Unique Positive Almost Periodic Solution for Discrete Nonlinear Delay Survival Red Blood Cells Model

Abstract

We obtain sufficient conditions which guarantee the global attractivity of solutions for nonlinear delay survival red blood cells model. Then, some criteria are established for the existence, uniqueness and global attractivity of positive almost periodic solutions of the almost periodic system.

1. Introduction

Recently, Saker [1] studied the existence and global attractivity of positive periodic solution for the following discrete nonlinear delay survival red blood cells model: 𝑥(𝑛+1)−𝑥(𝑛)=−𝛿(𝑛)𝑥(𝑛)+𝑃(𝑛)𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}, (1.1) where 𝛿(𝑛), 𝑃(𝑛), 𝑞(𝑛), 𝜏(𝑛) are nonnegative bounded sequence, 𝜏(𝑛)∈𝐍 for all 𝑛∈𝐙 and 0<𝛿(𝑛)<1. The dynamic behavios of (1.1) was investigated by many authors (see [1–18]) because of its biological and ecological significance. When 𝛿(𝑛), 𝑃(𝑛),𝑞(𝑛) and 𝜏(𝑛) are positive constants, the global attractivity of the positive equilibrium for equation 𝑥(𝑛+1)−𝑥(𝑛)=−𝛿𝑥(𝑛)+𝑃𝑒^{−𝑞𝑥(𝑛−𝜏)} (1.2) has been investigated by some authors (see [3, 15–18] and reference therein). Recently, Ma and Yu [17] obtained the following Theorem A.

Theorem 1 A. Suppose that 𝛿∈(0,1), 𝑃,𝑞∈[0,∞) and 𝜏 are nonnegative integer. If 𝑞𝑥(1−(1−𝛿)^𝜏+1)≤1, (1.3) then the unique positive equilibrium 𝑥 of (1.2) is a global attractor of all positive solutions of (1.2).

Equation (1.2) is the discrete analogue of equation 𝑥^′(𝑡)=−𝛿𝑥(𝑡)+𝑝exp(−𝑞𝑥(𝑡−𝜏)), (1.4) which was first used by Ważewska-Czyżewska and Lasota as a model for the survival of red blood cells in an animal in [19], see also [20]. Here, 𝑥(𝑡) denotes the number of red blood cells at time 𝑡, 𝛿 is the probability of death of a red blood cell, 𝑝 and 𝑞 are positive constants related to the production of red blood cells per unit time and 𝜏 is the time required to produce a red blood cell. Researching the behavior of the solution of (1.4) and its analogue was posed as open problems by Kocic and Ladas [10] as well as Györi and Ladas [6].

Because of seasonal variation (1.1) needs not to be exactly periodic but almost periodic instead. It is natural to ask if the results in [1] hold for the almost periodic case. It is a difficult problem in which significant difference appears in comparison with the periodic case, for example, contrary to periodic functions, there exists an almost periodic function 𝑥(𝑡) such that 𝑥(𝑡)>0 for all 𝑡∈ℝ and inf_𝑡∈ℝ𝑥(𝑡)=0.

One purpose of the present paper is to extend Theorem A to (1.1). The other purpose is to extend some results in [1] to the almost periodic case. In [1], the author use a fixed theorem to obtain the existence of positive periodic solution. The operator used in [1] depends on the period of system (1.1), therefore we cannot apply these topological tools to the almost periodic case. An important notion in almost periodic differential theory is the hull (cf. [21–23]). In this paper, we use essentially this notion to establish our results. We begin with some notations.

Let 𝜏=max_𝑛∈ℤ𝜏(𝑛), 𝐼={−𝜏,−𝜏+1,…,−1,0}, 𝐶={𝜑∶𝐼→ℝ} and 𝐶⁺={𝜑∈𝐶∶𝜑≥0,𝜑(0)>0}. For each 𝜑∈𝐶, we define the norm of 𝜑 as ‖𝜑‖=max_𝑠∈𝐼|𝜑(𝑠)|. Denote 𝑥_𝑛 the element of 𝐶 with 𝑥_𝑛=𝑥(𝑛+𝑠) for all 𝑠∈𝐼. For any bounded sequence 𝑒(𝑛), denote 𝑒_∗=liminf_𝑛→∞𝑒(𝑛) and 𝑒^∗=limsup_𝑛→∞𝑒(𝑛), ∏^𝑘_𝑗=𝑙𝑒(𝑗)=𝑒(𝑙)𝑒(𝑙+1)⋯𝑒(𝑘) if 𝑘≥𝑙 and ∏^𝑘_𝑗=𝑙𝑒(𝑗)=1 if 𝑘<𝑙. It is easy to see that, for any 𝜑∈𝐶⁺, there is a unique solution 𝑥(𝑛,0,𝜑) of (1.1) with 𝑥₀=𝜑 and 𝑥(𝑛,0,𝜑)>0 for all 𝑛∈ℕ. Now, we give some definitions.

