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Dynamical Properties in a FourthOrder Nonlinear Difference Equation
Advances in Difference Equations volume 2010, Article number: 679409 (2010)
Abstract
The rule of trajectory structure for fourthorder nonlinear difference equation , where and the initial values , is described clearly out in this paper. Mainly, the lengths of positive and negative semicycles of its nontrivial solutions are found to occur periodically with prime period 15. The rule is in a period. By utilizing this rule its positive equilibrium point is verified to be globally asymptotically stable.
1. Introduction
In this paper we consider the following fourthorder nonlinear difference equation:
where and the initial values
When , (1.1) becomes the trivial case , Hence, we will assume in the sequel that .
When , (1.1) is not a rational difference equation but a nonlinear one. So far, there have not been any effective general methods to deal with the global behavior of nonlinear difference equations of order greater than one. Therefore, to study the qualitative properties of nonlinear difference equations with higher order is theoretically meaningful.
In this paper, it is of key for us to find that the lengths of positive and negative semicycles of nontrivial solutions of (1.1) occur periodically with prime period 15 with the rule ,, , , , , , and in a period. With the help of this rule and utilizing the monotonicity of solution the positive equilibrium point of the equation is verified to be globally asymptotically stable.
Essentially, we derive the following results for solutions of (1.1).
Theorem CL.
The rule of the trajectory structure of (1.1) is that all of its solutions asymptotically approach its equilibrium; furthermore, any one of its solutions is either

(1)
eventually trivial

(2)
nonoscillatory and eventually negative (i.e., ) or

(3)
strictly oscillatory with the lengths of positive and negative semicycles periodically successively occurring with prime period 15 and the rule to be , , , , , , , in a period.
It follows from the results stated below that Theorem CL is true.
It is easy to see that the positive equilibrium of (1.1) satisfies
from which one can see that (1.1) has a unique equilibrium
In the following, we state some main definitions used in this paper.
Definition 1.1.
A positive semicycle of a solution of (1.1) consists of a "string" of terms , all greater than or equal to the equilibrium , with and such that
A negative semicycle of a solution of (1.1) consists of a "string" of term , all less than , with and such that
The length of a semicycle is the number of the total terms contained in it.
Definition 1.2.
A solution of (1.1) is said to be eventually trivial if is eventually equal to ; Otherwise, the solution is said to be nontrivial. A solution of (1.1) is said to be eventually positive (negative) if is eventually greater (less) than .
For the other concepts in this paper and related work, see [1–3] and [4–11], respectively.
2. Three Lemmas
Before drawing a qualitatively clear picture for the solutions of (1.1), we first establish three basic lemmas which will play key roles in the proof of our main results.
Lemma 2.1.
A solution of (1.1) is eventually trivial if and only if
Proof.
Sufficiency. Assume that (2.1) holds. Then it follows from (1.1) that the following conclusions hold:

(i)
if , then for ;

(ii)
if , then for ;

(iii)
if , then for ;

(iv)
if , then for .
Necessity. Conversely, assume that
Then one can show that
Assume the contrary that for some ,
Clearly,
which implies that , which contradicts (2.4).
Remark 2.2.
Lemma 2.1 actually demonstrates that a solution of (1.1) is eventually nontrivial if and only if
Therefore, if a solution is nontrivial, then for .
Lemma 2.3.
Let be a nontrivial positive solution of (1.1). Then the following conclusions are true:

(a)
for ;

