Lecture Notes in Differential Equations - Bruce E. Shapiro

218 LESSON 26. GENERAL EXISTENCE THEORY*
Figure 26.1: A sufficient condition for a bounded continuous function to
have a fixed point is that the range be a subset of the domain. A fixed
point occurs whenever the curve of f(t) intersects the line y = t.
b
S
a
a
b
Theorem 26.2 (Sufficient condition for fixed point). Suppose that
f(t) is a continuous function that maps its domain into a subset of itself,
i.e.,
f(t) : [a, b] ↦→ S ⊂ [a, b] (26.1)
Then f(t) has a fixed point in [a, b].
Proof. If f(a) = a or f(b) = b then there is a fixed point at either a or b.
So assume that both f(a) ≠ a and f(b) ≠ b. By assumption, f(t) : [a, b] ↦→
S ⊂ [a, b], so that
f(a) ≥ a and f(b) ≤ b (26.2)
Since both f(a) ≠ a and f(b) ≠ b, this means that
f(a) > a and f(b) < b (26.3)
Let g(t) = f(t) − t. Then g is continuous because f is continuous, and
furthermore,
g(a) = f(a) − a > 0 (26.4)
g(b) = f(b) − b < 0 (26.5)
Hence by the intermediate value theorem, g has a root r ∈ (a, b), where
g(r) = 0. Then
0 = g(r) = f(r) − r =⇒ f(r) = r (26.6)
i.e., r is a fixed point of f.