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

Ahmed Fadili,Hamid Bounit

doi:10.1155/2014/948597

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 aC0-semigroup. Next, a necessary and sufficient condition forLp-admissibilityp∈1,∞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).

Highlights

The purpose of this paper is to analyze conditions for the inversion formula and the Lp-admissibility for control operators for the solution of the following integrodifferential equation: t ẋ (t) = Ax (t) + ∫ k (t − s) Ax (s) ds + Bu (t), x (0) = x0 ∈ X, t ≥ 0, (1)which has a “big” intersection with the class of scalar Volterra integral equations
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

Summary

Introduction

This extends the results obtained for the semigroups’ case in [32] It was proved in [37] that, for scalar Volterra integral systems with a creep kernel, finite- and infinite-time L1-admissibility are equivalent to exponentially stable resolvent family, and if the state space X is reflexive finite-time and uniformly finite-time L1-admissibility are equivalent, extending wellknown results for semigroups. First we embed this class in a larger Cauchy system, a technique originating in Engel and Nagel [15, VI.7], in order to prove some results concerning the validity of the complex inversion formula. In a forthcoming work, we will consider a class of nonscalar kernels

Review on Resolvent Families

The Complex Inversion Formula and UMD Spaces

Integrodifferential Equation with Bounded Variation Kernels

Characterization of Admissibility