Global Exponential Stability of Positive Pseudo-Almost-Periodic Solutions for a Model of Hematopoiesis

Abstract

This paper presents a new generalized model of hematopoiesis with multiple time-varying delays. The main purpose of this paper is to study the existence and the global exponential stability of the positive pseudo almost periodic solutions, which are more general and complicated than periodic and almost periodic solutions. Under suitable assumptions, and by using fixed point theorem, sufficient conditions are given to ensure that all solutions of this model converge exponentially to the positive pseudo almost periodic solution for the considered model. These results improve and extend some known relevant results.

1. Introduction

As we all know, many phenomena in nature have oscillatory character and their mathematical models have led to the introduction of certain classes of functions to describe them. For example, the pseudo almost periodic functions are the natural generalization of the concept of almost periodicity. These are functions on the real numbers set that can be represented uniquely in the form 𝑓=ℎ+𝜑, where ℎ (the principal term) is an almost periodic function and 𝜑 (the ergodic perturbation) a continuous function whose mean vanishes at infinity. Note that there exists abundant literature on the topic (see, e.g., [1–6]). In a classic study of population dynamics, the following delay differential equation model 𝑥^′(𝑡)=−𝑎(𝑡)𝑥(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)1+𝑥^𝑛𝑡−𝜏_𝑖(𝑡), (1) where 𝑛 is a positive constant and 𝑎,𝑏_𝑖,𝜏_𝑖∶ℝ⟶(0,+∞)arecontinuousfunctionsfor𝑖=1,2,…,𝑚, (2) has been used by [7, 8] to describe the dynamics of hematopoiesis (blood cell production). As we known, (1) belongs to a class of biological systems and it (or its analogue equation) has attracted more attention to the problem of almost periodic solutions because of its extensively realistic significance. For example, some criteria ensuring the existence and stability of positive almost periodic solutions were established in [9–12] and the references cited therein. However, it is very difficult to study the global stability of positive pseudo almost periodic solution for (1). So far, no attention has been paid to the conditions for the global exponential stability on positive pseudo almost periodic solution of model (1) in terms of its coefficients. On the other hand, since the exponential convergent rate can be unveiled, the global exponential stability plays a key role in characterizing the behavior of dynamical system (see [13–15]). Thus, it is worthwhile to continue to investigate the existence and global exponential stability of positive pseudo almost periodic solutions of (1).

Motivated by the above discussions, in this paper, we consider the existence, uniqueness, and global exponential stability of positive pseudo almost periodic solutions of (1). Here in this paper, a new approach will be developed to obtain a delay-independent condition for the global exponential stability of the positive pseudo almost periodic solutions of (1), and the exponential convergent rate can be unveiled.

Throughout this paper, for 𝑖=1,2,…,𝑚, it will be assumed that 𝑎,𝑏_𝑖,𝜏_𝑖∶ℝ→(0,+∞) are continuous functions, and 𝑎⁻=inf_𝑡∈ℝ𝑎(𝑡)>0,𝑎⁺=sup_𝑡∈ℝ𝑎(𝑡),𝑏⁻_𝑖=inf_𝑡∈ℝ𝑏_𝑖(𝑡)>0,𝑏⁺_𝑖=sup_𝑡∈ℝ𝑏_𝑖(𝑡),𝑟=max_{1≤𝑖≤𝑚}sup_𝑡∈ℝ𝜏_𝑖(𝑡)>0. (3) Let ℝ₊ denote nonnegative real number space, let 𝐶=𝐶([−𝑟,0],ℝ) be the continuous functions space equipped with the usual supremum norm ‖⋅‖, and let 𝐶₊=𝐶([−𝑟,0],ℝ₊). If 𝑥(𝑡) is defined on [−𝑟+𝑡₀,𝜎) with 𝑡₀,𝜎∈ℝ, then we define 𝑥_𝑡∈𝐶 where 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) for all 𝜃∈[−𝑟,0].

