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Global behavior of solutions of the nonlinear difference equation $x_{n+1}=p_n+x_{n-1}/x_n$. (English) Zbl 1079.39005
The trichotomy results concerning the difference equation $$ x_{n+1}=p+x_{n-1}/x_n $$ are considered for the equation $$ x_{n+1}=p_n+x_{n-1}/x_n $$ with the initial conditions $x_{-1}\geq 0$, $x_0>0$ and $\{p_n\}_n$ a positive sequence with $\liminf_{n\rightarrow\infty}p_n=p\geq 0$, $\limsup_{n\rightarrow\infty}p_n=q<\infty$. If $p>0$ then $\{x_n\}_n$ is persistent and if $p>1$ then $\{x_n\}_n$ is bounded from above. Moreover, if $1<P\leq p_n\leq Q$ then the interval $[(PQ-1)/(Q-1),(PQ-1)/(P-1)]$ is a positive invariant set of the equation. If either $0<p_{2n+1}<1$ and $\lim_{n\rightarrow\infty}p_{2n+1}=0$ or $0<p_{2n}<1$ and $\lim_{n\rightarrow\infty}p_{2n}=0$ then there exist unbounded solutions to the equation. Some special cases of the equation are considered as applications.

39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
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