Asymptotic behavior of a linear delay difference equation

Abstract

Consider the linear delay difference equations \[ x n + 1 − x n = ∑ j = 1 m a j ( x n − k j − x n − l j ) , n = 0 , 1 , 2 , … {x_{n + 1}} - {x_n} = \sum \limits _{j = 1}^m {{a_j}({x_{n - {k_j}}} - {x_{n - {l_j}}}),\quad n = 0,1,2, \ldots } \] and \[ y n + 1 − y n = ∑ j = 1 k b j y n − j , n = 0 , 1 , 2 , … , {y_{n + 1}} - {y_n} = \sum \limits _{j = 1}^k {{b_j}{y_{n - j}},\quad n = 0,1,2, \ldots ,} \] where the coefficients a j {a_j} and b j {b_j} are real and k j {k_j} and l j {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.