Positive Almost Periodic Solution on a Nonlinear Differential Equation

Abstract

We study the following nonlinear equation 𝑑𝑥(𝑡)/𝑑𝑡=𝑥(𝑡)[𝑎(𝑡)−𝑏(𝑡)𝑥^𝛼(𝑡)−𝑓(𝑡,𝑥(𝑡))]+𝑔(𝑡), by using fixed point theorem, the sufficient conditions of the existence of a unique positive almost periodic solution for above system are obtained, by using the theories of stability, the sufficient conditions which guarantee the stability of the unique positive almost periodic solution are derived.

1. Introduction

Let us consider the following Logistic-type equation 𝑑𝑥(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑥𝑑𝑡=𝑎−𝑏²(𝑡)+𝑒(𝑡). (1.1) When the external perturbation 𝑒(𝑡)≡0 and 𝑎(𝑡),𝑏(𝑡) are positive constants, (1.1) is the typical Logistic equation. It was firstly introduced as a mathematical model for studying population dynamics and has become a classic topic in the textbooks on ordinary differential equations and its quality theory (see [1, 2]). By using the method of separation of variables and integration by partial fractions, we can get explicitly all the solutions of the typical Logistic equation and completely analyze the behavior of all the solutions. But when 𝑎(𝑡),𝑏(𝑡) are no longer constants and there exists some perturbation, the problem is not so simple as no explicit solutions can be found in general. In [2–6], the time-periodic case was considered. In [7], the existence of positive almost periodic solutions was considered when 𝑎(𝑡),𝑏(𝑡) are almost periodic and there is no perturbation. When 𝑎(𝑡),𝑏(𝑡) satisfy the assumption 𝛼≤𝑎(𝑡)≤𝐴,  𝛽≤𝑏(𝑡)≤𝐵, and there is no external perturbation (𝑒(𝑡)≡0), but there is a nonlinear perturbation 𝑞(𝑡,𝑥)𝑥, Nkashama [8] obtained the existence of bounded and positive almost periodic solutions. In [9], Zhu et al. considered the case that 𝑎(𝑡),𝑏(𝑡) satisfy the assumption 𝛼≤𝑎(𝑡)≤𝐴,  𝛽≤𝑏(𝑡)≤𝐵 and there exists an external perturbation 𝑒(𝑡), and they got the existence and uniqueness of positive periodic solutions and almost periodic solutions of (1.1).

In this paper, we consider the following more complex system: 𝑑𝑥(𝑡)(𝑡𝑎(𝑡)(𝑡)𝑥𝑑𝑡=𝑥)−𝑏^𝛼(𝑡)−𝑓(𝑡,𝑥(𝑡))+𝑔(𝑡), (1.2) where 𝛼>0,  𝑡∈𝑅, 𝑎(𝑡),𝑏(𝑡),𝑔(𝑡) are all continuous almost periodic functions, and 𝑓(𝑡,𝑥) is almost periodic in 𝑡 and uniformly with respect to 𝑥∈𝑅.

In this paper, we use the fixed point theorem and get the existence and uniqueness of positive almost periodic solution for (1.2), the stability of the unique positive almost periodic solution of (1.2) is also discussed, and some new results are obtained.

2. The Existence and Uniqueness of Positive Almost Periodic Solution

Before we start with our main results, for the sake of convenience, suppose that 𝑓 is a continuous bounded function, and we denote 𝑓_𝑀=sup_𝑡∈𝑅𝑓(𝑡) and 𝑓_𝐿=inf_𝑡∈𝑅𝑓(𝑡); first, we introduce some lemmas.

Lemma 2.1 (see [10]). Consider the following equation: 𝑑𝑥(𝑡)𝑥(𝑡)(𝑡)𝑑𝑡=𝑎+𝑏, (2.1) where 𝑎(𝑡),𝑏(𝑡) are continuous almost periodic functions; if Re𝑚(𝑎(𝑡))≠0, then (2.1) exists a unique almost periodic solution 𝜂(𝑡), and 𝜂(𝑡) can be written as follows: 𝜂(𝑡)=^𝑡−∞𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)𝑑𝑠,Re𝑚(𝑎(𝑡))<0,−_𝑡^+∞𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)𝑑𝑠,Re𝑚(𝑎(𝑡))>0, (2.2) where 𝑚(𝑎(𝑡))=lim_𝑇→+∞(1/𝑇)∫^𝑇₀𝑎(𝑡)𝑑𝑡,Re𝑚(𝑎(𝑡)) is the real part of 𝑚(𝑎(𝑡)).

