Research Article | Open Access

# Unique Positive Almost Periodic Solution for Discrete Nonlinear Delay Survival Red Blood Cells Model

**Academic Editor:**John Mallet-Paret

#### Abstract

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.

#### 1. Introduction

Recently, Saker [1] 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 [17] obtained the following Theorem A.

Theorem 1 A. *Suppose that , and are nonnegative integer. If
**
then the unique positive equilibrium of (1.2) is a global attractor of all positive solutions of (1.2).*

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 [19], see also [20]. 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 [10] as well as Györi and Ladas [6].

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 [1] 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 [1] to the almost periodic case. In [1], the author use a fixed theorem to obtain the existence of positive periodic solution. The operator used in [1] 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].

Corollary 2.2. *Assume that the conditions of Theorem 2.1 are satisfied, and , , and are -periodic with . Then (1.1) admits a positive -periodic solution.*

*Remark 2.3. *When , under the conditions of Corollary 2.2, Saker in [1, Theorem 2.1] proved that the conclusion of Corollary 2.2. Thus Corollary 2.2 extends Theorem 2.1 in [1].

Theorem 2.4. *Assume that the conditions of Theorem 2.1 are satisfied and
**
where
**
and is a positive solution of (1.1). Then every positive solution of (1.1) satisfies
*

Corollary 2.5. *Assume that the conditions of Theorem 2.1 are satisfied, and
**
where is the constant given in Theorem 2.1. Then every positive of (1.1) is globally attractive. *

*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 [25] 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 [26]). *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.*

The proof is similar to that of [21, Theorem 2.10.1]. We omit it here.

#### 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.

#### Acknowledgments

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).

#### References

- S. H. Saker, “Qualitative analysis of discrete nonlinear delay survival red blood cells model,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 2, pp. 471–489, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M.-P. Chen and J. S. Yu, “Oscillations of delay difference equations with variable coefficients,” in
*Proceedings of the 1st International Conference on Difference Equations (San Antonio, TX, 1994)*, S. N. Elaydi, J. R. Graef, G. Ladas, and A. C. Peterson, Eds., pp. 105–114, Gordon and Breach, Luxembourg, UK, 1995. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. A. El-Morshedy and E. Liz, “Convergence to equilibria in discrete population models,”
*Journal of Difference Equations and Applications*, vol. 11, no. 2, pp. 117–131, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. H. Erbe, H. Xia, and J. S. Yu, “Global stability of a linear nonautonomous delay difference equation,”
*Journal of Difference Equations and Applications*, vol. 1, no. 2, pp. 151–161, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. H. Erbe and B. G. Zhang, “Oscillation of discrete analogues of delay equations,”
*Differential and Integral Equations*, vol. 2, no. 3, pp. 300–309, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet - I. Györi and G. Ladas,
*Oscillation Theory of Delay Differential Equations with Applications*, Oxford Mathematical Monographs, Clarendon Press/Oxford University Press, New York, NY, USA, 1991. View at: Zentralblatt MATH | MathSciNet - A. F. Ivanov, “On global stability in a nonlinear discrete model,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 23, no. 11, pp. 1383–1389, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. C. Jiang and X. H. Tang, “Oscillation of nonlinear delay difference equations,”
*Journal of Computational and Applied Mathematics*, vol. 146, no. 2, pp. 395–404, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Karakostas, C. G. Philos, and Y. G. Sficas, “The dynamics of some discrete population models,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 17, no. 11, pp. 1069–1084, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. L. Kocic and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. View at: Zentralblatt MATH | MathSciNet - I. Kovácsvölgyi, “The asymptotic stability of difference equations,”
*Applied Mathematics Letters*, vol. 13, no. 1, pp. 1–6, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Kubiaczyk and S. H. Saker, “Oscillation and global attractivity in a discrete survival red blood cells model,”
*Applicationes Mathematicae*, vol. 30, no. 4, pp. 441–449, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. R. S. Kulenović, G. Ladas, and Y. G. Sficas, “Global attractivity in population dynamics,”
*Computers & Mathematics with Applications*, vol. 18, no. 10-11, pp. 925–928, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G. Ladas, C. Qian, P. N. Vlahos, and J. Yan, “Stability of solutions of linear nonautonomous difference equations,”
*Applicable Analysis*, vol. 41, no. 1–4, pp. 183–191, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-W. Li and S. S. Cheng, “Global attractivity in an RBC survival model of Wazewska and Lasota,”
*Quarterly of Applied Mathematics*, vol. 60, no. 3, pp. 477–483, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet - W.-T. Li and S. S. Cheng, “Asymptotic properties of the positive equilibrium of a discrete survival model,”
*Applied Mathematics and Computation*, vol. 157, no. 1, pp. 29–38, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Ma and J. Yu, “Global attractivity of ${x}_{n+1}=(1-\alpha {x}_{n})+\beta \mathrm{exp}\hspace{0.17em}(-\gamma {x}_{n-k})$,”
*Computers & Mathematics with Applications*, vol. 49, no. 9-10, pp. 1397–1402, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Q. Meng and J. Yan, “Global attractivity of delay difference equations,”
*Indian Journal of Pure and Applied Mathematics*, vol. 30, no. 3, pp. 233–242, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Ważewska-Czyżewska and A. Lasota, “Mathematical problems of the dynamics of a system of red blood cells,”
*Roczniki Polskiego Towarzystwa Matematycznego. Seria III*, vol. 6, pp. 23–40, 1976. View at: Google Scholar | Zentralblatt MATH | MathSciNet - O. Arino and M. Kimmel, “Stability analysis of models of cell production systems,”
*Mathematical Modelling*, vol. 7, no. 9–12, pp. 1269–1300, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. M. Fink,
*Almost Periodic Differential Equations*, vol. 377 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1974. View at: Zentralblatt MATH | MathSciNet - B. M. Levitan and V. V. Zhikov,
*Almost Periodic Functions and Differential Equations*, Cambridge University Press, Cambridge, UK, 1982. View at: Zentralblatt MATH | MathSciNet - C. Zhang,
*Almost Periodic Type Functions and Ergodicity*, Science Press, Beijing, China, 2003. View at: Zentralblatt MATH | MathSciNet - X. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,”
*Journal of Mathematical Analysis and Applications*, vol. 316, no. 1, pp. 161–177, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Wang and Z. Li, “Globally dynamical behaviors for a class of nonlinear functional difference equation with almost periodic coefficients,”
*Applied Mathematics and Computation*, vol. 190, no. 2, pp. 1116–1124, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. W.-H. So and J. S. Yu, “Global attractivity and uniform persistence in Nicholson's blowflies,”
*Differential Equations and Dynamical Systems*, vol. 2, no. 1, pp. 11–18, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

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.