Uniform $L^1 $ Behavior in Classes of Integrodifferential Equations with Completely Monotonic Kernels

Abstract

We find conditions on a family $\mathfrak{a}$ of completely monotone, locally integrable, nonconstant functions a which enable us to write the solution $u(t;a)$ of $u'(t) + \int_0^t {a(t - s)u(s)ds = 0} $, $u(0) = 1$, as $u(t;a) = - \int_{ - \varepsilon }^0 {e^{\sigma t} } d\mu (\sigma ;a) + u_1 (t;a)$, where $\mu ( \cdot ;a)$ is a finite nonnegative measure on $[ - \varepsilon ,0]$ and $|u_1 (t;a)| \leqslant Qe^{ - \varepsilon t} $ with Q, $\varepsilon $ positive constants independent of $a \in \mathfrak{a}$. This formula is then utilized to give conditions on the collection $\mathfrak{a}$ which ensure that $\rho (t)\sup _{a \in \mathfrak{a}} |u(t;a)| \to 0(t \to \infty )$ and $\int_0^\infty {\rho (t)\sup _{a \in \mathfrak{a}} |u(t;a)|dt < \infty } $ , where $\rho $ is a given weight function. These results can be combined with a resolvent formula to investigate the asymptotic behavior as $t \to \infty $ of solutions of certain integrodifferential equations in Hilbert space.