Research Article | Open Access
Unique Positive Almost Periodic Solution for Discrete Nonlinear Delay Survival Red Blood Cells Model
We obtain sufficient conditions which guarantee the global attractivity of solutions for nonlinear delay survival red blood cells model. Then, some criteria are established for the existence, uniqueness and global attractivity of positive almost periodic solutions of the almost periodic system.
Recently, Saker  studied the existence and global attractivity of positive periodic solution for the following discrete nonlinear delay survival red blood cells model: where , , , are nonnegative bounded sequence, for all and . The dynamic behavios of (1.1) was investigated by many authors (see [1–18]) because of its biological and ecological significance. When , and are positive constants, the global attractivity of the positive equilibrium for equation has been investigated by some authors (see [3, 15–18] and reference therein). Recently, Ma and Yu  obtained the following Theorem A.
Equation (1.2) is the discrete analogue of equation which was first used by Ważewska-Czyżewska and Lasota as a model for the survival of red blood cells in an animal in , see also . Here, denotes the number of red blood cells at time , is the probability of death of a red blood cell, and are positive constants related to the production of red blood cells per unit time and is the time required to produce a red blood cell. Researching the behavior of the solution of (1.4) and its analogue was posed as open problems by Kocic and Ladas  as well as Györi and Ladas .
Because of seasonal variation (1.1) needs not to be exactly periodic but almost periodic instead. It is natural to ask if the results in  hold for the almost periodic case. It is a difficult problem in which significant difference appears in comparison with the periodic case, for example, contrary to periodic functions, there exists an almost periodic function such that for all and .
One purpose of the present paper is to extend Theorem A to (1.1). The other purpose is to extend some results in  to the almost periodic case. In , the author use a fixed theorem to obtain the existence of positive periodic solution. The operator used in  depends on the period of system (1.1), therefore we cannot apply these topological tools to the almost periodic case. An important notion in almost periodic differential theory is the hull (cf. [21–23]). In this paper, we use essentially this notion to establish our results. We begin with some notations.
Let , , and . For each , we define the norm of as . Denote the element of with for all . For any bounded sequence , denote and , if and if . It is easy to see that, for any , there is a unique solution of (1.1) with and for all . Now, we give some definitions.
Definition 1.1 (see [22, 23]). A sequence 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 .
Denote set the hull of . It is easy to see that if is almost periodic, then for all ,
Definition 1.2. Let . Then is said to be almost periodic in uniformly on compact set of , if is continuous for each , and for any and every compact set , there is a constant such that in any interval of length there exists such that the inequality is satisfied for all and .
Definition 1.3. The positive solution to (1.1), in the sense that for all , is said to be global attractive, if for any ,
The remainder of this paper is organized as follows. The main results are given in Section 2 while the proofs are left to Section 4. In Section 3, we will give some lemmas needed in the proofs of main results.
2. Main Results
Theorem 2.1. Assume that and . Then, for any , the solution of (1.1) satisfies where is the limit of with and
The following corollary follows from [24, Theorems 1 and 2].
Remark 2.6. When , and , here , and are positive constants, we consider the global attractivity of the positive equilibrium for (1.2). In this case, and We see that (2.3) changes to (1.3) and Theorem 2.4 reproduces Theorem A.
Theorem 2.7. Assume that system (1.1) is almost periodic, that is, , , and are almost periodic, and the conditions of Corollary 2.5 are satisfied. Then there is a unique globally attractive positive almost periodic solution for (1.1).
Remark 2.8. Let , and be a positive constant such that Since , , , , , and we can see that there is unique globally attractive positive almost periodic solution for (1.1). Since we cannot obtain the existence of positive almost periodic solution of (1.1) by [25, Theorem 4.1]. Moreover, we should point out that (4.1) in  is not correct. To see this, we consider (1.2) with . Let . It is easy to see that the positive equilibrium of (1.2) is the unique zero point of . Since we see that . Thus we cannot conclude that for each positive solution of (1.2), Therefore the conclusions of [25, Theorem 4.1] may not hold.
3. Some Lemmas
Lemma 3.1 (see ). The following system of inequalities, where are real numbers, have exactly one solution .
Lemma 3.2. Assume that , is almost periodic sequence in uniformly on compact set of . If there is a solution () to the equation such that . then for each , equation has a solution which is defined on such that
Proof. Let and . Then is a compact subset of . For each , there is such that and uniformly on . By the diagonal process, we can choose such that as and satisfies (3.3) on . We want to prove that (3.4) holds.
For each , there is a such that For any fixed , there is such that for all . It follows from (3.5) that Setting and , we see that (3.4) holds. This completes the proof.
Lemma 3.3. Assume that , is almost periodic sequence in uniformly on compact set of . Let be a compact set of . If for all , equation admits a unique solution , defined on , whose range is in , then all these solutions are almost periodic.
4. The Proofs of Main Results
Proof of Theorem 2.1. Let for each . From (1.1), we have which prove that () by induction. For any , there is such that for , By the fact that we deduce that This implies By the fact that is arbitrary, we obtain that For each , there is such that By (1.1), (4.2), we obtain that It follows that for This implies that By induction we can see that It is easy to see that increase and decrease. Thus the limit of exists. Therefore we have This completes the proof of Theorem 2.1.
Proof of Theorem 2.4. Set . Then satisfies
The proof will be accomplished by showing that
We will prove that (4.13) holds in each of the following two cases.
Case 1. is nonoscillatory. Suppose that is eventually nonnegative. The case that is eventually nonpositive is similar and will be omitted. By (4.12) we see that is eventually decreasing. Thus the limit of exists. Let . Then . We claim that . Otherwise, there would exist such that It follows from (4.12) that This implies that By the fact that we see that , which contradicts the fact that is eventually nonnegative. Thus . Therefore (4.13) holds in this case.Case 2. is oscillatory. Let and . Then There exist positive sequences and such that , and for , for and for and where and . Let and . Since , by (4.12) we have We claim that We first assume that . Therefore we have . Thus (4.21) holds. Assume now that . By (4.20) we obtain that . This implies that . Thus (4.21) also holds.
Similarly, we can obtain that For each , there is such that for , Let . By (4.12) we obtain that This implies that It follows from (4.21) and (4.23)–(4.24) that for large enough Using (4.19), then by being arbitrary, this implies that Now we will prove that In fact, by (4.22)–(4.24), for large enough we have Note that is arbitrary. By (4.19) we can see that (4.28) holds. By Lemma 3.1 we see that . Thus, (4.13) also holds in this case. This completes the proof.
Proof of Theorem 2.7. We will prove our conclusion by applying Lemmas 3.1 and 3.2. For any , consider equation
It follows from Theorem 2.1 that for any ,
where , are the constants given in Theorem 2.1, and is the solution of (1.1) with . By the almost periodicity of (1.1) and Lemma 3.2, we can find a solution of (4.30) defined on whose range is in . Now we will prove that the solution which possesses the above properties is unique. Suppose that is another solution of (4.30) defined on whose range is in . Let . Then satisfies
We claim that . Otherwise, we would have
for some . Consider the following two cases.
Case 1. There is a such that (or 0) for all . Without loss of generality, we suppose that for all . Then for . Since is bounded on , we see that exists, denoted by . We claim that . Otherwise, there would exist such that Note that . It follows from (4.32) that This implies that Hence , which contradicts the fact that is bounded on . Thus . Therefore for all . It follows from (4.32) that for all , which contradicts (4.33).Case 2. There exist positive sequences and such that , and for , for and for . Let and . Then and . For each , there are such Suppose and . Let and . Set and . Similar to the argument in the proof of Theorem 2.4, we can obtain that Let . By the almost periodicity of , and , we have for all , Let . By (4.32) we obtain that This implies that Since for all , it follows from (4.39)–(4.40) that Thus, we have By the fact that is arbitrary, we obtain that Similarly, we can prove that Equations (4.45) and (4.46) produce that . Therefore for all . It follows from (4.32) that for all , which also contradicts (4.33). By Lemma 3.3 we see that (1.1) has a unique almost periodic solution whose range is in . By Corollary 2.5 we can see that is global attractivity. This also implies that the almost periodic solution of (1.1) is unique. This completes the proof.
The authors is greatly indebted to an anonymous referee for his/her valuable suggestions, which helped them to improve the exposition of the results. The research was partly supported by NSFC (10671021).
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Copyright © 2009 Xitao Yang and Siping Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.