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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.
MSC:
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
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