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

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.


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  = ℎ + , where ℎ (the principal term) is an almost periodic function and  (the ergodic perturbation) a continuous function whose mean vanishes at infinity.Note that there exists abundant literature on the topic (see, e.g., [1][2][3][4][5][6]).In a classic study of population dynamics, the following delay differential equation model where  is a positive constant and ,   ,   : R → (0, +∞) are continuous functions for  = 1, 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][10][11][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][14][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.

Preliminary Results
In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.
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 PAP 0 (R, R) as follows: A function  ∈ BC(R, R) is called pseudo almost periodic if it can be expressed as where ℎ ∈ AP(R, R) and  ∈ PAP 0 (R, R).The collection of such functions will be denoted by PAP(R, R).The functions ℎ and  in the above definition are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function .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 () = cos  + cos  + 1/(1 +  2 ) 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].
Proof.(1) For any  > 0, from the uniform continuity of  1 (⋅), we can choose a constant such that From the theory of almost periodic functions in [16,17], it follows that for  > 0, it is possible to find a real number  = () = (()) > 0, and for any interval with length , there exists a number  = () in this interval such that Combing ( 8) and ( 9), we obtain Abstract and Applied Analysis 3 (2) Set  =  − (); we get where which implies that  2 ( − ()) ∈ PAP 0 (R, R).
Definition 4 (see [16,17]).Let  ∈   and let () be an  ×  continuous matrix defined on .The linear system is said to admit an exponential dichotomy on  if there exist positive constants , and , projection , and the fundamental solution matrix () of ( 13 Lemma 5 (see [6,16]).Assume that () is an almost periodic matrix function and () ∈ (  ).If the linear system (13) admits an exponential dichotomy, then pseudo almost periodic system has a unique pseudo almost periodic solution (), and Lemma 6 (see [16,17]).Let   () be an almost periodic function on  and Then the linear system admits an exponential dichotomy on .
Lemma 8. Suppose that there exist two positive constants  and  such that Then, there exists   >  0 such that  <  (;  0 , ) < , ∀ ≥   . (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 () = (;  0 , ).We first claim that there exists Otherwise, Which, together with (19), implies that This yields that which contradicts the fact that () is positive and bounded on [ 0 , +∞).Hence, (21) holds.In the sequel, we prove that Suppose, for the sake of contradiction, there exists t ∈ ( # , +∞) such that Calculating the derivative of (), together with ( 19), (1), and (27), implies that which is a contradiction and implies that (26) holds.

Main Results
Theorem 9. Suppose that and there exist two positive constants  and  satisfying (19) and Then, there exists a unique positive pseudo almost periodic solution of (1) in the region Proof.Consider Υ : [0; 1] → R defined by Then, we have which implies that there exists a constant  ∈ (0, 1] such that For any  ∈ PAP(R, R), from (35), Remark 3, and the composition theorem of pseudo almost periodic functions [16], we have We next consider an auxiliary equation: Notice that [] > 0; it follows from Lemma 6 that the linear equation admits an exponential dichotomy on R. Thus, by Lemma 5, we obtain that the system (41) has exactly one pseudo almost periodic solution: Define a mapping  : PAP(R, R) → PAP(R, R) by setting Since  * = { |  ∈ PAP(R, R),  ≤ () ≤ , for all  ∈ R}, it is easy to see that  * is a closed subset of PAP(, ).For any  ∈  * , from (19), we have This implies that the mapping  is a self-mapping from  * to  * .Now, we prove that the mapping  is a contraction mapping on  * .In fact, for ,  ∈  * , we get ]            .
( This completes the proof of Theorem 10.

An Example
In this section, we present an example to check the validity of the results we obtained in the previous sections.
which imply that (63) satisfies the assumptions of Theorem 10.Therefore, (63) has a unique positive pseudo almost periodic solution  * (), which is globally exponentially stable with the exponential convergent rate  ≈ 0.01.The numerical simulation in Figure 1 strongly supports the conclusion.
Remark 11.We remark that the results in [9][10][11][12] give no opinions about global exponential convergence for the positive pseudo almost periodic solution.Thus, the results in [9][10][11][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.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.
then we define   ∈  where   () = ( + ) for all  ∈ [−, 0].Due to the biological interpretation of model (1), only positive solutions are meaningful and therefore admissible.Thus we just consider admissible initial conditions.