑

β_j(t^∗₀)¹

e²e^λr+β₁(t^∗₀)Me^−κ +β₁(t^∗₀)e^−κe^λr+H(t^∗₀)e^λr

U(t^∗₀) +¹ 2ηee^λt∗0

< −ηU(t^∗₀) +ηee^λt∗0,

(2.21)

which yields

e^λt∗0|u(t^∗₀)|=U(t^∗₀)<ee^λt∗0, and |u(t^∗₀)|<e. (2.22) For anyt>t^∗₀, with the same approach as that in deriving of (2.22), we can show

e^λt|u(t)|<ee^λt, and |u(t)|<e, (2.23) if U(t) =e^λt|u(t)|.

On the other hand, ifU(t)>e^λt|u(t)|andt>t^∗₀. We can choose t^∗₀ ≤ t3 <tsuch that U(t₃) =e^λt3|u(t₃)| and U(s)>e^λs|u(s)| for all s∈(t₃, t],

which, together with (2.23), yields

|u(t₃)|<_e.

With a similar argument as that in the proof of Case one, we can show that

U(s)≡U(t₃) is a constant for alls∈ (t₃, t]_{, (2.24)} which implies that

|u(t)|<e^−λtU(t) =e^−λtU(t₃) =|u(t₃)|e^{−λ(t−t3)}< e.

In summary, there must existN>max{t^∗₀, N₀, t₂}such that|u(t)| ≤eholds for allt> N.

The proof of Lemma 2.5is now complete.

3 Main results

In this section, we establish sufficient conditions on the existence, uniqueness, and global exponential stability of positive almost periodic solutions of (1.2).

Theorem 3.1. Under the assumptions of Lemma 2.5, equation (1.2)has at least one positive almost periodic solution x^∗(t). Moreover, x^∗(t)is globally exponentially stable, i.e., there exist constants Kϕ,x^∗

and t_ϕ,x^∗ such that

|x(t;t₀,ϕ)−x^∗(t)|< K_ϕ,x^∗e^−λt for all t>t_ϕ,x^∗, where λ is defined in(2.9).

Proof. Let v(t) = v(t;t0,ϕ^v) be a solution of equation (1.2) with initial conditions satisfying the assumptions in Lemma2.5. We also add the definition ofv(t)withv(t)≡v(t₀−r)for all t∈ (−_∞,t₀−r]. Set

e(k,t)

=−[a(t+t_k)−a(t)]v(t+t_k) +

∑

[β_j(t+t_k)−β_j(t)]v(t+t_k−τ_j(t+t_k))e^{−γj(t+tk)v(t+tk−τj(t+tk))} +

∑

β_j(t)^hv(t+t_k−τ_j(t+t_k))e^{−γj(t+tk)v(t+tk−τj(t+tk))}

−v(t−τ_j(t) +t_k)e^{−γj(t+tk)v(t−τj(t)+tk)}i +

∑

β_j(t)^hv(t−τ_j(t) +t_k)e^{−γj(t+tk)v(t−τj(t)+tk)}−v(t−τ_j(t) +t_k)e^{−γj(t)v(t−τj(t)+tk)}i +[β₁(t+t_k)−β₁(t)]v(t+t_k−τ₁(t+t_k))e^{−γ1(t+tk)x(t+tk)}

+β₁(t)^hv(t+t_k−τ₁(t+t_k))e^{−γ1(t+tk)v(t+tk)}−v(t−τ₁(t) +t_k)e^{−γ1(t)v(t+tk)}i

−[H(t+t_k)−H(t)]v(t+t_k−σ(t+t_k))

