On a Conjecture for a Higher-Order Rational Difference Equation
© Maoxin Liao et al. 2009
Received: 30 December 2008
Accepted: 14 March 2009
Published: 1 April 2009
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.
converges to its unique equilibrium , where and . Based on this fact, they put forward the following two conjectures.
with Then, the sequence converges to the unique equilibrium 1.
Then the sequence converges to the unique equilibrium 1.
Motivated by , Berenhaut et al. started with the investigation of the following difference equation for (see, [3, 4]). Among others, in  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 .
Some particular cases of (1.2) had been studied previously by Li in [6, 7], by using semicycle analysis similar to that in . The problem concerning periodicity of semicycles of difference equations was solved in very general settings by Berg and Stević in , partially motivated also by .
In the meantime, it turned out that the method used in  by Çinar et al. can be used in confirming Conjecture 1.2 (see also ). More precisely [11, 12] use Corollary 3 from  in solving similar problems. For example, Çinar et al. has shown, in an elegant way, that the main result in  is a consequence of Corollary 3 in . With some calculations it can be also shown that Conjecture 1.2 can be confirmed in this way (see ).
2. Preliminaries and Notations
Then we can rewrite (1.3) as
where is an odd integer and .
The following lemma can be obtained by simple calculations.
From the above inequalities, it follows that (2.6) holds. The proof is complete.
It follows that (2.9) holds. The proof is complete.
it follows from (2.25) and (2.18) that . The proof is complete.
3. Proof of Conjecture 1.2
Theorem 3.1 is a direct corollary of Lemmas 2.2 and 2.3.
which implies that (3.3) holds. The proof of Conjecture 1.2 is complete.
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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