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A solvable system of difference equations. (English) Zbl 07272993
Summary: In this paper, we show that the system of difference equations \[x_n={\frac{ay^p_{n-1}+b(x_{n-2}y_{n-1})^{p-1}}{cy_{n-1}+dx^{p-1}_{n-2}}},\ y_n={\frac{{\alpha}x^p_{n-1}+{\beta}(y_{n-2}x_{n-1})^{p-1}}{{\gamma}x_{n-1}+{\delta}y^{p-1}_{n-2}}}, \] \(n\in \mathbb{N}_0\) where the parameters \(a, b, c, d, \alpha, \beta, \gamma, \delta, p\) and the initial values \(x_{-2}, x_{-1}, y_{-2}, y_{-1}\) are real numbers, can be solved. Also, by using obtained formulas, we study the asymptotic behaviour of well-defined solutions of aforementioned system and describe the forbidden set of the initial values. Our obtained results significantly extend and develop some recent results in the literature.
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
39A10 Additive difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
Full Text: DOI
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