Dichotomic Behavior of Linear Difference Systems with Advanced Argument

In this paper, the difference equation with advanced argument yŽn 1. AŽn.yŽn. BŽn.yŽgŽn.. is considered. The sequence of advances gŽn.4 satisfies 1 gŽn. n N, where N is a fixed number. The matrices AŽn. are invertible, whereas, in general, matrices BŽn. are not. In this paper the notion of an ordinary dichotomy for a linear equation with advance is given. This construction relies on a variation of constants formula obtained for the nonhomogeneous equation yŽn 1. AŽn.yŽn. BŽn.yŽgŽn.. fŽn.and on the notion of admissibility of a pair of functional spaces. It is proven that these ordinary dichotomies are not destroyed by l1-perturbations. A theorem of existence of ordinary dichotomies is given