Research Article | Open Access
Global Exponential Stability of Positive Pseudo-Almost-Periodic Solutions for a Model of Hematopoiesis
This paper presents a new generalized model of hematopoiesis with multiple time-varying delays. The main purpose of this paper is to study the existence and the global exponential stability of the positive pseudo almost periodic solutions, which are more general and complicated than periodic and almost periodic solutions. Under suitable assumptions, and by using fixed point theorem, sufficient conditions are given to ensure that all solutions of this model converge exponentially to the positive pseudo almost periodic solution for the considered model. These results improve and extend some known relevant results.
As we all know, many phenomena in nature have oscillatory character and their mathematical models have led to the introduction of certain classes of functions to describe them. For example, the pseudo almost periodic functions are the natural generalization of the concept of almost periodicity. These are functions on the real numbers set that can be represented uniquely in the form , where (the principal term) is an almost periodic function and (the ergodic perturbation) a continuous function whose mean vanishes at infinity. Note that there exists abundant literature on the topic (see, e.g., [1–6]). In a classic study of population dynamics, the following delay differential equation model where is a positive constant and has been used by [7, 8] to describe the dynamics of hematopoiesis (blood cell production). As we known, (1) belongs to a class of biological systems and it (or its analogue equation) has attracted more attention to the problem of almost periodic solutions because of its extensively realistic significance. For example, some criteria ensuring the existence and stability of positive almost periodic solutions were established in [9–12] and the references cited therein. However, it is very difficult to study the global stability of positive pseudo almost periodic solution for (1). So far, no attention has been paid to the conditions for the global exponential stability on positive pseudo almost periodic solution of model (1) in terms of its coefficients. On the other hand, since the exponential convergent rate can be unveiled, the global exponential stability plays a key role in characterizing the behavior of dynamical system (see [13–15]). Thus, it is worthwhile to continue to investigate the existence and global exponential stability of positive pseudo almost periodic solutions of (1).
Motivated by the above discussions, in this paper, we consider the existence, uniqueness, and global exponential stability of positive pseudo almost periodic solutions of (1). Here in this paper, a new approach will be developed to obtain a delay-independent condition for the global exponential stability of the positive pseudo almost periodic solutions of (1), and the exponential convergent rate can be unveiled.
Throughout this paper, for , it will be assumed that are continuous functions, and Let denote nonnegative real number space, let be the continuous functions space equipped with the usual supremum norm , and let . If is defined on with , then we define where for all .
Due to the biological interpretation of model (1), only positive solutions are meaningful and therefore admissible. Thus we just consider admissible initial conditions. We write for an admissible solution of the admissible initial value problem (1) and (4). Also, let be the maximal right interval of existence of .
2. Preliminary Results
In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.
In this paper, denotes the set of bounded continued functions from to . Note that is a Banach space where denotes the sup norm .
Definition 1 (see [16, 17]). Let . is said to be almost periodic on if, for any , the set is relatively dense; that is, for any , it is possible to find a real number , and for any interval with length , there exists a number in this interval such that , for all .
We denote by the set of the almost periodic functions from to . Besides, the concept of pseudo almost periodicity (pap) was introduced by Zhang in the early nineties. It is a natural generalization of the classical almost periodicity. Precisely, define the class of functions as follows: A function is called pseudo almost periodic if it can be expressed as where and . The collection of such functions will be denoted by . The functions and in the above definition are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function . The decomposition given in definition above is unique. Observe that is a Banach space and is a proper subspace of since the function is pseudo almost periodic function but not almost periodic. It should be mentioned that pseudo almost periodic functions possess many interesting properties; we will need only a few of them and for the proofs we shall refer to .
Lemma 2. Let , and . Then(1); (2), if .
Proof. For any , from the uniform continuity of , we can choose a constant
From the theory of almost periodic functions in [16, 17], it follows that for , it is possible to find a real number , and for any interval with length , there exists a number in this interval such that
Combing (8) and (9), we obtain
which yields .
Set ; we get which implies that .
