Gabriele Gühring, Frank Räbiger, "Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations", Abstract and Applied Analysis, vol. 4, Article ID 153670, 26 pages, 1999. https://doi.org/10.1155/S1085337599000214
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 , 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 .
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