Lemma 2.2. Consider (1.2); if 𝑔(𝑡)≥0; then the domain 𝑅₊={𝑥∣𝑥>0} is a positive invariant with respect to (1.2).

Proof. Since 𝑔(𝑡)≥0, it follows that 𝑑𝑥(𝑡)(𝑡𝑎(𝑡)(𝑡)𝑥𝑑𝑡=𝑥)−𝑏^𝛼(𝑡)((𝑡(𝑡)(𝑡𝑎(𝑡)(𝑡)𝑥−𝑓𝑡,𝑥))+𝑔≥𝑥)−𝑏^𝛼(𝑡)−𝑓(𝑡,𝑥(𝑡)), (2.3) thus we have 𝑥(𝑡)≥𝑥𝑡₀exp^𝑡_𝑡0𝑎(𝑠)−𝑏(𝑠)𝑥^𝛼(𝑠)−𝑓(𝑠,𝑥(𝑠))𝑑𝑠, (2.4) the assertion is valid for all 𝑥(𝑡₀)>0. The proof is completed.

Theorem 2.3. Consider (1.2); 𝛼 is a constant and 0<𝛼<𝑏_𝐿/(𝑏_𝑀−𝑏_𝐿), 𝑎(𝑡),𝑏(𝑡),𝑔(𝑡) are continuous almost periodic functions, 𝑓(𝑡,𝑥) is a continuous almost periodic function in 𝑡∈𝑅 and uniformly with respect to 𝑥∈𝑅,𝑓(𝑡,0)≡0; if the following conditions hold:(1)𝑎_𝐿>0,  𝑏_𝐿>0,  𝑔_𝐿≥0,(2)|𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)|≤𝐿|𝑥−𝑦|,forall𝑡,𝑥,𝑦∈𝑅, where 𝐿 is a positive number,(3)𝐿𝛾/𝛼𝑎_𝐿𝛽^{(1+𝛼)/𝛼}+𝐿/𝑎_𝐿𝛽^1/𝛼+𝑔_𝑀𝛾^1/𝛼(1+𝛼)/𝛼𝑎_𝐿<1, where 𝛽=((1+𝛼)𝑏_𝐿−𝛼𝑏_𝑀)/((1+𝛼)(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼²𝑎_𝐿),  𝛾=((1+𝛼)[𝑏_𝑀(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼𝑎_𝐿𝑏_𝐿])/(𝑎_𝐿[(1+𝛼)(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼²𝑎_𝐿]),
then (1.2) exists a unique positive almost periodic solution 𝜙^∗(𝑡), and 1/𝛾^1/𝛼≤𝜙^∗(𝑡)≤1/𝛽^1/𝛼.

