Multiplicative perturbations of linear Volterra equations

Abstract:

We prove that the following problems are essentially equivalent: \[ \begin {array}{*{20}{c}} {{{[\operatorname {VO} ]}_{CT}}} & {\qquad \qquad u(t) = x + \int _0^t {a(t - s)CTu(s) ds,} } \\ {{{[\operatorname {VO} ]}_{TC}}} & {\qquad \qquad v(t) = y + \int _0^t {a(t - s)TCv(s) ds,} } \\ \end {array} \] where $T$ is an unbounded closed linear operator in a Banach space $X$ with dense domain $D(T),\;C$ is a bounded linear operator on $X$, 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.