© Dongmei Chen et al. 2009
Received: 19 April 2009
Accepted: 9 October 2009
Published: 13 October 2009
We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones.
Consider the following higher order difference equation:
It is easy to see that if one of the parameters is zero, then the equation is linear. If , then (1.1) can be reduced to a linear one by the change of variables . So in the sequel we always assume that the parameters and are positive real numbers.
El-Owaidy et al.  investigated the global asymptotical stability of zero equilibrium, the periodic character and the existence of unbounded solutions of (1.10).
On the other hand, when , (1.1) is just the discrete delay logistic model investigated in [4, ]. Therefore, it is both theoretically and practically meaningful to study (1.1).
Our aim in this paper is to extend and generalize the work in . That is, we will investigate the global behavior of (1.1), including the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence and asymptotic behavior of nonoscillatory solutions of the equation. Our results extend and generalize the corresponding ones of .
For the sake of convenience, we now present some definitions and known facts, which will be useful in the sequel.
Consider the difference equation
We need the following lemma.
2. Global Asymptotic Stability of Zero Equilibrium
In this section, we investigate global asymptotic stability of zero equilibrium of (1.3). We first have the following results.
The following statements are true.
When , if , then it is clear from (1.5) that every characteristic root satisfies , and so, by Lemma 1.1(ii), is unstable. If , then (1.6) has characteristic roots satisfying , which corresponds to a -dimension local stable manifold of (1.3), and characteristic roots satisfying , which corresponds to a -dimension local unstable manifold of (1.3).
If is odd and is even, then, regardless of or , correspondingly, the characteristic equation (1.8) or (1.9) always has one characteristic root lying the interval . It follows from Lemma 1.1(ii) that is unstable.
Now we state the main results in this section.
Theorem 2.3 includes [3, Theorem 3.3].
3. Existence of Eventual Period Two Solution
In this section, one studies the eventual nonnegative prime period two solutions of (1.3). A solution of (1.3) is said to be eventual prime periodic two solution if there exists an such that for and holds for all .
If , then we can derive from (3.1) that if or vice versa, which contradicts the assumption that is the eventual prime period two solution of (1.3). So, . Accordingly, and , which indicate that when or that and do not exist when , which are also impossible. Therefore, .
that is, and are two distinct positive roots of . From (3.4) we can see that does not have two distinct positive roots at all when , alternatively, (1.3) does not have the nonnegative prime period two solution at all when . Therefore, we assume in the following.
By simple calculation, one has
If , we can see for all . This means that is strictly increasing in the interval and hence the equation, cannot have two distinct positive roots. So, next we consider , which implies . Denote . We need to discuss several cases, respectively, as follows.
Let …, φ, ψ, φ, ψ,… be the eventual nonnegative prime period two solution of (1.3), then, it is eventually true that
Now let and set . Then the function, has at least two distinct positive roots. However, for any , which indicates that is strictly increasing in the interval . This implies that the function does not have two distinct positive roots at all in the interval . In turn, (1.3) does not have the prime period two solution when .
4. Existence of Oscillatory Solution
For the oscillatory solution of (1.3), we have the following results.
We will only prove the case where (4.1) holds. The case where (4.2) holds is similar and will be omitted. According to (1.3), one can see that
So, the proof follows by induction.
5. Existence of Unbounded Solution
With respect to the unbounded solutions of (1.3), the following results are derived.
then one has
The proof is complete.
Theorem 5.1 includes and generalizes [3, Theorem 3.5].
6. Existence and Asymptotic Behavior of Nonoscillatory Solution
In this section, we consider the existence and asymptotic behavior of nonoscillatory solution of (1.3). Because all solutions of (1.3) are nonnegative, relative to the zero equilibrium point , every solution of (1.3) is a positive semicycle, a trivial nonoscillatory solution! Thus, it suffices to consider the positive equilibrium when studying the nonoscillatory solutions of (1.3). At this time, .
Firstly, we have the following results.
A problem naturally arises: are there nonoscillatory solutions of (1.3)? Next, we will positively answer this question. Our result is as follows.
However (1.3) possesses asymptotic solutions with a single semicycle (positive semicycle or negative semicycle).
The main tool to prove this theorem is to make use of Berg' inclusion theorem . Now, for the sake of convenience of statement, we first state some preliminaries. For this, refer also to . Consider a general real nonlinear difference equation of order with the form
We now state the inclusion theorem .
Proof of Theorem 6.2.
Now, define again the function
Up to here, all conditions of Lemma 6.3 with and are satisfied. Accordingly, we see that, for arbitrary and for sufficiently large , say , (6.12) has a solution in the stripe , where and are as defined in (6.16). Because , for . Thus, (1.3) has a solution satisfying for . Since (1.3) is an autonomous equation, still is its solution, which evidently satisfies for . Therefore, the proof is complete.
The appropriate equation (6.12) is just the linearized equation of (1.3) associated with .
The existence and asymptotic behavior of nonoscillatory solution of special cases of (1.3) has not been found to be considered in the known literatures.
This work of the second author is partly supported by NNSF of China (Grant: 10771094) and the Foundation for the Innovation Group of Shenzhen University (Grant: 000133). Y. Wang work is supported by School Foundation of JiangSu Polytechnic University(Grant: JS200801).
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