Linear differential-difference equations in a banach space

A scheme for extending linear differential-difference equations from a Euclidean phase space into a Banach phase space. This is not a new idea. For example Travis and Web [9] have made this extension for autonomous systems involving partial differential equations of parabolic type. The technique presented in this discussion is based on the ability to express solutions of the problem in terms of certain transformations which are the analogues of the fundamental matrices of the finite dimensional theory (see e.g. [1] or [7]).