Asymptotic properties of mild solutions of nonautonomousevolution equations with applications to retarded differential equations

Abstract

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation ( d / d t ) u ( t ) = A u ( t ) + B ( t ) u ( t ) + f ( t ), t ∈ ℝ , where ( A , D ( A )) is a Hille‐Yosida operator on a Banach space X , B ( t ), t ∈ ℝ , is a family of operators in satisfying certain boundedness and measurability conditions and . The solutions of the corresponding homogeneous equations are represented by an evolution family . For various function spaces ℱ we show conditions on and f which ensure the existence of a unique solution contained in ℱ . In particular, if is p ‐periodic there exists a unique bounded solution u subject to certain spectral assumptions on U B ( p , 0), f and u . We apply the results to nonautonomous semilinear retarded differential equations. For certain p ‐periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of .