Positive kernels, fixed points, and integral equations

There is substantial literature going back to 1965 showing boundedness properties of solutions of the integro-differential equation \[ x'(t) = -\int^t_0 A(t-s) h(s,x(s))ds \] when $A$ is a positive kernel and $h$ is a continuous function using \[ \int^T_0 h(t,x(t))\int^t_0 A(t-s) h(s,x(s))ds dt \geq 0. \] In that study there emerges the pair: \[\text{Integro-differential equation and Supremum norm.} \] In this paper we study qualitative properties of solutions of integral equations using the same inequality and obtain results on $L^p$ solutions. That is, there occurs the pair: \[ \text{Integral equations and $L^p$ norm.}\] The paper also offers many examples showing how to use the $L^p$ idea effectively