Due to the biological interpretation of model (1), only positive solutions are meaningful and therefore admissible. Thus we just consider admissible initial conditions. 𝑥_𝑡0=𝜑,𝜑∈𝐶₊,𝜑(0)>0. (4) We write 𝑥_𝑡(𝑡₀,𝜑)(𝑥(𝑡;𝑡₀,𝜑)) for an admissible solution of the admissible initial value problem (1) and (4). Also, let [𝑡₀,𝜂(𝜑)) be the maximal right interval of existence of 𝑥_𝑡(𝑡₀,𝜑).

2. Preliminary Results

In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.

In this paper, BC(ℝ,ℝ) denotes the set of bounded continued functions from ℝ to ℝ. Note that (BC(ℝ,ℝ),‖⋅‖_∞) is a Banach space where ‖⋅‖_∞ denotes the sup norm ‖𝑓‖_∞∶=sup_𝑡∈ℝ|𝑓(𝑡)|.

Definition 1 (see [16, 17]). Let 𝑢(𝑡)∈BC(ℝ,ℝ). 𝑢(𝑡) is said to be almost periodic on ℝ if, for any 𝜀>0, the set 𝑇(𝑢,𝜀)={𝛿∶|𝑢(𝑡+𝛿)−𝑢(𝑡)|<𝜀forall𝑡∈ℝ} is relatively dense; that is, for any 𝜀>0, it is possible to find a real number 𝑙=𝑙(𝜀)>0, and for any interval with length 𝑙(𝜀), there exists a number 𝛿=𝛿(𝜀) in this interval such that |𝑢(𝑡+𝛿)−𝑢(𝑡)|<𝜀, for all 𝑡∈ℝ.

We denote by AP(ℝ,ℝ) the set of the almost periodic functions from ℝ to ℝ. Besides, the concept of pseudo almost periodicity (pap) was introduced by Zhang in the early nineties. It is a natural generalization of the classical almost periodicity. Precisely, define the class of functions PAP₀(ℝ,ℝ) as follows: 𝑓∈BC(ℝ,ℝ)∣lim_𝑇→+∞12𝑇^𝑇_−𝑇𝑓(𝑡)𝑑𝑡=0. (5) A function 𝑓∈BC(ℝ,ℝ) is called pseudo almost periodic if it can be expressed as 𝑓=ℎ+𝜑, (6) where ℎ∈AP(ℝ,ℝ) and 𝜑∈PAP₀(ℝ,ℝ). The collection of such functions will be denoted by PAP(ℝ,ℝ). The functions ℎ and 𝜑 in the above definition are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function 𝑓. The decomposition given in definition above is unique. Observe that (PAP(ℝ,ℝ),‖⋅‖_∞) is a Banach space and AP(ℝ,ℝ) is a proper subspace of PAP(ℝ,ℝ) since the function 𝜙(𝑡)=cos𝜋𝑡+cos𝑡+1/(1+𝑡²) is pseudo almost periodic function but not almost periodic. It should be mentioned that pseudo almost periodic functions possess many interesting properties; we will need only a few of them and for the proofs we shall refer to [16].

Lemma 2. Let 𝑥₁(⋅),𝜎(⋅)∈𝐴𝑃(ℝ,ℝ), 𝜎^′(⋅)∈𝐵𝐶(ℝ,ℝ) and 𝑥₂(⋅)∈𝑃𝐴𝑃₀(ℝ,ℝ). Then(1)𝑥₁(𝑡−𝜎(𝑡))∈𝐴𝑃(ℝ,ℝ); (2)𝑥₂(𝑡−𝜎(𝑡))∈𝑃𝐴𝑃₀(ℝ,ℝ), if (1−𝜎^′(𝑡))⁻=inf_𝑡∈ℝ(1−𝜎^′(𝑡))>0.