Definition 1.1 (see [22, 23]). A sequence 𝑓(𝑛) is said to be almost periodic, if for any 𝜖>0, there is a constant 𝑙(𝜖)>0 such that in any interval of length 𝑙(𝜖) there exists 𝜏∈ℤ such that the inequality |𝑓(𝑛+𝜏)−𝑓(𝑛)|<𝜖 (1.5) is satisfied for all 𝑛∈ℤ.

Denote set 𝐻(𝑓)={𝑔∶thereis{𝑛_𝑘}⊂ℤsuchthatlim_𝑛→∞𝑓(𝑛+𝑛_𝑘)=𝑔(𝑛)∀𝑛∈ℤ} (1.6) the hull of 𝑓. It is easy to see that if 𝑓(𝑛) is almost periodic, then for all 𝑓₁∈𝐻(𝑓), liminf_𝑛→∞𝑓(𝑛)=inf_𝑛∈ℤ𝑓(𝑛)=inf_𝑛∈ℤ𝑓₁(𝑛),limsup_𝑛→∞𝑓(𝑛)=sup_𝑛∈ℤ𝑓(𝑛)=sup_𝑛∈ℤ𝑓₁(𝑛). (1.7)

Definition 1.2. Let 𝑓∶ℤ×𝐶→ℝ. Then 𝑓 is said to be almost periodic in 𝑛∈ℤ uniformly on compact set of 𝐶, if 𝑓(𝑛,⋅) is continuous for each 𝑛∈ℤ, and for any 𝜖>0 and every compact set 𝐾⊂𝐶, there is a constant 𝑙(𝜖,𝐾)>0 such that in any interval of length 𝑙(𝜖,𝐾) there exists 𝜏 such that the inequality |𝑓(𝑛+𝜏,𝜑)−𝑓(𝑛,𝜑)|<𝜖 (1.8) is satisfied for all 𝑛∈ℤ and 𝜑∈𝐾.

Definition 1.3. The positive solution 𝑁(𝑛) to (1.1), in the sense that 𝑁(𝑛)>0 for all 𝑛≥0, is said to be global attractive, if for any 𝜑∈𝐶⁺, lim_𝑛→∞(𝑥(𝑛,0,𝜑)−𝑁(𝑛))=0. (1.9)

The remainder of this paper is organized as follows. The main results are given in Section 2 while the proofs are left to Section 4. In Section 3, we will give some lemmas needed in the proofs of main results.

2. Main Results

Theorem 2.1. Assume that 0<𝛿_∗<1 and 𝑃_∗>0. Then, for any 𝜑∈𝐶⁺, the solution 𝑥(𝑛,0,𝜑) of (1.1) satisfies 𝑢≤liminf_𝑛→∞𝑥(𝑛,0,𝜑)≤limsup_𝑛→∞𝑥(𝑛,0,𝜑)≤𝑣, (2.1) where (𝑢,𝑣) is the limit of {(𝑢_𝑛,𝑣_𝑛)} with 𝑢₀=0 and 𝑣_𝑘=𝑃^∗𝛿_∗𝑒^{−𝑞∗𝑢𝑘−1}𝑢,𝑘=1,2,…,_𝑘=𝑃_∗𝛿^∗𝑒^{−𝑞∗𝑣𝑘},𝑘=1,2,…. (2.2)

The following corollary follows from [24, Theorems 1 and  2].

Corollary 2.2. Assume that the conditions of Theorem 2.1 are satisfied, and 𝛿(𝑛), 𝑃(𝑛), 𝑞(𝑛) and 𝜏(𝑛) are 𝜔-periodic with 𝜔>0. Then (1.1) admits a positive 𝜔-periodic solution.

Remark 2.3. When 𝜏(𝑛)≡𝜔, under the conditions of Corollary 2.2, Saker in [1, Theorem  2.1] proved that the conclusion of Corollary 2.2. Thus Corollary 2.2 extends Theorem  2.1 in [1].

Theorem 2.4. Assume that the conditions of Theorem 2.1 are satisfied and 𝑎𝑞^∗≤1, (2.3) where 𝑎=limsup_𝑛→∞^𝑛+𝜏_𝑘=𝑛𝑃(𝑘)^𝑛+𝜏_𝑖=𝑘+1(1−𝛿(𝑖))𝑒^{−𝑞(𝑘)𝑥(𝑘−𝜏(𝑘))}, (2.4) and 𝑥(𝑛) is a positive solution of (1.1). Then every positive solution 𝑥(𝑛) of (1.1) satisfies lim_𝑛→∞[𝑥(𝑛)−𝑥(𝑛)]=0. (2.5)

Corollary 2.5. Assume that the conditions of Theorem 2.1 are satisfied, and 𝑞^∗limsup_𝑛→∞^𝑛+𝜏_𝑘=𝑛𝑃(𝑘)^𝑛+𝜏_𝑖=𝑘+1(1−𝛿(𝑖))𝑒^{−𝑞(𝑘)𝑢}≤1, (2.6) where 𝑢 is the constant given in Theorem 2.1. Then every positive of (1.1) is globally attractive.

