Original Research

Journal of Applied Mathematics and Computing

, Volume 44, Issue 1, pp 61-68

First online:

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

  • Taixiang SunAffiliated withCollege of Mathematics and Information Science, Guangxi University Email author 
  • , Xin WuAffiliated withCollege of Mathematics and Information Science, Guangxi University
  • , Qiuli HeAffiliated withCollege of Electrical Engineering, Guangxi University
  • , Hongjian XiAffiliated withDepartment of Mathematics, Guangxi College of Finance and Economics

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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).


Difference equation Positive solution Equilibrium Boundedness

Mathematics Subject Classification (2000)

37E25 37B20