- Research Article
- Open Access

# 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

https://doi.org/10.1155/2009/394635

© Maoxin Liao et al. 2009

**Received:**30 December 2008**Accepted:**14 March 2009**Published:**1 April 2009

## Abstract

This paper studies the global asymptotic stability for positive solutions to the higher order rational difference equation , where is odd and . Our main result generalizes several others in the recent literature and confirms a conjecture by Berenhaut et al., 2007.

## Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- General Setting

## 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

where is an odd integer and .

The following lemma can be obtained by simple calculations.

Lemma 2.1.

Lemma 2.2.

.

Proof.

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

(m+1)

- (1)
;

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

(m+1) .

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

## References

- Berenhaut KS, Foley JD, Stević S:
**The global attractivity of the rational difference equation**.*Applied Mathematics Letters*2007,**20**(1):54-58. 10.1016/j.aml.2006.02.022MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On the recursive sequence**.*Journal of Applied Mathematics & Computing*2005,**18**(1-2):229-234. 10.1007/BF02936567MATHMathSciNetView ArticleGoogle Scholar - Berenhaut KS, Foley JD, Stević S:
**The global attractivity of the rational difference equation**.*Proceedings of the American Mathematical Society*2007,**135**(4):1133-1140. 10.1090/S0002-9939-06-08580-7MATHMathSciNetView ArticleGoogle Scholar - Berenhaut KS, Foley JD, Stević S:
**The global attractivity of the rational difference equation**.*Proceedings of the American Mathematical Society*2008,**136**(1):103-110. 10.1090/S0002-9939-07-08860-0MATHMathSciNetView ArticleGoogle Scholar - Berenhaut KS, Stević S:
**The global attractivity of a higher order rational difference equation.***Journal of Mathematical Analysis and Applications*2007,**326**(2):940-944. 10.1016/j.jmaa.2006.02.087MATHMathSciNetView ArticleGoogle Scholar - Li X:
**Qualitative properties for a fourth-order rational difference equation.***Journal of Mathematical Analysis and Applications*2005,**311**(1):103-111. 10.1016/j.jmaa.2005.02.063MATHMathSciNetView ArticleGoogle Scholar - Li X:
**Global behavior for a fourth-order rational difference equation.***Journal of Mathematical Analysis and Applications*2005,**312**(2):555-563. 10.1016/j.jmaa.2005.03.097MATHMathSciNetView ArticleGoogle Scholar - Amleh AM, Kruse N, Ladas G:
**On a class of difference equations with strong negative feedback.***Journal of Difference Equations and Applications*1999,**5**(6):497-515. 10.1080/10236199908808204MATHMathSciNetView ArticleGoogle Scholar - Berg L, Stević S:
**Linear difference equations mod 2 with applications to nonlinear difference equations.***Journal of Difference Equations and Applications*2008,**14**(7):693-704. 10.1080/10236190701754891MATHMathSciNetView ArticleGoogle Scholar - Berg L, Stević S:
**Periodicity of some classes of holomorphic difference equations.***Journal of Difference Equations and Applications*2006,**12**(8):827-835. 10.1080/10236190600761575MATHMathSciNetView ArticleGoogle Scholar - Çinar C, Stević S, Yalçinkaya I:
**A note on global asymptotic stability of a family of rational equations.***Rostocker Mathematisches Kolloquium*2005, 59:41-49.MATHGoogle Scholar - Stević S:
**Global stability and asymptotics of some classes of rational difference equations.***Journal of Mathematical Analysis and Applications*2006,**316**(1):60-68. 10.1016/j.jmaa.2005.04.077MATHMathSciNetView ArticleGoogle Scholar - Kruse N, Nesemann T:
**Global asymptotic stability in some discrete dynamical systems.***Journal of Mathematical Analysis and Applications*1999,**235**(1):151-158. 10.1006/jmaa.1999.6384MATHMathSciNetView ArticleGoogle Scholar - Li X, Zhu D:
**Global asymptotic stability in a rational equation.***Journal of Difference Equations and Applications*2003,**9**(9):833-839. 10.1080/1023619031000071303MATHMathSciNetView ArticleGoogle Scholar - Aloqeily M:
**Global stability of a rational symmetric difference equation.**preprint, 2008Google Scholar - Gutnik L, Stević S:
**On the behaviour of the solutions of a second-order difference equation.***Discrete Dynamics in Nature and Society*2007,**2007:**-14.Google Scholar - Ladas G:
**A problem from the Putnam Exam.***Journal of Difference Equations and Applications*1998,**4**(5):497-499. 10.1080/10236199808808157MATHMathSciNetView ArticleGoogle Scholar - Putnam Exam The American Mathematical Monthly 1965, 734-736.Google Scholar
- Stević S:
**Asymptotics of some classes of higher-order difference equations.***Discrete Dynamics in Nature and Society*2007,**2007:**-20.Google Scholar - Stević S:
**Existence of nontrivial solutions of a rational difference equation.***Applied Mathematics Letters*2007,**20**(1):28-31. 10.1016/j.aml.2006.03.002MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**Nontrivial solutions of a higher-order rational difference equation.***Matematicheskie Zametki*2008,**84**(5):772-780.MathSciNetView ArticleGoogle Scholar - Sun T, Xi H:
**Global asymptotic stability of a higher order rational difference equation.***Journal of Mathematical Analysis and Applications*2007,**330**(1):462-466. 10.1016/j.jmaa.2006.07.096MATHMathSciNetView ArticleGoogle Scholar - Yang X, Sun F, Tang YY:
**A new part-metric-related inequality chain and an application.***Discrete Dynamics in Nature and Society*2008,**2008:**-7.Google Scholar - Yang X, Tang YY, Cao J:
**Global asymptotic stability of a family of difference equations.***Computers & Mathematics with Applications*2008,**56**(10):2643-2649. 10.1016/j.camwa.2008.04.032MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.