Resolvents of integral equations with continuous kernels

Abstract

In this paper we study in some detail two common forms of the resolvent equation for an integral equation $x(t)=a(t)-\int^t_0 C(t,s)x(s)ds$. While it is crucial, and commonly accepted, that there is a unique matrix $R$ satisfying both $R(t,s)=C(t,s)-\int^t_s R(t,u)C(u,s)du$ and $R(t,s)= C(t,s)-\int^t_s C(t,u)R(u,s)du$, the proofs of this in the literature seem complicated, fragmentary, and not particularly clear. We first give brief, clear, and simple proofs of those fundamental properties. We then use this information in two nonlinear perturbation problems.