Abstract

We present some results to the existence and uniqueness of the periodic solutions for the hematopoiesis models which are described by the functional differential equations with multiple delays. Our methods are based on the equivalent norm techniques and a new fixed point theorem in the continuous function space.

1. Introduction

In this paper, we aim to establish the existence and uniqueness result for the periodic solutions to the following functional differential equations with multiple delays: where , are -periodic functions on variable for and is a positive integer.

Recently, many authors investigate the dynamics for the various hematopoiesis models, which includ the attractivity and uniqueness of the periodic solutions. For examples, Mackey and Glass in [1] have built the following delay differential equation: where are positive constants, denotes the density of mature cells in blood circulation, and is the time between the production of immature cells in the bone marrow and their maturation for release in the circulating bloodstream; Liu et al. [2],Yang [3], Saker [4], Zaghrout et al. [5], and references therein, also investigate the attractivity and uniqueness of the periodic solutions for some hematopoiesis models.

This paper is organized as follows. In Section 2, we present two new fixed point theorems in continuous function spaces and establish the existence and uniqueness results for the periodic solutions of (1.1). An illustrative example to the hematopoiesis models is exhibited in the Section 3.

2. Fixed Point Theorems and Existence Results

2.1. Fixed Point Theorems

In this subsection, we will present two new fixed point theorems in continuous function spaces. More details about the fixed point theorems in continuous function spaces can be found in the literature [6–8] and references therein.

Let be a Banach space equipped with the norm . which denotes the Banach space consisting of all bounded continuous mappings from into with norm for .

Theorem 2.1. Let be a nonempty closed subset of and an operator. Suppose the following:
() there exist and such that for any,() there exist an and a positive bounded function such that Then has a unique fixed point in .

Proof. For any given , let , . By , we have Set , then we get
In order to prove that the sequence is a Cauchy sequence with respect to norm , we introduce an equivalent norm and show that is a Cauchy sequence with respect to the new one. Basing on the condition , we see that there are two positive constants and such that for all . Define the new norm by Then, Thus, the two norms and are equivalent.
Set , then we have for . By (2.13), we have Thus, This means is a Cauchy sequence with respect to norm . Therefore, also, is a Cauchy sequence with respect to norm . Thus, we see that has a limit point in , say . It is known that is the fixed point of in .
Suppose both and are the fixed points of , then , . Following the similar arguments, we prove that It is impossible. Thus the fixed point of is unique. This completes the proof of Theorem 2.1.

Let be a Banach space consisting of all -periodic functions in with the norm for . Then, following the similar arguments in Theorem 2.1, we deduce Theorem 2.2 which is a useful result for achieving the existence of periodic solutions of functional differential equations.

Theorem 2.2. Let be an operator. Suppose the following:
there exist and such that for any , where with and is a positive integer; there exist two constants and a positive function such that , and Then has a unique fixed point in .

Proof. For any given , let , . By , we have Set , then we get
Basing on the condition , we see that there are two positive constants and such that for all . Define the new norm by Then, Thus, the two norms and are equivalent.
Set , then we have for . By (2.13), we have Thus, This means is a Cauchy sequence with respect to norm . Therefore, is a Cauchy sequence with respect to norm . Therefore, we see that has a limit point in , say . It is easy to prove that is the fixed point of in . The uniqueness of the fixed point is obvious. This completes the proof of Theorem 2.2.

2.2. Existence and Uniqueness of the Periodic Solution

In order to show the existence of periodic solutions of (1.1), we assume that the function is fulfilling the following conditions:

there exist such that for any ,

for all , .

Theorem 2.3. Suppose and hold. Then the equation (1.1) has a unique -periodic solution in .

Proof. By direction computations, we see that is the -periodic solution if and only if is solution of the following integral equation: where .
Thus, we would transform the existence of periodic solution of (1.1) into a fixed point problem. Considering the map defined by, for , Then, is a -periodic solution of (1.1) if and only if is a fixed point of the operator in .
At this stage, we should check that fulfill all conditions of Theorem 2.2. In fact, for , by assumption , we have where , and .
Thus, the condition in Theorem 2.2 holds for , and .
On the other hand, we choose a constant such that . Take and for , then , and we have This implies the condition in Theorem 2.2 holds for .
Following Theorem 2.2, we conclude that the operator has a unique fixed point, say , in . Thus, (1.1) has a unique -periodic solution in . This completes the proof of Theorem 2.3.

3. Application to the Hematopoiesis Model

In this section, we consider the periodic solution of following hematopoiesis model with delays: where are -periodic functions, satisfies conditions , and is a real number .

Theorem 3.1. The delayed hematopoiesis model (3.1) has a unique positive -periodic solution.

Proof. Let , define the operator by It is easy to show that is welldefined. Furthermore, since the function with the bounded partial derivative then it is easy to prove that the condition holds. Following the similar arguments of Theorem 2.3, we claim that the operator has a unique fixed point in , which is the unique positive -periodic solution for equation (3.1). This completes the proof of Theorem 3.1.

Remark 3.2. Theorem 3.1 exhibits that the periodic coefficients hematopoiesis model admits a unique positive periodic solution without additional restriction. Also, Theorem 3.1 improves Theorem  2.1 in [2] and Corollary  1 in [3].