, Volume 44, Issue 1-2, pp 61-68
Date: 20 Apr 2013

On boundedness of solutions of the difference equation \(x_{n+1}=p+\frac{x_{n-1}}{x_{n}}\) for p<1

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Abstract

In this paper, we study the difference equation $$x_{n+1}=p+\frac{x_{n-1}}{x_n}, \quad n=0,1,\ldots, $$ where initial values x −1,x 0∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x −1,x 0)∈(0,+∞)×(0,+∞) such that the positive solutions \(\{x_{n}\}_{n=-1}^{\infty}\) are bounded. This answers the Open problem 4.8.11 proposed by Kulenovic and Ladas (Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, 2002).

Project supported by NNSF of China (11261005, 51267001) and NSF of Guangxi (2011GXNSFA018135, 2012GXNSFDA276040).