Multiplicative perturbations of linear Volterra equations

Rhandi, Abdelaziz

doi:10.1090/S0002-9939-1993-1169047-0

Multiplicative perturbations of linear Volterra equations

Author: Abdelaziz Rhandi
Journal: Proc. Amer. Math. Soc. 119 (1993), 493-501
MSC: Primary 47N20; Secondary 45D05, 47D03
DOI: https://doi.org/10.1090/S0002-9939-1993-1169047-0
MathSciNet review: 1169047
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Abstract: We prove that the following problems are essentially equivalent:

$\begin{displaymath}\begin{array}{*{20}{c}} {{{[\operatorname{VO} ]}_{CT}}} & {\q... ...uad v(t) = y + \int_0^t {a(t - s)TCv(s)\,ds,} } \\ \end{array} \end{displaymath}$

where

is an unbounded closed linear operator in a Banach space

with dense domain $D(T),\;C$ is a bounded linear operator on

, and $a \in L_{\operatorname{loc} }^1([0,\infty ),\mathbb{R})$ , which is exponentially bounded. We give some applications of our abstract theorem to second-order differential operators on the line.

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