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On a solvable nonlinear difference equation of higher order. (English) Zbl 1424.39023
Summary: In this paper we consider the following higher-order nonlinear difference equation \[ x_{n}=\alpha x_{n-k}+\frac{\delta x_{n-k}x_{n-\left( k+l\right)}}{\beta x_{n-\left( k+l\right)}+\gamma x_{n-l}},\ n\in \mathbb{N}_{0}, \] where \(k\) and \(l\) are fixed natural numbers, and the parameters \(\alpha \), \( \beta \), \(\gamma \), \(\delta \) and the initial values \(x_{-i}\), \(i=\overline{1,k+l}\), are real numbers such that \(\beta^{2}+\gamma^{2}\neq 0\). We solve the above-mentioned equation in closed form and considerably extend some results in the literature. We also determine the asymptotic behavior of solutions and the forbidden set of the initial values using the obtained formulae for the case \(l=1\).

MSC:
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
39A22 Growth, boundedness, comparison of solutions to difference equations
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