On the Positive Almost Periodic Solutions of a Class of Nonlinear Lotka-Volterra Type System with Feedback Control

Abstract

With the help of the variable substitution and applying the fixed point theorem, we derive the sufficient conditions which guarantee the existence of the positive almost periodic solutions for a class of Lotka-Volterra type system. The main results improve and generalize the former corresponding results.

1. Introduction

Denote𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)),𝑥_𝑡=(𝑥_1𝑡,𝑥_2𝑡,…,𝑥_𝑛𝑡),𝑥_𝑖𝑡(𝑠)=𝑥_𝑖(𝑡+𝑠)  (𝑖=1,2,…,𝑛),𝑠∈[−𝜏,0],𝜏 is a positive constant or 𝜏=+∞, the norm of a bounded continuous function space 𝐶([−𝜏,0],𝑅^𝑛) is defined as ‖𝜙‖=max_{𝑠∈[−𝜏,0]}|𝜙(𝑠)|, where |𝜙|=max{|𝜙_𝑖|,𝑖=1,2,…,𝑛},𝐶([−𝜏,0],𝑅₊)={𝜙∈𝐶([−𝜏,0],𝑅)∶forall𝜃∈[−𝜏,0],𝜙(𝜃)>0}.

We call an almost periodic function is positive if and only if each component has its positive infimum. Denote 𝐴𝑃(𝑅^𝑛)={𝑥(𝑡)∈𝑅^𝑛∶𝑥(𝑡) is a continuous almost periodic function on 𝑅^𝑛},𝐴𝑃(𝑅)={𝑥(𝑡)∈𝑅∶𝑥(𝑡) is a continuous almost periodic function on 𝑅},𝑇_𝛼𝑓(𝑡)=𝑔(𝑡)referstolim_𝑛→∞𝑓(𝑡+𝛼_𝑛)=𝑔(𝑡),where𝛼={𝛼_𝑛},{𝛼_𝑛} is a sequence of real numbers,𝑇(𝑓,𝜀)={𝜏∶|𝑓(𝑡+𝜏)−𝑓(𝑡)|<𝜀,𝑡∈𝑅},for more almost periodic monographs, see references [1, 2].

In [3], Teng first studied the existence of the almost periodic solutions for the following scalar equation: 𝑑𝑢𝛼(𝑡)(𝑡)𝑑𝑡=𝑢−𝛽𝑢, (1.1) where 𝑡∈𝑅,𝑢∈𝑅, and 𝛼(𝑡),𝛽(𝑡)∈𝐴𝑃(𝑅), then based on (1.1) and combined Schauder fixed point theorem, he studied the existence of the almost periodic solution for a class of Lotka-Volterra system 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑓_𝑖𝑡,𝑥_𝑡,(𝑖=1,2,…,𝑛), (1.2) where 𝑎_𝑖(𝑡),𝑏_𝑖(𝑡) are continuous almost periodic functions,𝑓_𝑖(𝑡,𝜙) is a continuous almost periodic function in 𝑡 uniformly with respect to 𝜙∈𝐶([−𝜏,0],𝑅^𝑛), and 𝑓_𝑖(𝑡,0)≡0(𝑖=1,2,…,𝑛). The author of [4] inherited in the method and ideas in [3], and promoted its conclusions to the following system with feedback control𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑓_𝑖𝑡,𝑥_𝑡,𝑢_𝑡,𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+ℎ_𝑖𝑡,𝑥_𝑖𝑡, (1.3) where 𝑢_𝑖(𝑡) is the control variable, 𝑖=1,2,…,𝑛.

The authors of [5] used transformation techniques and fixed point theorem and studied a time-delay system with feedback control which is much wider then the system (1.3),𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑓_𝑖𝑡,𝑥_𝑡,𝑢_𝑡,𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+ℎ_𝑖𝑡,𝑥_𝑡. (1.4) In the case of non-Lipschitz condition, they gave a sufficient condition of the existence of the almost periodic solutions for the system (1.4).

However, in the real world, the competition between species is not always shown by the linear relationship, while shown by a certain degree of nonlinearity, therefore, studying the following system becomes more realistic and necessary, over this paper, we study the system which is more extensive than the system (1.4) as follows:𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥^𝛼𝑖_𝑖(𝑡)−𝑓_𝑖𝑡,𝑥_𝑡,𝑢_𝑡,𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+ℎ_𝑖𝑡,𝑥_𝑡, (1.5) where 𝑢_𝑖(𝑡) is the control variable, 𝛼_𝑖 is a positive constant, 𝑖=1,2,…,𝑛, by using Schauder’s fixed point theorem, we get the sufficient conditions of the existence of the almost periodic solution for the system (1.5), and by using the contraction mapping principle, we give the conclusion of the existence of a unique almost periodic solution for the system (1.5) in the one-dimensional case, some new results are obtained.

2. Some Related Lemmas

Lemma 2.1 (see [6]). For the equation 𝑑𝑥(𝑡)𝑥(𝑡)(𝑡),𝑑𝑡=𝑎+𝑏 (2.1) where 𝑎(𝑡),𝑏(𝑡)∈𝐴𝑃(𝑅) and are continuous, if  Re𝑚(𝑎(𝑡))≠0, then (2.1) exists a unique almost periodic solution 𝜂(𝑡),mod(𝜂)⊂mod(𝑎,𝑏), and 𝜂(𝑡) can be written as follows 𝜂(𝑡)=^𝑡−∞𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)(𝑎(𝑡−𝑑𝑠,Re𝑚))<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 (see [2]). Suppose 𝑓(𝑡),𝑔(𝑡)∈𝐴𝑃(𝑅), then the following conditions are equivalent.(1)mod(𝑓)⊃mod(𝑔); (2) for any 𝜀>0,∃𝛿>0,such that 𝑇(𝑓,𝛿)⊂𝑇(𝑔,𝜀);(3)𝑇_𝛼𝑓 exists which implies 𝑇_𝛼𝑔 exists (any sense);(4)𝑇_𝛼𝑓=𝑓, implies 𝑇_𝛼𝑔=𝑔(any sense);(5)𝑇_𝛼𝑓=𝑓, implies there is 𝛼^′⊂𝛼,so that 𝑇_𝛼′𝑔=𝑔(any sense). Consider the equation 𝑑𝑥(𝑡)(𝑡𝑎(𝑡)(𝑡)𝑥𝑑𝑡=𝑥)−𝑏^𝛼(𝑡), (2.3) where 𝛼>0,𝑎(𝑡),𝑏(𝑡)∈𝐴𝑃(𝑅) and are continuous on 𝑅.