(b)
for .
Proof.
In view of (1.1), we can see that
from which inequalities (a) and (b) follow. So the proof is complete.
Lemma 2.4.
There exist nonoscillatory solutions of (1.1), which must be eventually negative. There do not exist eventually positive nonoscillatory solutions of (1.1).
Proof.
Consider a solution of (1.1) with , , and . We then know from Lemma 2.3(a) that for . So, this solution is just a nonoscillatory solution, and furthermore, eventually negative. Suppose that there exist eventually positive nonoscillatory solutions of (1.1). Then, there exists a positive integer such that for . Thereout, for , . This contradicts Lemma 2.3(a). So, there do not exist eventually positive nonoscillatory solutions of (1.1), as desired.
3. Main Results and Their Proofs
First we analyze the structure of the semicycles of nontrivial solutions of (1.1). Here we confine us to consider the situation of the strictly oscillatory solution of (1.1).
Theorem 3.1.
Let be any strictly oscillatory solution of (1.1). Then, the lengths of positive and negative semicycles of the solution periodically successively occur with prime period 15. And in a period, the rule is , , , , , , , .
Proof.
By Lemma 2.3(a), one can see that the length of a positive semicycle is not larger than 4, whereas, the length of a negative semicycle is at most 3. Based on the strictly oscillatory character of the solution, we see, for some integer , that one of the following four cases must occur.
Case 1.
, , , .
Case 2.
, , , .
Case 3.
, , , .
Case 4.
, , , .
If Case 1 occurs, it follows from Lemma 2.3(a) that , ,, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,, which means that the rule for the lengths of positive and negative semicycles of the solution of (1.1) to successively occur is
If Case 2 happens, then Lemma 2.3(a) tells us that , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,. This shows that the rule for the numbers of terms of positive and negative semicycles of the solution of (1.1) to successively occur still is
If Case 3 happens, then Lemma 2.3(a) implies that , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , This shows that the rule for the numbers of terms of positive and negative semicycles of the solution of (1.1) to successively occur still is
If Case 4 happens, then it is to be seen from Lemma 2.3(a) that , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,. This shows that the rule for the numbers of terms of positive and negative semicycles of the solution of (1.1) to successively occur still is
Hence, the proof is complete.
Now, we present the global asymptotical stable results for (1.1).
Theorem 3.2.
Assume that . Then the unique positive equilibrium of (1.1) is globally asymptotically stable.
Proof.
When , (1.1) is trivial. So, we only consider the case and prove that the positive equilibrium point of (1.1) is both locally asymptotically stable and globally attractive. The linearized equation of (1.1) about the positive equilibrium is
By virtue of [3, Remark , page 13], is locally asymptotically stable. It remains to be verified that every positive solution of (1.1) converges to as . Namely, we want to prove that
If the initial values of the solutions satisfy (2.1), that is to say, the solution is a trivial solution, then Lemma 2.1 says that the solution is eventually equal to and of course (3.2) holds.
If the solution is a nontrivial solution, then we can further divide the solution into two cases.

(a)
nonoscillatory solution;

(b)
oscillatory solution.
If Case happens, then it follows from Lemma 2.3 that the solution is actually an eventually negative one. According to Lemma 2.3(b), we see that , , and are eventually increasing and bounded from the upper by the constant . So the limits
exist and are finite. Noting that
and taking the limits on both sides of the above equalities, respectively, one may obtain
Solving these equations, we get , which shows that (3.2) is true.
If case (b) happens, the solution is strictly oscillatory.
Consider now to be strictly oscillatory about the positive equilibrium point of (1.1). By virtue of Theorem 3.1, one understands that the lengths of positive and negative semicycles of the solution periodically successively occur, and in a period, the rule is
For simplicity, for some integer , we denote by the terms of a positive semicycle of length four, followed by negative semicycle with length three, then a positive semicycle , a negative semicycle , a positive semicycle , a negative semicycle , a positive semicycle , and a negative semicycle . Namely, the rule for the lengths of negative and positive semicycles to occur successively can be periodically expressed as follows:, , , , , , , , and
From Lemma 2.3(b), we may immediately obtain the following results:

(i)
; ;

(ii)
; .
Also, the following inequalities hold:

(iii)
; ;

(iv)
; ;

(v)
;

(vi)
; ; ;

(vii)
; .
In fact, from the observation that
we know that the first inequality in (iii) is true. The other inequalities in (iii)–(vi) can be similarly proved. Noticing that and from that the observation
we know that the first inequality in (vii) holds. The other inequality in (vii) can be analogously proved.
Combining the above inequalities, one can derive that
It follows from (3.8) that is decreasing with lower bound 1. So, the limit
exists and is finite. Accordingly, by view of (3.8), we obtain
It is easy to see from (3.9) that is decreasing with lower bound 1. So, the limit
exists and is finite. Thereout, in light of (3.9), one has
It follows from (3.10) that is decreasing with lower bound 1. So, the limit
exists and is finite. Accordingly, by view of (3.10), we obtain
Taking the limits on both sides of , one has , which gives rise to . We further obtain from (i) and (3.12) that
Hence, . Therefore,
It is easy to derive from (v) that . Noticing that , one can see that .
Similarly, taking the limits on both sides of , one has . Finally, by taking the limits on both sides of , one has .
Up to now, we have shown that
So, the proof for Theorem 3.2 is complete.
Remark 3.3.
One can see from the process of proofs stated previously that these results in this paper also hold for .
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Acknowledgment
This work is partly supported by NNSF of China (grant: 10771094) and the Foundation for the Innovation Group of Shenzhen University (grant: 000133).
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Chen, Y., Li, X. Dynamical Properties in a FourthOrder Nonlinear Difference Equation. Adv Differ Equ 2010, 679409 (2010). https://doi.org/10.1155/2010/679409
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Keywords
 Functional Equation
 Equilibrium Point
 Nontrivial Solution
 Clear Picture
 Stable Result