Mathematics

Picard on the Picard Method

1

Let us take a single first-order equation

then, setting

when
one can establish the fundamental existence theorem for this equation. To this end, consider the equations

effecting each quadrature2 in such a way that for

one has
The problem is to prove that, as n → ∞, yn tends to a limit y which represents the desired integral provided that x remains in the neighborhood of x0. We assume that the function F(x, y) is continuous and defined [on
moreover, that one can determine a positive constant k such that3

we also assume that the function and the variables are real.

Let

when x and y remain between the indicated limits. One will have

Let ρ be a quantity nearly equal to a; y1 will stay within the desired limits if

and it is evident that the same will be true for
Letting δ denote a quantity nearly equal to ρ, we will suppose that . . .

251

Putting

we then have

and all the z vanish at

One has
and generally,

Hence, writing

one sees that yn tends to a limit if

As a decreasing geometric progression, the series

will be convergent. Thus yn converges to a limit y when . . .

δ being the smallest of the quantities a, b/M, 1/k.

In this interval, y evidently represents a continuous function of x. Thus one has

and, as yn and

tend to y, it follows that

and hence

that is, the limit y satisfies the differential equation. Thus, the existence of the solution has been established. One can evidently employ the same type of proof if F is an analytic function of the complex variables z and w.

1 E. Picard, "Mémoire sur la théorie des équations aux derivées partielles et la méthode des approximations successives," J. math. pures appl. 6 (1890), 145–210 [198–200].

2 What is meant is, choosing each constant of integration.

3 Picard here uses the Lipschitz condition of Selection 48.