In this paper, we show that the following higher-order system of nonlinear difference equations, xn = x n−k y n−k−l y n−l (an + bnx n−k y n−k−l) , yn = y n−k x n−k−l x n−l (αn + βny n−k x n−k−l) , n ∈ N0, where k, l ∈ N , (an) n∈N 0 , (bn) n∈N 0 , (αn) n∈N 0 , (βn) n∈N 0 and the initial values x−i, y−i , i = 1, k + l , are real numbers, can be solved and some results in the literature can be extended further. Also, by using these obtained formulas, we investigate the asymptotic behavior of well-defined solutions of the above difference equations system for the case k = 2, l = k .

We show that the next di¤erence equations system x n+1 = anx n k+1 y n k yn n + n+1 ; y n+1 = bny n k+1 x n k xn n + n+1 ; n 2 N 0 ; where N 0 = N [ f0g, the sequences (an) n2N 0 , (bn) n2N 0 , (n) n2N 0 , (n) n2N 0 are two periodic and the initial conditions x i , y i i 2 f0; 1; : : : ; kg, are nonzero real numbers, can be solved. Also, we investigate the behavior of solutions of above mentioned system when a 0 = b 1 and a 1 = b 0 .

In this paper we show that the system of difference equations x n = ay n−k + dy n−k x n−(k+l) bx n−(k+l) + cy n−l , y n = αx n−k + δx n−k y n−(k+l) βy n−(k+l) + γx n−l , where n ∈ N 0 , k and l are positive integers, the parameters a, b, c, d, α, β, γ, δ are real numbers and the initial values x −j , y −j , j = 1, k + l, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case l = 1 and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

In this paper, we show that the following three-dimensional system of difference equations x n + 1 = y n x n − 2 a x n − 2 + b z n − 1 , y n + 1 = z n y n − 2 c y n − 2 + d x n − 1 , z n + 1 = x n z n − 2 e z n − 2 + f y n − 1 , n ∈ N 0 , $$\begin{equation*} x_{n+1}=\frac{y_{n}x_{n-2}}{ax_{n-2}+bz_{n-1}}, \quad y_{n+1}=\frac{z_{n}y_{n-2}}{cy_{n-2}+dx_{n-1}}, \quad z_{n+1}=\frac{x_{n}z_{n-2}}{ez_{n-2}+fy_{n-1}}, \quad n\in \mathbb{N}_{0}, \end{equation*}$$ where the parameters a, b, c, d, e, f and the initial values x −i , y −i , z −i , i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given.

In this paper, we show that the system of difference equations can be solved in the closed form. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system.