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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A NOTE ON DISCRETE MAXIMAL REGULARITY FOR FUNCTIONAL DIFFERENCE EQUATIONS WITH INFINITE DELAY | A NOTE ON DISCRETE MAXIMAL REGULARITY FOR FUNCTIONAL DIFFERENCE EQUATIONS WITH INFINITE DELAY CLAUDIO CUEVAS AND CLAUDIO VIDAL Received 4 October 2005 Accepted 1 November 2005 Dedicated to Juan Cuevas Gonzalez Using exponential dichotomies we get maximal regularity for retarded functional difference equations. Applications on Volterra difference equations with infinite delay are shown. Copyright 2006 C. Cuevas and C. Vidal. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The maximal regularity problem for the discrete time evolution equations has been recently considered by Blunck 4 5 . Discrete maximal regularity properties appears not to be considered in the literature before the paper 5 . The continuous maximal regularity problem for time evolution equations is well-know see 1 4 5 19 20 and the reference contained therein . In the present paper we are concerned in the study of maximal regularity for the following homogeneous retarded linear functional equation x n 1 L n xn n n0 0 1.1 where L N n0 X Cr is a bounded linear map with respect to the second variable denotes an abstract phase space that we will explain briefly later for the basic theory of phase spaces the reader is referred to the book by Hino et al. 14 x. denotes the -valued function defined by n - xn and N n0 denotes the set n e N n n0 . The abstract phase spaces was introduced by Hale and Kato 13 for studying qualitative theory of functional differential equations with unbounded delay. The idea of considering phase spaces for studying qualitative properties of functional difference equations was used first by Murakami 18 for study some spectral properties of the solution operator for linear Volterra difference systems and then by Elaydi et al. 12 for study asymptotic equivalence of bounded solutions of a homogeneous .