Remark 2.6. When 𝛿(𝑛)≡𝛿, 𝑃(𝑛)≡𝑃 and 𝑞(𝑛)≡𝑞, here 𝛿, 𝑃 and 𝑞 are positive constants, we consider the global attractivity of the positive equilibrium 𝑥 for (1.2). In this case, 𝛿𝑥=𝑃𝑒^−𝑞𝑥 and 𝑎=limsup_𝑛→∞^𝑛+𝜏_𝑘=𝑛𝑃(𝑘)^𝑛+𝜏_𝑖=𝑘+1(1−𝛿(𝑖))𝑒^{−𝑞(𝑘)𝑥(𝑘−𝜏(𝑘))}=𝑃𝑒^−𝑞𝑥1−(1−𝛿)^𝜏+1𝛿=𝑥(1−(1−𝛿)^𝜏+1). (2.7) We see that (2.3) changes to (1.3) and Theorem 2.4 reproduces Theorem A.

Theorem 2.7. Assume that system (1.1) is almost periodic, that is, 𝛿(𝑡), 𝑃(𝑛), 𝑞(𝑛) and 𝜏(𝑛) are almost periodic, and the conditions of Corollary 2.5 are satisfied. Then there is a unique globally attractive positive almost periodic solution 𝑝(𝑛) for (1.1).

Remark 2.8. Let 𝑃(𝑛)=1/10+(1/20)sin𝑛, 𝑞(𝑛)=𝜏(𝑛)=2 and 𝛿(𝑛)=𝛿 be a positive constant such that 3𝑒10^{−(1/10𝛿)𝑒−3/10𝛿}1−(1−𝛿)³𝛿≤1,3(1−𝛿)𝛿>1. (2.8) Since 𝑃^∗=3/20, 𝑃_∗=1/20, 𝑞^∗=𝑞_∗=2, 𝑣₁=𝑃^∗/𝛿=3/20𝛿, 𝑢>𝑢₁=(𝑃_∗/𝛿)𝑒^{−𝑞∗𝑣1}=(1/20𝛿)𝑒^−3/10𝛿,, 𝜏=sup_𝑛∈𝐙𝜏(𝑛)=2 and 𝑞^∗limsup_𝑛→∞^𝑛+𝜏_𝑘=𝑛𝑃(𝑘)^𝑛+𝜏_𝑖=𝑘+1(1−𝛿(𝑖))𝑒^{−𝑞(𝑘)𝑢}≤3𝑒10^{−(1/10𝛿)𝑒−3/10𝛿}1−(1−𝛿)³𝛿≤1, (2.9) we can see that there is unique globally attractive positive almost periodic solution 𝑝(𝑛) for (1.1). Since 𝑃^∗𝑞^∗(1−𝛿)𝛿=3(1−𝛿)𝛿>1, (2.10) we cannot obtain the existence of positive almost periodic solution of (1.1) by [25, Theorem  4.1]. Moreover, we should point out that (4.1) in [25] is not correct. To see this, we consider (1.2) with 𝑃(1−𝛿)/𝛿<ln𝑃/𝑞<1. Let 𝑓(𝑥)=−𝛿𝑥+𝑃𝑒^−𝑞𝑥. It is easy to see that the positive equilibrium 𝑥 of (1.2) is the unique zero point of 𝑓. Since 𝑓(ln𝑃𝑞)=1−𝛿ln𝑃𝑞>1−𝛿>0, (2.11) we see that 𝑥>ln𝑃/𝑞>𝑃(1−𝛿)/𝛿. Thus we cannot conclude that for each positive solution 𝑥(𝑛) of (1.2), limsup_𝑛→∞𝑥(𝑛)≤𝛼∶=𝑃(1−𝛿)𝛿. (2.12) Therefore the conclusions of [25, Theorem  4.1] may not hold.

3. Some Lemmas

Lemma 3.1 (see [26]). The following system of inequalities, 𝑦≤𝑒^−𝑥−1,𝑥≥𝑒^−𝑦−1, (3.1) where 𝑥≤𝑦 are real numbers, have exactly one solution 𝑥=𝑦=0.

