Colloquium publications.

LECTURE III.

IMPLICIT FUNCTIONAL EQUATIONS*
~ 1. THE METHOD OF SUCCESSIVE APPROXIMATIONS
35. An Introductory Theorem. An implicit functional equation which is easily solvable is the following:
b
(1) p(x) = F[(p(s) x],
a
where, besides depending on the function;p and the variable x,
the functional constituting the right-hand member may depend
upon other functions f, g,... and other variables y, z,..,
appearing parametrically. The equation (1) may be solved
immediately by the method of successive approximations, and
in fact serves, with the conditions imposed upon it, rather as a
convenient formulation of that method than as a theorem of
explicit value.
Consider a class L of limited functions ~p(x), which contains
the limit function of any uniformly convergent sequence of its
functions pn(x). Such a class is for instance the totality of continuous functions, in numerical value less than or equal to a given
constant.
In regard to the functional F we assume:
(i) If (p(x) is a function of L, then F[sp x] is a function of L.
(ii) The functional F satisfies a Cauchy-Lipschitz condition:
* This lecture is based on the following references:
Volterra, Lecons sur les fonctions de lignes, Paris (1913), chapter 4.
Hadamard, Lecons sur le calcul des variations, Paris (1910), chapter 7, Book II.
Riesz, Les operations fonctionelles lineaires, Annales Scientifiques de 'tcole
Normale Superieure, vol. 31 (1914), pp. 9-14.
Lebesgue, Sur l'integrale de Stieltjes et sur les operations fonctionelles lineaires,
Comptes Rendus, vol. 150 (1910), pp. 86-88.
Evans, Some general types of functional equations, Proceedings of the Fifth
International Congress of Mathematicians, Cambridge (1912), vol. 1,
pp. 385-396.
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