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Dynamics and solutions of a fifth-order nonlinear difference equation. (English) Zbl 1417.39004
Summary: The main objective of this paper is to study the behavior of the rational difference equation of the fifth-order $$y_{n + 1} = \alpha y_n + \beta y_n y_{n - 3} /(A y_{n - 4} + B y_{n - 3})$$, $$n = 0,1, \ldots$$, where $$\alpha, \beta, A$$, and $$B$$ are real numbers and the initial conditions $$y_{- 4}, y_{- 3}, y_{- 2}, y_{- 1}$$ and $$y_0$$ are positive real numbers such that $$A y_{- 4} + B y_{- 3} \neq 0$$. Also, we obtain the solution of some special cases of this equation.