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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 ${x}_{-1}\ge 0$, ${x}_{0}>0$ and ${\left\{{p}_{n}\right\}}_{n}$ a positive sequence with ${lim inf}_{n\to \infty }{p}_{n}=p\ge 0$, ${lim sup}_{n\to \infty }{p}_{n}=q<\infty$. If $p>0$ then ${\left\{{x}_{n}\right\}}_{n}$ is persistent and if $p>1$ then ${\left\{{x}_{n}\right\}}_{n}$ is bounded from above. Moreover, if $1 then the interval $\left[\left(PQ-1\right)/\left(Q-1\right),\left(PQ-1\right)/\left(P-1\right)\right]$ is a positive invariant set of the equation. If either $0<{p}_{2n+1}<1$ and ${lim}_{n\to \infty }{p}_{2n+1}=0$ or $0<{p}_{2n}<1$ and ${lim}_{n\to \infty }{p}_{2n}=0$ then there exist unbounded solutions to the equation. Some special cases of the equation are considered as applications.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations