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On a Conjecture for a Higher-Order Rational Difference Equation
Advances in Difference Equations volume 2009, Article number: 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 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
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 –(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 Higher-Order 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
Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- General Setting