On a two-dimensional nonlinear system of difference equations close to the bilinear system

Abstract

We consider the two-dimensional nonlinear system of difference equations xn = xn−k ayn−l + byn−(k+l) cyn−l + dyn−(k+l), yn = yn−k αxn−l + βxn−(k+l) γxn−l + δxn−(k+l), for n ∈ N0, where the delays k and l are two natural numbers, and the initial values x−j, y−j, 1 ≤ j ≤ k+l, and the parameters a, b, c, d, α, β, γ, δ are real numbers. We show that the system of difference equations is solvable by presenting a method for finding its general solution in detail. Bearing in mind that the system of equations is a natural generalization of the corresponding one-dimensional difference equation, whose special cases appear in the literature from time to time, our main result presented here also generalizes many results therein.