On the asymptotics of solutions of Volterra integral equations (Q1061971)

Consider the complex-valued, linear Volterra equation \[ (*)\quad v(t)- \int^{t}_{t_ 0}G(t,\tau)v(\tau)d\tau =v_ 0(t),\quad t\geq t_ 0. \] It is known that the solution of this equation can be expressed in terms of a variation of constants formula \[ v(t)=v_ 0(t)- \int^{t}_{t_ 0}R(t,\tau)v_ 0(\tau)d\tau,\quad t\geq t_ 0, \] where R often is called the resolvent of -G. In this paper it is assumed throughout that for some constants C, c, \(\alpha\) and \(\beta\), \(| G(t,\tau)| \leq C(t/\tau)^{\beta}\tau^{\alpha -1}\), \(t_ 0\leq \tau \leq t\) and \(| G(t,t)| \geq ct^{\alpha -1}\), \(t\geq t_ 0\). Under these (plus some other) assumptions, estimates are given on R which imply that the solution v of (*) grows with at most polynomial rate if \(v_ 0\) do so. The results of this paper are modelled after a particular problem in scattering theory.