Proof. (1) For any 𝜀>0, from the uniform continuity of 𝑥₁(⋅), we can choose a constant (𝜀)<𝜀0<𝛿=𝛿2 (7) such that 𝑥₁𝑡^′−𝑥₁𝑡^′′<𝜀2,∀𝑡^′,𝑡^′′∈ℝ,𝑡^′−𝑡^′′<𝛿. (8) From the theory of almost periodic functions in [16, 17], it follows that for 𝛿>0, it is possible to find a real number 𝑙=𝑙(𝛿)=𝑙(𝛿(𝜀))>0, and for any interval with length 𝑙, there exists a number 𝜏=𝜏(𝜀) in this interval such that |𝜎(𝑡+𝜏)−𝜎(𝑡)|<𝛿,𝑥₁(𝑡+𝜏)−𝑥₁(𝑡𝜀)<𝛿<2,∀𝑡∈ℝ. (9) Combing (8) and (9), we obtain 𝑥₁(𝑡+𝜏−𝜎(𝑡+𝜏))−𝑥₁((𝑡≤𝑥𝑡−𝜎))₁(𝑡+𝜏−𝜎(𝑡+𝜏))−𝑥₁((𝑡+𝑥𝑡+𝜏−𝜎))₁(𝑡+𝜏−𝜎(𝑡))−𝑥₁((𝑡<𝜀𝑡−𝜎))2+𝜀2=𝜀,∀𝑡∈ℝ. (10) which yields 𝑥₁(𝑡−𝜎(𝑡))∈AP(ℝ,ℝ).
(2) Set 𝑠=𝑡−𝜎(𝑡); we get 10≤2𝑇^𝑇_−𝑇𝑥₂((𝑡=1𝑡−𝜎))𝑑𝑡2𝑇₋₍₍^{𝑇−𝜎(𝑇)}_{𝑇−𝜎−𝑇))}𝑥₂(𝑠1)1−𝜎^′(𝑡)≤1𝑑𝑠1−𝜎^′(𝑡)⁻𝑇+𝜎⁺𝑇12(𝑇+𝜎⁺)^𝑇+𝜎+_{−(𝑇+𝜎}⁺₎𝑥₂(𝑠)𝑑𝑠,where𝜎⁺=sup_𝑡∈ℝ𝜎(𝑡), (11) which implies that 𝑥₂(𝑡−𝜎(𝑡))∈PAP₀(ℝ,ℝ).

Remark 3. Set 𝑥(⋅)=𝑥₁(⋅)+𝑥₂(⋅) with 𝑥₁(⋅)∈AP(ℝ,ℝ) and 𝑥₂(⋅)∈PAP₀(ℝ,ℝ). It follows from Lemma 2 that 𝑥(𝑡−𝜎(𝑡))∈PAP(ℝ,ℝ),if1−𝜎^′(𝑡)⁻𝜎(⋅)()>0,∈APℝ,ℝ,𝜎^′(⋅)∈BC(ℝ,ℝ). (12)

Definition 4 (see [16, 17]). Let 𝑥∈𝑅^𝑛 and let 𝑄(𝑡) be an 𝑛×𝑛 continuous matrix defined on 𝑅. The linear system 𝑥^′(𝑡)=𝑄(𝑡)𝑥(𝑡) (13) is said to admit an exponential dichotomy on 𝑅 if there exist positive constants 𝑘, and 𝛼, projection 𝑃, and the fundamental solution matrix 𝑋(𝑡) of (13) satisfying 𝑋(𝑡)𝑃𝑋⁻¹(𝑠)≤𝑘𝑒^{−𝛼(𝑡−𝑠)}𝑋(𝑡)𝑋for𝑡≥𝑠,)(𝐼−𝑃⁻¹(𝑠)≤𝑘𝑒^{−𝛼(𝑠−𝑡)}for𝑡≤𝑠. (14)

Lemma 5 (see [6, 16]). Assume that 𝑄(𝑡) is an almost periodic matrix function and 𝑔(𝑡)∈𝑃𝐴𝑃(𝑅^𝑛). If the linear system (13) admits an exponential dichotomy, then pseudo almost periodic system 𝑥^′(𝑡)=𝑄(𝑡)𝑥+𝑔(𝑡) (15) has a unique pseudo almost periodic solution 𝑥(𝑡), and 𝑥(𝑡)=^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝑠)𝑔(𝑠)−𝑑𝑠_𝑡+∞𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝑠)𝑔(𝑠)𝑑𝑠. (16)

Lemma 6 (see [16, 17]). Let 𝑐_𝑖(𝑡) be an almost periodic function on 𝑅 and 𝑀𝑐_𝑖=lim_𝑇→+∞1𝑇_𝑡^𝑡+𝑇𝑐_𝑖(𝑠)𝑑𝑠>0,𝑖=1,2,…,𝑛. (17) Then the linear system 𝑥^′(𝑡)=diag−𝑐₁(𝑡),−𝑐₂(𝑡),…,−𝑐_𝑛(𝑡)𝑥(𝑡) (18) admits an exponential dichotomy on 𝑅.

