Open Access

On a Conjecture for a Higher-Order Rational Difference Equation

Advances in Difference Equations20092009:394635

DOI: 10.1155/2009/394635

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.

1. Introduction

In 2007, Berenhaut et al. [1] proved that every solution of the following rational difference equation
(1.1)

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

Conjecture.

Suppose that and that atisfies
(1.2)

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

Conjecture.

Suppose that is odd and , and define . If satisfies
(1.3)
with , where
(1.4)

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 [1624].

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, 57] and other literature.

2. Preliminaries and Notations

Observe that
(2.1)
Define function as follows:
(2.2)

Then we can rewrite (1.3) as

(2.3)

or

(2.4)

where is an odd integer and .

The following lemma can be obtained by simple calculations.

Lemma 2.1.

Let be defined by (2.2). Then
(2.5)

Lemma 2.2.

Assume that . If , then
(2.6)
where
(2.7)

.

Proof.

Since is symmetric in , we can assume, without loss of generality, that . Then there are possible cases:
  1. (1)

     
  2. (2)

     
  3. (3)

     
  4. (4)

     

(m+1)

And, for the above cases –(m+1), by the monotonicity of , in turn, we may get
  1. (1)

    ;

     
  2. (2)

     
  3. (3)

     
  4. (4)

     

(m+1) .

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

Lemma 2.3.

Assume that . Then
(2.8)
(2.9)

.

Proof.

For , it is easy to see that
(2.10)
which yields
(2.11)
and so
(2.12)
It follows that (2.8) holds. Similarly, for , it is easy to see that
(2.13)
which yields
(2.14)

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

Lemma 2.4.

Let
(2.15)
where
(2.16)
Assume that Then
(2.17)

Proof.

By induction, we easily show that
(2.18)
It follows from Lemma 2.3 that
(2.19)
Hence, by (2.15) and (2.18), we have
(2.20)
Equation (2.20) implies that the limits and exist, and
(2.21)
It follows from (2.16) that
(2.22)
. Let in (2.15), we have
(2.23)
It follows that there exist such that
(2.24)
From (2.24), we have
(2.25)
Since
(2.26)

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

3. Proof of Conjecture 1.2

Theorem 3.1.

Suppose that and that
(3.1)
Then the solution of (1.3) satisfies
(3.2)

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

Proof.

Let be a solution of (1.3) with . We need to prove that
(3.3)
Choose and such that
(3.4)
In view of Theorem 3.1, we have
(3.5)
Let , and be defined as in Lemma 2.4. Then by (3.5) and Lemma 2.2, we have
(3.6)
That is
(3.7)
By (3.7) and Lemma 2.2, we obtain
(3.8)
That is
(3.9)
Repeating the above procedure, in general, we can obtain
(3.10)
By Lemma 2.4, we have
(3.11)

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

(1)
School of Mathematical Sciences and Computing Technology, Central South University
(2)
School of Mathematics and Physics, University of South China
(3)
College of Science, Hunan Institute of Engineering

References

  1. 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
  2. Stević S:On the recursive sequence . Journal of Applied Mathematics & Computing 2005,18(1-2):229-234. 10.1007/BF02936567MATHMathSciNetView ArticleGoogle Scholar
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. Aloqeily M: Global stability of a rational symmetric difference equation. preprint, 2008Google Scholar
  16. 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
  17. Ladas G: A problem from the Putnam Exam. Journal of Difference Equations and Applications 1998,4(5):497-499. 10.1080/10236199808808157MATHMathSciNetView ArticleGoogle Scholar
  18. Putnam Exam The American Mathematical Monthly 1965, 734-736.Google Scholar
  19. Stević S: Asymptotics of some classes of higher-order difference equations. Discrete Dynamics in Nature and Society 2007, 2007:-20.Google Scholar
  20. 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
  21. Stević S: Nontrivial solutions of a higher-order rational difference equation. Matematicheskie Zametki 2008,84(5):772-780.MathSciNetView ArticleGoogle Scholar
  22. 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
  23. 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
  24. 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

© Maoxin Liao et al. 2009

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.