Functions of shift operator and their applications to difference equations - Chaikovs'kyi, A. V. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
We study the representation for functions of shift operator acting upon bounded sequences of elements of a Banach space. An estimate is obtained for the bounded solution of a linear difference equation in the Banach space. For two types of differential equations in Banach spaces, we present sufficient conditions for their bounded solutions to be limits of bounded solutions of the corresponding difference equations and establish estimates for the rate of convergence.