A solvable system of difference equations
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In this paper, we show that the system of difference equations \begin{equation*} x_{n}=\frac{ay_{n-1}^{p}+b\left( x_{n-2}y_{n-1}\right) ^{p-1}}{ cy_{n-1}+dx_{n-2}^{p-1}},\ y_{n}=\frac{\alpha x_{n-1}^{p}+\beta \left( y_{n-2}x_{n-1}\right) ^{p-1}}{\gamma x_{n-1}+\delta y_{n-2}^{p-1}}, \end{equation*} $n \in \mathbb{N}_{0}$ 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.
Abstract
In this paper, we show that the system of difference equations \begin{equation*} x_{n}=\frac{ay_{n-1}^{p}+b\left( x_{n-2}y_{n-1}\right) ^{p-1}}{ cy_{n-1}+dx_{n-2}^{p-1}},\ y_{n}=\frac{\alpha x_{n-1}^{p}+\beta \left( y_{n-2}x_{n-1}\right) ^{p-1}}{\gamma x_{n-1}+\delta y_{n-2}^{p-1}}, \end{equation*} $n \in \mathbb{N}_{0}$ 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.
참고문헌 (28)
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