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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.

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
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