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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 321328, 10 pages
http://dx.doi.org/10.1155/2014/321328
Research Article

-Almost Periodic Functions and an Application to a Lasota-Wazewska Model on Time Scales

1Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
2School of Mathematics and Computer Science, Yunnan Nationalities University, Kunming, Yunnan 650091, China

Received 10 February 2014; Accepted 4 May 2014; Published 19 May 2014

Academic Editor: Shiping Lu

Copyright © 2014 Li Yang et al. 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.

Abstract

We first give the definition and some properties of -almost periodic functions on time scales. Then, as an application, we are concerned with a class of Lasota-Wazewska models on time scales. By means of the fixed point theory and differential inequality techniques on time scales, we obtain some sufficient conditions ensuring the existence and global exponential stability of -almost periodic solutions for the considered model. Our results are essentially new when or . Finally, we present a numerical example to show the feasibility of obtained results.

1. Introduction

Between the years 1923 and 1926, Harald Bohr found a theory of almost periodic real (and complex) functions. Several generalizations and classes of almost periodic functions have been introduced in the literature, including pseudo-almost periodic functions and almost automorphic functions [1, 2].

The author of [3] initiated the study on -almost periodic functions, which turns out to be one of the most important generalizations of the concept of almost periodic functions in the sense of Bohr. This generalization relies on the requirement that a given function and its derivatives up to the th order inclusively are almost periodic in the sense of Bohr. Many properties of such functions with real values are given in [3, 4]. Recently, -almost periodic functions have attracted more and more attention. For example, the authors of [5] extended the study on -almost periodicity to functions , where is a Banach space; in [6], the authors proved the existence of -almost periodic solutions for some ordinary differential equations by using the exponential dichotomy approach. For more results on -almost periodic functions, we refer readers to [79] and references therein.

On the other hand, the theory of time scales, which was introduced by Hilger [10] in his Ph.D. thesis in order to unify continuous and discrete analysis, has recently received lots of attention. The study of dynamic equations on time scales helps avoid proving results twice, once for differential equations and once for difference equations. Many authors obtained a lot of good results on the study on dynamic equations on time scales (see [1117] and reference therein). In [18], the authors proposed the concept of almost periodic time scales and the definition of almost periodic functions. They extended the study on almost periodicity to functions , where is an almost periodic time scale. However, to the best of our knowledge, there is no paper published on the existence of -almost periodic solutions of dynamic equations on time scales.

Motivated by the above discussion, in this paper, we first give the definition and some properties of -almost periodic functions on time scales. As an application, we are concerned with the existence and global exponential stability of -almost periodic solutions for the following Lasota-Wazewska model on time scales: where , is an almost periodic time scale, denotes the number of red blood cells at time , is the rate of the red blood cells, and describe the production of red blood cells per unite time, and is the time required to produce a red blood cell and satisfies for , . There is extensive literature concerning oscillation, global attractivity, periodicity, almost periodicity, and Hopf bifurcation of Lasota-Wazewska model, which was proposed to describe the survival of red blood cells in animals [19]. We refer readers to [2025] and references therein for results on Lasota-Wazewska models.

Due to the biological meaning of (1), we just consider the following initial conditions: where is bounded, . Throughout this paper, we denote .

2. Preliminaries

In this section, we introduce some definitions and state some preliminary results.

Definition 1 (see [10]). Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess are defined, respectively, by

Definition 2 (see [10]). A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .

Definition 3 ([26]). A function is rd-continuous provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions will be denoted by .

Definition 4 ([26]). A function is called regressive if for all . If is regressive function, then the generalized exponential function is defined by with the cylinder transformation

Definition 5 (see [26]). A function is called regressive provided for all ; is called positively regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by and the set of all positively regressive functions and rd-continuous functions will be denoted by .

Lemma 6 (see [26]). Assume that are two regressive functions; then (i) and ;(ii);(iii);(iv).

Lemma 7 (see [26]). Let be -differentiable functions on ; then (i) for any constants ;(ii).

Lemma 8 (see [26] (Leibniz formula)). Let be the set consisting of all possible strings of length , containing exactly times and times . If exists for all , then

Lemma 9 (see [26]). Suppose that ; then (i), for all ;(ii)if for all ,  , then for all .

Lemma 10 (see [26]). If and , then

Lemma 11 (see [26]). Let be right-dense continuous and regressive, and . Then the unique solution of the initial value problem is given by

Definition 12 (see [18]). A time scale is called an almost periodic time scale if
Throughout this paper, we restrict our results on almost periodic time scales. We first recall some definitions and lemmas on almost periodic functions on almost periodic time scales, which can be found in [18].

Definition 13 (see [18]). Let be an almost periodic time scale. A function is said to be almost periodic on , if, for any , the set is relatively dense in ; that is, for any , there exists a constant such that each interval of length contains at least one such that The set is called the -translation set of , and is called the -translation number of .

Lemma 14 (see [18]). If is an almost periodic function, then is bounded on .

Lemma 15 (see [18]). If are almost periodic functions, then are also almost periodic.

Lemma 16 (see [18]). If is almost periodic, then is almost periodic if and only if is bounded.

Lemma 17 (see [18]). If is almost periodic and is uniformly continuous on the value field of , then is almost periodic.

Definition 18 (see [27]). Let and be an   -continuous matrix on ; the linear system is said to admit an exponential dichotomy on if there exist positive constants , projection , and the fundamental solution matrix of (13) satisfying where is a matrix norm on ; that is, if , then we can take .

