International Journal of Differential Equations

On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems

Ahmed Fadili and Hamid Bounit

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Abstract

This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates a C0-semigroup. Next, a necessary and sufficient condition for Lp-admissibility p1, of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).

Article information

Source
Int. J. Differ. Equ., Volume 2014 (2014), Article ID 948597, 13 pages.

Dates
Received: 13 January 2014
Revised: 13 April 2014
Accepted: 26 April 2014
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1484881403

Digital Object Identifier
doi:10.1155/2014/948597

Mathematical Reviews number (MathSciNet)
MR3219411

Zentralblatt MATH identifier
1291.44001

Citation

Fadili, Ahmed; Bounit, Hamid. On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems. Int. J. Differ. Equ. 2014 (2014), Article ID 948597, 13 pages. doi:10.1155/2014/948597. https://projecteuclid.org/euclid.ijde/1484881403


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