#### Abstract

The difference equation is studied and some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable, are obtained.

#### 1. Introduction

The delay differential equation was first proposed by Wazewska-Czyzewska and Lasota [1] as a model for the survival of red blood cell in an animal. Here, denotes the number of red blood cells at time , is the probability of death of red blood cells, and are positive constants which are related to the production of red blood cells, and is the time which is required to produce a red blood cell. The oscillation and global attractivity of (1) were studied by Győri and Ladas [2] and Li and Cheng [3], while the bifurcation and the direction of the stability were investigated by Song et al. [4]. Xu and Li [5] and Liu [6] considered its generalization with several delays and obtained sufficient conditions for the global stability of survival blood cells model with several delays and piecewise constant argument.

Research on the oscillation and global stability of the discrete analogue of (1), that is, for the equation where was proposed by Kocić and Ladas [7] as an open problem.

Kubiaczyk and Saker [8] investigated the oscillation of (2) about its positive equilibrium point , where is the unique solution of the equation and showed that every solution of (2) oscillates about if

Meng and Yan [9] investigated the global attractivity of the positive equilibrium point and showed that is a global attractor of all positive solutions of (2) if where .

Zeng and Shi [10] established another condition for global attractivity of and showed that is a global attractor of all positive solutions of (2) if

Obviously, the condition (7) improves (6).

Kubiaczyk and Saker [8] also considered (2) when and proved that is a global attractor of all positive solutions of (2) provided that

Ma and Yu [11] proved that is a global attractor of all solutions of (2) if

By (2), we have So, if (7) holds, then we have , which implies that . Hence, (8) is satisfied. But, the converse is not true. So, the condition (8) improves (7).

In addition, we can also easily see that the conditions (7) and are equivalent to the condition .

For the system with delay, many authors deemed that arbitrary finite number of discrete delays is more appropriate than the single discrete delay; see [12–14] and the references cited therein.

Stemming from the above discussion, the difference equation in the following form will be studied in this paper: where Besides, we denote that , , .

Obviously, the case is the form of (2). Besides, if , , then, the corresponding solution of (10) is positive, and (10) has a unique positive equilibrium point , which satisfies

The aim of this paper is to investigate the oscillation and the global asymptotic stability of (10).

#### 2. Some Lemmas

Lemma 1 (see [7, page 6]). *Assume that and with , . Let be sequences of positive numbers such that
**
Suppose that the linear difference inequality
**
has an eventually positive solution. Then, the difference equation
**
has a positive solution.*

Lemma 2 (see [7, page 5]). *Consider the linear homogeneous difference equation
**
where is a nonnegative integer and , . Then, the following statements are equivalent:*(a)*every solution of (16) oscillates;*(b)*the characteristic equation of (16)
**has no positive roots.*

Lemma 3 (see [7, page 12]). *Assume that and is a nonnegative integer. Then, is a sufficient condition for the asymptotic stability of the difference equation
*

Lemma 4. *Assume that (11) holds, and is a solution of (10) with positive initial conditions . Then,
*

*Proof. *Clearly, we have , for . So by (10), we can find that
Define a sequence by
Obviously,
So, we have

Lemma 5. *Assume that (11) holds, and
**Then, the positive equilibrium of (10) is locally asymptotically stable.*

*Proof. *To prove that the positive equilibrium is locally asymptotically stable, it suffices to prove that the zero solution of the linear equation of (10) is locally asymptotically stable. The linearized equation associated with (10) about positive equilibrium is
which satisfies
Then, by Lemma 3, the positive equilibrium solution of (10) is locally asymptotically stable.

Lemma 6 (see [15]). *The following system of inequalities,
**
with being real numbers, have exactly one solution .*

#### 3. Main Results

Theorem 7. *Assume that (11) holds, and
**
Then, every positive solution of (10) oscillates about the positive equilibrium .*

*Proof. *Assume for the sake of contradiction that (10) has a positive solution which does not oscillate about . We assume that eventually. If eventually, the proof is similar and will be omitted. So, there exists an such that for , and consequently for , where .

From Lemma 4, we have as a bounded sequence. In the following, we will claim that
Otherwise, let
Then, and there exists a subsequence such that
Equation (10) can be reformulated in the form
Then, from (31) and (32), we find that
So, we obtain
which is a contradiction. Accordingly, (29) holds.

Set
By the assumption , we have that is an eventually positive solution of the difference equation
which can also be rewritten in the form
where , .

By some simple calculations and (29), we get

One can easily see that the hypothesis of Lemma 1 is satisfied and so the linear equation
has an eventually positive solution.

Let be an eventually positive solution of (39); then is an eventually positive solution of
Let
be the characteristic polynomial of (40). Now, we prove that , for .

If , then obviously . Else if , we have

Therefore, the characteristic equation of (40)
has no positive roots.

According to Lemma 2, (40) has no nonoscillatory solution.

This is a contradiction. The proof is completed.

Theorem 8. *Assume that (11) holds, and
**
Then, the positive equilibrium of (10) is a global attractor of all positive solutions of (10).*

*Proof. *To prove that the positive equilibrium is a global attractor of all positive solutions of (10), it suffices to show that (29) holds.

We will prove that (29) holds in each of the following two cases.*Case 1 (** is nonoscillatory).* Let be eventually positive. The case that is eventually positive is similar and will be omitted. So,there exists an such that for , and consequently for , where .

From Lemma 4, we have as a bounded sequence. Assume for the sake of contradiction that (29) is not satisfied. Let
Then, and there exists a subsequence such that
It follows from (10) that
So, we obtain
which is a contradiction. Accordingly, (29) holds.*Case 2 (** is strictly oscillatory).* To show that (29) holds, it suffices to prove that holds, when is a strictly oscillatory solution of (36).

To this end, let
be the th positive semicycle of followed by the th negative semicycle
Let , be the extreme values in these two semicycles with the smallest possible indices and . Then, we claim that
In the following, we will prove that (51) holds for positive semicycles, while for negative semicycles, the proof is similar and will be omitted. Assume for the sake of contradiction that the first inequality in (51) is not true. Then, and the terms are in a positive semicycle. Because of , (36) renders
So, we have
So there exists at least a s.t. , which contradicts that is in the positive semicycle. So, (51) is true. Noting that is bounded from Lemma 4, we can let

To prove that holds, it is sufficient to show that .

From (54), it follows that if is given, then there exists such that
Equation (36) can be reformulated in the form
Multiplying (56) by and then summing up from to for being sufficiently large, we get
From (55) and , we have
So,
By using (54), is arbitrary and ; we get
From the assumption of the theorem, we have
By the same trick as in proving (61), we can prove that
Therefore, by Lemma 6, we can get ; that is, , which implies that is a global attractor of all positive solutions of (10).

By Lemma 3 and Theorem 8, we can get the following result.

Theorem 9. *Suppose that (11) holds and that
**
Then, the positive equilibrium is globally asymptotically stable.*

*Remark 10. *From Theorem 7, it is clear that if the condition (28) holds, then the oscillation condition for as established by Kubiaczyk and Saker [8] is already satisfied.

*Remark 11. *When , the condition of Theorem 8 is independent from the argument .

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (G61074068, G61034007, G61174036, G61374065, and G61374002), the Fund for the Taishan Scholar Project of Shandong Province, the Natural Science Foundation of Shandong Province (ZR2010FM013), and the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01).