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Abstract and Applied Analysis
Volume 7, Issue 12, Pages 637-661
http://dx.doi.org/10.1155/S1085337502208015

Bounded solutions of nonlinear Cauchy problems

FB Mathematik, Universität Essen, Essen D-45117, Germany

Received 28 February 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.

Abstract

For a given closed and translation invariant subspace Y of the bounded and uniformly continuous functions, we will give criteria for the existence of solutions uY to the equation u(t)+A(u(t))+ωu(t)f(t),t, or of solutions u asymptotically close to Y for the inhomogeneous differential equation u(t)+A(u(t))+ωu(t)f(t),t>0,u(0)=u0, in general Banach spaces, where A denotes a possibly nonlinear accretive generator of a semigroup. Particular examples for the space Y are spaces of functions with various almost periodicity properties and more general types of asymptotic behavior.