Two Tauberian theorems for nonconvolution Volterra integral operators

Abstract

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