Proof. Let 𝑢(𝑡)=1/𝑥^𝛼(𝑡), since we only consider positive solutions of (1.2), by Lemma 2.2, it follows that 𝑥(𝑡)=𝑢^−1/𝛼(𝑡), then (1.2) can be written as follows: 𝑑𝑢(𝑡)𝑢(𝑡)(𝑡)(𝑡)𝑓𝑑𝑡=−𝛼𝑎+𝛼𝑏+𝛼𝑢𝑡,𝑢^−1/𝛼(𝑡)−𝛼𝑢^{(1+𝛼)/𝛼}(𝑡)𝑔(𝑡). (2.5) Define 𝐵=𝜑(𝑡);𝜑(𝑡)∈𝐴𝑃(𝑅),𝛽≤𝜑(𝑡)≤𝛾, (2.6) where 𝐴𝑃(𝑅) denotes the set of almost periodic functions in 𝑅, the norm is defined as ‖𝜑‖=sup_𝑡∈𝑅|𝜑(𝑡)|, thus (𝐵,‖⋅‖) is a Banach space, given any 𝜑(𝑡)∈𝐵, consider the following equation: 𝑑𝑢(𝑡)𝑢(𝑡)(𝑡)(𝑡)𝑓𝑑𝑡=−𝛼𝑎+𝛼𝑏+𝛼𝜑𝑡,𝜑^−1/𝛼(𝑡)−𝛼𝜑^{(1+𝛼)/𝛼}(𝑡)𝑔(𝑡), (2.7) since 𝑎_𝐿>0, it follows that 𝑚(𝑎(𝑡))>0, thus 𝑚[−𝛼𝑎(𝑡)]<0, from (2.7), by virtue of Lemma 2.1, we can know that (2.7) has a unique almost periodic solution 𝜂(𝑡), it can be written as follows: 𝜂(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)−𝜑^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠. (2.8) Now, define an operator 𝑇 as follows: 𝑇𝜑(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)−𝜑^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠, (2.9) note that 𝑇𝜑(𝑡)≤𝛼^𝑡_−∞𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}𝑏(𝑠)+𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)𝑑𝑠≤𝛼^𝑡_−∞𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}|𝑏(𝑠)|+𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)−𝑓(𝑠,0)𝑑𝑠≤𝛼^𝑡_−∞𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}|𝑏(𝑠)|+𝐿𝜑(𝑠)𝜑^−1/𝛼(𝑠≤1)𝑑𝑠𝑎_𝐿𝑏_𝑀+𝛾𝐿𝛽^−1/𝛼. (2.10) Since the condition 𝐿𝛾/𝛼𝑎_𝐿𝛽^{(1+𝛼)/𝛼}+𝐿/𝑎_𝐿𝛽^1/𝛼+𝑔_𝑀𝛾^1/𝛼(1+𝛼)/𝛼𝑎_𝐿<1 holds, it follows that 𝐿𝛾<𝛼𝑎_𝐿𝛽^{(1+𝛼)/𝛼}; thus, we have (𝑡)≤1𝑇𝜑𝑎_𝐿𝑏_𝑀+𝛼𝑎_𝐿𝛽=𝛾. (2.11) Also 𝑇𝜑(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)−𝜑^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠≥𝛼^𝑡_−∞𝑒^{−𝛼𝑎𝑀(𝑡−𝑠)}𝑏_𝐿−𝛾𝐿𝛽^−1/𝛼−𝛾^{(1+𝛼)/𝛼}𝑔(𝑠)𝑑𝑠. (2.12) Since 𝐿𝛾/𝛼𝑎_𝐿𝛽^{(1+𝛼)/𝛼}+𝐿/𝑎_𝐿𝛽^1/𝛼+𝑔_𝑀𝛾^1/𝛼(1+𝛼)/𝛼𝑎_𝐿<1, it follows that 𝑔(𝑡)≤𝛼𝑎_𝐿/(1+𝛼)𝛾^1/𝛼; thus, we