Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes

Carlos Lizama; Marina Murillo-Arcila

doi:10.3934/dcds.2020020

Article Contents

2020, Volume 40, Issue 1: 509-528. Doi: 10.3934/dcds.2020020

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Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes

Carlos Lizama^1, , and
Marina Murillo-Arcila^2,

1.
Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Las Sophoras 173, Santiago, Estación Central, Santiago, Chile
2.
Universitat Jaume I, Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Campus del Riu Sec s/n, 12071 Castelló, Spain

^* Corresponding author: Carlos Lizama
^* Corresponding author: Carlos Lizama

Received: February 28, 2019

Revised: June 30, 2019

Published: October 2019

The first author is supported by FONDECYT grant 1180041

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Abstract

In this paper we investigate conditions for maximal regularity of Volterra equations defined on the Lebesgue space of sequences $ \ell_p(\mathbb{Z}) $ by using Blünck's theorem on the equivalence between operator-valued $ \ell_p $-multipliers and the notion of $ R $-boundedness. We show sufficient conditions for maximal $ \ell_p-\ell_q $ regularity of solutions of such problems solely in terms of the data. We also explain the significance of kernel sequences in the theory of viscoelasticity, establishing a new and surprising connection with schemes of approximation of fractional models.

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Mathematics Subject Classification: Primary: 49M25, 49N60; Secondary: 42A45, 47A58.

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