Formulas of the solutions of a non-autonomous difference equation and two systems of difference equations
In this work, we explicitly solve the following: * A higher-order non-autonomous difference equation: equation* x_n+1 = α_n x_n-k + β_n x_n x_n-1 … x_n-k+1 , equation* where n N_0, k N, the sequences (α_n)_n N_0 and (β_n)_n N_0 are real, and the initial values x_-k, x_-k+1, …, x_0 are nonzero real numbers. * A three-dimensional system of second-order difference equations: equation* x_n+1 = a_1 y_n-1 z_n-1 a x_n-1 + b y_n-1 + c z_n-1 , y_n+1 = a_2 x_n-1 z_n-1 a x_n-1 + b y_n-1 + c z_n-1 , equation* equation* z_n+1 = a_3 x_n-1 y_n-1 a x_n-1 + b y_n-1 + c z_n-1 , equation* where n N_0, the parameters a, b, c, a_1, a_2, a_3 are real numbers, and the initial values x_-1, x_0, y_-1, y_0, z_-1, z_0 are nonzero real numbers. * A three-dimensional system of first-order difference equations: equation* x_n+1 = a_1 y_n z_n a x_n + b y_n + c z_n , y_n+1 = a_2 x_n z_n a x_n + b y_n + c z_n , z_n+1 = a_3 x_n y_n a x_n + b y_n + c z_n , equation* where n N_0, the parameters a, b, c, a_1, a_2, a_3 are real numbers, and the initial values x_0, y_0, z_0 are nonzero real numbers. , number=2, journal=Journal of Innovative Applied Mathematics and Computational Sciences, author=Zabat, Hiba and Touafek, Nouressadat and Dekkar, Imane, year=2024, month=Dec., pages=188–200