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Dynamics for Nonlinear Difference Equation
Advances in Difference Equations volume 2009, Article number: 235691 (2009)
Abstract
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 nonoscillatory solutions of the equation. Our results extend and generalize the known ones.
1. Introduction
Consider the following higher order difference equation:
where , the parameters and are nonnegative real numbers and the initial conditions and are nonnegative real numbers such that
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.
The change of variables reduces (1.1) into the following equation:
where .
Note that is always an equilibrium point of (1.3). When , (1.3) also possesses the unique positive equilibrium .
The linearized equation of (1.3) about the equilibrium point is
so, the characteristic equation of (1.3) about the equilibrium point is either, for ,
or, for
The linearized equation of (1.3) about the positive equilibrium point has the form
with the characteristic equation either, for ,
or, for
When , , (1.1) has been investigated in [1–4]. When , , (1.1) reduces to the following form:
ElOwaidy et al. [3] 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 [3]. 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 [3].
For the sake of convenience, we now present some definitions and known facts, which will be useful in the sequel.
Consider the difference equation
where is a positive integer, and the function has continuous partial derivatives.
A point is called an equilibrium of (1.11) if
That is, for is a solution of (1.11), or equivalently, is a fixed point of .
The linearized equation of (1.11) associated with the equilibrium point is
We need the following lemma.

(i)
If all the roots of the polynomial equation
lie in the open unit disk , then the equilibrium of (1.11) is locally asymptotically stable.

(ii)
If at least one root of (1.11) has absolute value greater than one, then the equilibrium of (1.11) is unstable.
For the related investigations for nonlinear difference equations, see also [7–11] and the references cited therein.
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.
Lemma 2.1.
The following statements are true.
(a)If , then the equilibrium point of (1.3) is locally asymptotically stable.
(b)If , then the equilibrium point of (1.3) is unstable. Moreover, for , (1.3) has a dimension local stable manifold and a dimension local unstable manifold.
(c)If , is odd and is even, then the positive equilibrium point of (1.3) is unstable.
Proof.

(a)
When , it is clear from (1.5) and (1.6) that every characteristic root satisfies or , and so, by Lemma 1.1(i), is locally asymptotically stable.

(b)
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).

(c)
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.
Remark 2.2.
Lemma 2.1(a) includes and improves [3, Theorem 3.1(i)]. Lemma 2.1(b) and (c) include and generalize [3, Theorem 3.1(ii) and (iii)], respectively.
Now we state the main results in this section.
Theorem 2.3.
Assume that , then the equilibrium point of (1.3) is globally asymptotically stable.
Proof.
We know from Lemma 2.1 that the equilibrium point of (1.3) is locally asymptotically stable. It suffices to show that for any nonnegative solution of (1.3).
Since
converges for any . Let , then
Thereout, one has
that is,
which implies . The proof is over.
Remark 2.4.
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 .
Theorem 3.1.

(a)
Assume is odd and is even, then (1.3) possesses eventual prime period two solutions if and only if .

(b)
Assume is odd and is odd, then (1.3) possesses eventual prime period two solutions if and only if .

(c)
Assume is even and is even. Then the necessary condition for (1.3) to possess eventual prime period two solutions is and .

(d)
Assume is even and is odd. Then, (1.3) has no eventual prime period two solutions.
Proof.

(a)
If (1.3) has the eventual nonnegative prime period two solution then, we eventually have and . Hence,
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, .
Conversely, if , then choose the initial conditions such as and , or such as and . We can see by induction that is the prime period two solution of (1.3).

(b)
Let be the eventual prime period two solution of (1.3), then, it holds eventually that and . Hence,
If , then . This is impossible. So . Moreover, and or and , that is, is the prime period two solution of (1.3).

(c)
Assume that (1.3) has the eventual nonnegative prime period two solution then eventually
Obviously, implies or vice versa. This is impossible. So . It is easy to see from (3.3) that and satisfy the equation
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.
Let in (3.4), then the equation has at least two distinct positive roots.
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.
Case 1.
It holds that . Then for all , hence, is convex. Again, . So it is impossible for to have two distinct positive roots.
Case 2.
It holds that and . Then, for , and so ; for , and so . At this time, one always has . Then cannot have two distinct positive roots.
Case 3.
It holds that , and . Then, for , and so and hence , that is, has no solutions for ; for , , that is, is convex for . Noticing , it is also impossible for to have two distinct positive roots for .
Case 4.
It holds that , and . This is only case where could have two distinct positive roots, which implies and .