Lemma 3.2. Assume that 𝑓∶ℤ×𝐶→ℝ, is almost periodic sequence in 𝑛 uniformly on compact set of 𝐶. If there is a solution 𝑥(𝑛) (𝑛≥0) to the equation 𝑥(𝑛+1)=𝑓(𝑛,𝑥_𝑛), (3.2) such that 𝐴≤liminf_𝑛→∞𝑥(𝑡)≤limsup_𝑛→∞𝑥(𝑡)≤𝐵. then for each 𝑔∈𝐻(𝑓), equation 𝑥(𝑛+1)=𝑔(𝑛,𝑥_𝑛) (3.3) has a solution 𝑝(𝑛) which is defined on ℤ such that 𝐴≤𝑝(𝑛)≤𝐵,𝑛∈ℤ. (3.4)

Proof. Let 𝐷=sup_𝑛∈ℕ|𝑥(𝑛)| and 𝑆={𝜑∈𝐶∶‖𝜑‖≤𝐷}. Then 𝑆 is a compact subset of 𝐶. For each 𝑔∈𝐻(𝑓), there is {𝑛_𝑘}⊂ℕ such that 𝑛_𝑘→∞ and 𝑓(𝑛+𝑛_𝑘,𝜑)→𝑔(𝑛,𝜑) uniformly on ℤ×𝑆. By the diagonal process, we can choose {𝑛_𝑘𝑗}⊂{𝑛_𝑘} such that 𝑥(𝑛+𝑛_𝑘𝑗)→𝑝(𝑛) as 𝑗→∞ and 𝑝(𝑛) satisfies (3.3) on ℤ. We want to prove that (3.4) holds.
For each 𝜖>0, there is a 𝑇>0 such that 𝐴−𝜖≤𝑥(𝑛)≤𝐵+𝜖,𝑛≥𝑇. (3.5) For any fixed 𝑛∈ℤ, there is 𝐽>0 such that 𝑛+𝑛_𝑘𝑗>𝑇 for all 𝑗>𝐽. It follows from (3.5) that 𝐴−𝜖≤𝑥(𝑛+𝑛_𝑘𝑗)≤𝐵+𝜖,𝑗≥𝐽. (3.6) Setting 𝑗→∞ and 𝜖→0, we see that (3.4) holds. This completes the proof.

Lemma 3.3. Assume that 𝑓∶ℤ×𝐶→ℝ, is almost periodic sequence in 𝑛 uniformly on compact set of 𝐶. Let 𝑆 be a compact set of ℝ. If for all 𝑔∈𝐻(𝑓), equation 𝑥(𝑛+1)=𝑔(𝑛,𝑥_𝑛) (3.7) admits a unique solution 𝑝_𝑔(𝑛), defined on ℤ, whose range is in 𝑆, then all these solutions 𝑝_𝑔(𝑛) are almost periodic.

The proof is similar to that of [21, Theorem 2.10.1]. We omit it here.

4. The Proofs of Main Results

Proof of Theorem 2.1. Let 𝑥(𝑛)=𝑥(𝑛,0,𝜑) for each 𝜑∈𝐶⁺. From (1.1), we have 𝑥(1)=(1−𝛿(0))𝑥(0)+𝑃(0)𝑒^{−𝑞(0)𝑥(−𝜏(0))}≥(1−𝛿(0))𝑥(0)>0, (4.1) which prove that 𝑥(𝑛)>0 (𝑛=1,2,…) by induction. For any 𝜖∈(0,𝛿_∗), there is 𝑁>0 such that for 𝑛>𝑁, 𝛿_∗−𝜖<𝛿(𝑛)<𝛿^∗+𝜖,𝑃_∗−𝜖<𝑃(𝑛)<𝑃^∗𝑞+𝜖,_∗−𝜖<𝑞(𝑛)<𝑞^∗+𝜖. (4.2) By the fact that 𝑥(𝑛)>0 we deduce that 𝑥(𝑛+1)≤(1−𝛿_∗+𝜖)𝑥(𝑛)+𝑃^∗+𝜖,𝑛>𝑁. (4.3) This implies 𝑥(𝑛)≤(1−𝛿_∗+𝜖)^{𝑛−𝑁−1}𝑃𝑥(𝑁+1)+^∗+𝜖𝛿_∗−𝜖[(1−(1−𝛿_∗+𝜖)^{𝑛−𝑁−1}],𝑛>𝑁. (4.4) By the fact that 𝜖 is arbitrary, we obtain that limsup_𝑛→∞𝑃𝑥(𝑛)≤^∗𝛿_∗=𝑣₁. (4.5) For each 𝜖∈(0,𝛿_∗), there is 𝑁₁>𝑁 such that 𝑥(𝑛)<𝑣₁+𝜖,𝑛≥𝑁₁. (4.6) By (1.1), (4.2), we obtain that 𝑥(𝑛+1)≥(1−𝛿^∗−𝜖)𝑥(𝑛)+(𝑃_∗−𝜖)𝑒^{−(𝑞∗+𝜖)(𝑣1+𝜖)},𝑛≥𝑁₁+𝜏. (4.7) It follows that for 𝑛≥𝑁₁+𝜏𝑥(𝑛)≥(1−𝛿^∗−𝜖)^{𝑛−𝑁1−𝜏}𝑥(𝑁₁𝑃+𝜏)+^∗−𝜖𝛿^∗𝑒+𝜖^{−(𝑞∗+𝜖)(𝑣1+𝜖)}[(1−(1−𝛿^∗−𝜖)^{𝑛−𝑁1−𝜏}]. (4.8) This implies that liminf_𝑛→∞𝑃𝑥(𝑛)≥^∗𝛿^∗𝑒^{−𝑞∗𝑣1}=𝑢₁. (4.9) By induction we can see that 𝑢_𝑘≤liminf_𝑛→∞𝑥(𝑛)≤limsup_𝑛→∞𝑥(𝑛)≤𝑣_𝑘,𝑘=1,2,…. (4.10) It is easy to see that {𝑢_𝑘} increase and {𝑣_𝑘} decrease. Thus the limit of {(𝑢_𝑛,𝑣_𝑛)} exists. Therefore we have 𝑢≤liminf_𝑛→∞𝑥(𝑛)≤limsup_𝑛→∞𝑥(𝑛)≤𝑣. (4.11) This completes the proof of Theorem 2.1.