Lemma 7 (see [11, Lemma 2.3]). Every solution 𝑥(𝑡;𝑡₀,𝜑) of (1) and (4) is positive and bounded on [𝑡₀,𝜂(𝜑)), and 𝜂(𝜑)=+∞.

Lemma 8. Suppose that there exist two positive constants 𝜅 and 𝑀 such that 𝑀>𝜅,sup_𝑡∈𝑅−𝑎(𝑡)𝑀+^𝑚_𝑖=1𝑏_𝑖(𝑡)<0,inf_𝑡∈𝑅−𝑎(𝑡)𝜅+^𝑚_𝑖=1𝑏_𝑖(𝑡)1+𝑀^𝑛>0. (19) Then, there exists 𝑡_𝜑>𝑡₀ such that 𝜅<𝑥𝑡;𝑡₀,𝜑<𝑀,∀𝑡≥𝑡_𝜑. (20)

Proof. This Lemma can be proven in a similar way to that in Lemma 2.2 of [12]. But for convenience of reading, we give the proof as follows. Let 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝜑). We first claim that there exists 𝑡^#∈[𝑡₀,+∞) such that 𝑥𝑡^#<𝑀. (21) Otherwise, 𝑥(𝑡)≥𝑀,∀𝑡∈𝑡₀,+∞, (22) Which, together with (19), implies that 𝑥^′(𝑡)=−𝑎(𝑡)𝑥(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)1+𝑥^𝑛𝑡−𝜏_𝑖(𝑡)≤−𝑎(𝑡)𝑀+^𝑚_𝑖=1𝑏_𝑖(𝑡)1+𝑀^𝑛≤−𝑎(𝑡)𝑀+^𝑚_𝑖=1𝑏_𝑖(𝑡)≤sup_𝑡∈𝑅−𝑎(𝑡)𝑀+^𝑚_𝑖=1𝑏_𝑖(𝑡)<0,∀𝑡≥𝑡₀+𝑟. (23) This yields that 𝑥(𝑡)=𝑥𝑡₀+𝑟+^𝑡_𝑡0+𝑟𝑥^′(𝑠)𝑡𝑑𝑠≤𝑥₀+𝑟+sup_𝑡∈𝑅−𝑎(𝑡)𝑀+^𝑚_𝑖=1𝑏_𝑖(𝑡×𝑡)𝑡−₀+𝑟,∀𝑡≥𝑡₀+𝑟. (24) Thus lim_𝑡→+∞𝑥(𝑡)=−∞, (25) which contradicts the fact that 𝑥(𝑡) is positive and bounded on [𝑡₀,+∞). Hence, (21) holds. In the sequel, we prove that 𝑥(𝑡)<𝑀,∀𝑡∈𝑡^#,+∞. (26) Suppose, for the sake of contradiction, there exists 𝑡̃∈(𝑡^#,+∞) such that 𝑥𝑡̃=𝑀,𝑥(𝑡)<𝑀,∀𝑡∈𝑡^#,𝑡̃. (27) Calculating the derivative of 𝑥(𝑡), together with (19), (1), and (27), implies that 0≤𝑥^′𝑡̃=−𝑎𝑡̃𝑥𝑡̃+^𝑚_𝑖=1𝑏_𝑖𝑡̃1+𝑥^𝑛𝑡̃−𝜏_𝑖𝑡̃≤−𝑎𝑡̃𝑀+^𝑚_𝑖=1𝑏_𝑖𝑡̃<0, (28) which is a contradiction and implies that (26) holds.
We finally show that 𝑙=liminf_𝑡→∞𝑥(𝑡)>𝜅. By way of contradiction, we assume that 0≤𝑙≤𝜅. By the fluctuation lemma [18, Lemma A.1.], there exists a sequence {𝑡_𝑘}_𝑘≥1 such that 𝑡_𝑘⟶+∞,𝑥𝑡_𝑘⟶liminf_𝑡→+∞𝑥(𝑡),𝑥^′𝑡_𝑘⟶0as𝑘⟶+∞. (29) Since {𝑥_𝑡𝑘} is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that 𝑥_𝑡𝑘⟶𝜑^∗(𝑘⟶+∞)forsome𝜑^∗∈𝐶₊. (30) Moreover, 𝜑^∗(0)=𝑙≤𝜑^∗(𝜃)≤𝑀for𝜃∈[−𝑟,0). (31) Without loss of generality, we assume that all 𝑎(𝑡_𝑘), 𝑏_𝑖(𝑡_𝑘), and 𝜏_𝑖(𝑡_𝑘) are convergent to 𝑎^∗, 𝑏^∗_𝑖, and 𝜏^∗_𝑖, respectively. This can be achieved because of almost periodicity. It follows from 𝑥^′𝑡_𝑘=−𝑎𝑡_𝑘𝑥𝑡_𝑘+^𝑚_𝑖=1𝑏_𝑖𝑡_𝑘1+𝑥^𝑛𝑡_𝑘−𝜏_𝑖𝑡_𝑘 (32) that (taking limits) 0=−𝑎^∗𝑙+^𝑚_𝑖=1𝑏^∗_𝑖1+𝜑^∗−𝜏^∗_𝑖^𝑛≥−𝑎^∗𝑙+^𝑚_𝑖=1𝑏^∗_𝑖1+𝑀^𝑛≥−𝑎^∗𝜅+^𝑚_𝑖=1𝑏^∗_𝑖1+𝑀^𝑛≥inf_𝑡∈𝑅−𝑎(𝑡)𝜅+^𝑚_𝑖=1𝑏_𝑖(𝑡)1+𝑀^𝑛>0, (33) is a contradiction. This proves that 𝑙>𝜅. Hence, from (26), we can choose 𝑡_𝜑>𝑡₀ such that 𝜅<𝑥𝑡;𝑡₀,𝜑<𝑀,∀𝑡≥𝑡_𝜑. (34) This ends the proof of Lemma 8.