Lemma 19 (see [18]). If the linear system (13) admits an exponential dichotomy, then the following system has a solution as follows: where is the fundamental solution matrix of (13).

Lemma 20 (see [18]). Let be a function on , where ,  ,  for all and . Then the linear system admits an exponential dichotomy on .

3. -Almost Periodic Functions on Time Scales

In this section, we will state the definition and prove some properties of -almost periodic functions on time scales.

We denote by the space of all functions which have a continuous th -derivative on and by the subspace of consisting of such functions satisfying , where denotes the th -derivative of and . It is not difficult to verify that is a Banach space with the norm .

Definition 21. Let be an almost periodic time scale. A function is said to be -almost periodic on , if, for any , the set is relatively dense in ; that is, for any , there exists a constant such that each interval of length contains at least one such that

Remark 22. We denote by the set of all -almost periodic functions from to . In particular, we denote by , which is the set of all almost periodic functions from to .

Theorem 23. if and only if , .

Proof. Assume that ; then, for any , there exists a constant such that in any interval of length there exists such that Hence, for , we have that which means that ,  . On the other hand, if , , then, for any , there exists a constant such that, in any interval of length , there exists such that Therefore, for any , there exists a constant such that in any interval of length there exists such that that is, . This completes the proof.

Theorem 24. If , then is bounded on .

Proof. Since , it follows from Theorem 23 that , . By Lemma 14, there exist positive constants such that . Hence, , which implies that is bounded on . This completes the proof.

Theorem 25. If , , then are all -almost periodic on . Moreover, if , , then .

Proof. Since the proofs of are similar to that of , we only prove that . Since , it follows from Theorem 23 that , . By Lemma 15, , . Hence, , , which means .

Theorem 26. If , then if and only if is bounded.

Similar to the proofs of Theorem 2.7 in [5] and Theorems 3.13 and 3.14 in [18], we have the following theorem, which is an analogue of Bochner's criterion for the case of -almost periodicity on time scales.

Theorem 27. A function if and only if for every sequence there exists a subsequence such that converges uniformly in , .

Definition 28. Let be an almost periodic time scale. A functions set is said to be equi-almost periodic on , if, for any , there exists a constant such that each interval of length contains at least one such that for all

Definition 29. Let be an almost periodic time scale. A functions set is said to be equi--almost periodic on , if for any there exists a constant such that each interval of length contains at least one such that for all

Similar to Theorem 2.2 in [7], we have the following theorem.

Theorem 30. Let be an almost periodic time scale. For a functions set is precompact if and only if is precompact, equicontinuous, and equi-almost periodic, where , .

Definition 31. Let be a -almost periodic solution of (1) with initial value . If there exist positive constants with and such that any solution of (1) with initial value satisfies where , , then the solution is said to be globally exponentially stable.

4. -Almost Periodic Solutions of (1)

In this section, we will state and prove the sufficient conditions for the existence and global exponential stability of -almost periodic solutions of (1).

Set ,   with the norm ; then is a Banach space. For convenience, for a -almost periodic function , we denote .

Lemma 32. Assume that Then, for every solution of (1) and (2), we have that for are positive and bounded on .

Proof. By Lemma 11, we have where , . First, we prove By way of contradiction, assume that (28) does not hold. Then, there exists and the first time such that
Then, we can obtain which is a contradiction and hence (28) holds. On the other hand, for , we have that which implies that is bounded on . This completes the proof.

Theorem 33. Let hold. Suppose further that then there is a unique -almost periodic solution of (1) in , where is a constant satisfying .

Proof. For any , we consider the following -almost periodic system: where . Since and , it follows from Lemma 19 that the linear system admits an exponential dichotomy on . Thus, (32) has a -almost periodic solution , where
Define a map on by where . It is obvious that is a Banach space with the norm . At first, we show that is a self-mapping from to . For any , we have Therefore, we have that . Hence, the mapping is a self-mapping from to . Next, we prove that the mapping is a contraction mapping on . For any , , since we have that Hence, we obtain
Noting that , we see that is a contraction mapping on . By the fixed point theorem in Banach space, has a unique fixed point in , which implies that (1) has a unique -almost periodic solution in . This completes the proof.

Theorem 34. Let and hold. Suppose further that then the -almost periodic solution of (1) is globally exponentially stable.

Proof. According to Theorem 33, we know that (1) has a -almost periodic solution with initial condition . Suppose that is an arbitrary solution of (1) with initial condition . Denote , . Then, it follows from (1) that where . The initial condition of (40) is By Lemma 11 and (40), for , we have
Take a constant with such that . Let be a constant satisfying By and , it is easy to verify that and in view of for , we have We claim that To prove this claim, we show that, for any , the following inequality holds: which implies that, for , we have By way of contradiction, assume that (46) does not hold. Firstly, we consider the following two cases.
Case One. (47) is not true. Then there exists and such that Hence, there must be a constant such that In view of (42), we have which is a contradiction.
Case Two. (48) is not true. Then there exists and such that Hence, there must be a constant such that In view of (42), we have which is a contradiction. Therefore, (46) holds. Let ; then (45) holds. Hence, we have that that is, the -almost periodic solution of (1) is globally exponentially stable. This completes the proof.

5. An Example

In this section, we present an example to illustrate the feasibility of our results obtained in previous sections.

Example 1. Let . Consider the following Lasota-Wazewska model on an almost periodic time scale : in which we take coefficients as follows:

By calculating, we have It can be verified that all conditions of Theorems 33 and 34 are satisfied. Therefore, (57) has a -almost periodic solution, which is globally exponentially stable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant no. 11361072.

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