Proof of Theorem 2.4. Set 𝑧(𝑛)=𝑥(𝑛)−𝑥(𝑛). Then 𝑧(𝑛) satisfies 𝑧(𝑛+1)−𝑧(𝑛)=−𝛿(𝑛)𝑧(𝑛)+𝑃(𝑛)𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}(𝑒^{−𝑞(𝑛)𝑧(𝑛−𝜏(𝑛))}−1). (4.12) The proof will be accomplished by showing that lim_𝑛→∞𝑧(𝑛)=0. (4.13) We will prove that (4.13) holds in each of the following two cases.
Case 1. {𝑧(𝑛)} is nonoscillatory. Suppose that {𝑧(𝑛)} is eventually nonnegative. The case that {𝑧(𝑛)} is eventually nonpositive is similar and will be omitted. By (4.12) we see that {𝑧(𝑛)} is eventually decreasing. Thus the limit of {𝑧(𝑛)} exists. Let 𝑏=lim_𝑛→∞𝑧(𝑛). Then 𝑏≥0. We claim that 𝑏=0. Otherwise, there would exist 𝑁>0 such that 𝑏2<𝑧(𝑛)<3𝑏2,𝑛≥𝑁. (4.14) It follows from (4.12) that 𝑏𝑧(𝑛+1)−𝑧(𝑛)<−2𝛿(𝑛),𝑛≥𝑁+𝜏. (4.15) This implies that 𝑏𝑧(𝑛)<−2^𝑛_{𝑘=𝑁+𝜏}𝛿(𝑘)+𝑧(𝑁+𝜏),𝑛≥𝑁+𝜏. (4.16) By the fact that 𝛿_∗>0 we see that lim_𝑛→∞𝑧(𝑛)=−∞, which contradicts the fact that {𝑧(𝑛)} is eventually nonnegative. Thus 𝑏=0. Therefore (4.13) holds in this case.Case 2. {𝑧(𝑛)} is oscillatory. Let 𝜆=liminf_𝑛→∞𝑧(𝑛) and 𝜇=limsup_𝑛→∞𝑧(𝑛). Then −∞<𝜆≤0≤𝜇<∞. (4.17) There exist positive sequences {𝑝_𝑖} and {𝑞_𝑖} such that lim_𝑖→∞𝑝_𝑖=∞, 𝑝_𝑖<𝑞_𝑖<𝑝_𝑖+1, (4.18)𝑧(𝑝_𝑖)<0 and 𝑧(𝑞_𝑖)>0 for 𝑖∈ℕ, 𝑧(𝑘)≥0 for 𝑝_𝑖<𝑘≤𝑞_𝑖 and 𝑧(𝑘)≤0 for 𝑞_𝑖<𝑘≤𝑝_𝑖+1 and limsup_𝑖→∞𝐴_𝑖=𝜇,liminf_𝑖→∞𝑎_𝑖=𝜆, (4.19) where 𝐴_𝑖=max{𝑧(𝑗)∣𝑝_𝑖≤𝑗≤𝑞_𝑖} and 𝑎_𝑖=min{𝑧(𝑗)∣𝑞_𝑖≤𝑗≤𝑝_𝑖+1}. Let 𝑀_𝑖=min{𝑗∣𝑝_𝑖≤𝑗≤𝑞_𝑖,𝑧(𝑗)=𝐴_𝑖} and 𝑚_𝑖=min{𝑗∣𝑞_𝑖≤𝑗≤𝑝_𝑖+1,𝑧(𝑗)=𝑎_𝑖}. Since 𝑧(𝑀_𝑖)>𝑧(𝑀_𝑖−1), by (4.12) we have 𝛿(𝑀_𝑖−1)𝑧(𝑀_𝑖−1)<𝑃(𝑀_𝑖−1)𝑒^{−𝑞(𝑀𝑖−1)𝑥(𝑀𝑖−1−𝜏(𝑀𝑖−1))}(𝑒^{−𝑞(𝑛)𝑧(𝑀𝑖−1−𝜏(𝑀𝑖−1))}−1). (4.20) We claim that 𝑀_𝑖−1−𝑝_𝑖≤𝜏. (4.21) We first assume that 𝑧(𝑀_𝑖−1)<0. Therefore we have 𝑀_𝑖−1≤𝑝_𝑖. Thus (4.21) holds. Assume now that 𝑧(𝑀_𝑖−1)≥0. By (4.20) we obtain that 𝑧(𝑀_𝑖−1−𝜏(𝑀_𝑖−1))<0. This implies that 𝑀_𝑖−1−𝑝_𝑖≤𝜏(𝑀_𝑖−1)≤𝜏. Thus (4.21) also holds.
