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On a Conjecture for a HigherOrder Rational Difference Equation
Advances in Difference Equations volumeÂ 2009, ArticleÂ number:Â 394635 (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
converges to its unique equilibrium , where and . Based on this fact, they put forward the following two conjectures.
Conjecture.
Suppose that and that atisfies
with Then, the sequence converges to the unique equilibrium 1.
Conjecture.
Suppose that is odd and , and define . If satisfies
with , where
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 SteviÄ‡ in [9], partially motivated also by [10].
In the meantime, it turned out that the method used in [11] by Ã‡inar et al. can be used in confirming Conjecture 1.2 (see also [12]). More precisely [11, 12] use Corollaryâ€‰â€‰3 from [13] in solving similar problems. For example, Ã‡inar et al. has shown, in an elegant way, that the main result in [14] is a consequence of Corollaryâ€‰â€‰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 [16â€“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, 5â€“7] and other literature.
2. Preliminaries and Notations
Observe that
Define function as follows:
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.
Let be defined by (2.2). Then
Lemma 2.2.
Assume that . If , then
where
.
Proof.
Since is symmetric in , we can assume, without loss of generality, that . Then there are possible cases:
 (1)
 (2)
 (3)
 (4)
(m+1)
And, for the above cases â€“(m+1), by the monotonicity of , in turn, we may get

(1)
;
 (2)
 (3)
 (4)
(m+1).
From the above inequalities, it follows that (2.6) holds. The proof is complete.
Lemma 2.3.
Assume that . Then
.
Proof.
For , it is easy to see that
which yields
and so
It follows that (2.8) holds. Similarly, for , it is easy to see that
which yields
It follows that (2.9) holds. The proof is complete.
Lemma 2.4.
Let
where
Assume that Then
Proof.
By induction, we easily show that
It follows from Lemma 2.3 that
Hence, by (2.15) and (2.18), we have
Equation (2.20) implies that the limits and exist, and
It follows from (2.16) that
. Let in (2.15), we have
It follows that there exist such that
From (2.24), we have
Since
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
Then the solution of (1.3) satisfies
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
Choose and such that
In view of Theorem 3.1, we have
Let , and be defined as in Lemma 2.4. Then by (3.5) and Lemma 2.2, we have
That is
By (3.7) and Lemma 2.2, we obtain
That is
Repeating the above procedure, in general, we can obtain
By Lemma 2.4, we have
which implies that (3.3) holds. The proof of Conjecture 1.2 is complete.
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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).
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Liao, M., Tang, X. & Xu, C. On a Conjecture for a HigherOrder Rational Difference Equation. Adv Differ Equ 2009, 394635 (2009). https://doi.org/10.1155/2009/394635
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DOI: https://doi.org/10.1155/2009/394635