Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

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 ℒ ( D ( A ) ¯ , X ) satisfying certain boundedness and measurability conditions and f ∈ L loc 1 ( ℝ , X ) . The solutions of the corresponding homogeneous equations are represented by an evolution family ( U B ( t , s ) ) t ≥ s . For various function spaces ℱ we show conditions on ( U B ( t , s ) ) t ≥ s and f which ensure the existence of a unique solution contained in ℱ . In particular, if ( U B ( t , s ) ) t ≥ s 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 ( U B ( t , s ) ) t ≥ s .