I'm interested in the historical development of applications of successive approximations methods (Picard's iterative method for example) for demonstrating the existence of solutions to integral equations. That is, an equation of the form $ \phi(x)=\int_{a}^{b} K(x, s, \phi(s)) \;d s $ where the function $K$ is known and $\phi$ is the unknown.
More precisely, I would like to know who was the first mathematician to have this idea to show existence of solution for integral equations using successive approximation methods.
My guess is that the first to apply methods of successive approximations to show existence of solutions to integral equations is Volterra. But I'm not sure about that.