A note on continuous dependence of solutions of Volterra integral equations

Abstract:

The topologies induced by two families of seminorms on a vector space of functions $g:{R^ + } \times {R^ + } \times {E^n} \to {E^n}$ are compared. It is found that the continuous dependence of solutions of the Volterra equation $x(t) = f(t) + \int {_0^tg(t,s,x(s))ds}$ does not hold for the weaker topology. This result corrects an error in the book of Miller, Benjamin, Menlo Park, Calif., 1971.