- Research Article
- Open Access

# Dynamical Properties in a Fourth-Order Nonlinear Difference Equation

- Yunxin Chen
^{1}and - Xianyi Li
^{2}Email author

**2010**:679409

https://doi.org/10.1155/2010/679409

© Y. Chen and X. Li. 2010

**Received:**21 October 2009**Accepted:**29 March 2010**Published:**9 May 2010

## Abstract

The rule of trajectory structure for fourth-order 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.

## Keywords

- Functional Equation
- Equilibrium Point
- Nontrivial Solution
- Clear Picture
- Stable Result

## 1. Introduction

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 semi-cycles 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.

- (1)
eventually trivial

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

- (3)
strictly oscillatory with the lengths of positive and negative semi-cycles 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.

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.

The length of a semi-cycle 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.

Proof.

- (i)
if , then for ;

- (ii)
if , then for ;

- (iii)
if , then for ;

- (iv)
if , then for .

which implies that , which contradicts (2.4).

Remark 2.2.

Therefore, if a solution is nontrivial, then for .

Lemma 2.3.

- (a)
for ;

- (b)
for .

Proof.

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 non-oscillatory 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 non-oscillatory 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 non-oscillatory solutions of (1.1), as desired.

## 3. Main Results and Their Proofs

First we analyze the structure of the semi-cycles 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 semi-cycles 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 semi-cycle is not larger than 4, whereas, the length of a negative semi-cycle 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 semi-cycles 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 semi-cycles 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 semi-cycles 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 semi-cycles 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.

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.

- (a)
non-oscillatory solution;

- (b)
oscillatory solution.

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 semi-cycles 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 semi-cycle of length four, followed by negative semi-cycle with length three, then a positive semi-cycle , a negative semi-cycle , a positive semi-cycle , a negative semi-cycle , a positive semi-cycle , and a negative semi-cycle . Namely, the rule for the lengths of negative and positive semi-cycles to occur successively can be periodically expressed as follows: , , , , , , , , and

- (i)
; ;

- (ii)
; .

- (iii)
; ;

- (iv)
; ;

- (v)
;

- (vi)
; ; ;

- (vii)
; .

we know that the first inequality in (vii) holds. The other inequality in (vii) can be analogously proved.

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 .

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 .

## Declarations

### Acknowledgment

This work is partly supported by NNSF of China (grant: 10771094) and the Foundation for the Innovation Group of Shenzhen University (grant: 000133).

## Authors’ Affiliations

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