Stability properties of linear Volterra equations (Q756077)

The stability properties of the linear Volterra equation \(\dot x(t)=A(t)x(t)+\int^{t}_{0}B(t,s)x(s)ds,\) in an n-dimensional Euclidean space is studied. Here the functions A(t) and \(B(t,t+s)\) are almost periodic in t and satisfy some other, more technical, conditions. The main result is that the null solution is uniformly asymptotically stable if and only if \(\sup_{t\geq \sigma \geq 0}\{| R(t,\sigma)| +\int^{t}_{\sigma}| R(t,s)| ds\}<\infty,\) where R(t,s) is the resolvent.