 
Summary: SIAM J. MATH. ANAL.
Vol. 5, No. 5, October 1974
AN EXISTENCE THEOREM FOR ABEL INTEGRAL EQUATIONS*
KENDALL E. ATKINSON$
Abstract. An existence and smoothness theorem is given for the Abel integral equation
oK(s, t)f(t)(s tP) dt g(s), 0 < =< T, with given p > 0 and 0< a < 1. Particular attention is
given to the behavior of g(s) andf(s) about 0.
1. Introduction. Consider the Abel integral equation
; K(s, t)f(t) dt
(1.1)
(sp
tp)
g(s), O < s <= T,
with given p > 0 and 0 < e < 1. To avoid degeneracy, we shall assume K(s, s) 4:0
for 0 __< s =< T. This is a classical equation, and it is obtained from a variety of
mathematical and physical problems;see the bibliography of Noble [7].
In the past this equation has been examined case by case (for example, see
Schmeidler [8] and the references in [7]). The methods of analysis were usually
constructive or explicit, and the numerical analysis of (1.1) was usually based on
these methods. Within the last few years, direct numerical methods for (1.1) have