−H(t)v(t+t_k−σ(t+t_k))−v(t−σ(t) +t_k), t ∈R,

(3.1)

where{t_k}is any sequence of real numbers. By Lemma2.3, the solutionv(t)is bounded and κ<v(t)< M for all t ≥tϕ^v, (3.2) which implies that the right side of (1.2) is also bounded, and v⁰(t)is a bounded function on [t₀−r,+_∞). Thus, in view of the fact thatv(t)≡v(t₀−r)fort∈ (−_∞,t₀−r], we obtain that v(t)is uniformly continuous onR. Then, from the almost periodicity ofa,H,σ, τ_j,γ_j and β_j, we can select a sequence{t_k} →+_{∞such that}

|a(t+t_k)−a(t)| ≤ ¹_k, |H(t+t_k)−H(t)| ≤ ¹_k,

|τ_j(t+t_k)−τ_j(t)| ≤ ¹_k, |σ(t+t_k)−σ(t)| ≤ ¹_k

|β_j(t+t_k)−β_j(t)| ≤ ¹_k, |γ_j(t+t_k)−γ_j(t)| ≤ ¹_k, |e(k,t)| ≤ ¹_k







for allj,t. (3.3)

Since {v(t+t_k)}⁺_k=^∞₁ is uniformly bounded and equiuniformly continuous, by the Ascoli–

Arzelà lemma and diagonal selection principle, we can choose a subsequence {t_kj} of {t_k}, such that v(t+t_kj) (for convenience, we still denote by v(t+t_k)) uniformly converges to a continuous functionx^∗(t)on any compact set ofR, and

κ≤ x^∗(t)≤ M for all t ∈R. (3.4)

Now, we prove that x^∗(t) is a solution of (1.2). In fact, for anyt ≥ t₀ and ∆t ∈ R, from (3.3), we have

x^∗(t+_∆t)−x^∗(t)

= lim

k→+_∞[v(t+_∆t+t_k)−v(t+t_k)]

= _lim

k→+_∞ Z _t+∆t

t

n−a(µ+t_k)v(µ+t_k) +

∑

β_j(µ+t_k)v(µ+t_k−τ_j(µ+t_k))e^{−γj(µ+tk)v(µ+tk−τj(µ+tk))} +β₁(µ+t_k)v(µ+t_k−τ₁(µ+t_k))e^{−γ1(µ+tk)v(µ+tk)}

−H(µ+t_k)v(µ+t_k−σ(µ+t_k))^odµ

= lim

k→+_∞ Z _t+∆t

t

n−a(µ)v(µ+t_k) +

∑

β_j(µ)v(µ+t_k−τ_j(µ))e^{−γj(µ)v(µ+tk−τj(µ))} +β₁(µ)v(µ+t_k−τ₁(µ))e^{−γ1(µ)v(µ+tk)}

−H(µ)v(µ−σ(µ) +t_k) +e(k,µ)^odµ

=

Z _t+∆t

t

n−a(µ)x^∗(µ) +

∑

β_j(µ)x^∗(µ−τ_j(µ))e^{−γj(µ)x∗(µ−τj(µ))} +β₁(µ)x^∗(µ−τ₁(µ))e^{−γ1(µ)x∗(µ)}−H(µ)x^∗(µ−σ(µ))^odµ + lim

k→+_∞ Z _t+_∆t

t e(k,µ)dµ

=

Z _t+_∆t t

n−a(µ)x^∗(µ) +

∑

β_j(µ)x^∗(µ−τ_j(µ))e^{−γj(µ)x∗(µ−τj(µ))} +β₁(µ)x^∗(µ−τ₁(µ))e^{−γ1(µ)x∗(µ)}−H(µ)x^∗(µ−σ(µ))^odµ,

(3.5)

wheret+_∆t≥t₀. Consequently, (3.5) implies that d

dt{x^∗(t)}= −a(t)x^∗(t) +

∑

β_j(t)x^∗(t−τ_j(t))e^{−γj(t)x∗(t−τj(t))} +β₁(t)x^∗(t−τ₁(t))e^{−γ1(t)x∗(t)}−H(t)x^∗(t−σ(t))_.

(3.6)

Therefore, x^∗(t)is a solution of (1.2).

