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Global asymptotic stability for a higher order nonlinear rational difference equations. (English) Zbl 1110.39011
Consider the rational difference equation $$x_{n+1}=\frac{ax_{n}+bx_{n-k}}{A+Bx_{n}}~\ \ ,~\ n=0,1,\dots \tag {$*$}$$ where $a~,b~,A,B$ are positive real numbers, $k\geq 1$ is a positive integer, and the initial conditions $x_{-k},~\dots,~x_{-1},~x_{0}$ are nonnegative real numbers. The authors solve an open problem posed by {\it M. R. S. Kulenović} and {\it G. Ladas} [Dynamics of second order rational difference equations, Chapman & Hall / CRC, Boca Raton, FL (2002; Zbl 0981.39011), p. 129]. They prove the following {Theorem}: (a) If $b\leq A-a,$ then the zero equilibrium of Eq. ($*$) is globally asymptotically stable. (b) If $A-a<b<A+a,$ then the positive equilibrium of Eq. ($*$) is globally asymptotically stable. The boundedness, periodic character, invariant intervals of all nonnegative solutions of ($*$) are investigated.

MSC:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
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Full Text: DOI
References:
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