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
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 , where is a Hille-Yosida operator on a Banach space , 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 we show conditions on and which ensure the existence of a unique solution contained in . In particular, if is -periodic there exists a unique bounded solution subject to certain spectral assumptions on and . We apply the results to nonautonomous semilinear retarded differential equations. For certain -periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of .