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Almost automorphic solutions for semilinear boundary differential equations


Authors: S. Boulite, L. Maniar and G. M. N'Guérékata
Journal: Proc. Amer. Math. Soc. 134 (2006), 3613-3624
MSC (2000): Primary 34A05, 34G20, 47A55
DOI: https://doi.org/10.1090/S0002-9939-06-08423-1
Published electronically: June 12, 2006
MathSciNet review: 2240674
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Abstract: In this work, we use the extrapolation methods to study the existence and uniqueness of almost automorphic solutions to the semilinear boundary differential equation

\begin{displaymath}(SBDE)\;\;\;\begin{cases} \begin{array}{lll} x'(t) & = & A_mx... ... Lx(t)&=&g(t,x(t)),\;\;t\in \mathbb{R}, \end{array}\end{cases}\end{displaymath}      

where $ A:=A_m\vert\ker L$ generates a hyperbolic $ C_0$-semigroup on a Banach space $ X$ and $ h,g$ are almost automorphic functions which take values in $ X$ and a ``boundary space'' $ \partial X$, respectively. These equations are an abstract formulation of partial differential equations with semilinear terms at the boundary, such as population equations, retarded differential equations and boundary control systems. An application to retarded differential equations is given.


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Additional Information

S. Boulite
Affiliation: Department of Mathematics, Faculty of Sciences Semlalia, B.P. 2390, Marrakesh, Morocco
Email: sboulite@ucam.ac.ma

L. Maniar
Affiliation: Department of Mathematics, Faculty of Sciences Semlalia, B.P. 2390, Marrakesh, Morocco
Email: maniar@ucam.ac.ma

G. M. N'Guérékata
Affiliation: Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, Maryland 21251
Email: gnguerek@morgan.edu.

DOI: https://doi.org/10.1090/S0002-9939-06-08423-1
Keywords: Almost automorphic functions, semilinear boundary differential equations, retarded differential equations, hyperbolic semigroups, extrapolation space, Dirichlet map
Received by editor(s): June 13, 2005
Received by editor(s) in revised form: July 6, 2005
Published electronically: June 12, 2006
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society

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