On linear Volterra integral equations of convolution type

Abstract:

Let A be the set of all complex-valued locally integrable functions defined on $[0, + \infty )$, and let T be the topology for A determined by the seminorms ${t_r}(f) = \smallint _0^r {|f(x)|dx}$ for $r = 1,2, \cdots$ , so that A is a topological algebra under pointwise addition, complex scalar multiplication, and Laplace convolution. Then the map $f \to f’$ from each element to its quasi-inverse is a homeomorphism of (A, T) onto itself. For each f, g in A the equation $v = f + g ^\ast v$ has a unique solution in A which depends T-continuously on f, g, and is the T-limit of Picard approximations. The set of all f in A with $f’$ in ${L^1}[0, + \infty )$ is a set of first category in (A, T) but an open subset of A with the metric ${\left \| {f - g} \right \|_1}$. For each series $\sum \nolimits _{n = 1}^\infty {{p_n}{z^n}}$ converging in some neighborhood of $z = 0$, and each element f in A, the series $\sum \nolimits _{n = 1}^\infty {{p_n}{f^{ \ast n}}}$ converges in T to some element ${p^ \ast }(f)$ in A.