3. Main Results

Theorem 9. Suppose that 𝑎,𝜏_𝑖∈𝐴𝑃(ℝ,ℝ),𝜏^′_𝑖(⋅)∈𝐵𝐶(ℝ,ℝ),𝑏_𝑖∈𝑃𝐴𝑃(ℝ,ℝ),inf_𝑡∈ℝ1−𝜏^′_𝑖(𝑡)>0,𝑖=1,2,…,𝑚, (35) and there exist two positive constants 𝜅 and 𝑀 satisfying (19) and sup_𝑡∈𝑅−𝑎(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑛4𝜅<0. (36) Then, there exists a unique positive pseudo almost periodic solution of (1) in the region 𝐵^∗={𝜑∣𝜑∈𝑃𝐴𝑃(ℝ,ℝ),𝜅≤𝜑(𝑡)≤𝑀,𝑓𝑜𝑟𝑎𝑙𝑙𝑡∈ℝ}.

Proof. Consider Υ∶[0;1]→ℝ defined by Υ(𝑢)=sup_𝑡∈𝑅−𝑎(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑛𝑒4𝜅^𝑢,𝑢∈[0,1]. (37) Then, we have Υ(0)=sup_𝑡∈𝑅−𝑎(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑛4𝜅<0, (38) which implies that there exists a constant 𝜍∈(0,1] such that Υ(𝜍)=sup_𝑡∈𝑅−𝑎(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑛𝑒4𝜅^𝜍<0. (39)
For any 𝜙∈PAP(ℝ,ℝ), from (35), Remark 3, and the composition theorem of pseudo almost periodic functions [16], we have ^𝑚_𝑖=1𝑏_𝑖(𝑡)1+𝜙^𝑛𝑡−𝜏_𝑖(𝑡)∈PAP(ℝ,ℝ). (40) We next consider an auxiliary equation: 𝑥^′(𝑡)=−𝑎(𝑡)𝑥(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)1+𝜙^𝑛𝑡−𝜏_𝑖(𝑡). (41) Notice that 𝑀[𝑎]>0; it follows from Lemma 6 that the linear equation 𝑥^′(𝑡)=−𝑎(𝑡)𝑥(𝑡) (42) admits an exponential dichotomy on ℝ. Thus, by Lemma 5, we obtain that the system (41) has exactly one pseudo almost periodic solution: 𝑥^𝜙(𝑡)=^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎(𝑢)𝑑𝑢𝑚}_𝑖=1𝑏_𝑖(𝑠)1+𝜙^𝑛𝑡−𝜏_𝑖(𝑠)𝑑𝑠. (43) Define a mapping 𝑇∶PAP(ℝ,ℝ)→PAP(ℝ,ℝ) by setting 𝑇𝜙(𝑡)=𝑥^𝜙(𝑡),∀𝜙∈PAP(ℝ,ℝ). (44)
