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International Journal of Differential Equations
Volume 2014, Article ID 948597, 13 pages
http://dx.doi.org/10.1155/2014/948597
Research Article

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

Department of Mathematics, Faculty of Sciences, Ibn Zohr University, BP 8106, 8000 Agadir, Morocco

Received 13 January 2014; Revised 13 April 2014; Accepted 26 April 2014; Published 1 June 2014

Academic Editor: J. M. A. M. van Neerven

Copyright © 2014 Ahmed Fadili and Hamid Bounit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -semigroup. Next, a necessary and sufficient condition for -admissibility 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).