Lemma 2.3. If (2.3) satisfies one of the following conditions
(1)𝑚(𝑎(𝑡))>0,𝑚(𝑏(𝑡))>0,𝑏(𝑡)≥0(2)𝑚(𝑎(𝑡))<0,𝑚(𝑏(𝑡))<0,𝑏(𝑡)≤0,

then (2.3) exists a unique positive almost periodic solution 𝜙(𝑡), and mod(𝜙)⊂mod(𝑎,𝑏).

Proof. Let 𝑥^−𝛼(𝑡)=𝑢(𝑡), then (2.3) can be changed as follows: 𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)(𝑡)𝑑𝑡=−𝛼𝑎+𝛼𝑏, (2.4) by the condition 𝑚(𝑎(𝑡))>0,𝛼>0, we have Re𝑚(−𝛼𝑎(𝑡))<0, also by the condition 𝑚(𝑏(𝑡))>0,𝑏(𝑡)≥0, according to Lemma 2.1, it follows that (2.4) exists a unique positive almost periodic solution 𝜂(𝑡), and 𝜂(𝑡) can be written as follows: 𝜂(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)𝑑𝑠. (2.5)
Next, we prove that mod(𝜂)⊂mod(𝑎,𝑏). If there is a sequence 𝛽^′ such that 𝑇_𝛽′𝑎(𝑡)=𝑎(𝑡),𝑇_𝛽′𝑏(𝑡)=𝑏(𝑡) are convergent uniformly on 𝑅, then there exists a sequence 𝛽⊂𝛽^′ such that 𝑇_𝛽𝜂(𝑡)=𝜁(𝑡) is convergent uniformly on 𝑅, and 𝜁(𝑡) is also an almost periodic solution of (2.4), by the uniqueness of the almost periodic solution of (2.4), we can get 𝑇_𝛽𝜂(𝑡)=𝜂(𝑡), by Lemma 2.2, it follows mod(𝜂)⊂mod(𝑎,𝑏). Since 𝑥^−𝛼(𝑡)=𝑢(𝑡), (2.3) exists a unique positive almost periodic solution 𝜙(𝑡), and it can be written as follows 𝜙(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)𝑑𝑠^−1/𝛼, (2.6) by (2.6), we can easily get mod(𝜙)⊂mod(𝑎,𝑏).
If the condition (2) holds, similarly, we can prove that (2.3) exists a unique positive almost periodic solution 𝜙(𝑡), it can be written as follows 𝜙(𝑡)=−𝛼_𝑡^+∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)𝑑𝑠^−1/𝛼, (2.7) and mod(𝜙)⊂mod(𝑎,𝑏). This is the end of the proof of Lemma 2.3.

Lemma 2.4 (see [2]). Suppose that an almost periodic sequence {𝑓_𝑛(𝑡)} is convergent uniformly on any compact set of 𝑅,  𝑓(𝑡) is an almost periodic function, and mod(𝑓_𝑛)⊂mod(𝑓)(𝑛=1,2,…), then {𝑓_𝑛(𝑡)} is convergent uniformly on 𝑅.

3. The Conclusion of the N-dimentional System

Consider the following equation 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥^𝛼𝑖_𝑖(𝑡), (3.1) where 𝛼_𝑖>0,𝑎_𝑖(𝑡),𝑏_𝑖(𝑡)∈𝐴𝑃(𝑅),𝑖=1,2,…,𝑛.

By Lemma 2.3, it follows if one of the following conditions holds(𝐴₁)𝑚(𝑎_𝑖(𝑡))>0,𝑚(𝑏_𝑖(𝑡))>0,𝑏_𝑖(𝑡)≥0(𝐴₂)𝑚(𝑎_𝑖(𝑡))<0,𝑚(𝑏_𝑖(𝑡))<0,𝑏_𝑖(𝑡)≤0,

then (3.1) exists a unique positive almost periodic solution 𝑥⁰(𝑡)=(𝑥⁰₁(𝑡),𝑥⁰₂(𝑡),…,𝑥⁰_𝑛(𝑡)), we can easily get if (𝐴₁) holds, then 𝑥⁰_𝑖(𝑡)=𝛼_𝑖^𝑡_−∞𝑒^{−𝛼𝑖∫𝑡𝑠𝑎𝑖(𝜏)𝑑𝜏}𝑏_𝑖(𝑠)𝑑𝑠^{−1/𝛼𝑖}, (3.2) if (𝐴₂) holds, then 𝑥⁰_𝑖(𝑡)=−𝛼_𝑖𝑡^+∞𝑒^{−𝛼𝑖∫𝑡𝑠𝑎𝑖(𝜏)𝑑𝜏}𝑏_𝑖(𝑠)𝑑𝑠^{−1/𝛼𝑖}. (3.3) Now, consider (1.5), suppose(𝐵₁)𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖(𝑡) are continuous almost periodic functions,𝑓_𝑖(𝑡,𝜙,𝜓) is continuous for all variables, and is almost periodic in 𝑡 uniformly with respect to (𝜙,𝜓)∈𝐶([−𝜏,0],𝑅^𝑛)×𝐶([−𝜏,0],𝑅^𝑛),  ℎ_𝑖(𝑡,𝜙) is continuous for all variables, and is almost periodic in 𝑡 uniformly with respect to 𝜙∈𝐶([−𝜏,0],𝑅^𝑛);(𝐵₂)ℎ_𝑖(𝑡,𝜙)>0,  for all 𝜙_𝑖>0,𝑡∈𝑅(𝑖=1,2,…,𝑛);(𝐵₂)ℎ_𝑖(𝑡,𝜙)<0,  for all 𝜙_𝑖>0,𝑡∈𝑅(𝑖=1,2,…,𝑛);(𝐵₃)𝑚(𝑎_𝑖(𝑡))>0,𝑚(𝑏_𝑖(𝑡))>0,𝑏_𝑖(𝑡)≥0,𝑚(𝑐_𝑖(𝑡))>0(𝑖=1,2,…,𝑛); (𝐵₃)𝑚(𝑎_𝑖(𝑡))<0,𝑚(𝑏_𝑖(𝑡))<0,𝑏_𝑖(𝑡)≤0,𝑚(𝑐_𝑖(𝑡))<0(𝑖=1,2,…,𝑛).