have 𝑇𝜑(𝑡)≥𝛼^𝑡_−∞𝑒^{−𝛼𝑎𝑀(𝑡−𝑠)}𝑏_𝐿−𝛾𝐿𝛽^−1/𝛼−𝛾^{(1+𝛼)/𝛼}𝑔(𝑠≥1)𝑑𝑠𝑎_𝑀𝑏_𝐿−𝛼𝛽𝑎_𝐿−𝛼𝑎_𝐿𝛾1+𝛼=𝛽. (2.13) Hence 𝑇𝜑(𝑡)∈𝐵; therefore, 𝑇∶𝐵→𝐵. For any 𝜑(𝑡),𝜓(𝑡)∈𝐵, it follows that 𝑇𝜑(𝑡)−𝑇𝜓(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)−𝜑^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)−𝜓(𝑠)𝑓𝑠,𝜓^−1/𝛼(𝑠)+𝜓^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠=𝛼)𝑑𝑠_𝑡−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)−𝜓(𝑠)𝑓𝑠,𝜓^−1/𝛼(𝑠)+𝜓(𝑠)𝑓𝑠,𝜓^−1/𝛼(𝑠)−𝜓(𝑠)𝑓𝑠,𝜓^−1/𝛼(𝑠)+𝑔(𝑠)𝜓^{(1+𝛼)/𝛼}(𝑠)−𝜑^{(1+𝛼)/𝛼}(𝑠)𝑑𝑠≤𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝜑(𝑠)𝑓𝑠,𝜑^−1/𝛼(𝑠)−𝑓𝑠,𝜓^−1/𝛼(𝑠)+𝜑(𝑠)−𝜓(𝑠)𝑓𝑠,𝜓^−1/𝛼(𝑠)+𝑔(𝑠)𝜑^{(1+𝛼)/𝛼}(𝑠)−𝜓^{(1+𝛼)/𝛼}(𝑠)𝑑𝑠≤𝛼^𝑡_−∞𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}𝐿𝜑(𝑠)𝜑^−1/𝛼(𝑠)−𝜓^−1/𝛼(𝑠)+𝐿𝜓^−1/𝛼(𝑠𝜑(𝑠)(𝑠+𝑔(𝑠𝜑)−𝜓))^{(1+𝛼)/𝛼}(𝑠)−𝜓^{(1+𝛼)/𝛼}(𝑠)𝑑𝑠. (2.14) According to mean value theorem, we can get 𝑇𝜑(𝑡)−𝑇𝜓(𝑡)≤𝛼^𝑡_−∞𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}𝐿𝜑(𝑠−1)𝛼𝜉^{(−1/𝛼)−1}𝜑(𝑠)−𝜓(𝑠)+𝐿𝜓^−1/𝛼𝜑(𝑠)(𝑠+𝑔(𝑠−𝜓))1+𝛼𝛼𝜁^1/𝛼𝜑(𝑠)−𝜓(𝑠)𝑑𝑠, (2.15) where 𝜑(𝑠)<𝜉,  𝜁<𝜓(𝑠) or 𝜓(𝑠)<𝜉,  𝜁<𝜑(𝑠). Notice that ‖𝜑(𝑡)‖≤𝛾,  ‖𝜁^1/𝛼‖≤𝛾^1/𝛼,  ‖𝜓^−1/𝛼(𝑡)‖≤𝛽^−1/𝛼,  ‖𝜉^{−(1/𝛼)−1}‖≤𝛽^{−(1/𝛼)−1}; it follows that (𝑡)(𝑡≤1𝑇𝜑−𝑇𝜓)𝑎_𝐿1𝐿𝛾𝛼𝛽^{−(1/𝛼)−1}𝜑−𝜓+𝐿𝛽^−1/𝛼𝜑−𝜓+𝑔_𝑀1+𝛼𝛼𝛾^1/𝛼=1𝜑−𝜓𝑎_𝐿1𝐿𝛾𝛼𝛽^{−(1/𝛼)−1}+𝐿𝛽^−1/𝛼+𝑔_𝑀1+𝛼𝛼𝛾^1/𝛼𝜑−𝜓. (2.16) Note that 𝐿𝛾/𝛼𝑎_𝐿𝛽^{(1+𝛼)/𝛼}+𝐿/𝑎_𝐿𝛽^1/𝛼+𝑔_𝑀𝛾^1/𝛼(1+𝛼)/𝛼𝑎_𝐿<1, and 𝐿 is a positive number; it follows that 10<𝑎_𝐿1𝐿𝛾𝛼𝛽^{−(1/𝛼)−1}+𝐿𝛽^−1/𝛼+𝑔_𝑀1+𝛼𝛼𝛾^1/𝛼<1, (2.17) therefore, 𝑇 is a contraction mapping on 𝐵, that is to say, 𝑇 has a unique fixed point on 𝐵, the unique fixed point is the unique positive almost periodic solution 𝜙(𝑡) of (2.5), and 𝛽≤𝜙(𝑡)≤𝛾. Notice that 𝑢(𝑡)=1/𝑥^𝛼(𝑡), and, by Lemma 2.2, (1.2) has a unique positive almost periodic solution 𝜙^∗(𝑡)=[𝜙(𝑡)]^−1/𝛼, and 1/𝛾^1/𝛼≤𝜙^∗(𝑡)≤1/𝛽^1/𝛼. This completes the proof of Theorem 2.3.