Since 𝐵^∗={𝜑∣𝜑∈PAP(ℝ,ℝ),𝜅≤𝜑(𝑡)≤𝑀,forall𝑡∈ℝ}, it is easy to see that 𝐵^∗ is a closed subset of PAP(𝑅,𝑅). For any 𝜙∈𝐵^∗, from (19), we have 𝑥^𝜙(𝑡)≤^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎(𝑢)𝑑𝑢𝑚}_𝑖=1𝑏_𝑖(𝑠≤)𝑑𝑠_𝑡−∞𝑒^{−∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎(𝑠)𝑥𝑀𝑑𝑠=𝑀,∀𝑡∈𝑅,_𝜙(𝑡)≥^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎(𝑢)𝑑𝑢𝑚}_𝑖=1𝑏_𝑖(𝑠)1+𝑀^𝑛≥𝑑𝑠_𝑡−∞𝑒^{−∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎(𝑠)𝜅𝑑𝑠=𝜅,∀𝑡∈𝑅. (45)(46) This implies that the mapping 𝑇 is a self-mapping from 𝐵^∗ to 𝐵^∗. Now, we prove that the mapping 𝑇 is a contraction mapping on 𝐵^∗. In fact, for 𝜑,𝜓∈𝐵^∗, we get 𝑇𝜑−𝑇𝜓_∞=sup_𝑡∈𝑅𝑇𝜑(𝑡)−𝑇𝜓(𝑡)=sup_𝑡∈𝑅^𝑡_−∞𝑒^{−∫𝑡𝑠𝑚𝑎(𝑢)𝑑𝑢}_𝑖=1𝑏_𝑖(𝑠1)1+𝜑^𝑛𝑡−𝜏_𝑖(𝑠)−11+𝜓^𝑛𝑠−𝜏_𝑖(𝑠)𝑑𝑠. (47) In view of (39), (45), (46), and (47), from the inequality 11+𝑥^𝑛−11+𝑦^𝑛=−𝑛𝜃^𝑛−11+𝜃^𝑛2≤𝑥−𝑦𝑛𝜃^𝑛−12√𝜃^𝑛2≤𝑛𝑥−𝑦,4𝜅𝑥−𝑦 (48) where 𝑥,𝑦∈[𝜅,𝑀] and 𝜃 lies between 𝑥 and 𝑦, we have 𝑇𝜑−𝑇𝜓_∞≤sup_𝑡∈𝑅^𝑡_−∞𝑒^{−∫𝑡𝑠𝑚𝑎(𝑢)𝑑𝑢}_𝑖=1𝑏_𝑖(𝑠)𝑛×𝜑4𝜅𝑠−𝜏_𝑖(𝑠)−𝜓𝑠−𝜏_𝑖(𝑠)𝑑𝑠≤sup_𝑡∈𝑅^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎(𝑠)𝑒^−𝜍×𝜑𝑠−𝜏_𝑖(𝑠)−𝜓𝑠−𝜏_𝑖(𝑠)𝑑𝑠≤𝑒^−𝜍𝜑−𝜓_∞. (49) Noting that 𝑒^−𝜍<1, it is clear that the mapping 𝑇 is a contraction on 𝐵^∗. Using Theorem 0.3.1 of [19], we obtain that the mapping 𝑇 possesses a unique fixed point 𝜑^∗∈𝐵^∗, 𝑇𝜑^∗=𝜑^∗. By (41), 𝜑^∗ satisfies (1). So 𝜑^∗ is a positive pseudo almost periodic solution of (1) in 𝐵^∗. The proof of Theorem 9 is now complete.

