Dynamics of a rational difference equation

In this paper, we investigate the periodic character, invariant intervals, oscillation and global stability of all positive solutions of the equationxn+1=pxn+xn−kq+xn−k,n=0,1,…,where the parameters p and q and the initial conditions x−k,…,x0 are nonnegative real numbers. As might be expected, the two cases p⩽q and p>q give rise to different dynamic behaviors. In particular, our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2001].