On linear volterra integral equations of convolution type

Abstract

Let A be the set of all complex-valued locally integrable functions defined on [0, +∞), and let T be the topology for A determined by the seminorms tr(f)=∫0r| f(x)|dx for r=1, 2, · · ·, so that A is a topological algebra under pointwise addition, complex scalar multiplication, and Laplace convolution. Then the map f→f’from each element to its quasi-inverse is a homeomprphism of (A, T) onto itself. For each f, g in A the equation v=f+ g *v has a unique solution in A which depends T-continuously on fig, and is the T-Iimit of Picard approximations. The set of all f in A with f’ in Ll[0, +∞) is a set of first category in (A, T) but an open subset of A with the metric \\f–g\\ i. For each series Σn=1∞pnZnconverging in some neighborhood of z=0, and each element f in A, the series Σn=1∞pnf*nconverges in TXo some element p*(f) in A. © American Mathematical Society 1972.