Asymptotic behavior of a linear delay difference equation

Abstract:

Consider the linear delay difference equations \[ {x_{n + 1}} - {x_n} = \sum \limits _{j = 1}^m {{a_j}({x_{n - {k_j}}} - {x_{n - {l_j}}}),\quad n = 0,1,2, \ldots } \] and \[ {y_{n + 1}} - {y_n} = \sum \limits _{j = 1}^k {{b_j}{y_{n - j}},\quad n = 0,1,2, \ldots ,} \] 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.