Asymptotic behavior of a linear delay difference equation

Consider the linear delay difference equations x n+1 −x n =Σ m j=1 a j (x n−kj −x n−lj ), n=0,1,0,... and y n+1 −y n = k j=1 b j y n−j , n=0,1,2,..., where the coefficients a j and b j are real and k j and l j are nonnegative integers. In this note we describe, in terms of the initial conditions, the asymptotic behavior of solutions of these equations in several cases when the characteristic equation has a dominant real root. Some of the results extend to systems of equations