Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 12
Existence of Solutions*
In this section we will prove the fundamental existence theorem. We will
defer the proof of the uniqueness until section 13. Since we will prove
the theorem using the method of successive approximations, the following
statement of the fundamental existence theorem is really just a re-wording
of theorem 11.2 with the references to uniqueness and Picard iteration
removed. By proving that the Picard iterations converge to the solution, we
will, in effect, be proving that a solution exists, which is why the reference
to Picard iteration is removed.
Theorem 12.1 (Fundamental Existence Theorem). Suppose that
f(t, y) and ∂f(t, y)/∂y are continuous in some rectangle R defined by
Then the initial value problem
t 0 − a ≤ t ≤ t 0 + a (12.1)
y 0 − b ≤ y ≤ y 0 + b (12.2)
⎫
dy
= f(t, y) ⎬
dt
⎭
y(t 0 ) = y 0
(12.3)
has a solution in some interval
t 0 − a ≤ t 0 − h ≤ t ≤ t 0 + h ≤ t 0 + a (12.4)
∗ Most of the material in this section can be omitted without loss of continuity with
the remainder of the notes. Students should nevertheless familiarize themselves with the
statement of theorem 12.1.
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