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Abstract and Applied Analysis
Volume 3, Issue 3-4, Pages 425-436
http://dx.doi.org/10.1155/S1085337598000645

Almost periodic mild solutions of a class of partial functional differential equations

1Department of Mathematics, University of Augsburg, Augsburg D-86135, Germany
2Department of Mathematics, The University of Electro-Communications, Chofugaoka 1-5-1, Chofu, Tokyo 182, Japan

Received 10 May 1998

Copyright © 1998 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

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form (*)x=f(t,x,xt). To this end, we first associate with every almost periodic semilinear equation (**)x=F(t,x). a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (**) as fixed points of the semigroup. Those results are then carried over to equation (*). The main results are stated under accretiveness conditions of the function f in terms of x and Lipschitz conditions with respect to xt.