Similarly, we can obtain that 𝑚_𝑖−1−𝑞_𝑖≤𝜏. (4.22) For each 𝜖>0, there is 𝑁>0 such that for 𝑛>𝑁, 𝑞_∗−𝜖<𝑞(𝑛)<𝑞^∗+𝜖,𝜆−𝜖<𝑧(𝑛)<𝜇+𝜖,_{𝑛+𝜏𝑘=𝑛}𝑃(𝑘)^𝑛+𝜏_𝑖=𝑘+1(1−𝛿(𝑖))𝑒^{−𝑞(𝑘)𝑥(𝑘−𝜏(𝑘))}<𝑎+𝜖, (4.23) Let 𝑧(𝑛)=𝑦(𝑛)∏^𝑛−1_𝑖=0(1−𝛿(𝑖)). By (4.12) we obtain that 𝑦(𝑛+1)=𝑦(𝑛)+(^𝑛_𝑖=0(1−𝛿(𝑖)))⁻¹𝑃(𝑛)𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}(𝑒^{−𝑞(𝑛)𝑧(𝑛−𝜏(𝑛))}−1). (4.24) This implies that 𝑦(𝑀_𝑖)=𝑦(𝑝_𝑖)+^𝑀𝑖−1_{𝑛=𝑝𝑖}(^𝑛_𝑗=0(1−𝛿(𝑗)))⁻¹𝑃(𝑛)𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}(𝑒^{−𝑞(𝑛)𝑧(𝑛−𝜏(𝑛))}−1). (4.25) It follows from (4.21) and (4.23)–(4.24) that for 𝑖 large enough 𝑧(𝑀_𝑖)=^𝑀𝑖−1_𝑗=0(1−𝛿(𝑗))×{𝑦(𝑝_𝑖)+^𝑀𝑖−1_{𝑛=𝑝𝑖}(^𝑛_𝑗=0(1−𝛿(𝑗)))⁻¹𝑃(𝑛)𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}(𝑒^{−𝑞(𝑛)𝑧(𝑛−𝜏(𝑛))}−1)}≤(𝑒^{−(𝑞∗+𝜖)(𝜆−𝜖)}−1)^𝑀𝑖−1_{𝑛=𝑝𝑖}𝑃(𝑛)^𝑀𝑖−1_𝑗=𝑛+1(1−𝛿(𝑗))𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}≤(𝑎+𝜖)(𝑒^{−(𝑞∗+𝜖)(𝜆−𝜖)}−1). (4.26) Using (4.19), then by 𝜖 being arbitrary, this implies that 𝜇≤𝑎(𝑒^{−𝑞∗𝜆}−1). (4.27) Now we will prove that 𝜆≥𝑎(𝑒^{−𝑞∗𝜇}−1). (4.28) In fact, by (4.22)–(4.24), for 𝑖 large enough we have 𝑧(𝑚_𝑖)=^𝑚𝑖−1_𝑗=0(1−𝛿(𝑗)){𝑦(𝑞_𝑖)+^𝑚𝑖−1_{𝑛=𝑞𝑖}(^𝑛_𝑗=0(1−𝛿(𝑗)))⁻¹𝑃(𝑛)𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}(𝑒^{−𝑞(𝑛)𝑧(𝑛−𝜏(𝑛))}−1)}≥(𝑒^{−(𝑞∗+𝜖)(𝜇+𝜖)}−1)^𝑚𝑖−1_{𝑛=𝑞𝑖}𝑃(𝑛)^𝑚𝑖−1_𝑗=𝑛(1−𝛿(𝑗))𝑒^{−𝑞(𝑛)𝑥(𝑛−𝜏(𝑛))}≥(𝑎+𝜖)(𝑒^{−(𝑞∗+𝜖)(𝜇+𝜖)}−1). (4.29) Note that 𝜖 is arbitrary. By (4.19) we can see that (4.28) holds. By Lemma 3.1 we see that 𝜆=𝜇=0. Thus, (4.13) also holds in this case. This completes the proof.