3. The Uniqueness of the Solution with Initial Value Problem

Consider (2.5); if given initial value 𝑢(𝑡₀)=𝑢₀, and 𝛽≤𝑢₀≤𝛾, then we have the following theorem.

Theorem 3.1. Consider (2.5); 𝛼 is a constant and 0<𝛼<𝑏_𝐿/(𝑏_𝑀−𝑏_𝐿),𝑎(𝑡),𝑏(𝑡),𝑔(𝑡) are continuous functions, 𝑓(𝑡,𝑥) is a continuous function in 𝑡∈𝑅 and uniformly with respect to 𝑥∈𝑅,𝑓(𝑡,0)≡0; if the following conditions hold:(1)𝑎_𝐿>0,  𝑏_𝐿>0,  𝑔_𝐿≥0,(2)|𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)|≤𝐿|𝑥−𝑦|,forall𝑡,𝑥,𝑦∈𝑅, where 𝐿 is a positive number,(3)𝐿𝛾/𝛼𝑎_𝐿𝛽^{(1+𝛼)/𝛼}+𝐿/𝑎_𝐿𝛽^1/𝛼+𝑔_𝑀𝛾^1/𝛼(1+𝛼)/𝛼𝑎_𝐿<1, where 𝛽=(1+𝛼)𝑏_𝐿−𝛼𝑏_𝑀/(1+𝛼)(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼²𝑎_𝐿,  𝛾=((1+𝛼)[𝑏_𝑀(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼𝑎_𝐿𝑏_𝐿])/(𝑎_𝐿[(1+𝛼)(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼²𝑎_𝐿]),
then (2.5) exists a unique continuous solution 𝑢(𝑡) with initial value 𝑢(𝑡₀)=𝑢₀ and 𝛽≤𝑢(𝑡)≤𝛾(𝑡≥𝑡₀).

Proof. The initial value problem of (2.5) is equivalent to the solution of the following integral equation: 𝑢(𝑡)=𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝑢(𝑠)𝑓𝑠,𝑢^−1/𝛼(𝑠)−𝑢^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠. (3.1) Define an operator 𝑇 as follows: 𝑇𝑢(𝑡)=𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝑢(𝑠)𝑓𝑠,𝑢^−1/𝛼(𝑠)−𝑢^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠. (3.2) The following proof is similar as the proof of Theorem 2.3, so we omit it here.

4. The Stability of the Positive Almost Periodic Solution

Theorem 4.1. Consider (1.2); 𝛼 is a constant and 0<𝛼<𝑏_𝐿/(𝑏_𝑀−𝑏_𝐿),𝑎(𝑡),𝑏(𝑡),𝑔(𝑡) are continuous almost periodic functions, 𝑓(𝑡,𝑥) is a continuous almost periodic function in 𝑡∈𝑅 and uniformly with respect to 𝑥∈𝑅; if the following conditions hold:(1)𝑎_𝐿>0,  𝑏_𝐿>0,  𝑔_L≥0,(2)|𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)|≤𝐿|𝑥−𝑦|,forall𝑡,𝑥,𝑦∈𝑅, where 𝐿 is a positive number,(3)𝐿𝛾/𝛼𝑎_𝐿𝛽^{(1+𝛼)/𝛼}+𝐿/𝑎_𝐿𝛽^1/𝛼+𝑔_𝑀𝛾^1/𝛼(1+𝛼)/𝛼𝑎_𝐿<1, where 𝛽=((1+𝛼)𝑏_𝐿−𝛼𝑏_𝑀)/((1+𝛼)(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼²𝑎_𝐿),  𝛾=((1+𝛼)[𝑏_𝑀(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼𝑎_𝐿𝑏_𝐿])/(𝑎_𝐿[(1+𝛼)(𝑎_𝑀+𝛼𝑎_𝐿)+𝛼²𝑎_𝐿]),
then there exists a region Ω={(𝑡,𝑥);1/𝛾^1/𝛼≤𝑥≤1/𝛽^1/𝛼,𝑡≥𝑡₀}, and, in this region, the unique positive almost periodic solution of (1.2) is uniformly asymptotically stable.

