Positive Almost Periodic Solution for a Model of Hematopoiesis with Infinite Time Delays and a Nonlinear Harvesting Term
A generalized model of Hematopoiesis with infinite time delays and a nonlinear harvesting term is investigated. By utilizing a fixed point theorem of the differential equations and constructing a suitable Lyapunov functional, we establish some conditions which guarantee the existence of a unique positive almost periodic solution and the exponential convergence of the system. Finally, we give an example to illustrate the effectiveness of our results.
Recently, the existence problem of periodic solutions or almost periodic solutions has been one of the most attracting topics in the qualitative theory of functional differential equations because it is applied extensively in the physical sciences. Neutral functional differential equations arise in many areas of applied mathematics; it has received considerable attention in the recent decades. There have been many remarkable works (see, [1–5]). It is well known that there has been considerable interest in the existence of almost periodic solutions of functional differential equations [3–9].
There are many works about Hematopoiesis model due to its real applications [10–17]. Wang and Zhang  proved the existence, nonexistence, and uniqueness of positive almost periodic solution of the model of Hematopoiesis: which was used by Mackey and Glass  as a model for the dynamics of Hematopoiesis ( blood cells production). On one hand, as pointed out in , in the real world, the delays in differential equations of populations, ecology, and dynamic problems are usually infinite time delays; for example, Zhou et al.  studied positive almost periodic solutions for a class of Lasota-Wazewska model with infinite time delays; On the other hand, assuming that a harvesting function is a function of the delayed estimate of the true ecological and dynamic models, Zhou et al.  presented an overview of the results on the classical Nicholsons proposed Nicholsons, blowflies model with a linear harvesting term. The investigation of biological dynamics with harvesting is an important subject in the exploitation of biological resources which is related to the optimal management of renewable resources [21, 22].
Motivated by the above, the main aim of this paper is to obtain the conditions for the existence and exponential convergence of the positive almost solutions for the following model of Hematopoiesis with infinite time delays and a nonlinear harvesting term:
Under proper assumptions, we will obtain a unique positive almost periodic solution of (2) by using the method of a fixed point theorem. A model with infinite time delays and nonlinear harvesting terms is more realistic than that with discrete time delays and linear harvesting term. To our best knowledge, there are some works about Hematopoiesis model with discrete delays and a linear harvesting term [21–23]. Apparently, our results are more meaningful in reality, so it is worthwhile continuing to study the existence and convergence of positive almost periodic solutions of (2).
For convenience, we will introduce some notations: where and are bounded continuous and uniformly bounded continuous functions, respectively. The following conditions (H1) hold:
In (2), throughout this paper, the following hypotheses conditions hold:(H2) and are almost periodic; there exists a compact subset such that is almost;(H3) are uniformly almost periodic and there exists a constant such that (H4) the delay kernel and .
Let be the bounded continuous functions space. is equipped with usual supremum norm . If is continuous and defined on with , then, for all , we define for all . In view of the actual applications of system (2), the admissible initial conditions , and are needed. We write for an admissible solution of (2) with the above initial conditions. We let be the maximal rightinterval of existence of .
The rest of the paper is organized as follows. In Section 2, we shall introduce some definitions and Lemmas which play important roles in the proofs of our main results. In Section 3, some conditions for the existence of positive almost periodic solutions and that all solutions converge exponentially to a positive almost solution of (2) are established. In Section 4, we give an example to illustrate the effectiveness of our results.
Now let us state the following definitions and lemmas, which will be useful in proving our main results.
Definition 1 (see ). A function is said to be almost periodic, if for any there is a constant such that in any interval of length there exists such that the inequality is satisfied for all . The number is called an -translation number of .
Definition 2 (see [7, 9]). A function is said to be almost periodic in uniformly for , if for any and for each compact subject of there is a constant , in any interval of length ; there exists such that the inequality is satisfied for all .
Lemma 3 (see ). If is an almost periodic, in uniformly for , is almost periodic, and for all , with being a compact subset of , then is almost periodic.
Definition 4 (see [7, 9]). Let and be an continuous matrix defined on . The linear system is said to admit an exponential dichotomy on if there exist positive constants , a projection , and the fundamental solution matrix of (8) satisfying
3. Main Results
In this section, we state and prove our main results concerning the existence of positive almost solutions of (2). Let denote the set of almost periodic function on . For any , we define ; obviously, is a Banach space.
For any , considering an auxiliary equation,
Let us define the operator by where
It is clear that (2) has one unique positive almost periodic solution if and only if the operator has exactly one fixed point. We set and suppose that
Lemma 7. If (H1) holds and . Then, for , the solution of (2) satisfies , for all and .
Proof. Set . Let be an interval such that for all . We prove
By method of contradiction, if (17) does not hold, there exists such that and for all . Calculating the derivative of and by (H1), we have that
which is a contradiction. So it implies that (17) holds.
Next we prove that
Otherwise, there exists such that and for all .
So from (17), we get , for . Calculating the derivative of , together with (H1) and (17), we obtain that which is a contradiction and implies that (19) holds. From Theorem in , we have that . The proof of Lemma 7 is completed.
Theorem 8. Assume that (H1)–(H4) and . Then, there exists a unique positive almost periodic solution of (2) in the region .
Proof. For any , by (H2)-(H3) and Lemma 6, we know that (14) is an almost periodic differential equation. Since , notice that . From Lemmas 3 and 5, we obtain that the system (2) has exactly one almost periodic solution:
Define a mapping
It is easy to see that is a closed subset of . For any , from (H1) and (14), we obtain that Thus, the mapping is a self-mapping from to . Next, we show that the mapping is a contraction mapping on . Actually, for any , we have According to and (23), we know that is a contraction on . Thus, has a unique fixed point such that . By (14), satisfies (2); then (2) exists as a unique positive almost periodic solution in . The proof of Theorem 8 is complete.
Proof. Let and , where . So we have
According to Lemma 7, we get that is positive and bounded on and for all .
Define a continuous function by setting Then, we have which implies that there exists a constant such that
We consider the Lyapunov function Calculating the upper right derivative of along the solution of (24), we obtain that
We claim that Contrarily, there must exist such that which implies that Because and for .
By (29) and (32), we have Obviously, (33) contradicts with (25). Hence, (28) holds. It follows that , for all . Thus, the proof is now completed.
Remark 10. Obviously, we consider system (2) with the nonlinear harvesting term and infinite time delays of (2), and the results of the references therein cannot be applied to prove the existence and exponential convergence of solutions of (2).
4. An Example
In this section, we give an example to illustrate the results obtained in the previous sections.
Example 11. Consider the following Hematopoiesis model with infinite delays and nonlinear harvesting term: We know that It is not difficult to obtain that satisfies Lipschitz condition and take . It follows from Theorem 9 that Hematopoiesis model (34) has a unique positive almost periodic solution . Moreover, if , then the solutions of (34) converge exponentially to as .
Remark 12. It is obvious that the delays in (34) are infinite time delays and a nonlinear harvesting term. This implies that the results of this paper are new and they complement previously known results [14–17].
The authors would like to express sincere appreciation to the editor and referee for their valuable comments and good suggestions which have led to an improvement in the presentation of the paper. Research is supported by the National Natural Science Foundation of China (11071001) and Anhui Province Natural Science Foundation (1208085MA13).
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