Following the front of the 𝑥⁰(𝑡) defined, suppose that (𝐵₁),(𝐵₂),(𝐵₃) or (𝐵₁),(𝐵₂),(𝐵₃) hold, and denoted the unique positive almost periodic solution of the following equation 𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+ℎ_𝑖𝑡,𝑥⁰_𝑡, (3.4) by 𝑢⁰(𝑡)=(𝑢⁰₁(𝑡),𝑢⁰₂(𝑡),…,𝑢⁰_𝑛(𝑡)),where 𝑢⁰_𝑖(𝑡)=^𝑡−∞𝑒^{∫𝑡𝑠(−𝑐𝑖(𝜏))𝑑𝜏}ℎ_𝑖𝑠,𝑥⁰_𝑠𝑑𝑠,Re𝑚𝑐_𝑖(𝑡)−>0,_𝑡+∞𝑒^{∫𝑡𝑠(−𝑐𝑖(𝜏))𝑑𝜏}ℎ_𝑖𝑠,𝑥⁰_𝑠𝑑𝑠,Re𝑚𝑐_𝑖(𝑡)<0. (3.5) For any (𝜙,𝜓)∈𝐶([−𝜏,0],𝑅^𝑛)×𝐶([−𝜏,0],𝑅^𝑛), we  define 𝐹₁𝑡,𝜙,𝜓=𝑎₁(𝑡)−𝑏₁(𝑡)𝜙^𝛼1₁(0)−𝑓₁𝑡,𝜙,𝜓,…,𝑎_𝑛(𝑡)−𝑏_𝑛(𝑡)𝜙^𝛼𝑛_𝑛(0)−𝑓_𝑛,𝐻𝑡,𝜙,𝜓₁𝑡,𝜙,𝜓=−𝑐₁(𝑡)𝜓₁(0)+ℎ₁𝑡,𝜙,…,−𝑐_𝑛(𝑡)𝜓_𝑛(0)+ℎ_𝑛𝑋𝑡,𝜙₁=𝜙(𝑡),𝜓(𝑡)∈𝐴𝑃𝑅^2𝑛∶mod𝜙,𝜓⊂mod𝐹₁,𝐻₁, (3.6)

then 𝑋₁ is a Banach space with the model of ‖(𝜙(𝑡),𝜓(𝑡))‖=sup_𝑡∈𝑅(|𝜙(𝑡)|+|𝜓(𝑡)|),construct a bounded closed convex set 𝐺₁ as follows:𝐺₁=𝜙,𝜓∈𝑋₁∶0≤𝜙_𝑖(𝑡)≤𝑥⁰_𝑖(𝑡),0≤𝜓_𝑖(𝑡)≤𝑢⁰_𝑖(𝑡). (3.7)(𝐵₄) For any (𝜙,𝜓)∈𝐺₁ and 𝑡∈𝑅, 0≤𝑓_𝑖(𝑡,𝜙_𝑡,𝜓_𝑡)≤𝑓_𝑖(𝑡,𝑥⁰_𝑡,𝑢⁰_𝑡),0≤ℎ_𝑖(𝑡,𝜙_𝑡)≤ℎ_𝑖(𝑡,𝑥⁰_𝑡),𝑚(𝑎_𝑖(𝑡)−𝑓_𝑖(𝑡,𝑥⁰_𝑡,𝑢⁰_𝑡))>0,(𝑖=1,2,…,𝑛)(𝐵₄) For any (𝜙,𝜓)∈𝐺₁ and 𝑡∈𝑅,0≥𝑓_𝑖(𝑡,𝜙_𝑡,𝜓_𝑡)≥𝑓_𝑖(𝑡,𝑥⁰_𝑡,𝑢⁰_𝑡),0≥ℎ_𝑖(𝑡,𝜙_𝑡)≥ℎ_𝑖(𝑡,𝑥⁰_𝑡),𝑚(𝑎_𝑖(𝑡)−𝑓_𝑖(𝑡,𝑥⁰_𝑡,𝑢⁰_𝑡))<0,(𝑖=1,2,…,𝑛).

Theorem 3.1. If (𝐵₁), (𝐵₂), (𝐵₃), (𝐵₄) or (𝐵₁), (𝐵₂), (𝐵₃), (𝐵₄) hold, then (1.5) exists at least a positive almost periodic solution in 𝐺₁.

