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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 H. L. S. Kulenvić and 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 Multiplicative and other generalized difference equations, e.g., of Lyness type
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