Function bounds for solutions of Volterra equations and exponential asymptotic stability (Q880306)

The main purpose of this paper is to obtain functions that bound the solutions of the scalar linear Volterra integro-differential equation \[ x'(t)=-a(t)x(t)+\int_0^tb(t-s)x(s)ds, \] where \(a,b:[0,\infty)\to {\mathbb R}\) are continuous functions and \(b\) is nonnegative. In order to achieve this purpose an extension of Gronwall's inequality is derived. The obtained bound functions are then used to find conditions that drive its solutions to zero, and to obtain conditions under which the zero solution is exponentially asymptotically stable.