Proof. If (𝐵₁), (𝐵₂), (𝐵₃) hold, for any (𝜙,𝜓)∈𝐺₁, consider the following equation𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑓_𝑖𝑡,𝜙_𝑡,𝜓_𝑡−𝑏_𝑖(𝑡)𝑥^𝛼𝑖_𝑖(𝑡),(𝑖=1,2,…,𝑛) (3.8) by the condition (𝐵₄), it follows 𝑚(𝑎_𝑖(𝑡)−𝑓_𝑖(𝑡,𝜙_𝑡,𝜓_𝑡))>0, by Lemma 2.3,it follows that (3.8) exists a unique positive almost periodic solution𝑍_𝜙𝜓(𝑡)=(𝑍_𝜙𝜓1(𝑡),𝑍_𝜙𝜓2(𝑡),…,𝑍_𝜙𝜓𝑛(𝑡)), and it can be written as follows 𝑍_𝜙𝜓𝑖(𝑡)=𝛼_𝑖^𝑡_−∞𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝜙𝜏,𝜓𝜏))𝑑𝜏}𝑏_𝑖(𝑠)𝑑𝑠^{−1/𝛼𝑖},(𝑖=1,2,…,𝑛), (3.9) and mod(𝑍_𝜙𝜓𝑖(𝑡))⊂mod(𝑎_𝑖(𝑡)−𝑓_𝑖(𝑡,𝜙_𝑡,𝜓_𝑡),𝑏_𝑖(𝑡)), since 𝑓_𝑖(𝜏,𝜙_𝜏,𝜓_𝜏)−𝑎_𝑖(𝜏)>−𝑎_𝑖(𝜏), thus by (3.2) and (3.9), we can get 0<𝑍_𝜙𝜓𝑖(𝑡)≤𝑥⁰_𝑖(𝑡),𝑖=1,2,…,𝑛, on the other hand, since mod(𝑎_𝑖(𝑡)−𝑓_𝑖(𝑡,𝜙_𝑡,𝜓_𝑡),𝑏_𝑖(𝑡))⊂mod(𝐹₁), it follows mod(𝑍_𝜙𝜓(𝑡))⊂mod(𝐹₁)_∘.
Consider the following equation𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+ℎ_𝑖𝑡,𝜙_𝑡, (3.10) by Lemma 2.1, (3.10) exists a unique positive almost periodic solution 𝑉_𝜙(𝑡)=𝑉_𝜙1(𝑡),𝑉_𝜙2(𝑡),…,𝑉_𝜙𝑛(𝑡), (3.11) and it can be written as follows: 𝑉_𝜙𝑖(𝑡)=^𝑡_−∞𝑒^{∫𝑡𝑠(−𝑐𝑖(𝜏))𝑑𝜏}ℎ_𝑖𝑠,𝜙_𝑠𝑑𝑠,Re𝑚𝑐_𝑖(𝑡)>0, (3.12) also by the condition 0≤ℎ_𝑖(𝑡,𝜙_𝑡)≤ℎ_𝑖(𝑡,𝑥⁰_𝑡),(𝑖=1,2,…,𝑛), the first formula of (3.5) and (3.12), we have 0<𝑉_𝜙𝑖(𝑡)≤𝑢⁰_𝑖(𝑡), and mod(𝑉_𝜙𝑖(𝑡))⊂mod(𝑐_𝑖(𝑡),h_i(𝑡,𝜙_𝑡)), also mod(𝑐_𝑖(𝑡),ℎ_𝑖(𝑡,𝜙_𝑡))⊂mod(𝐻₁), thus mod(𝑉_𝜙𝑖(𝑡))⊂mod(𝐻₁), therefore, we can get mod(𝑍_𝜙𝜓,𝑉_𝜙)⊂mod(𝐹₁,𝐻₁), hence we can define 𝑃∶𝐺₁⟶𝐺₁,𝑃𝜙,𝜓=𝑍_𝜙𝜓,𝑉_𝜙. (3.13)
Now we are committed to prove the continuity of𝑃 in 𝐺₁. Suppose {(𝜙^𝑘,𝜓^𝑘)⊂𝐺₁}, and when 𝑘→∞,(𝜙^𝑘,𝜓^𝑘)→(𝜙,𝜓)∈𝐺₁, define 𝑆={𝜙^𝑘,𝜓^𝑘}∪{(𝜙,𝜓)}, then 𝑆 is a compact set of 𝑋₁, since 𝑚(𝑎_𝑖(𝑡)−𝑓_𝑖(𝜏,𝑥⁰_𝜏,𝑢⁰_𝜏))>0,𝑖=1,2,…,𝑛, there exist the positive numbers 𝜆and𝐻, such that when 𝑡≥𝑠,𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝑥0𝜏,𝑢0𝜏))𝑑𝜏}≤𝐻𝑒^{𝜆(𝑠−𝑡)},𝑖=1,2,…,𝑛,similarly, since 𝑚(𝑐_𝑖(𝑡))>0,𝑖=1,2,…,𝑛, there exist the positive numbers 𝜇 and 𝐼, such that when 𝑡≥𝑠,𝑒^{∫𝑡𝑠(−𝑐𝑖(𝜏))𝑑𝜏}≤𝐼𝑒^{𝑢(𝑠−𝑡)},  for any (𝜙,𝜓)∈𝐺₁,itfollows0≤𝑓_𝑖(𝑡,𝜙_𝑡,𝜓_𝑡)≤𝑓_𝑖(𝑡,𝑥⁰_𝑡,𝑢⁰_𝑡).
Therefore, by the mean value theorem 𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝜙𝑘𝜏,𝜓𝑘𝜏))𝑑𝜏}−𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝜙𝜏,𝜓𝜏))𝑑𝜏}=𝛼_𝑖^𝑡_𝑠𝑓_𝑖𝜏,𝜙^𝑘_𝜏,𝜓^𝑘_𝜏−𝑓_𝑖𝜏,𝜙_𝜏,𝜓_𝜏𝑑𝜏𝑒^𝜉≤𝛼_𝑖^𝑡_𝑠𝑓_𝑖𝜏,𝜙^𝑘_𝜏,𝜓^𝑘_𝜏−𝑓_𝑖𝜏,𝜙_𝜏,𝜓_𝜏𝑑𝜏𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝑥0𝜏,𝑢0𝜏))𝑑𝜏}≤(𝑡−𝑠)𝛼_𝑖𝑓_𝑖𝑡,𝜙^𝑘_𝑡,𝜓^𝑘_𝑡−𝑓_𝑖𝑡,𝜙_𝑡,𝜓_𝑡𝐻𝑒^{𝜆(𝑠−𝑡)},𝑡≥𝑠,𝑖=1,2,⋯,𝑛. (3.14) Where,𝜉≤max−𝛼_𝑖^𝑡_𝑠𝑎_𝑖(𝜏)−𝑓_𝑖𝜏,𝜙^𝑘_𝜏,𝜓^𝑘_𝜏𝑑𝜏,−𝛼_𝑖^𝑡_𝑠𝑎_𝑖(𝜏)−𝑓_𝑖𝜏,𝜙_𝜏,𝜓_𝜏𝑑𝜏≤−𝛼_𝑖^𝑡_𝑠𝑎_𝑖(𝜏)−𝑓_𝑖𝜏,𝑥⁰_𝜏,𝑢⁰_𝜏𝑑𝜏. (3.15) Also by the mean value theorem, we can obtain the following: 𝑃_𝑖𝜙^𝑘,𝜓^𝑘−𝑃_𝑖=𝑍𝜙,𝜓_{𝜙𝑘𝜓𝑘𝑖}(𝑡)−𝑍_𝜙𝜓𝑖(𝑡),𝑉_𝜙𝑘𝑖(𝑡)−𝑉_𝜙𝑖(𝑡)=−𝑍_{𝜙𝑘𝜓𝑘𝑖}(𝑡)^−𝛼𝑖−𝑍_𝜙𝜓𝑖(𝑡)^−𝛼𝑖𝛼_𝑖𝜁^−𝛼𝑖_𝑖⁻¹,𝑉_𝜙𝑘𝑖(𝑡)−𝑉_𝜙𝑖(𝑡), (3.16) where, 𝜁_𝑖 is between the positive almost periodic function 𝑍_{𝜙𝑘𝜓𝑘𝑖}(𝑡) and 𝑍_𝜙𝜓𝑖(𝑡), thus 0<𝜁_𝑖≤𝑥⁰_𝑖(𝑡). (3.17) By (3.9), (3.12), and (3.14), it follows that −𝑍_{𝜙𝑘𝜓𝑘𝑖}(𝑡)^−𝛼𝑖−𝑍_𝜙𝜓𝑖(𝑡)^−𝛼𝑖𝛼_𝑖𝜁^−𝛼𝑖_𝑖⁻¹=1𝜁^−𝛼𝑖_𝑖^{−1𝑡−∞}𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝜙𝑘𝜏,𝜓𝑘𝜏))𝑑𝜏}−𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝜙𝜏,𝜓𝜏))𝑑𝜏}𝑏_𝑖(𝑠)≤1𝑑𝑠𝜁^−𝛼𝑖_𝑖^{−1𝑡−∞}𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝜙𝑘𝜏,𝜓𝑘𝜏))𝑑𝜏}−𝑒^{−𝛼𝑖∫𝑡𝑠(𝑎𝑖(𝜏)−𝑓𝑖(𝜏,𝜙𝜏,𝜓𝜏))𝑑𝜏}𝑏_𝑖(𝑠≤𝐻)𝑑𝑠𝑥⁰_𝑖(𝑡)^{−𝛼𝑖−1}𝛼_𝑖𝑏_𝑖(𝑡)𝑓_𝑖𝑡,𝜙^𝑘_𝑡,𝜓^𝑘_𝑡−𝑓_𝑖𝑡,𝜙_𝜏,𝜓_𝜏^𝑡_−∞(𝑡−𝑠)𝑒^{𝜆(𝑠−𝑡)}≤𝐻𝑑𝑠𝜆²𝑥⁰_𝑖(𝑡)^𝛼𝑖+1𝛼_𝑖𝑏_𝑖(𝑡)𝑓_𝑖𝑡,𝜙^𝑘_𝑡,𝜓^𝑘_𝑡−𝑓_𝑖𝑡,𝜙_𝑡,𝜓_𝑡,𝑉_𝜙𝑘𝑖(𝑡)−𝑉_𝜙𝑖(𝑡=_𝑡−∞)𝑒^{∫𝑡𝑠(−𝑐𝑖(𝜏))𝑑𝜏}ℎ_𝑖𝑠,𝜙^𝑘_𝑠𝑑𝑠−^𝑡_−∞𝑒^{∫𝑡𝑠(−𝑐𝑖(𝜏))𝑑𝜏}ℎ_𝑖𝑠,𝜙_𝑠≤𝑑𝑠_𝑡−∞𝑒^{∫𝑡𝑠(−𝑐𝑖(𝜏))𝑑𝜏}ℎ_𝑖𝑠,𝜙^𝑘_𝑠−ℎ_𝑖𝑠,𝜙_𝑠≤𝐼𝑑𝑠𝜇ℎ_𝑖𝑡,𝜙^𝑘_𝑡−ℎ_𝑖𝑡,𝜙_𝑡, (3.18)thus we have 𝑃_𝑖𝜙^𝑘,𝜓^𝑘−𝑃_𝑖≤𝐻𝜙,𝜓𝜆²𝑥⁰_𝑖(𝑡)^𝛼𝑖+1𝛼_𝑖𝑏_𝑖(𝑡)𝑓_𝑖𝑡,𝜙^𝑘_𝑡,𝜓^𝑘_𝑡−𝑓_𝑖𝑡,𝜙_𝑡,𝜓_𝑡+𝐼𝜇ℎ_𝑖𝑡,𝜙^𝑘_𝑡−ℎ_𝑖𝑡,𝜙_𝑡. (3.19) Hence 𝑃𝜙^𝑘,𝜓^𝑘≤𝐻−𝑃𝜙,𝜓𝜆²𝑥⁰(𝑡)^𝛼+1‖𝛼‖‖𝑏(𝑡)‖𝑓𝑡,𝜙^𝑘_𝑡,𝜓^𝑘_𝑡−𝑓𝑡,𝜙_𝑡,𝜓_𝑡+𝐼𝜇ℎ𝑡,𝜙^𝑘_𝑡−ℎ𝑡,𝜙_𝑡. (3.20) In addition, taking into account that 𝑓(𝑡,𝜙,𝜓),ℎ(𝑡,𝜙) are continuous uniformly on 𝑅×𝑆, when 𝑘→∞, it follows ‖𝑓(𝑡,𝜙^𝑘_𝑡,𝜓^𝑘_𝑡)−𝑓(𝑡,𝜙_𝑡,𝜓_𝑡)‖→0,‖ℎ(𝑡,𝜙^𝑘_𝑡)−ℎ(𝑡,𝜙_𝑡)‖→0, thus when 𝑘→∞, we have 𝑃𝜙^𝑘,𝜓^𝑘−𝑃𝜙,𝜓⟶0, (3.21) therefore, 𝑃 is continuous.

