Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

53

namely, if yi and ~2 are any two functions of L, then a constant
M can be found, M < 1, such that the following condition holds:
b b b b
(2) max IF[(pl(s) x] - F[(p2(s)I x] - M max | Si() -- 2(X) |,
a a a a
b
where max I s(x) denotes the upper bound of a function s(x)
a
in the range ab.
Under these conditions, the class L contains one and only one
solution of equation (1).
To construct this solution we take 0po, any particular function
in L, and write
Pn(x) = F[pn-llx], n = 1, 2, 3, *-.
Then the function
(p(x) = lim <p(x)
n=o0
is in L, as is seen at once from the uniform convergence of the
series
<Po + (01 - <Po) + (<P2- ) + * * *,
and is a solution of (1), since
b b b
max I p - F[(p x] max p - <pn+ + max I F[pn ix] - F[p Ix].
a a a
If there were two solutions sp and p' we should have, by
equation (2):
b b
max | so- ps' I M max I o- o'I,
a a
which is a contradiction, since M < 1.
We have the corollary, that if L contains a continuous function, and if F[sp I x] represents a continuous function of x when its
argument <p(x) represents a continuous function of x, then the
unique solution of (1) is continuous.
36. The Case of a Variable Upper Limit. If the functional F[s I x] depends upon sp only for values between a and x, we are able to make use of a
property of prolongation, which, speaking generally, makes less restrictive the
convergence condition (ii) imposed on M in Art. 35. In particular, if F consists of terms independent of sp plus a term whose variation is of the form