Theorem 10. Under the assumptions of Theorem 9, (1) has at least one positive pseudo almost periodic solution 𝑥^∗(𝑡). Moreover, 𝑥^∗(𝑡) is globally exponentially stable; that is, there exist constants 𝐾_𝜑,𝑥∗, 𝑡_𝜑,𝑥∗, and 𝜆>0 such that 𝑥𝑡;𝑡₀,𝜑−𝑥^∗(𝑡)<𝐾_𝜑,𝑥∗𝑒^−𝜆𝑡,∀𝑡>𝑡_𝜑,𝑥∗. (50)

Proof. By Theorem 9, (1) has a positive pseudo almost periodic solution; say 𝑥^∗(𝑡). It suffices to show that 𝑥^∗(𝑡) is globally exponentially stable. Define a continuous function Γ(𝑢) by setting Γ(𝑢)=sup_𝑡∈𝑅−[𝑎(𝑡)−𝑢]+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑛𝑒4𝜅^𝑟𝑢,𝑢∈[0,1]. (51) Then, we have Γ(0)=sup_𝑡∈𝑅−𝑎(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑛4𝜅<0, (52) which implies that there exist two constants 𝜂>0 and 𝜆∈(0,1] such that Γ(𝜆)=sup_𝑡∈𝑅−[𝑎(𝑡)−𝜆]+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑛𝑒4𝜅^𝜆𝑟<−𝜂<0. (53)
Let 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝜑) and 𝑦(𝑡)=𝑥(𝑡)−𝑥^∗(𝑡), where 𝑡∈[𝑡₀−𝑟,+∞). Then 𝑦^′(𝑡)=−𝑎(𝑡)𝑦(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡1)1+𝑥^𝑛𝑡−𝜏_𝑖(𝑡)−11+𝑥^∗𝑛𝑡−𝜏_𝑖(𝑡). (54)
It follows from Lemma 8 that there exists 𝑡_𝜑,𝑥∗>𝑡₀ such that 𝜅≤𝑥(𝑡),𝑥^∗(𝑡)≤𝑀,∀𝑡∈𝑡_𝜑,𝑥∗−𝑟,+∞. (55)
We consider the Lyapunov functional 𝑉(𝑡)=𝑦(𝑡)𝑒^𝜆𝑡. (56) Calculating the upper left derivative of 𝑉(𝑡) along the solution 𝑦(𝑡) of (54), we have 𝐷⁻(𝑉(𝑡(𝑡𝑦(𝑡𝑒))≤−𝑎))^𝜆𝑡+^𝑚_𝑖=1𝑏_𝑖(𝑡)×11+𝑥^𝑛𝑡−𝜏_𝑖(𝑡)−11+𝑥^∗𝑛𝑡−𝜏_𝑖(𝑡)𝑒^𝜆𝑡+𝜆𝑦(𝑡)𝑒^𝜆𝑡=−(𝑎(𝑡)−𝜆)𝑦(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)×11+𝑥^𝑛𝑡−𝜏_𝑖(𝑡)−11+𝑥^∗𝑛𝑡−𝜏_𝑖(𝑡)𝑒^𝜆𝑡,∀𝑡>𝑡_𝜑,𝑥∗. (57)
We claim that 𝑉(𝑡)=𝑦(𝑡)𝑒^𝜆𝑡<𝑒^{𝜆𝑡∗𝜑,𝑥}max_{𝑡𝑡∈0−𝑟,𝑡∗𝜑,𝑥}𝑥(𝑡)−𝑥^∗(𝑡)+1∶=𝐾_𝜑,𝑥∗,∀𝑡>𝑡_𝜑,𝑥∗. (58) Contrarily, there must exists 𝑡_∗>𝑡_𝜑,𝑥∗ such that 𝑉𝑡_∗=𝐾_𝜑,𝑥∗,𝑉(𝑡)<𝐾_𝜑,𝑥∗,∀𝑡∈𝑡₀−𝑟,𝑡_∗. (59) Together with (48), (57), and (59), we obtain 0≤𝐷⁻𝑉𝑡_∗≤−𝑎𝑡_∗−𝜆𝑦𝑡_∗+^𝑚_𝑖=1𝑏_𝑖𝑡_∗×11+𝑥^𝑛𝑡_∗−𝜏_𝑖𝑡_∗−11+𝑥^∗𝑛𝑡_∗−𝜏_𝑖𝑡_∗𝑒^𝜆𝑡∗≤−𝑎𝑡_∗−𝜆𝑦𝑡_∗𝑒^𝜆𝑡∗+^𝑚_𝑖=1𝑏_𝑖𝑡_∗×𝑛𝑒4𝜅^{𝜆𝜏𝑖(𝑡∗)}𝑒^{𝜆(𝑡∗−𝜏𝑖(𝑡∗))}𝑦𝑡_∗−𝜏_𝑖𝑡_∗≤−𝑎𝑡_∗−𝜆+^𝑚_𝑖=1𝑏_𝑖𝑡_∗𝑛𝑒4𝜅^𝜆𝑟𝐾_𝜑,𝑥∗. (60) Thus, 0≤−𝑎𝑡_∗−𝜆+^𝑚_𝑖=1𝑏_𝑖𝑡_∗𝑛𝑒4𝜅^𝜆𝑟, (61) which contradicts (53). Hence, (58) holds. It follows that 𝑦(𝑡)<𝐾_𝜑,𝑥∗𝑒^−𝜆𝑡∀𝑡>𝑡_𝜑,𝑥∗. (62) This completes the proof of Theorem 10.

