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Journal of Applied Mathematics and Stochastic Analysis
Volume 15 (2002), Issue 2, Pages 105-114

Periodic solutions of non-densely defined delay evolution equations

1Faculty des Sciences Semlalia, Departement de Mathematiques, Marrakech BP 2390, Morocco
2James Madison University, Department of Mathematics, Harrisonburg 22807, VA, USA

Received 1 July 2000; Revised 1 May 2001

Copyright © 2002 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.


We study the finite delay evolution equation {x'(t)=Ax(t)+F(t,xt),t0,x0=ϕC([r,0],E), where the linear operator A is non-densely defined and satisfies the Hille-Yosida condition. First, we obtain some properties of “integral solutions” for this case and prove the compactness of an operator determined by integral solutions. This allows us to apply Horn's fixed point theorem to prove the existence of periodic integral solutions when integral solutions are bounded and ultimately bounded. This extends the study of periodic solutions for densely defined operators to the non-densely defined operators. An example is given.