Abstract and Applied Analysis
Volume 4 (1999), Issue 3, Pages 169-194
doi:10.1155/S1085337599000214

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

Abstract

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(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 (UB(t,s))t≥s. For various function spaces ℱ we show conditions on (UB(t,s))t≥s and f which ensure the existence of a unique solution contained in ℱ. In particular, if (UB(t,s))t≥s is p-periodic there exists a unique bounded solution u subject to certain spectral assumptions on UB(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 (UB(t,s))t≥s.