4. An Example

In this section, we present an example to check the validity of the results we obtained in the previous sections.

Example 1. Consider the following model of hematopoiesis with multiple time-varying delays:𝑥^′(𝑡)(𝑡)+1=−1.3𝑥212+2cos√2𝑡+111001+𝑡²×11+𝑥𝑡−2𝑒^{(1/10)cos𝑡}+12×12+2||+1sin𝑡11001+𝑡²11+𝑥𝑡−2𝑒^{(1/10)sin𝑡}. (63) Obviously 𝑎⁺=𝑎⁻=1.3,𝑏⁻₁=𝑏⁻₂=1,𝑏⁺₁=𝑏⁺₂=1.26,𝑛=1,𝑟=2𝑒^1/10. (64) Let 𝜅=0.5 and 𝑀=2. Then −𝑎⁻𝑀+𝑏⁺₁+𝑏⁺₂=−0.08<0,−𝑎⁺𝑏𝜅+⁻¹+𝑏⁻₂=11+𝑀60>0,−𝑎⁻+𝑏⁺₁+𝑏⁺₂𝑛14𝜅=−1.3+2.52×2=−0.04<0, (65) which imply that (63) satisfies the assumptions of Theorem 10. Therefore, (63) has a unique positive pseudo almost periodic solution 𝑥^∗(𝑡), which is globally exponentially stable with the exponential convergent rate 𝜆≈0.01. The numerical simulation in Figure 1 strongly supports the conclusion.

Figure 1 

Numerical solution 𝑥(𝑡) of (63) for initial value 𝜑(𝑡)≡0.2,1,1.9.

Remark 11. We remark that the results in [9–12] give no opinions about global exponential convergence for the positive pseudo almost periodic solution. Thus, the results in [9–12] and the references therein cannot be applied to prove the global exponential stability of positive pseudo almost periodic solution for (63). This implies that the results of this paper are new and they complement previously known results.

Conflict of Interests

The author declares no conflict of interests. She also declares that she has no financial or personal relationships with other people or organizations that can inappropriately influence her work. There are no professional or other personal interests of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this paper.

Acknowledgments

The author would like to express the sincere appreciation to the editor and reviewer for their helpful comments in improving the presentation and quality of the paper. In particular, the author expresses the sincere gratitude to Professor Zhibin Chen and Professor Bingwen Liu for the helpful discussion when this revision work was being carried out. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grant nos. Y6110436 and LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant no. Z201122436).