Proof. From Theorem 3.1, we know when 𝛽≤𝑢₀≤𝛾, (2.5) exists a unique solution 𝑢(𝑡) with initial value 𝑢(𝑡₀)=𝑢₀, and 𝛽≤𝑢(𝑡)≤𝛾, since the transformation 𝑢(𝑡)=1/𝑥^𝛼(𝑡), it follows that 𝑥(𝑡)=1/𝑢^1/𝛼(𝑡), so (1.2) exists a unique solution with initial value 𝑥(𝑡₀)=1/𝑢₀^1/𝛼, and 1/𝛾^1/𝛼≤𝑥(𝑡)≤1/𝛽^1/𝛼. It is easy to know that, the conditions of Theorem 4.1 are met with Theorem 2.3, so (2.5) exists a unique positive almost periodic solution 𝜙(𝑡); from (2.9), the unique positive almost periodic solution 𝜙(𝑡) of (2.5) can be expressed by integral equation as follows: 𝜙(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜙(𝑠)𝑓𝑠,𝜙^−1/𝛼(𝑠)−𝜙^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠=𝛼^𝑡0_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜙(𝑠)𝑓𝑠,𝜙^−1/𝛼(𝑠)−𝜙^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜙(𝑠)𝑓𝑠,𝜙^−1/𝛼(𝑠)−𝜙^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠=𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜙(𝑠)𝑓𝑠,𝜙^−1/𝛼(𝑠)−𝜙^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠. (4.1) The solution of (2.5) with initial value 𝑢(𝑡₀)=𝑢₀ is given as follows: 𝑢(𝑡)=𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝑢(𝑠)𝑓𝑠,𝑢^−1/𝛼(𝑠)−𝑢^{(1+𝛼)/𝛼}(𝑠)𝑔(𝑠)𝑑𝑠, (4.2) from (4.1) and (4.2), when 𝑡≥𝑡₀, we have 𝑢(𝑡)−𝜙(𝑡)=𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑓𝑠,𝑢^−1/𝛼(𝑠)𝑢(𝑠)(𝑠)(𝑠𝑓−𝜙+𝜙)𝑠,𝑢^−1/𝛼(𝑠)−𝑓𝑠,𝜙^−1/𝛼(𝑠)+𝑔(𝑠)𝜙^{(1+𝛼)/𝛼}(𝑠)−𝑢^{(1+𝛼)/𝛼}(𝑠)𝑑𝑠≤𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑓𝑠,𝑢^−1/𝛼(𝑠)𝑢(𝑠)(𝑠+𝜙(𝑠𝑓−𝜙))𝑠,𝜙^−1/𝛼(𝑠)−𝑓𝑠,𝑢^−1/𝛼(𝑠)+𝑔(𝑠)𝜙^{(1+𝛼)/𝛼}(𝑠)−𝑢^{(1+𝛼)/𝛼}(𝑠)𝑑𝑠≤𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝐿𝑢^−1/𝛼(𝑠)𝑢(𝑠)−𝜙(𝑠)+𝐿𝜙(𝑠)𝜙^−1/𝛼(𝑠)−𝑢^−1/𝛼(𝑠+𝑔(𝑠𝜙))^{(1+𝛼)/𝛼}(𝑠)−𝑢^{(1+𝛼)/𝛼}(𝑠)𝑑𝑠. (4.3) According to mean value theorem, we can get 