Existence and Roughness of the Exponential Dichotomy for Skew-Product Semiflow in Banach Spaces

In this paper we introduce a concept of exponential dichotomy for skew-product semiflow in infinite dimensional Banach spaces which is an extension of the classic concept for evolution operators. This concept is used to study the roughness property of the skew-product semiflow. Also, we introduce the concept of discrete skew-product and give a necessary and sufficient condition for this discrete skew-product to have a Discrete Dichotomy. After that, we give necessary and sufficient conditions for the existence of exponential dichotomy for skew-product semiflow. Moreover we prove that the exponential dichotomy for skew-product semiflow is not destroyed by small perturbation. Finally, we apply these results to parabolic partial differential equations and functional differential equations.