Then to prove 𝑃 is relatively compact in 𝐺₁. By the boundedness of𝐺₁ and (3.8), (3.10), we can obtain that there exists a positive number 𝐴 such that ‖𝑑(𝑃(𝜙,𝜓))/𝑑𝑡‖≤𝐴,forall(𝜙,𝜓)∈𝐺₁, thus 𝑃(𝐺₁) is uniformly bounded and equicontinuous on 𝑅, by the theorem of Ascoli-arzela, for any sequence {(𝜉^𝑘,𝜁^𝑘)}in 𝐺₁, there exist a subsequences (also denoted by {(𝜉^𝑘,𝜁^𝑘)}) such that {(𝜉^𝑘,𝜁^𝑘)} is convergent uniformly in any compact set of 𝑅, also combined with Lemma 2.4,{(𝜉^𝑘,𝜁^𝑘)}is convergent uniformly on 𝑅, that is to say 𝑃is relatively compact in 𝐺₁. According to Schauder^'s fixed point theorem, 𝑃 exists as a fixed point in 𝐺₁, that is to say (1.5) exists at least a positive almost periodic solution in 𝐺₁. Similarly, when (𝐵₁), (𝐵₂), (𝐵₃), (𝐵₄) hold, we can prove that (1.5) exists at least a positive almost periodic solution in 𝐺₁.

