Solutions structures for some systems of fractional difference equations
OMA-Vol. 3 (2019), Issue 1, pp. 51-61 Open Access Full-Text PDF
M. B. Almatrafi
Abstract: It is a well-known fact that the majority of rational difference equations cannot be solved theoretically. As a result, some scientific experts use manual iterations to obtain the exact solutions of some of these equations. In this paper, we obtain the fractional solutions of the following systems of difference equations:
$$
x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ \ \
y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( \pm 1\pm y_{n-1}x_{n-3}\right) }
,\ \ \ n=0,1,…,
$$
where the initial data \(x_{-3},\ x_{-2},\ x_{-1},\ \)\ \(
x_{0},\ y_{-3},\ y_{-2},\ y_{-1}\) and \(\ \ y_{0}\;\) are arbitrary non-zero real numbers. All solutions will be depicted under specific initial conditions.
$$
x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ \ \
y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( \pm 1\pm y_{n-1}x_{n-3}\right) }
,\ \ \ n=0,1,…,
$$
where the initial data \(x_{-3},\ x_{-2},\ x_{-1},\ \)\ \(
x_{0},\ y_{-3},\ y_{-2},\ y_{-1}\) and \(\ \ y_{0}\;\) are arbitrary non-zero real numbers. All solutions will be depicted under specific initial conditions.
