# On a Conjecture for a Higher-Order Rational Difference Equation

- Maoxin Liao
^{1, 2}Email author, - Xianhua Tang
^{1}and - Changjin Xu
^{1, 3}

**2009**:394635

**DOI: **10.1155/2009/394635

© Maoxin Liao et al. 2009

**Received: **30 December 2008

**Accepted: **14 March 2009

**Published: **1 April 2009

## Abstract

## 1. Introduction

converges to its unique equilibrium , where and . Based on this fact, they put forward the following two conjectures.

Conjecture.

with Then, the sequence converges to the unique equilibrium 1.

Conjecture.

Then the sequence converges to the unique equilibrium 1.

Motivated by [2], Berenhaut et al. started with the investigation of the following difference equation for (see, [3, 4]). Among others, in [3] they used a transformation method, which has turned out to be very useful in studying (1.1) and (1.2) as well as in confirming Conjecture 1.1; see [5].

Some particular cases of (1.2) had been studied previously by Li in [6, 7], by using semicycle analysis similar to that in [8]. The problem concerning periodicity of semicycles of difference equations was solved in very general settings by Berg and Stević in [9], partially motivated also by [10].

In the meantime, it turned out that the method used in [11] by Çinar et al. can be used in confirming Conjecture 1.2 (see also [12]). More precisely [11, 12] use Corollary 3 from [13] in solving similar problems. For example, Çinar et al. has shown, in an elegant way, that the main result in [14] is a consequence of Corollary 3 in [13]. With some calculations it can be also shown that Conjecture 1.2 can be confirmed in this way (see [15]).

Some other related results can be found in [16–24].

In this paper, we will prove that Conjecture 1.2 is correct by using a new method. Obviously, our results generalize the corresponding works in [1, 5–7] and other literature.

## 2. Preliminaries and Notations

Then we can rewrite (1.3) as

or

The following lemma can be obtained by simple calculations.

Lemma 2.1.

Lemma 2.2.

Proof.

- (1)
- (2)
- (3)
- (4)

From the above inequalities, it follows that (2.6) holds. The proof is complete.

Lemma 2.3.

Proof.

It follows that (2.9) holds. The proof is complete.

Lemma 2.4.

Proof.

it follows from (2.25) and (2.18) that . The proof is complete.

## 3. Proof of Conjecture 1.2

Theorem 3.1.

Theorem 3.1 is a direct corollary of Lemmas 2.2 and 2.3.

Proof.

which implies that (3.3) holds. The proof of Conjecture 1.2 is complete.

## Declarations

### Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and many valuable comments and suggestions that greatly improved the presentation of this work. This work is supported partly by NNSF of China (Grant: 10771215, 10771094), Project of Hunan Provincial Youth Key Teacher and Project of Hunan Provincial Education Department (Grant: 07C639).

## Authors’ Affiliations

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