(d)
Let …, φ, ψ, φ, ψ,… be the eventual nonnegative prime period two solution of (1.3), then, it is eventually true that
It is easy to see that and . So, we have
that is, and are two distinct positive roots of . Obviously, when , the has no positive roots.
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.
Theorem 4.1.
Assume , is odd and is even. Then, there exist solutions of (1.3) which oscillate about with semicycles of length one.
Proof.
We only prove the case where (the proof of the case where is similar and will be omitted). Choose the initial values of (1.3) such that
or
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.
Theorem 5.1.
Assume , is odd, and is even, then (1.3) possesses unbounded solutions.
Proof.
We only prove the case where (the proof of the case where is similar and will be omitted). Choose the initial values of (1.3) such that
In the following, assume . From the proof of Theorem 4.1, one can see that when is odd and that for even. Together with
It is derived that
So, is decreasing for odd whereas is increasing for even. Let
then one has
(1) for odd and for even,
(2).
Now, either for some even in which case the proof is complete, or for all even . We shall prove that this latter is impossible. In fact, we prove that for all even .
Assume for some even , then one has, by (5.2), . Noticing (1), one hence further gets . However is odd, according to (1), . This is a contradiction.
Therefore, for any even . Accordingly, are unbounded subsequences of this solution of (1.3) for even . Simultaneously, for odd , we get
The proof is complete.
Remark 5.2.
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.
Theorem 6.1.
Every nonoscillatory solution of (1.3) with respect to approaches .
Proof.
Let be any one nonoscillatory solution of (1.3) with respect to . Then, there exists an such that
or
We only prove the case where (6.1) holds. The proof for the case where (6.2) holds is similar and will be omitted. According to (6.1), for , one has
So, is decreasing for with upper bound . Hence, exists and is finite. Denote
Then . Taking limits on both sides of (1.3), we can derive
which shows and completes this proof.
A problem naturally arises: are there nonoscillatory solutions of (1.3)? Next, we will positively answer this question. Our result is as follows.
Theorem 6.2.
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 [12]. Now, for the sake of convenience of statement, we first state some preliminaries. For this, refer also to [13]. Consider a general real nonlinear difference equation of order with the form
where , . Let and be two sequences satisfying and as . Then (maybe under certain additional conditions), for any given , there exist a solution of (6.6) and an such
Denote
which is called an asymptotic stripe. So, if , then it is implied that there exists a real sequence such that and for .
We now state the inclusion theorem [12].
Lemma 6.3.
Let be continuously differentiable when , for , and . Let the partial derivatives of satisfy
as uniformly in for , , as far as . Assume that there exist a sequence and constants such that
for as , and suppose there exists an integer , with , such that
Then, for sufficiently large , there exists a solution of (6.6) satisfying (6.7).
Proof of Theorem 6.2.
We only prove the case where (the proof of the case where is similar and will be omitted). Put ( is denoted into for short). Then (1.3) is transformed into
An approximate equation of (6.12) is
provided that as . The general solution of (6.13) is
where and are the roots of the polynomial
Obviously, . So, has a positive root lying in the interval (0, 1). Now, choose the solution for this . For some , define the sequences and , respectively, as follows:
Obviously, and as .
Now, define again the function
Then the partial derivatives of w.r.t. , respectively, are
When , . So, , , as uniformly in for , .
Moreover, from the definition of the function and (6.17) and (6.18), after some calculation, we find
Now choose . Noting
we have Again,
where
Therefore, one has
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.
Remark 6.4.
If we take in (6.16), then . At this time, (1.3) possesses solutions which remain below its equilibrium for all , that is, (1.3) has solutions with a single negative semicycle.
Remark 6.5.
The appropriate equation (6.12) is just the linearized equation of (1.3) associated with .
Remark 6.6.
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.
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Acknowledgments
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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Chen, D., Li, X. & Wang, Y. Dynamics for Nonlinear Difference Equation . Adv Differ Equ 2009, 235691 (2009). https://doi.org/10.1155/2009/235691
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DOI: https://doi.org/10.1155/2009/235691
Keywords
 Asymptotic Behavior
 Difference Equation
 Positive Root
 Global Behavior
 Global Asymptotical Stability