A spectral countability condition for almost automorphy of solutions of differential equations
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- by Nguyen Van Minh, Toshiki Naito and Gaston Nguerekata PDF
- Proc. Amer. Math. Soc. 134 (2006), 3257-3266 Request permission
Abstract:
We consider the almost automorphy of bounded mild solutions to equations of the form \begin{equation*} (*)\quad \qquad \qquad \qquad \qquad \qquad dx/dt = A(t)x + f(t) \quad \qquad \qquad \qquad \qquad \qquad \qquad \end{equation*} 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.References
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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
- ORCID: 0000-0001-5765-7175
- Email: gnguerek@jewel.morgan.edu
- Received by editor(s): May 18, 2005
- Published electronically: May 12, 2006
- Communicated by: Carmen C. Chicone
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 3257-3266
- MSC (2000): Primary 47D06; Secondary 34G10, 45M05
- DOI: https://doi.org/10.1090/S0002-9939-06-08528-5
- MathSciNet review: 2231910