Remark 3. Set with and . It follows from Lemma 2 that
Definition 4 (see [16, 17]). Let and let be an continuous matrix defined on . The linear system is said to admit an exponential dichotomy on if there exist positive constants , and , projection , and the fundamental solution matrix of (13) satisfying
Lemma 5 (see [6, 16]). Assume that is an almost periodic matrix function and . If the linear system (13) admits an exponential dichotomy, then pseudo almost periodic system has a unique pseudo almost periodic solution , and
Lemma 8. Suppose that there exist two positive constants and such that Then, there exists such that
Proof. This Lemma can be proven in a similar way to that in Lemma 2.2 of . But for convenience of reading, we give the proof as follows. Let . We first claim that there exists such that
Which, together with (19), implies that
This yields that
which contradicts the fact that is positive and bounded on . Hence, (21) holds. In the sequel, we prove that
Suppose, for the sake of contradiction, there exists such that
Calculating the derivative of , together with (19), (1), and (27), implies that
which is a contradiction and implies that (26) holds.
We finally show that . By way of contradiction, we assume that . By the fluctuation lemma [18, Lemma A.1.], there exists a sequence such that Since is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that Moreover, Without loss of generality, we assume that all , , and are convergent to , , and , respectively. This can be achieved because of almost periodicity. It follows from that (taking limits) is a contradiction. This proves that . Hence, from (26), we can choose such that This ends the proof of Lemma 8.
3. Main Results
Proof. Consider defined by
Then, we have
which implies that there exists a constant such that
For any , from (35), Remark 3, and the composition theorem of pseudo almost periodic functions , we have We next consider an auxiliary equation: Notice that ; it follows from Lemma 6 that the linear equation admits an exponential dichotomy on . Thus, by Lemma 5, we obtain that the system (41) has exactly one pseudo almost periodic solution: Define a mapping by setting
Since , it is easy to see that is a closed subset of . For any , from (19), we have This implies that the mapping is a self-mapping from to . Now, we prove that the mapping is a contraction mapping on . In fact, for , we get In view of (39), (45), (46), and (47), from the inequality where and lies between and , we have Noting that , it is clear that the mapping is a contraction on . Using Theorem of , we obtain that the mapping possesses a unique fixed point , . By (41), satisfies (1). So is a positive pseudo almost periodic solution of (1) in . The proof of Theorem 9 is now complete.
Theorem 10. Under the assumptions of Theorem 9, (1) has at least one positive pseudo almost periodic solution . Moreover, is globally exponentially stable; that is, there exist constants , , and such that
Proof. By Theorem 9, (1) has a positive pseudo almost periodic solution; say . It suffices to show that is globally exponentially stable. Define a continuous function by setting
Then, we have
which implies that there exist two constants and such that
Let and , where . Then
It follows from Lemma 8 that there exists such that
We consider the Lyapunov functional Calculating the upper left derivative of along the solution of (54), we have
We claim that Contrarily, there must exists such that Together with (48), (57), and (59), we obtain Thus, which contradicts (53). Hence, (58) holds. It follows that This completes the proof of Theorem 10.
4. An Example
In this section, we present an example to check the validity of the results we obtained in the previous sections.
Example 1. Consider the following model of hematopoiesis with multiple time-varying delays: Obviously Let and . Then which imply that (63) satisfies the assumptions of Theorem 10. Therefore, (63) has a unique positive pseudo almost periodic solution , which is globally exponentially stable with the exponential convergent rate . The numerical simulation in Figure 1 strongly supports the conclusion.
Remark 11. We remark that the results in [9–12] give no opinions about global exponential convergence for the positive pseudo almost periodic solution. Thus, the results in [9–12] and the references therein cannot be applied to prove the global exponential stability of positive pseudo almost periodic solution for (63). This implies that the results of this paper are new and they complement previously known results.
Conflict of Interests
The author declares no conflict of interests. She also declares that she has no financial or personal relationships with other people or organizations that can inappropriately influence her work. There are no professional or other personal interests of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this paper.
The author would like to express the sincere appreciation to the editor and reviewer for their helpful comments in improving the presentation and quality of the paper. In particular, the author expresses the sincere gratitude to Professor Zhibin Chen and Professor Bingwen Liu for the helpful discussion when this revision work was being carried out. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grant nos. Y6110436 and LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant no. Z201122436).
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