Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 26
General Existence
Theory*
In this section we will show that convergence of Picard Iteration is the
equivalent of finding the fixed point of an operator in a general linear vector
space. This allows us to expand the scope of the existence theorem to initial
value problems involving differential equations of any order as well systems
of differential equations. This section is somewhat more theoretical and
may be skipped without any loss of continuity in the notes.
Before we look at fixed points of operators we will first review the concept
of fixed points of functions.
Definition 26.1. Fixed Point of a Function. Let f : R ↦→ R. A number
a ∈ R is called a fixed point of f if f(a) = a.
Example 26.1. Find the fixed points of the function f(x) = x 4 + 2x 2 +
x − 3.
x = x 4 + 2x 2 + x − 3
0 = x 4 + 2x 2 − 3
= (x − 1)(x + 1)(x 2 + 3)
Hence the real fixed points are x = 1 and x = −1.
A function f : R ↦→ R has a fixed point if and only if its graph intersects
with the line y = x. If there are multiple intersections, then there are
multiple fixed points. Consequently a sufficient condition is that the range
of f is contained in its domain (see figure 26.1).
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