The Initial Value Problem y' = f(x, y); y(c) = d 43

We state the following existence theorem without proof. This theorem is

due to the Italian mathematician and logician Giuseppe Peano (1858-1932).

FUNDAMENTAL EXISTENCE THEOREM

Let R = {(x,y)I a< x < f3 and 'Y < y < o} where a, (3, ry, and o a re

finite real constants. If f(x, y) is a continuous function of x and y in the

finite rectangle Rand if (c, d) ER, then there exists a solution to the initial

value problem y' = f(x, y); y(c) = d on some interval I = (c - h , c + h)

where I is a subinterval of the interval (a , (3).

The geometry of the fundamental existence theorem is depicted in Fig-
ure 2.5. The theorem, itself, states a fairly simple condition- namely, con-
tinuity off (x, y)- which guarantees the existence of a solution to the initial
value problem (1) y' = f(x, y); y(c) = d. However, neither the theorem
nor its proof provides a method for producing the solution or for satisfacto-
rily calculating the value of h which, in t heory, determines the interval on
which the solution exists. The theorem simply states that there is an interval

(c - h, c + h), which is a subinterval of (a, (3), on which t he solution exists.

But the length of the interval is not specified. As the following example il-
lustrates, there are instances in which the interval of existence depends to
a greater extent upon the initial condition y(c) = d than it does upon the
function f(x, y).

y

8 - - -

R

~

I I

(c,d) I

d

I

y

c-h c c+h f3 x

Figure 2.5 Geometry of the Fundamental Existence Theorem