Vector-Valued Tauberian Theorems and Asymptotic Behavior of Linear Volterra Equations

The asymptotic behavior of the solutions of linear Volterra equations in a Banach space X of the form \[ ( * ) u(t) = f(t) + ∫_0^t {a(t - τ )} Au(τ )d(τ ), t 0\] is studied, in particular that of the resolvent $S(t)$ for $( * )$; here a ∈ L_loc¹ (R_ + ) and A is a closed linear operator in X with dense domain. A complete characterization of the existence of lim t → 0 S(t)x = Px for all x ∈ X in the sense of Abel is obtained, and the nature of the ergodic limit P is studied. By means of vector-valued Tauberian theorems for the Laplace transform, a general result on convergence of $S(t)$ in the strong sense is derived. Several examples are given which illustrate this result, and also an application to the theory of linear viscoelasticity is presented.