Dichotomic behavior of linear difference systems with advanced argument

In this paper, the difference equation with advanced argument y(n + 1) = A( n)y(n) + B(n)y(g(n)) is considered. The sequence of advances {g(n)} satisfi es 1 less than or equal to g(n) - n less than or equal to N, where N is a f ixed number. The matrices A(n) are invertible, whereas, in general, matrice s B(n) are not. In this paper the notion of an ordinary dichotomy for a lin ear equation with advance is given. This construction relies on a variation of constants formula obtained for the nonhomogeneous equation y(n + 1) = A (n)y(n) f B(n)y(g(n)) + f(n) and on the notion of admissibility of a pair o f functional spaces. It is proven that these ordinary dichotomies are not d estroyed by l(1)-perturbations. A theorem of existence of ordinary dichotom ies is given. (C) 2000 Academic Press.