Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Almost automorphic solutions of evolution equations

Author(s): Toka Diagana; Gaston Nguerekata; Nguyen Van Minh
Journal: Proc. Amer. Math. Soc. 132 (2004), 3289-3298.
MSC (2000): Primary 34G10; Secondary 43A60
Posted: June 18, 2004
MathSciNet review: 2073304
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Abstract | References | Similar articles | Additional information

Abstract: This paper is concerned with the existence of almost automorphic mild solutions to equations of the form

$\begin{displaymath}\dot u(t)= Au(t)+f(t),\tag*{$(*)$ }\end{displaymath}$

where

generates a holomorphic semigroup and

is an almost automorphic function. Since almost automorphic functions may not be uniformly continuous, we introduce the notion of the uniform spectrum of a function. By modifying the method of sums of commuting operators used in previous works for the case of bounded uniformly continuous solutions, we obtain sufficient conditions for the existence of almost automorphic mild solutions to

in terms of the imaginary spectrum of

and the uniform spectrum of

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