Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line

Abstract

Consider an evolution family U = ( U ( t , s ) ) t ⩾ s ⩾ 0 on a half-line R + and an integral equation u ( t ) = U ( t , s ) u ( s ) + ∫ s t U ( t , ξ ) f ( ξ ) d ξ . We characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of L p type, the Lorentz spaces L p , q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.