Solvability of a system of higher order nonlinear difference equations

Abstract

In this paper we show that the system of difference equations xn = ayn−k + dyn−kxn−(k+l) bxn−(k+l) + cyn−l , yn = αxn−k + δxn−kyn−(k+l) βyn−(k+l) + γxn−l , where n ∈ N0, 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.