Lecture Notes in Differential Equations - Bruce E. Shapiro

219
In the case just proven, there may be multiple fixed points. If the derivative
is sufficiently bounded then there will be a unique fixed point.
Theorem 26.3 (Condition for a unique fixed point). Let f be a
continuous function on [a, b] such that f : [a, b] ↦→ S ⊂ (a, b), and suppose
further that there exists some postive constant K < 1 such that
Then f has a unique fixed point in [a, b].
|f ′ (t)| ≤ K, ∀t ∈ [a, b] (26.7)
Proof. By theorem 26.2 a fixed point exists. Call it p,
p = f(p) (26.8)
Suppose that a second fixed point q ∈ [a, b], q ≠ p also exists, so that
Hence
q = f(q) (26.9)
|f(p) − f(q)| = |p − q| (26.10)
By the mean value theorem there is some number c between p and q such
that
f ′ f(p) − f(q)
(c) = (26.11)
p − q
Taking absolute values,
f(p) − f(q)
∣ p − q ∣ = |f ′ (c)| ≤ K < 1 (26.12)
and thence
|f(p) − f(q)| < |p − q| (26.13)
This contradicts equation 26.10. Hence our assumption that a second,
different fixed point exists must be incorrect. Hence the fixed point is
unique.
Theorem 26.4 (Fixed Point Iteration Theorem). Let f be as defined
in theorem 26.3, and p 0 ∈ (a, b). Then the sequence of numbers
⎫
p 1 = f(p 0 )
p 2 = f(p 1 )
⎪⎬
.
(26.14)
p n = f(p n−1 )
⎪⎭
.
converges to the unique fixed point of f in (a, b).