A Source Book in Classical Analysis

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Author: E. Picard  | Date: 1890

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.

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