Colloquium publications.

104

THE CAMBRIDGE COLLOQUIUM.

Associated with (7) we have the equation
(8) (a(x) = f(x) - X f (s)dsAA(xs),
and the theorem:
THEOREM. If a(xs) is continuous in x for every value of s and x,
and is of uniformly limited variation in s for x in ab, and if f(x) is
continuous in ab, then (2) has one and only one continuous solution,
and it may be written in the form (8), provided X is taken small
enough ( l1/Tl).
In fact, (8) follows from (2) by (7) in the same manner exactly,
as in the corresponding theory of the ordinary integral equation;*
and vice versa, (2) follows from (8). Hence our problem is
reduced to finding a function AA(xs), continuous in x for every
value of s, and of uniformly limited variation in s for x in ab,
which satisfies both of the equations (7). This function is
given by the formula
0
where ai is the ith iterated kernel:
rb
ai(xs) = b o(ss)dso-l(xs) i > 0,
ao(xs) a= (xs).
These last formulae may be rewritten for i > 0, in the form:
fb
(9') ai(xs) = aj(s's)ds'aijl(xs), j = O, 1, **, i - 1,
va
as we see by (6). As is verified by mathematical induction, all
these functions are continuous in x for every s; moreover they
are all functions of uniformly limited variation in s. We have
in fact from (5') the inequality
Tai c ( Ta) i
* See for instance Bocher, Introduction to the Study of Integral Equations,
Cambridge (Eng.) (1909), pp. 21, 22.