New results on the positive pseudo almost periodic solutions for a generalized model of hematopoiesis

The main purpose of this paper is to study the existence and global exponential stability of the positive pseudo almost periodic solutions for a generalized model of hematopoiesis with multiple time-varying delays. By using the exponential dichotomy theory and fixed point theorem, some sufficient conditions are given to ensure that all solutions of this model converge exponentially to the positive pseudo almost periodic solution, which improve and extend some known relevant results. Moreover, an example and its numerical simulation are given to illustrate the theoretical results.


Introduction
In a classic study of population dynamics, the following delay differential equation model has been used by [4,9] to describe the dynamics of hematopoiesis (blood cell production).
In medical terms, x(t) denotes the density of mature cells in blood circulation, the cells are lost from the circulation with a rate a(t) at time t, the flux f (x(t − τ i (t))) = b i (t)x m (t−τ i (t)) 1+x n (t−τ i (t)) of the cells into the circulation from the stem cell compartment depends on x(t − τ i (t)) at time t − τ i (t), and τ i (t) is the time delay between the production of immature cells in the bone marrow and their maturation for release in circulating bloodstreams at time t.As we know, the existence of almost periodic, asymptotically almost periodic, pseudo almost periodic solutions Email: hongzhang320@aliyun.com 2 H. Zhang are among the most attractive topics in qualitative theory of differential equations due to their applications, especially in biology, economics and physics [3,17].The concept of pseudo almost periodicity, which is the central subject in this paper, was introduced by Zhang [14,15,16] in the early nineties.Besides, equation (1.1) belongs to a class of biological systems and it (or its analogue equation) has been attracted more attention on problem of almost periodic solutions and pseudo 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 [1,2,8,10,18] and the references cited therein.Most recently, the author in [10] also obtained a new sufficient condition of the existence and uniqueness of positive pseudo almost periodic solution for equation (1.1) with m = 0, which can be described as follows: .
However, to the best of our knowledge, there exist few results on the existence and exponential stability of the positive pseudo almost periodic solutions of (1.1) without m = 0. On the other hand, since the exponential convergence rate can be unveiled, the global exponential convergence plays a key role in characterizing the behavior of dynamical system (see [6,7,11]).Thus, it is worthwhile to continue to investigate the existence and global exponential stability of positive pseudo almost periodic solutions of (1.1) without m = 0. Motivated by the above discussions, in this paper we consider the existence, uniqueness and global exponential stability of positive pseudo almost periodic solutions of (1.1).Here in this present 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 (1.1), and the exponential convergence rate can be unveiled.Throughout this paper, for i = 1, 2, . . ., K, it will be assumed that a : R → (0, +∞) is an almost periodic function, b i , τ i : R → [0, +∞) are pseudo almost periodic functions, and Let R + denote the space of nonnegative real numbers, C = C([−r, 0], R) be the space of continuous functions equipped with the usual supremum norm • , and let The initial conditions associated with (1.1) are defined as follows: We denote by x t (t 0 , ϕ)(x(t; t 0 , ϕ)) an admissible solution of admissible initial value problem (1.1) and (1.4).Also, let [t 0 , η(ϕ)) be the maximal right-interval of the existence of x t (t 0 , ϕ).
we can choose a positive constant κ such that and Moreover, from (1.5), (1.6) implies there exists a constant κ such that (1.7)

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. ([3, 17]).Let u(t) ∈ BC(R, R).The function 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)| < ε for all t ∈ R} is relatively dense, i.e., for any ε > 0, it is possible to find a real number l = l(ε) > 0, with the property that 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. Define the class of functions PAP 0 (R, R) as follows: where h ∈ AP(R, R) and ϕ ∈ PAP 0 (R, R).The collection of such functions will be denoted by PAP(R, R).

Definition 2.3 ([3, 17]
).Let x ∈ R l and Q(t) be an l × l continuous matrix defined on R. The linear system is said to admit an exponential dichotomy on R if there exist positive constants k, α, projection P and the fundamental solution matrix X(t) of (2.1) satisfying H. Zhang

Lemma 2.4 ([17]
). Assume that Q(t) is an almost periodic matrix function and g(t) ∈ PAP(R, R l ).If the linear system (2.1) admits an exponential dichotomy, then the pseudo almost periodic system has a unique pseudo almost periodic solution x(t), and Lemma 2.5 ([3, 17]).Let c i (t) be an almost periodic function on R and Then the linear system Then, there exists t ϕ > t 0 such that κ < x(t; t 0 , ϕ) < M for all t ≥ t ϕ . ( Set Then, we get the following lemma. Lemma 2.8.B * is a closed subset of PAP(R, R).
Proof.Suppose that {x p } +∞ p=1 ⊆ B * satisfies lim Obviously, x ∈ PAP(R, R), and K 1 ≤ x(t) ≤ K 2 , for all t ∈ R. We next show that x is uniformly continuous on R. In fact, for any ε > 0, from (2.6), we can choose p > 0 such that Then, equation (1.1) has at least one positive pseudo almost periodic solution x * (t).
Proof.Consider Υ : [0; 1] → R defined by Then, from (3.1), we have which implies that there exists a constant ς ∈ (0, 1] such that According to Corollary 5.4 in [17, p. 58] and the composition theorem of pseudo almost periodic functions, we have We next consider an auxiliary equation Notice that M[a] > 0, it follows from Lemma 2.4 that the linear equation admits an exponential dichotomy on R. Thus, by Lemma 2.4, we obtain that the system (3.3) has exactly one pseudo almost periodic solution

H. Zhang
Define a mapping T : PAP(R, R) −→ PAP(R, R) by setting For any φ ∈ B, from (2.4), we have Moreover, from Remark 1.1, we get , where m < n, and which yield −∞ e − t s a(u) du a(s)κ ds = κ, for all t ∈ R. (3.9)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 (3.10) Pseudo almost periodic model of hematopoiesis 7 From the differential mid-value theorem, we have and where x, y ∈ [κ, M], θ lies between x and y.Then, (3.10), (3.11) and (3.12) yield Noting that e −ς <  [8] considered the almost periodic solution problem of (1.1) with almost periodic coefficients and delays under the assumption of (2.4).Noting that the pseudo almost periodic functions is a natural generalization of the concept of almost periodicity, it is obvious that the main results in [8] are special cases of our results.

An example
In this section, we present an example to check the validity of our results we obtained in the previous sections.pseudo almost periodic solution for equation (1.1) with m > 1.Whether or not our results and method in this paper are available for this case, it is an open interesting problem and we leave it as our work in the future.
[8]it is clear that the mapping T is a contraction on B. Using Theorem 0.3.1 of[5], we obtain that the mapping T possesses a unique fixed point ϕ * ∈ B with Tϕ * = ϕ * .By (3.3), ϕ * satisfies (1.1).So ϕ * is a positive pseudo almost periodic solution of (1.1) in B. The proof of Theorem 3.1 is completed.By using Lemma 2.7, the proof of global exponential stability of x * (t) is similar to that of Theorem 3.2 in[8], and we obtain the following theorem.Under the assumptions of Theorem 3.1, the pseudo almost periodic solution x * (t) of equation (1.1) is globally exponentially stable.Most recently, B. Liu − 2e −t 6 sin 2 t )