Secondly, we prove thatx^∗(t)is an almost periodic solution of (1.2). From Lemma2.5, for any ε > 0, there exists l = l(ε) > 0, such that every interval [_α,α+l] contains at least one numberδ for which there existsN >0 satisfing

|v(t+δ)−v(t)| ≤ε for all t> N. (3.7) Then, for any fixed s ∈ R, we can find a sufficiently large positive integer N₁ > N such that for any k> N₁,

s+t_k >N, |v(s+t_k+δ)−v(s+t_k)| ≤_{ε. (3.8)}

Letk→+∞, we obtain

|x^∗(s+δ)−x^∗(s)| ≤ε,

which implies thatx^∗(t)is an almost periodic solution of equation (1.2).

Finally, we prove thatx^∗(t)is globally exponentially stable.

Letx(t) =x(t;t0,ϕ)andy(t) =x(t)−x^∗(t), wheret ∈[t0−r,+_∞). Then y⁰(_t) = −a(_t)_y(_t)

+

∑

β_j(t)^hx(t−τ_j(t))e^{−γj(t)x(t−τj(t))}−x^∗(t−τ_j(t))e^{−γj(t)x∗(t−τj(t))}i +β₁(t)^hx(t−τ₁(t))e^{−γ1(t)x(t)}−x^∗(t−τ₁(t))e^{−γ1(t)x∗(t)}i

−H(t)y(t−σ(t)).

(3.9)

It follows from Lemma2.4that there existst_ϕ,ϕ^∗ >t₀such that

κ≤x(t), x^∗(t)≤ M for all t∈[tϕ,ϕ^∗−r, +_∞). (3.10) We consider the Lyapunov functional

V(t) =|y(t)|e^λt. (3.11)

Calculating the upper left derivative ofV(t)along the solutiony(t)of (3.9), we have D⁻(V(t))≤ −a(t)|y(t)|e^λt

+

∑

β_j(t)|x(t−τ_j(t))e^{−γj(t)x(t−τj(t))}−x^∗(t−τ_j(t))e^{−γj(t)x∗(t−τj(t))}|e^λt +β₁(t)|x(t−τ₁(t))e^{−γ1(t)x(t)}−x^∗(t−τ₁(t))e^{−γ1(t)x∗(t)}|e^λt

+H(t)|y(t−σ(t))|e^λt+λ|y(t)|e^λt, for all t >t_ϕ,ϕ^∗.

(3.12)

We claim that

V(t) =|y(t)|e^λt< e^λtϕ,ϕ∗

t∈[t₀max−r,t_ϕ,ϕ∗]|x(t)−x^∗(t)|+1

=:K_ϕ,ϕ^∗ for all t >t_ϕ,ϕ^∗. (3.13) Contrarily, there must existt∗ >tϕ,ϕ^∗ such that

V(t∗) =K_ϕ,ϕ^∗ and V(t)<K_ϕ,ϕ^∗ for all t∈ [t₀−r, t∗). (3.14) Since

κ≤γ_j(t∗)x(t∗−τ_j(t∗)), γ_j(t∗)x^∗(t∗−τ_j(t∗))≤γ⁺_j M≤_eκ, j=1, 2, . . . ,m.

Together with (1.5), (1.6), (1.7), (2.18), (2.19), (3.12) and (3.14), we obtain 0≤ D⁻(V(t∗))

≤ −a(t∗)|y(t∗)|e^λt∗ +

∑

β_j(t∗)x(t∗−τ_j(t∗))e^{−γj(t∗)x(t∗−τj(t∗))}−x^∗(t∗−τ_j(t∗))e^{−γj(t∗)x∗(t∗−τj(t∗))} e^λt∗ +β₁(t∗)

x(t∗−τ₁(t∗))e^{−γ1(t∗)x(t∗)}−x^∗(t∗−τ₁(t∗))e^{−γ1(t∗)x∗(t∗)} e^λt∗ +H(t∗)|y(t∗−σ(t∗))|e^λt∗+λ|y(t∗)|e^λt∗