Two Tauberian theorems for nonconvolution Volterra integral operators

Abstract:

Two sets of sufficient conditions on the kernel $k(t,s)$ are given so that one can prove that if $x$ is a bounded function such that \[ \lim \limits _{\begin {array}{*{20}{c}} {t \to \infty } \\ {\tau \to 0} \\ \end {array} } \left | {x(t + \tau ) - x(t)} \right | = 0\quad {\text {and}}\quad \lim \limits _{t \to \infty } \int _0^t {k(t,s)x(s)ds} \] exists, then ${\lim _{t \to \infty }}x(t)$ exists.