Integral equations, Volterra equations, and the remarkable resolvent: contractions

Abstract

This paper concerns several variants of an integral equation $$ x(t)=a(t)-\int^t_0 C(t,s) x(s)ds,$$ a resolvent $$ R(t,s),$$ and a variation-of-parameters formula $$ x(t)=a(t)-\int^t_0 R(t,s) a(s)ds $$ with special accent on the case in which $a(t)$ is unbounded. We use contraction mappings to establish close relations between $a(t)$ and $\int^t_0R(t,s) a(s)ds$.