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Almost automorphic solutions of evolution equations


Authors: Toka Diagana, Gaston Nguerekata and Nguyen Van Minh
Journal: Proc. Amer. Math. Soc. 132 (2004), 3289-3298
MSC (2000): Primary 34G10; Secondary 43A60
DOI: https://doi.org/10.1090/S0002-9939-04-07571-9
Published electronically: June 18, 2004
MathSciNet review: 2073304
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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 $A$ generates a holomorphic semigroup and $f$ 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 $A$and the uniform spectrum of $f$.


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Additional Information

Toka Diagana
Affiliation: Department of Mathematics, Howard University, 2441 6th Street N.W., Washington D.C. 20059
Email: tdiagana@howard.edu

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

Nguyen Van Minh
Affiliation: Department of Mathematics, Hanoi University of Science, Khoa Toan, Dai Hoc Khoa Hoc Tu Nhien, 334 Nguyen Trai, Hanoi, Vietnam
Address at time of publication: Department of Mathematics, State University of West Georgia, Carrollton, Georgia 30118
Email: nvminh@netnam.vn, ngvminh@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-04-07571-9
Keywords: Analytic semigroup, almost automorphic solution, uniform spectrum, sums of commuting operators
Received by editor(s): July 16, 2003
Published electronically: June 18, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society