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Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces

Published online by Cambridge University Press:  21 August 2013

Carlos Lizama  and
Rodrigo Ponce
Carlos Lizama
Affiliation:
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile (carlos.lizama@usach.cl)
Rodrigo Ponce
Affiliation:
Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile (rponce@inst-mat.utalca.cl)
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Abstract

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Let A and M be closed linear operators defined on a complex Banach space X and let a ∈ L1(ℝ+) be a scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equation

with initial condition Mu(0) = Mu(2π), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel–Lizorkin and Lebesgue vector-valued function spaces.

Keywords

differential equations with delayoperator-valued Fourier multipliersR-boundednessUMD spacesBesov vector-valued spacesLebesgue vector-valued spacesPrimary 35K9045N0545D05Secondary 34G1034K13
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 853 - 871
Copyright
Copyright © Edinburgh Mathematical Society 2013 

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