Dynamics for a class of general hematopoiesis model with periodic coefficients

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Abstract

Sufficient conditions are obtained for the existence and global attractivity of a unique positive periodic solution x˜(t) of (∗)x(t)=-a(t)x(t)+b(t)1+xn(t-τ(t)),t>0,where n > 1, a and b are continuous positive periodic function. Also, some sufficient conditions are established for oscillation of all positive solutions of (∗) about x˜(t). For the proof of existence and uniqueness of x˜(t), the method used here is better than contraction mapping principle.

Introduction

The purpose of the present paper is to investigate the existence and global attractivity of unique positive solution x˜(t) of the following equation with periodic coefficients:x(t)=-a(t)x(t)+b(t)1+xn(t-τ(t)),n>1.Meanwhile, the oscillation of every positive solution of (1.1) about x˜(t) will be also studied. It is necessary to consider behaviors of all positive solutions of (1.1). In fact, (1.1) is one of generations of hematopoiesis modelsx(t)=-ax(t)+b1+xn(t-τ),n>0,which (after some transformations) was first proposed by Mackey and Glass [10] to describe some physiological control systems. The global attractivity of positive steady solution K of (1.2) and the oscillatory behavior of all positive solution about K have been studied. See Karakostas et al. [5], Kuang [7], Saker [11] and Zaghrout et al. [16], for instance. For further investigation in this area, for example, the delay differential equationsx(t)=-a(t)x(t)+b(t)xm(t-kω)1+xn(t-kω),where m = 0,1 and n > m, a(t) and b(t) are positive ω-periodic functions, andx(t)=-a(t)x(t)+b(t)0K(s)11+xn(t-s)ds,n>0,where K: [0, ∞)  [0, ∞), and a(t) and b(t) are positive ω-periodic functions. In [12], the author not only obtained that (1.3) had a unique positive periodic solution x˜(t) under some assumptions when k = 0 but also studied oscillation of all positive solutions of (1.3) about x˜(t) and global attractivity of x˜(t). And some sufficient conditions have been obtained by Yang and Weng [15] for the existence and global attractivity of a positive periodic solution of (1.4).

As far as we know, though the existence of positive periodic solution of (1.1) has been already done by Jiang and Wei [4] and Wan et al. [13], other behaviors of solutions of (1.1) have never been studied. Motivated by [2], [6], [9], in this paper we shall investigate the existence and uniqueness of positive periodic solution x˜(t) of (1.1) by using a fixed theorem in cone which is different from that used in [4], [13], [14] and also show that the method used here is better than contraction mapping principle. And we shall prove that if τ(t)  τ, then x˜(t) is a global attractor and give some sufficient conditions to guarantee that every positive solution oscillates about x˜(t).

Note that if τ(t) =  in (1.1), then (1.1) reduces to (1.3) for m = 0 and n > 1. For this case, our main results complement that in [12].

Throughout this paper, in (1.1), we always suppose that a(t), b(t) and τ(t) are positive continuous ω-periodic functions on R.

For convenience, we also need to introduce a few notations. LetG(t,s)=exp(tsa(r)dr)exp(0ωa(r)dr)-1,s[t,t+ω],N=G(t,t)=mint[0,ω],s[t,t+ω]{G(t,s)}maxt[0,ω],s[t,t+ω]{G(t,s)}=G(t,t+ω)=M,h=maxt[0,ω]h(t),h=mint[0,ω]h(t),andh=0ωh(s)ds,where h(t) is a continuous ω-periodic function on R.

In view of the actual applications of (1.1), we shall only consider the solutions of (1.1) with initial conditionx(s)=ϕ(s)fors[-τ,0],ϕC([-τ,0],[0,)),ϕ(0)>0.

Section snippets

Some definitions and lemmas

The proofs of the main results in our paper are based on an application of fixed point theorem in cone (see [3]). To make use of fixed point theorem in cone, firstly, we need to introduce some definitions and lemmas. Let X be a real Banach space, P is a cone of X. The semi-order induced by the cone P is denoted by “⩽”. That is, x  y if and only if y  x  P for any x, y  P.

Definition 2.1

A cone P of X is said to be normal if there exists a positive constant δ such that ∥x + y  δ for any x, y  P, ∥x = y = 1.

Definition 2.2

P is a cone of

Main results

In this section, we shall prove that (1.1) has a unique positive ω-periodic solution x˜(t) which is globally asymptotically stable. Moreover, we shall establish some sufficient conditions under which every positive solution of (1.1) oscillates about x˜(t).

Theorem 3.1

If Mb1n-1n holds, then (1.1) has a unique positive ω-periodic solution x˜(t). Furthermore,xn-x˜0(n),where xn = Axn1 (n = 1, 2, 3, ) for any initial x0  P.

Proof

Firstly, from Lemma 2.2 and (2.1), it is clear that A satisfies condition (i) of Lemma 2.1

References (16)

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