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 quadrature^{2} in such a way that for
one has
The problem is to prove that, as
n → ∞,
y_{n} tends to a limit
y which represents the desired integral provided that
x remains in the neighborhood of
x_{0}. We assume that the function
F(
x, y) is continuous and defined [on
moreover, that one can determine a positive constant
k such that
^{3}
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; y_{1} 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 y_{n} tends to a limit if
As a decreasing geometric progression, the series
will be convergent. Thus y_{n} 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 y_{n} 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.