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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A spectral countability condition for almost automorphy of solutions of differential equations


Authors: Nguyen Van Minh, Toshiki Naito and Gaston Nguerekata
Journal: Proc. Amer. Math. Soc. 134 (2006), 3257-3266
MSC (2000): Primary 47D06; Secondary 34G10, 45M05
Published electronically: May 12, 2006
MathSciNet review: 2231910
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Abstract: We consider the almost automorphy of bounded mild solutions to equations of the form

$\displaystyle (*)\quad\qquad\qquad\qquad\qquad\qquad dx/dt = A(t)x + f(t) \quad\qquad\qquad\qquad\qquad\qquad\qquad $

with (generally unbounded) $ \tau$-periodic $ A(\cdot )$ and almost automorphic $ f(\cdot )$ in a Banach space $ \mathbb{X}$. Under the assumption that $ \mathbb{X}$ does not contain $ c_0$, the part of the spectrum of the monodromy operator associated with the evolutionary process generated by $ A(\cdot )$ on the unit circle is countable. We prove that every bounded mild solution of $ (*)$ on the real line is almost automorphic.


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Additional Information

Nguyen Van Minh
Affiliation: Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118
Email: vnguyen@westga.edu

Toshiki Naito
Affiliation: Department of Mathematics, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
Email: naito@e-one.uec.ac.jp

Gaston Nguerekata
Affiliation: Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, Maryland 21251
Email: gnguerek@jewel.morgan.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08528-5
PII: S 0002-9939(06)08528-5
Keywords: Evolution equation, mild solution, almost automorphy, uniform spectrum
Received by editor(s): May 18, 2005
Published electronically: May 12, 2006
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society