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 + x R + x El -* El are compared. It is found that the continuous dependence of solutions of the Volterra equation x(t) f(t) + 5 , g(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. The purpose of this paper is to give a counterexample to a proposition of R. K. Miller, concerning continuous dependence of solutions of Volterra integral equations. Consider the nonlinear Volterra equation x(t) = f(t) + f g(t, s, x(s)) ds, (E) where f E C(R +; E) (that is, f is a continuous function from R + = [0, oo) to the n-dimensional Euclidean space) and g is a function defined on R + X R + x En with values in En. In his book [2] Miller defines a vector space g of functions g: R + x R + x En--> En which satisfy certain hypotheses. Then he defines a family of seminorms on g (see (2) below) and proves, roughly speaking, that the solutions x of (E) depend continuously on the functionsf and g, where the functionsf belong to C(R +; En) topologized with the topology of uniform convergence on compact subsets and the functions g belong to 9 topologized with the locally convex vector topology induced by the seminorms mentioned above (cf. Theorem II.4.2 and its corollaries [2, pp. 108-116]). Miller gives an alternative characterization of the seminorms (see (4) below). In this note I will prove that this characterization is erroneous and that the seminorms on the right side of (4) lead to a topology on g for which the continuity theorems in [2] do not hold. The exact definition of g is given in [2, pp. 106-108]. For the purpose of this note it suffices to know that 9 contains all functions of the type g(t, s, x) = a(t s)h(s, x), (1) where h is a continuous function R + x En---En, and a is a locally integrable real valued function which vanishes on (oo, 0). The seminorms on 9 are defined in the following way (cf. [2, p. 107]): For every g E 9, every positive real number K and every bounded set B in Et, p(g, K, B) is defined by p(g, K, B) = sup{f Ig(t, s, p(s))I ds|t E[0, K], q E C([0, K]; B)}, (2) Received by the editors May 9, 1980. AMS (MOS) subject classifications (1970). Primary 45D05.