On boundedness of solutions of the difference equation x_(n+1) = p + (x_(n?1))/(x_n)for p <1

摘要

In this paper, we study the difference equation x_(n+1) = p + (x_(n?1))/(x_n), n= 0, 1, . . ., where initial values x_(?1), x_0 ∈ (0,+∞) and 0<1, and obtain the set of all initial values (x_(?1), x_0) ∈ (0,+∞) × (0,+∞) such that the positive solutions {xn}_n=?1~∞ 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).