On the Existence and Exponential Attractivity of a Unique Positive Almost Periodic Solution to an Impulsive Hematopoiesi
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On the Existence and Exponential Attractivity of a Unique Positive Almost Periodic Solution to an Impulsive Hematopoiesis Model with Delays Trinh Tuan Anh1 · Tran Van Nhung2 · Le Van Hien1
Received: 10 September 2014 / Revised: 2 January 2015 / Accepted: 22 January 2015 / Published online: 15 August 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
Abstract In this paper, a generalized model of hematopoiesis with delays and impulses is considered. By employing the contraction mapping principle and a novel type of impulsive delay inequality, we prove the existence of a unique positive almost periodic solution of the model. It is also proved that, under the proposed conditions in this paper, the unique positive almost periodic solution is globally exponentially attractive. A numerical example is given to illustrate the effectiveness of the obtained results. Keywords Hematopoiesis model · Almost periodic solution · Impulsive systems Mathematics Subject Classification (2010) 34A37 · 34K25 · 34K45 · 92B05
1 Introduction The nonlinear delay differential equation x(t) ˙ = −ax(t) +
b 1 + x n (t
− τ)
,
n > 0,
Le Van Hien
[email protected] Trinh Tuan Anh [email protected] Tran Van Nhung [email protected] 1
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam
2
Vietnam State Council for Professor Promotion, No. 1, Dai Co Viet, Hanoi, Vietnam
(1.1)
338
T. T. Anh et al.
where a, b, τ are positive constants, proposed by Mackey and Glass [17], has been used as an appropriate model for the dynamics of hematopoiesis (blood cells production) [5, 7, 17]. In medical terms, x(t) denotes the density of mature cells in blood circulation at time t and τ is the time delay between the production of immature cells in the bone marrow and their maturation for release in circulating bloodstream. As we may know, the periodic or almost periodic phenomena are popular in various natural problems of real-world applications [2, 5, 6, 9, 10, 16, 19, 22, 23]. In comparing with periodicity, almost periodicity is more frequent in nature and much more complicated in studying for such model [20, 21]. On the other hand, many dynamical systems which describe the real phenomena depend on the history as well as undergo abrupt changes in their states. This kind of models are best described by impulsive delay differential equations [3, 18, 20]. A great deal of effort from researchers has been devoted to the study of existence and asymptotic behavior of almost periodic solutions of (1.1) and its generalizations due to their extensively realistic significance. We refer the reader to [8, 13, 15, 21, 24, 25] and the references therein. Particularly, in [21], Wang and Zhang investigated the existence, nonexistence, and uniqueness of positive almost periodic solution of the following model x(t) ˙ = −a(t)x(t) +
b(t)x(t − τ (t)) , 1 + x n (t − τ (t))
n > 1,
(1.2)
by using a new fixed point theorem in the cone. Very recent
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