Continuous Dependence of Solutions of Volterra Integral Equations

The nonlinear Volterra integral equation \[ x(t) = f(t) + \int_0^t {g(t,s,x(s))ds} \] is considered. We discuss topologies on the collection of functions g such that the solution of the equation varies continuously with the data $gf$ and f, where the topology on f is the uniform convergence on compact intervals. We give a necessary and sufficient condition (on such a topology) for the continuous dependence to hold. In a particular case where a Lipschitz condition is added we show that there exists a smallest topology which satisfies the condition, and characterize it.