An Existence Theorem for Abel Integral Equations

Abstract

An existence and smoothness theorem is given for the Abel integral equation $\int _0^s K(s,t)f(t)(s^p - t^p )^{ - \alpha } dt = g(s)$, $0 < s \leqq T$, with given $p > 0$ and $0 < \alpha < 1$. Particular attention is given to the behavior of $g(s)$ and $f(s)$ about $s = 0$.