Proof of Theorem 2.7. We will prove our conclusion by applying Lemmas 3.1 and 3.2. For any (𝛿₁,𝑃₁,𝑞₁,𝜏₁)∈𝐻((𝛿,𝑃,𝑞,𝜏)), consider equation 𝑥(𝑛+1)−𝑥(𝑛)=−𝛿₁(𝑛)𝑥(𝑛)+𝑃₁(𝑛)𝑒^{−𝑞1(𝑛)𝑥(𝑛−𝜏1(𝑛))}. (4.30) It follows from Theorem 2.1 that for any 𝜑∈𝐶⁺, 𝑢≤liminf_𝑛→∞𝑥(𝑛,0,𝜑)≤limsup_𝑛→∞𝑥(𝑛,0,𝜑)≤𝑣, (4.31) where 𝑢, 𝑣 are the constants given in Theorem 2.1, and 𝑥(𝑛,0,𝜑) is the solution of (1.1) with 𝑥₀=𝜑. By the almost periodicity of (1.1) and Lemma 3.2, we can find a solution 𝑁(𝑛) of (4.30) defined on ℤ whose range is in [𝑢,𝑣]. Now we will prove that the solution which possesses the above properties is unique. Suppose that 𝑁(𝑛) is another solution of (4.30) defined on ℤ whose range is in [𝑢,𝑣]. Let 𝑧(𝑛)=𝑁(𝑛)−𝑁(𝑛). Then 𝑧(𝑛) satisfies 𝑧(𝑛+1)−𝑧(𝑛)=−𝛿₁(𝑛)𝑧(𝑛)+𝑃₁(𝑛)𝑒^{−𝑞1(𝑛)𝑁(𝑛−𝜏(𝑛))}(𝑒^{−𝑞1(𝑛)𝑧(𝑛−𝜏1(𝑛))}−1). (4.32) We claim that 𝑧(𝑛)≡0. Otherwise, we would have 𝑧(𝜁)≠0 (4.33) for some 𝜁∈ℤ. Consider the following two cases.
Case 1. There is a 𝑇<𝜁 such that 𝑧(𝑛)≥0 (or ≤0) for all 𝑛≤𝑇. Without loss of generality, we suppose that 𝑧(𝑛)≥0 for all 𝑛≤𝑇. Then 𝑧(𝑛+1)≤𝑧(𝑛) for 𝑛≤𝑇. Since 𝑧(𝑛) is bounded on ℤ, we see that lim_{𝑛→−∞}𝑧(𝑛) exists, denoted by 𝑙. We claim that 𝑙=0. Otherwise, there would exist 𝑁<𝑇 such that 𝑙2<𝑧(𝑛)<3𝑙2,𝑛<𝑁. (4.34) Note that 𝛿_1∗=𝛿_∗=inf_𝑛∈ℤ𝛿(𝑛). It follows from (4.32) that 𝑙𝑧(𝑛+1)−𝑧(𝑛)<−2𝛿_∗,𝑛<𝑁. (4.35) This implies that 𝑙𝑧(𝑛)<𝑧(𝑛−𝑘)−𝑘2𝛿_∗∀𝑘∈ℕ,𝑛<𝑁. (4.36) Hence lim_{𝑛→−∞}𝑧(𝑛)=+∞, which contradicts the fact that 𝑧(𝑛) is bounded on ℤ. Thus 𝑙=0. Therefore 𝑧(𝑛)=0 for all 𝑛≤𝑇. It follows from (4.32) that 𝑧(𝑛)=0 for all 𝑛∈ℤ, which contradicts (4.33).Case 2. There exist positive sequences {𝑝_𝑖} and {𝑞_𝑖} such that lim_𝑖→∞𝑝_𝑖=−∞, 𝑝_𝑖+1<𝑞_𝑖<𝑝_𝑖, (4.37)𝑧(𝑝_𝑖)<0 and 𝑧(𝑞_𝑖)>0 for 𝑖∈ℕ, 𝑧(𝑘)≥0 for 𝑝_𝑖+1<𝑘≤𝑞_𝑖 and 𝑧(𝑘)≤0 for 𝑞_𝑖<𝑘≤𝑝_𝑖 . Let 𝑑=sup_𝑛≤𝑝1𝑧(𝑛) and 𝑐=inf_𝑛≤𝑝1𝑧(𝑛). Then 𝑑>0 and 𝑐<0. For each 𝜖∈(0,min(−𝑐,𝑑)), there are 