On a Conjecture for a Higher-Order Rational Difference Equation

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 [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

(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)
• (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

(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.