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

Original Research

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,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).

Keywords

Difference equation Positive solution Equilibrium Boundedness 

Mathematics Subject Classification (2000)

37E25 37B20 

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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

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