State of the Problem

The equations of the type

$$\frac{dy}{dx}=f(x,y),$$

whose solutions were found, in the preceding chapter, by the application of elementary processes, are integrable on account of the fact that they belong to certain simple classes. In general, however, an equation of the type in question is not amenable to so elementary a treatment, and in many cases the investigator is obliged to have recourse to a method of numerical approximation. The theoretical question therefore arises as to whether a solution does exist, either in general or under particular restrictions. Researches into this question have brought to light a group of theorems known as existence-theorems, the more important of which will be studied in the present chapter.¹

Let $(x_0,y_0)$ be a particular pair of values assigned to the real variables $(x,y)$ such that within a rectangular domain $D$ surrounding the point $(x_0,y_0)$ and defined by the inequalities

$$|x-x_0|\le a,|y-y_0|\le b,$$

$f(x,y)$ is a one-valued continuous² function of $x$ and $y$.

Let $M$ be the upper bound of $|f(x,y)|$ in $D$ and let $h$ be the smaller of $a$ and $b/M$. If , the more stringent restriction

$$|x-x_0|\le h$$

is imposed upon $x$. (Fig. 1.)

Yet another condition must be satisfied by $f(x,y)$, namely that, if $(x,y)$ and $(x,Y)$ be two points within $D$, of the same abscissa, then

where $K$ is a constant. This is known as the Lipschitz condition.³

Then, these conditions being satisfied, there exists a unique continuous function of $x$, say $y(x)$, defined for all values of $x$ such that , which satisfies the differential equation and reduces to $y_0$ when $x=x_0$.

Two entirely distinct proofs of this existence theorem will now be given, known respectively as the Method of Successive Approximations and the Cauchy-Lipschitz Method.

Footnotes

1. See also Chap. XII., where the question is discussed from the point of view of the theory of functions of a complex variable. ↩

2. $f(x,y)$ is a continuous function of $x$ and $y$ in $D$ if, given an arbitrarily small positive number $ϵ$, a number $\delta$ can be determined such that , provided that $(x,y)$ and $(x+h,y+k)$ are in $D$ and , . It is important to note that $h$ and $k$ vary independently. ↩

3. It will be seen, as the theory develops, that it is only necessary that the Lipschitz condition should hold in the smaller region , . ↩