Lp-solutions of singular integro-differential equations

We study a variety of scalar integro-differential equations with singular kernels including linear, nonlinear, and resolvent equations. The first result involves a type of existence theorem which uses a fixed point mapping defined by the integro-differential equation itself and produces a unique solution with a continuous derivative in a very simple way. We then construct a Liapunov functional yielding qualitative properties of solutions. The work answers questions raised by Volterra in 1928, by Levin in 1963, and by Grimmer and Seifert in 1975. Previous results had produced bounded solutions from bounded perturbations. Our results mainly concern integrable solutions from integrable perturbations.