𝑢(𝑡)−𝜙(𝑡)≤𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝐿𝑢^−1/𝛼(𝑠𝑢(𝑠)(𝑠𝜙(𝑠−1)−𝜙)+𝐿)𝛼𝜉^{−(1/𝛼)−1}𝜙(𝑠)(𝑠)+𝑔(𝑠−𝑢)1+𝛼𝛼𝜁^1/𝛼𝜙(𝑠)−𝑢(𝑠)𝑑𝑠, (4.4) where 𝜙(𝑠)<𝜉,  𝜁<𝑢(𝑠) or  𝑢(𝑠)<𝜉,  𝜁<𝜙(𝑠). Notice that |𝜙|≤𝛾,  |𝜉^{−1−(1/𝛼)}|≤𝛽^{−1−(1/𝛼)},  |𝑢^−1/𝛼|≤𝛽^−1/𝛼,  |𝜁^1/𝛼|≤𝛾^1/𝛼; it follows that 𝑢(𝑡)−𝜙(𝑡)≤𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}𝐿𝛽^−1/𝛼𝑢(𝑠)(𝑠1−𝜙)+𝐿𝛾𝛼𝛽^{−(1/𝛼)−1}𝑢(𝑠)(𝑠+𝑔(𝑠−𝜙))1+𝛼𝛼𝛾^1/𝛼𝑢(𝑠)−𝜙(𝑠)𝑑𝑠≤𝑒^{∫𝑡𝑡0(−𝛼𝑎(𝜏))𝑑𝜏}𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}𝐿𝛽^−1/𝛼1+𝐿𝛾𝛼𝛽^{−(1/𝛼)−1}+𝑔_𝑀1+𝛼𝛼𝛾^1/𝛼𝑢(𝑠)−𝜙(𝑠)𝑑𝑠≤𝑒^{−𝛼𝑎𝐿(𝑡−𝑡0)}𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{−𝛼𝑎𝐿(𝑡−𝑠)}𝐿𝛽^−1/𝛼1+𝐿𝛾𝛼𝛽^{−(1/𝛼)−1}+𝑔_𝑀1+𝛼𝛼𝛾^1/𝛼𝑢(𝑠)−𝜙(𝑠)𝑑𝑠. (4.5) Multiplying both sides of the above inequality by 𝑒^{𝛼𝑎𝐿(𝑡−𝑡0)}, we have 𝑒^{𝛼𝑎𝐿(𝑡−𝑡0)}𝑢(𝑡)−𝜙(𝑡)≤𝑢𝑡₀−𝜙𝑡₀+𝛼^𝑡_𝑡0𝑒^{𝛼𝑎𝐿(𝑠−𝑡0)}𝐿𝛽^−1/𝛼1+𝐿𝛾𝛼𝛽^{−(1/𝛼)−1}+𝑔_𝑀1+𝛼𝛼𝛾^1/𝛼𝑢(𝑠)−𝜙(𝑠)𝑑𝑠. (4.6) According to Bellman’s inequality, we can obtain 𝑒^{𝛼𝑎𝐿(𝑡−𝑡0)}𝑢(𝑡)−𝜙(𝑡)≤𝑢𝑡₀−𝜙𝑡₀𝑒^{∫𝑡𝑡0[𝛼𝐿𝛽−1/𝛼+𝐿𝛾𝛽−(1/𝛼)−1+𝑔𝑀(1+𝛼)𝛾1/𝛼]𝑑𝑠}. (4.7) Multiplied both sides of the above inequality by 𝑒^{−𝛼𝑎𝐿(𝑡−𝑡0)}, we get 𝑢(𝑡)−𝜙(𝑡)≤𝑢𝑡₀−𝜙𝑡₀𝑒^{(−𝛼𝑎𝐿+𝛼𝐿𝛽−1/𝛼+𝐿𝛾𝛽−(1/𝛼)−1+𝑔𝑀(1+𝛼)𝛾1/𝛼)(𝑡−𝑡0)}. (4.8) Notice that 𝑢(𝑡)=1/𝑥^𝛼(𝑡), hence 𝑥(𝑡)=𝑢^−1/𝛼(𝑡), and 𝜙^∗(𝑡)=𝜙^−1/𝛼(𝑡) is the unique positive almost periodic solution of (1.2), so we have 𝑥(𝑡)−𝜙^∗(𝑡)=𝑢^−1/𝛼(𝑡)−𝜙^−1/𝛼(𝑡=−1)𝛼𝜁₁^{−(1/𝛼)−1}𝑢(𝑡)(𝑡≤1−𝜙)𝛼𝛽^{(1+𝛼)/𝛼}𝑢𝑡₀−𝜙𝑡₀𝑒^{(−𝛼𝑎𝐿+𝛼𝐿𝛽−1/𝛼+𝐿𝛾𝛽−(1/𝛼)−1+𝑔𝑀(1+𝛼)𝛾1/𝛼)(𝑡−𝑡0)}, (4.9) where 𝑢(𝑡)<𝜁₁<𝜙(𝑡) or 𝜙(𝑡)<𝜁₁<𝑢(𝑡), note that the condition (3) holds, it follows that −𝛼𝑎_𝐿+𝛼𝐿𝛽^−1/𝛼+𝐿𝛾𝛽^{−(1/𝛼)−1}+𝑔_𝑀(1+𝛼)𝛾^1/𝛼<0, (4.10) therefore, there must be a small positive constant 𝛿 such that the following equality holds: 𝑥(𝑡)−𝜙^∗(𝑡≤1)𝛼𝛽^{(1+𝛼)/𝛼}𝑢𝑡₀−𝜙𝑡₀𝑒^{−𝛿(𝑡−𝑡0)}. (4.11) From (4.11), we know that the unique positive almost periodic solution 𝜙^∗(𝑡) of (1.2) is uniformly asymptotically stable.