The limits of the solutions of a nonautonomous linear delay difference equation

Consider the system of linear delay difference equations x(n+1) - x(n) = Sigma (m)(j=1) A(j)(n) (x(n-kj) - x(n-lj)), n= 0, 1, 2, ... , where the coefficients A(j)(n) are square matrices and k(j) and l(j) are no nnegative integers. In this note, we show that if the coefficients are "sma ll", then every solution of the above equation tends to a constant vector a s n --> infinity and the value of the limit can be characterized by a speci al solution of the matrix equation Yn = I - Sigma (m)(j=1) Sigma (i=n+lj)(n+kj-1) Y(i+1)A(j)(i), n = 0, 1, 2, ..., and the initial conditions. (C) 2001 Elsevier Science Ltd. All rights reser ved.