Abstract and Applied Analysis
Volume 4 (1999), Issue 3, Pages 169-194

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

Gabriele Gühring and Frank Räbiger

Universität Tübingen, Mathematisches Institut, Auf der Morgenstelle 10, Tübingen 72076, Germany

Received 18 May 1999

Copyright © 1999 Gabriele Gühring and Frank Räbiger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(t)+B(t)u(t)+f(t),t, where (A,D(A)) is a Hille-Yosida operator on a Banach space X,B(t),t, is a family of operators in (D(A)¯,X) satisfying certain boundedness and measurability conditions and fLloc1(,X). The solutions of the corresponding homogeneous equations are represented by an evolution family (UB(t,s))ts. For various function spaces we show conditions on (UB(t,s))ts and f which ensure the existence of a unique solution contained in . In particular, if (UB(t,s))ts 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))ts.