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 equationx"n"+"1=px"n+x"n"-"kq+x"n"-"k,n=0,1,...,where the parameters p and q and the initial conditions x"-"k,...,x"0 are nonnegative real numbers. As might be expected, the two cases 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].

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