Journal of Applied Mathematics and Computing

, Volume 44, Issue 1, pp 61–68

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

Original Research

DOI: 10.1007/s12190-013-0680-2

Cite this article as:
Sun, T., Wu, X., He, Q. et al. J. Appl. Math. Comput. (2014) 44: 61. doi:10.1007/s12190-013-0680-2


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,x0∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x−1,x0)∈(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 equationPositive solutionEquilibriumBoundedness

Mathematics Subject Classification (2000)


Copyright information

© Korean Society for Computational and Applied Mathematics 2013

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceGuangxi UniversityNanningP.R. China
  2. 2.College of Electrical EngineeringGuangxi UniversityNanningP.R. China
  3. 3.Department of MathematicsGuangxi College of Finance and EconomicsNanningP.R. China