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) - ∫0t K (t, s) x (s) d s and suppose that the kernel K and forcing function a depend on some parameters ε{lunate} ∈ Rd. We prove that, under suitable conditions, the solutions depend on ε{lunate} as smoothly the functions a and K. 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. © 2010 Elsevier Ltd. All rights reserved.

Publication Title

Nonlinear Analysis, Theory, Methods and Applications