Perron Conditions for Pointwise and Global Exponential Dichotomy of Linear Skew-Product Flows

Abstract

The aim of this paper is to give necessary and sufficient conditions for pointwise and uniform exponential dichotomy of linear skew-product flows. We shall obtain that the pointwise exponential dichotomy of a linear skew-product flow is equivalent to the pointwise admissibility of the pair \((C_0 ({\mathbf{R}},X),C_0 ({\mathbf{R}},X)).\) As a consequence, we prove that a linear skew-product flow π on \(E = X \times \Theta \) is uniformly exponentially dichotomic if and only if the pair \((C_0 ({\mathbf{R}},X),C_0 ({\mathbf{R}},X))\) is uniformly admissible for π.