𝑛⁰,𝑛₀≤𝑝₁ such 𝑧(𝑛⁰)>𝑑−𝜖,𝑧(𝑛₀)<𝑐+𝜖. (4.38) Suppose 𝑛⁰∈[𝑝_𝑗,𝑞_𝑗−1] and 𝑛₀∈[𝑞_𝑖,𝑝_𝑖]. Let 𝐶=max{𝑧(𝑘)∣𝑝_𝑗≤𝑘≤𝑞_𝑗−1} and 𝑐=min{𝑧(𝑘)∣𝑞_𝑖≤𝑘≤𝑝_𝑖}. Set 𝑀=min{𝑘∣𝑝_𝑗≤𝑘≤𝑞_𝑗−1,𝑧(𝑘)=𝐶} and 𝑚=min{𝑘∣𝑞_𝑖≤𝑘≤𝑝_𝑖,𝑧(𝑘)=𝑐}. Similar to the argument in the proof of Theorem 2.4, we can obtain that 𝑀−1−𝑝_𝑗≤𝜏,𝑚−1−𝑞_𝑖≤𝜏. (4.39) Let 𝑎=limsup_𝑛→∞∑^𝑛+𝜏_𝑘=𝑛+1𝑃(𝑘)∏^𝑛+𝜏_𝑖=𝑘(1−𝛿(𝑖))𝑒^{−𝑞(𝑘)𝑢}. By the almost periodicity of 𝛿(𝑛), 𝑃(𝑛) and 𝑞(𝑛), we have for all 𝑛∈ℤ, 𝑞_∗≤𝑞₁(𝑛)≤𝑞^∗,_{𝑛+𝜏𝑘=𝑛}𝑃₁(𝑘)^𝑛+𝜏_𝑖=𝑘+1(1−𝛿₁(𝑖))𝑒^{−𝑞1(𝑘)𝑢}≤𝑎. (4.40) Let 𝑧(𝑛)=𝑦(𝑛)∏^𝑛−1_𝑖=0(1−𝛿₁(𝑖)). By (4.32) we obtain that 𝑦(𝑛+1)=𝑦(𝑛)+(^𝑛_𝑖=0(1−𝛿₁(𝑖)))⁻¹𝑃₁(𝑛)𝑒^{−𝑞1(𝑛)𝑁(𝑛−𝜏1(𝑛))}(𝑒^{−𝑞1(𝑛)𝑧(𝑛−𝜏1(𝑛))}−1). (4.41) This implies that 𝑦(𝑀)=𝑦(𝑝_𝑗)+^𝑀𝑖−1_{𝑛=𝑝𝑗}(^𝑛_𝑘=0(1−𝛿₁(𝑘)))⁻¹𝑃₁(𝑛)𝑒^{−𝑞1(𝑛)𝑁(𝑛−𝜏1(𝑛))}(𝑒^{−𝑞1(𝑛)𝑧(𝑛−𝜏1(𝑛))}−1). (4.42) Since 𝑐≤𝑧(𝑛)≤𝑑 for all 𝑛≤𝑝₁, it follows from (4.39)–(4.40) that 𝑑−𝜖<𝑧(𝑛⁰=)<𝑧(𝑀)_{𝑀−1𝑘=0}(1−𝛿₁(𝑘))×{𝑦(𝑝_𝑗)+^𝑀−1_{𝑛=𝑝𝑗}(^𝑛_𝑘=0(1−𝛿₁(𝑘)))⁻¹𝑃₁(𝑛)𝑒^{−𝑞1(𝑛)𝑁(𝑛−𝜏1(𝑛))}(𝑒^{−𝑞1(𝑛)𝑧(𝑛−𝜏1(𝑛))}−1)}≤(𝑒^{−𝑞∗𝑐}−1)^𝑀𝑖−1_{𝑛=𝑝𝑗}𝑃₁(𝑛)^𝑀−1_𝑘=𝑛+1(1−𝛿₁(𝑘))𝑒^{−𝑞1(𝑛)𝑢}≤𝑎(𝑒^{−𝑞∗𝑐}−1). (4.43) Thus, we have 𝑑−𝜖≤𝑎(𝑒^{−𝑞∗𝑐}−1). (4.44) By the fact that 𝜖 is arbitrary, we obtain that 𝑑≤𝑎(𝑒^{−𝑞∗𝑐}−1). (4.45) Similarly, we can prove that 𝑐≥𝑎(𝑒^{−𝑞∗𝑑}−1). (4.46) Equations (4.45) and (4.46) produce that 𝑐=𝑑=0. Therefore 𝑧(𝑛)=0 for all 𝑛≤𝑝₁. It follows from (4.32) that 𝑧(𝑛)=0 for all 𝑛∈ℤ, which also contradicts (4.33). By Lemma 3.3 we see that (1.1) has a unique almost periodic solution 𝑝(𝑛) whose range is in [𝑢,𝑣]. By Corollary 2.5 we can see that 𝑝(𝑛) is global attractivity. This also implies that the almost periodic solution of (1.1) is unique. This completes the proof.

Acknowledgments

The authors is greatly indebted to an anonymous referee for his/her valuable suggestions, which helped them to improve the exposition of the results. The research was partly supported by NSFC (10671021).