On the inversion and admissibility for a class of Volterra integro-differential problems

Abstract

Abstract We study the Volterra integro-differential problems of convolution kernel type from two perspectives: complex inversion formula and the admissibility in the Salamon–Weiss sense (which leads us to consider unbounded control operators). Due to the hybrid Cauchy–Volterra context of these problems, the control operators need to have the admissibility property. First, we show the validity of the inverse formula of the Laplace transform for the resolvent operators associated with some class of Volterra integro-differential problems of convolution type in Banach spaces with the leading operator generating a $C_{0}$-semigroup. This extends and improves the results in Hille & Phillips (1957, 347–348, Theorem 11.6.1), Driouich & El-Mennaoui (1999, 72, Theorem 1), Cioranescu & Lizama (2003, 188–189, Proposition 2), Haase (2008, 84, 81–82, Theorems 4.2 and 4.1) and Fadili & Bounit (2014, 7–8, Proposition 8), including the strong version for this class on UMD spaces. Second, the sufficient or/and necessary conditions for $L^{p}$-admissibility with $p\in (1,+\infty )$ of the control operators are given in terms of the UMD geometric property of its underlying control spaces for a wide class of Volterra integro-differential problems, which provides a generalization of a result known to hold for the standard Cauchy problems shown in Bounit et al. (2010, Proposition 3.2) and the standard Volterra integro-differential problems shown in Fadili & Bounit (2014, 9–10, Theorem 12).