Remark 3.2. When 𝛼_𝑖≡1, then (1.5) turns into (1.4), obviously, Theorem 3.1 is the generalization of Theorems 2.1 and 2.2 in [5];

Remark 3.3. In Theorem 2.2 of [5], it requires the condition𝑓_𝑖(𝑡,0,0)≡0, in this paper, we do not require 𝑓_𝑖(𝑡,0,0)≡0, this can be seen in the weak conditions we get the similar results, thus Theorem 3.1 extends the results of Theorem 2.2  of the paper [5].

4. The Conclusions Of The One-dimentional System

Here we will discuss the system (1.5) in the one-dimensional case𝑑𝑥(𝑡)(𝑡)𝑎(𝑡)(𝑡)𝑥𝑑𝑡=𝑥−𝑏^𝛼(𝑡)−𝑓𝑡,𝑥_𝑡,𝑢_𝑡,𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)𝑑𝑡=−𝑐+ℎ𝑡,𝑥_𝑡, (4.1) where 𝛼>0,𝑡∈𝑅,𝑎(𝑡),𝑏(𝑡),𝑐(𝑡)are continuous almost periodic functions, 𝑓(𝑡,𝜙,𝜓) is almost periodic in t uniformly with respect to (𝜙,𝜓)∈𝐶([−𝜏,0],𝑅₊)×𝐶([−𝜏,0],𝑅₊), and 𝑓(𝑡,0,0)≡0,ℎ(𝑡,𝜙) is almost periodic in t uniformly with respect to 𝜙∈𝐶([−𝜏,0],𝑅₊), and ℎ(𝑡,0)≡0. For convenience, we introduce the following notations. For a continuous bounded function 𝑓(𝑡), denote 𝑓_𝐿=inf_𝑡∈𝑅𝑓(𝑡),𝑓_𝑀=sup_𝑡∈𝑅𝑓(𝑡),(𝐶₁)𝑎_𝐿>0,𝑏_𝐿>0,𝑐_𝐿>0; (𝐶₂)𝑓(𝑡,𝜙,𝜓)≥0,ℎ(𝑡,𝜙)≥0for𝜙≥0,𝜓≥0and𝑡∈𝑅;(𝐶₃)|𝑓(𝑡,𝑥₁,𝑦₁)−𝑓(𝑡,𝑥₂,𝑦₂)|≤𝐿₁(|𝑥₁−𝑥₂|+|𝑦₁−𝑦₂|),|ℎ(𝑡,𝑥₁)−ℎ(𝑡,𝑥₂)|≤𝐿₂|𝑥₁−𝑥₂|

for 𝑡∈𝑅,𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑅₊,𝐿₁,𝐿₂ are positive constants; (𝐶₄)𝐿₁𝛾₁/𝛼𝑎_𝐿𝛽^1/𝛼+1+𝐿₁/𝑎_𝐿𝛽^1/𝛼+𝐿₁𝛾₂/𝑎_𝐿+𝐿₂/𝛼𝑐_𝐿𝛽^1/𝛼+1+𝐿₁𝛾₁/𝑎_𝐿<1,  where,𝛽=min{𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1),[𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1)]^𝛼/2𝛼+1},𝛾₁=(𝑏_𝑀+𝛼𝑎_𝐿(𝛽+𝛼𝛽^2+(1/𝛼)))/𝑎_𝐿,𝛾₂=𝛼𝛽.

In the next paper, we suppose that initial values (𝑥(𝜃),𝑢(𝜃)) of system (4.1) satisfy 𝑥(𝜃)≥0,𝑢(𝜃)≥0,𝜃∈[−𝜏,0], for the solution (𝑥(𝑡),𝑢(𝑡)) of the system (4.1), we only consider its positive initial value, that is to say 𝑥(0)>0,𝑢(0)>0. (4.2) It is not difficult for us to see the solution (𝑥(𝑡),𝑢(𝑡)) of (4.1) with the positive initial value is always positive.

Theorem 4.1. If (𝐶₁),(𝐶₂),(𝐶₃),(𝐶₄) hold, then (4.1) exists a unique positive almost periodic solution.

