Positive Almost Periodic Solution for a Model of Hematopoiesis with Infinite Time Delays and a Nonlinear Harvesting Term

and Applied Analysis 3 For any u ∈ B, considering an auxiliary equation, x 󸀠 (t) = −a (t) x (t)


Introduction
Recently, the existence problem of periodic solutions or almost periodic solutions has been one of the most attracting topics in the qualitative theory of functional differential equations because it is applied extensively in the physical sciences.Neutral functional differential equations arise in many areas of applied mathematics; it has received considerable attention in the recent decades.There have been many remarkable works (see, [1][2][3][4][5]).It is well known that there has been considerable interest in the existence of almost periodic solutions of functional differential equations [3][4][5][6][7][8][9].
There are many works about Hematopoiesis model due to its real applications [10][11][12][13][14][15][16][17].Wang and Zhang [16] proved the existence, nonexistence, and uniqueness of positive almost periodic solution of the model of Hematopoiesis: () = − ()  () +  () 1 +   ( −  ()) ,  > 1, which was used by Mackey and Glass [18] as a model for the dynamics of Hematopoiesis ( blood cells production).On one hand, as pointed out in [19], in the real world, the delays in differential equations of populations, ecology, and dynamic problems are usually infinite time delays; for example, Zhou et al. [20] studied positive almost periodic solutions for a class of Lasota-Wazewska model with infinite time delays; On the other hand, assuming that a harvesting function is a function of the delayed estimate of the true ecological and dynamic models, Zhou et al. [21] presented an overview of the results on the classical Nicholsons proposed Nicholsons, blowflies model with a linear harvesting term.The investigation of biological dynamics with harvesting is an important subject in the exploitation of biological resources which is related to the optimal management of renewable resources [21,22].Motivated by the above, the main aim of this paper is to obtain the conditions for the existence and exponential convergence of the positive almost solutions for the following model of Hematopoiesis with infinite time delays and a nonlinear harvesting term: ( Under proper assumptions, we will obtain a unique positive almost periodic solution of (2) by using the method of a fixed point theorem.A model with infinite time delays and nonlinear harvesting terms is more realistic than that with discrete time delays and linear harvesting term.To our best knowledge, there are some works about Hematopoiesis model with discrete delays and a linear harvesting term [21][22][23].Apparently, our results are more meaningful in reality, so it is worthwhile continuing to study the existence and convergence of positive almost periodic solutions of (2).
The rest of the paper is organized as follows.In Section 2, we shall introduce some definitions and Lemmas which play important roles in the proofs of our main results.In Section 3, some conditions for the existence of positive almost periodic solutions and that all solutions converge exponentially to a positive almost solution of (2) are established.In Section 4, we give an example to illustrate the effectiveness of our results.

Preliminaries
Now let us state the following definitions and lemmas, which will be useful in proving our main results.
Definition 1 (see [9]).A function  :  →   is said to be almost periodic, if for any  > 0 there is a constant () > 0 such that in any interval of length () there exists  such that the inequality is satisfied for all  ∈ .The number  is called an -translation number of ().
Definition 2 (see [7,9]).A function  :  ×   →   is said to be almost periodic in  uniformly for  ∈   , if for any  > 0 and for each compact subject  of   there is a constant () > 0, in any interval of length (); there exists  such that the inequality is satisfied for all (, ) ∈  × .
Lemma 6 (see [9]).If the linear system (8) admits an exponential dichotomy, then the almost periodic system has a unique almost periodic solution (), and

Main Results
In this section, we state and prove our main results concerning the existence of positive almost solutions of (2).Let  denote the set of almost periodic function on .For any  ∈ , we define ‖‖  := sup ∈ |()|; obviously,  is a Banach space.
It is easy to see that  * is a closed subset of .For any  ∈  * ,  ∈ , from (H1) and ( 14), we obtain that Thus, the mapping  is a self-mapping from  * to  * .Next, we show that the mapping  is a contraction mapping on  * .Actually, for any , V ∈  * , we have According to   +  ∑  =1  +  ≤  − and ( 23), we know that  is a contraction on  * .Thus,  has a unique fixed point  * ∈  * such that  * =  * .By (14),  * satisfies (2); then (2) exists as a unique positive almost periodic solution  * in  * .The proof of Theorem 8 is complete.
Remark 10.Obviously, we consider system (2) with the nonlinear harvesting term and infinite time delays of (2), and the results of the references therein cannot be applied to prove the existence and exponential convergence of solutions of (2).

An Example
In this section, we give an example to illustrate the results obtained in the previous sections.