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Abstract and Applied Analysis
Volume 1, Issue 4, Pages 351-380
http://dx.doi.org/10.1155/S108533759600019X

Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations

1Department of Mathematics, University of Augsburg, D-86135, Augsburg, Germany
2Department of Mathematics, University of Tübingen, D-72076, Tübingen, Germany

Received 25 July 1996

Copyright © 1996 Hindawi Publishing Corporation. 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.

Abstract

This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so-called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional differential equations.