A contraction mapping proof of the smooth dependence on parameters of solutions to Volterra integral equations

We consider the linear Volterra equation x(t)=a(t)−∫0tK(t,s)x(s)ds and suppose that the kernel KK and forcing function aa depend on some parameters ϵ∈Rdϵ∈Rd. We prove that, under suitable conditions, the solutions depend on ϵϵ as smoothly the functions aa and KK. The proof is based on the contraction mapping principle and the variational equation. Though our conditions are not the most generally possible, they nonetheless include many important examples.