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Dynamics of a higher order rational difference equation. (English) Zbl 1158.39301
Summary: We will investigate a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic character, the character of semicycles and global asymptotic stability of all positive solutions of $$x_{n+1}=\frac{\beta x_n+\gamma x_{n-k}}{Bx_n+Cx_{n-k}}\,,\quad n=0,1,\dots\,.$$ It is worth to mention that our results solve the open problem proposed by {\it H. L. S. Kulenvić} and {\it G. Ladas} in their monograph [Dynamics of second order rational difference equations: with open problems and conjectures, Chapman & Hall/CRC, Boca Raton (2002; Zbl 0981.39011)].

##### MSC:
 39A20 Generalized difference equations
##### Keywords:
recursive sequence; global asymptotic stability
Full Text:
##### References:
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