Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

71

Ib
s(x) -J G(xx')p(x')dx'
(39) b
= y(X) - G(xx') (x')dx' - H[p, fx]
and denote the right-hand member by O[(, f x]. The differential of 0,
(39') d[ko, flx]= - f G(xx') - H4[X,, fixx'] }A (x')dx',
may be written for brevity in the form
f R[,, f I xx']A, (')x',
a
and satisfies the inequality
b
l ae - dOe < e max a(p(x) \,
a
where the e is the e of (37).
Equation (39) may be taken as an integral equation in sp(x),
the right-hand member being regarded for the moment as a
known function of x. It yields then the result
(40) s(x) = [o, f jx]
where
fa
(40') 4p, f Ix] = [o, f x] - fr(')o, fl x']dx',
and r(xx') is the kernel resolvent for G(xx'). Moreover, we can
deduce (39) from (40) by the resolution of the inverse integral
equation, so that (39) and (40) are fully equivalent. Equation
(40) is however of the form of (1) Art. 1, and satisfies the conditions there imposed.
We have in fact for dA, the formula
Jb
d,= J dx'At(x') { R[p, f xx']
(41) a
- r(x, x")R[, Ylx"x']dx"}
and for A,4 the inequality
1A4 - dl < e max A(x) {1 + r(xx') I'},
a a