Dynamics for a class of general hematopoiesis model with periodic coefficients
Introduction
The purpose of the present paper is to investigate the existence and global attractivity of unique positive solution of the following equation with periodic coefficients:Meanwhile, the oscillation of every positive solution of (1.1) about 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 modelswhich (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 equationswhere m = 0,1 and n > m, a(t) and b(t) are positive ω-periodic functions, andwhere 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 under some assumptions when k = 0 but also studied oscillation of all positive solutions of (1.3) about and global attractivity of . 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 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 is a global attractor and give some sufficient conditions to guarantee that every positive solution oscillates about .
Note that if τ(t) = kω 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. Letwhere 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 condition
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 which is globally asymptotically stable. Moreover, we shall establish some sufficient conditions under which every positive solution of (1.1) oscillates about . Theorem 3.1 If holds, then (1.1) has a unique positive ω-periodic solution . Furthermore,where xn = Axn−1 (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)
- et al.
Existence and global attractivity of unique positive periodic solution for a Lasota–Wazewska model
Nonlinear Analysis
(2006) Oscillation and global attractivity in hematopoiesis model with delay time
Applied Mathematics and Computation
(2003)Oscillation and global attractivity in hematopoiesis model with periodic coeffecients
Applied Mathematics and Computation
(2003)- et al.
A new existence theory for positive periodic solutions to functional differential equations
Computers and Mathematics with Applications
(2004) Positive periodic solutions of functional differential equations
Journal of Differential Equations
(2004)- et al.
Oscillations and global attractivity in delay differential equations of population dynamics
Applied Mathematics and Computation
(1996) Stability and Oscillations in Delay Differential Equations of Population Dynamics
(1992)- et al.
Global attractivity and level crossing in model of hematopoiesis
Bulletin of the Institute of Mathematics, Academia Sinica
(1994)
Cited by (21)
Global asymptotic stability for a periodic delay hematopoiesis model with impulses
2020, Applied Mathematical ModellingExistence regions of positive periodic solutions for a discrete hematopoiesis model with unimodal production functions
2019, Applied Mathematical ModellingCitation Excerpt :A hematopoiesis model with the monotonically decreasing production function (1.3) and a hematopoiesis model with the unimodal production function (1.4) are called D-type and U-type, respectively. We can refer to [8–17] and [9–11,13,18–22] for the study of hematopoiesis models of D-type and of U-type, respectively. The main themes of those studies are as follows:
A new result on the existence of positive almost periodic solution for generalized hematopoiesis model
2018, Journal of Mathematical Analysis and ApplicationsCitation Excerpt :Due to the effects of various seasonal factors such as weather, food supplies, mating habits, and so on, it is reasonable to consider periodic or almost periodic biological system. Many authors [12,14,16] explored the several properties of Eq. (1.1) or Eq. (1.2) with periodic coefficients, and the standard tools used to obtain their results consist of contraction mapping principle, Schauder fixed point theorems or Mawhin's continuation theorem. However, almost periodicity is more frequent and general than periodicity in realistic requirements, especially, several time-dependent parameters with different periods are considered from the biological point of view.
Existence of positive almost periodic solutions to the hematopoiesis model
2016, Applied Mathematics and ComputationCitation Excerpt :Since its introduction in the literature, the hematopoiesis model has gained a lot of attention due to its applications in our daily lives. Recently, there have been extensive contributions on the existence of periodic and almost periodic solutions to the hematopoiesis model, see, e.g., [2–10] and the references therein. [16]
Mackey-Glass model of hematopoiesis with monotone feedback revisited
2013, Applied Mathematics and ComputationCitation Excerpt :□ The oscillation part of Theorem 6.1 coincides with [27] which, unlike Theorem 6.1, does not contain nonoscillation results. [3]
Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis
2013, Applied Mathematics Letters