Uniform asymptotic stability via Liapunov-Razumikhin technique

Abstract

The Liapunov-Razumikhin technique is applied to obtain the uniform asymptotic stability for linear integrodifferential equations in Hilbert spaces, \[ x ′ ( t ) = A [ x ( t ) + ∫ # t F ( t − s ) x ( s ) d s ] , t ≥ t 0 ≥ 0 ( # = 0 or − ∞ ) , x’(t) = A\left [ {x(t) + \int _\# ^t {F(t - s)x(s)\,ds} } \right ],\quad t \geq {t_0} \geq 0(\# = 0\;{\text {or}} - \infty ), \] which occur in viscoelasticity and in heat conduction for materials with memory.