Proof. Let 𝑦(𝑡)=1/𝑥^𝛼(𝑡), then (4.1) can be changed into the following equation: 𝑑𝑦(𝑡)(𝑡)𝑦(𝑡)(𝑡)(𝑡)f𝑑𝑡=−𝛼𝑎+𝛼𝑏+𝛼𝑦𝑡,𝑦_𝑡^−1/𝛼,𝑢_𝑡,𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)𝑑𝑡=−𝑐+ℎ𝑡,𝑦_𝑡^−1/𝛼. (4.3)
For(𝜙,𝜓)∈𝐶([−𝜏,0],𝑅₊)×𝐶([−𝜏,0],𝑅₊), define 𝐹𝑡,𝜙,𝜓=−𝑎(𝑡)𝜙(0)+𝑏(𝑡)+𝑓𝑡,𝜙^−1/𝛼𝜙(0),𝐻(𝑡)𝜓(0),𝜓𝑡,𝜙,𝜓=−𝑐+ℎ𝑡,𝜙^−1/𝛼,𝑋=𝜙,𝜓∈𝐴𝑃𝑅²∶𝛽≤𝜙≤𝛾₁,0<𝜓≤𝛾₂,mod𝜙,𝜓⊂mod(𝐹,𝐻), (4.4) then 𝑋 is a Banach space with the model of ‖𝜙,𝜓‖=sup_𝑡∈𝑅(|𝜓(𝑡)|+|𝜓(𝑡)|), for all(𝜙,𝜓)∈𝑋, consider the following equation: 𝑑𝑦(𝑡)(𝑡)𝑦(𝑡)(𝑡)(𝑡)𝑓𝑑𝑡=−𝛼𝑎+𝛼𝑏+𝛼𝜙𝑡,𝜙_𝑡^−1/𝛼,𝜓_𝑡, (4.5) by Lemma 2.1, it follows that (4.5) exists a unique positive almost periodic solution 𝑍_𝜙𝜓(𝑡), it can be written as follows:𝑍_𝜙𝜓(𝑡)=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)+𝜙(𝑠)𝑓𝑠,𝜙_𝑠^−1/𝛼,𝜓_𝑠𝑑𝑠, (4.6) similar to the proof of Lemma 2.3 we can obtain mod𝑍_𝜙𝜓⊂mod𝑎(𝑡),𝑏(𝑡)+𝑓𝑡,𝜙_𝑡^−1/𝛼,𝜓𝜙(𝑡)⊂mod(𝐹), (4.7) in addition, 𝑍_𝜙𝜓(𝑡)≥𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏_𝐿−𝜙(𝑠)𝑓𝑠,𝜙_𝑠^−1/𝛼,𝜓_𝑠𝑑𝑠=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏_𝐿−𝜙(𝑠)𝑓𝑠,𝜙_𝑠^−1/𝛼,𝜓_𝑠−𝑓(𝑠,0,0)𝑑𝑠≥𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏_𝐿−𝐿₁𝜙(𝑠)𝜙_𝑠^−1/𝛼+𝜓_𝑠≥𝑏𝑑𝑠_𝐿−𝛾₁𝐿₁𝛽^−1/𝛼+𝛾₂𝑎_𝑀. (4.8) Note that the condition (𝐶₄) holds, thus we have 𝐿₁<𝛼𝑎_𝐿𝛽^1+(1/𝛼)/𝛾₁, hence 𝑍_𝜙𝜓(𝑡)≥𝑏_𝐿−𝛾₁𝛼𝑎_𝐿𝛽^1+1/𝛼/𝛾₁𝛽^−1/𝛼+𝛼𝛽𝑎_𝑀≥𝑏_𝐿−𝛼𝑎_𝐿𝛽^1+1/𝛼𝛽^−1/𝛼+𝛼𝛽𝑎_𝑀=𝑏_𝐿−𝛼𝑎_𝐿𝛽−𝛼²𝑎_𝐿𝛽^2+(1/𝛼)𝑎_𝑀. (4.9) Note that 𝛽=min{𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1),[𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1)]^{𝛼/(2𝛼+1)}}, if 0<𝑏_𝐿/(𝑎_𝑀(𝛼²+𝛼+1))≤1, then 𝛽=𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1), thus we can get 𝑍_𝜙𝜓(𝑡)≥𝑏_𝐿−𝛼𝑎_𝐿𝛽−𝛼²𝑎_𝐿𝛽^2+(1/𝛼)𝑎_𝑀≥𝑏_𝐿−𝛼𝑎_𝑀𝛽−𝛼²𝑎_𝑀𝛽𝑎_𝑀=𝛽, (4.10) if 𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1)>1, then 𝛽=[𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1)]^{𝛼/(2𝛼+1)}, therefore 𝑍_𝜙𝜓(𝑡)≥𝑏_𝐿−𝛼𝑎_𝐿𝛽−𝛼²𝑎_𝐿𝛽^2+(1/𝛼)𝑎_𝑀≥𝑏_𝐿−𝛼𝑎_𝑀𝛽^2+(1/𝛼)−𝛼²𝑎_𝑀𝛽^2+(1/𝛼)𝑎_𝑀=𝑏_𝐿𝑎_𝑀𝛼²+𝛼+1≥𝛽. (4.11) No matter what 0<𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1)≤1 or 𝑏_𝐿/𝑎_𝑀(𝛼²+𝛼+1)>1, we always have 𝑍_𝜙𝜓(𝑡)≥𝛽. (4.12) Also we have 𝑍_𝜙𝜓(𝑡)≤𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}|𝑏(𝑠)|+𝜙(𝑠)𝑓𝑠,𝜙_𝑠^−1/𝛼,𝜓_𝑠𝑑𝑠=𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}|𝑏(𝑠)|+𝜙(𝑠)𝑓𝑠,𝜙_𝑠^−1/𝛼,𝜓_𝑠−𝑓(𝑠,0,0)𝑑𝑠≤𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}|𝑏(𝑠)|+𝜙(𝑠)𝐿₁𝜙_𝑠^−1/𝛼+𝜓_𝑠≤𝑏𝑑𝑠_𝑀+𝜙(𝑡)𝐿₁𝜙_𝑡^−1/𝛼+𝜓_𝑡𝑎_𝐿≤𝑏_𝑀+𝛾₁𝐿₁𝛽^−1/𝛼+𝛾₂𝑎_𝐿≤𝑏_𝑀+𝛾₁𝛼𝑎_𝐿𝛽^1+1/𝛼/𝛾₁𝛽^−1/𝛼+𝛾₂𝑎_𝐿=𝑏_𝑀+𝛼𝑎_𝐿𝛽^1+1/𝛼𝛽^−1/𝛼+𝛼𝛽𝑎_𝐿=𝛾₁. (4.13) Consider the following equation: 𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)𝑑𝑡=−𝑐+ℎ𝑡,𝜙_𝑡^−1/𝛼. (4.14) By Lemma 2.1 we can see that (4.14) has a unique positive almost periodic solution 𝑉_𝜙(𝑡), it can be written as follows 𝑉_𝜙(𝑡)=^𝑡_−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}ℎ𝑠,𝜙_𝑠^−1/𝛼𝑑𝑠. (4.15) Similar to the proof of Lemma 2.3, we can