Asymptotically ordinary linear Volterra difference equations with infinite delay

Abstract

• A linear Volterra difference equation with infinite delay and sufficiently small coefficients is shown to be asymptotically equivalent to a linear ordinary difference equation. • The coefficient matrix of the corresponding ordinary difference equation can be written as a limit of successive approximations. • The eigenvalues of the approximating matrices converge to the characteristic roots of the Volterra difference equation at an exponential rate. A class of linear Volterra difference equations with infinite delay is considered. It is shown that if the coefficient matrices are sufficiently small, then the Volterra difference equation is asymptotically equivalent to a linear ordinary difference equation at infinity. The coefficient matrix of the ordinary difference equation is a solution of an associated matrix equation which can be obtained by successive approximations. The eigenvalues of the approximating matrices converge exponentially to the characteristic roots of the Volterra difference equation. As a corollary, an efficient new method is obtained for the computation of the characteristic roots with an explicit error estimate.