On boundedness of solutions of the difference equation \(x_{n+1}=p+\frac{x_{n-1}}{x_{n}}\) for p<1
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).
boundedness difference equation positive solution equilibrium 37e25 37b20