get mod𝑉_𝜙(𝑡)⊂mod𝑐(𝑡),ℎ𝑡,𝜙_𝑡^−1/𝛼(𝐻),𝑉⊂mod_𝜙(𝑡)≤^𝑡_−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}ℎ𝑠,𝜙_𝑠^−1/𝛼=𝑑𝑠_𝑡−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}ℎ𝑠,𝜙_𝑠^−1/𝛼−ℎ(𝑠,0)𝑑𝑠≤𝐿₂^𝑡_−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}𝜙_𝑠^−1/𝛼≤𝐿𝑑𝑠₂𝛽^−1/𝛼𝑐_𝐿. (4.16) Note that the condition (𝐶₄) holds, it follows that𝐿₂<𝛼𝑐_𝐿𝛽^1+(1/𝛼), hence 𝑉_𝜙(𝑡)≤𝛼𝑐_L𝛽^1+1/𝛼𝛽^−1/𝛼𝑐_𝐿=𝛼𝛽=𝛾₂. (4.17) Now, define a mapping 𝑇𝜙,𝜓(𝑡)=𝑇₁𝜙,𝜓(𝑡),𝑇₂𝜙,𝜓(𝑡)=𝑍_𝜙𝜓(𝑡),𝑉_𝜙(𝑡). (4.18) From the above discussion we can see 𝑇(𝜙,𝜓)(𝑡)∈𝑋, thus 𝑇∶𝑋→𝑋, for given any (𝜙₁,𝜓₁),(𝜙₂,𝜓₂)∈𝑋, we have 𝑇𝜙₁,𝜓₁−𝑇𝜙₂,𝜓₂=𝑇₁𝜙₁,𝜓₁−𝑇₁𝜙₂,𝜓₂,𝑇₂𝜙₁,𝜓₁−𝑇₂𝜙₂,𝜓₂=𝑍_𝜙1𝜓1(𝑡)−𝑍_𝜙2𝜓2(𝑡),𝑉_𝜙1(𝑡)−𝑉_𝜙2(𝑡)=sup_𝑡∈𝑅𝛼^𝑡_−∞e^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝜙₁(𝑠)𝑓𝑠,𝜙^−1/𝛼_1𝑠,𝜓_1𝑠−𝜙₂(𝑠)𝑓𝑠,𝜙^−1/𝛼_2𝑠,𝜓_2𝑠+𝑑𝑠_𝑡−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}ℎ𝑠,𝜙^−1/𝛼_1𝑠−ℎ𝑠,𝜙^−1/𝛼_2𝑠𝑑𝑠≤sup_𝑡∈𝑅𝛼^t_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝜙₁(𝑠)𝑓𝑠,𝜙^−1/𝛼_1𝑠,𝜓_1𝑠−𝑓𝑠,𝜙^−1/𝛼_2𝑠,𝜓_2𝑠+𝑓𝑠,𝜙^−1/𝛼_2𝑠,𝜓_2𝑠𝜙₁(𝑠)−𝜙₂(𝑠+)𝑑𝑠_𝑡−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}ℎ𝑠,𝜙^−1/𝛼_1𝑠−ℎ𝑠,𝜙^−1/𝛼_2𝑠𝑑𝑠≤sup_𝑡∈𝑅𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝐿₁𝜙₁(𝑠)𝜙^−1/𝛼_1𝑠−𝜙^−1/𝛼_2𝑠+𝜓_1𝑠−𝜓_2𝑠+𝐿₁𝜙^−1/𝛼_2𝑠+𝜓_2𝑠𝜙₁(𝑠)−𝜙₂(𝑠)𝑑𝑠+𝐿₂^𝑡_−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}𝜙^−1/𝛼_1𝑠−𝜙^−1/𝛼_2𝑠𝑑𝑠. (4.19) By the mean value theorem (where 𝜉 of the following formula is between 𝜙_1𝑠 and 𝜙_2𝑠), the above formula can be changed as follows: 𝑇𝜙₁,𝜓₁−𝑇𝜙₂,𝜓₂≤sup_𝑡∈𝑅𝛼^t_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝐿₁𝜙₁(𝑠1)𝛼𝜉^{(−1/𝛼)−1}𝜙_1𝑠−𝜙_2𝑠+𝜓_1𝑠−𝜓_2𝑠+𝐿₁𝜙^−1/𝛼_2𝑠+𝜓_2𝑠𝜙₁(𝑠)−𝜙₂(𝑠)+𝐿₂^𝑡_−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}1𝛼𝜉^{(−1/𝛼)−1}𝜙_1𝑠−𝜙_2𝑠𝑑𝑠≤sup_𝑡∈𝑅𝛼^𝑡_−∞𝑒^{−𝛼∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝐿₁𝛾₁1𝛼𝛽^{−1/𝛼−1}𝜙_1𝑠−𝜙_2𝑠+𝜓_1𝑠−𝜓_2𝑠+𝐿₁𝛽^(−1/𝛼)+𝛾₂𝜙₁(𝑠)−𝜙₂(𝑠)𝑑𝑠+𝐿₂^𝑡_−∞𝑒^{−∫𝑡𝑠𝑐(𝜏)𝑑𝜏}1𝛼𝛽^{(−1/𝛼)−1}𝜙_1𝑠−𝜙_2𝑠≤1𝑑𝑠𝑎_𝐿𝐿₁𝛾₁1𝛼𝛽^{(−1/𝛼)−1}𝜙_1𝑡−𝜙_2𝑡+𝐿₁𝛾₁𝜓_1𝑡−𝜓_2𝑡+𝐿₁𝛽^(−1/𝛼)+𝛾₂𝜙₁(𝑡)−𝜙₂(𝑡+1)𝑐_𝐿𝐿₂1𝛼𝛽^{(−1/𝛼)−1}𝜙_1𝑡−𝜙_2𝑡=1𝑎_𝐿𝐿₁𝛾₁1𝛼𝛽^{−1/𝛼−1}+1𝑎_𝐿𝐿₁𝛽^−1/𝛼+1𝑎_𝐿𝐿₁𝛾₂+1𝑐_𝐿𝐿₂1𝛼𝛽^{(−1/𝛼)−1}𝜙₁−𝜙₂+1𝑎_𝐿𝐿₁𝛾₁𝜓₁−𝜓₂≤1𝑎_𝐿𝐿₁𝛾₁1𝛼𝛽^{(−1/𝛼)−1}+1𝑎_𝐿𝐿₁𝛽^−1/𝛼+1𝑎_𝐿𝐿₁𝛾₂+1𝑐_𝐿𝐿₂1𝛼𝛽^{(−1/𝛼)−1}+1𝑎_𝐿𝐿₁𝛾₁×𝜙₁,𝜓₁−𝜙₂,𝜓₂ (4.20) notice that the condition (𝐶₄) holds, it follows 𝑇𝜙₁,𝜓₁−𝑇𝜙₂,𝜓₂<𝜙₁,𝜓₁−𝜙₂,𝜓₂, (4.21) by (4.21), 𝑇 is a contraction mapping, thus 𝑇 exists a unique fixed point in 𝑋, this fixed point is the only positive almost periodic solution(𝜑^∗(𝑡),𝑢^∗(𝑡)) of (4.3), and 𝛽≤𝜑^∗(𝑡)≤𝛾₁,0<𝑢^∗(𝑡)≤𝛾₂. note that 𝑦(𝑡)=1/𝑥^𝛼(𝑡),  𝑥(𝑡)=𝑢^−1/𝛼(𝑡), thus (4.1) exists a positive almost periodic solution (𝜓^∗(𝑡),𝑢^∗(𝑡)), and 1/(𝛾₁^1/𝛼)≤𝜓^∗(𝑡)≤1/𝛽^1/𝛼,0<𝑢^∗(𝑡)≤𝛾₂. This completes the proof of Theorem 4.1.

Acknowledgments

This paper is supported by the National Nature Science Foundation of China (70871056) and Jiangsu Province Innovation Project of Graduate Education (1221190037).