The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 2. SUCCESSIVE APPROXIMATIONS 33
Proof. By theorem 2.1 a fixed point exists. Call it p,
p = f(p) (2.26)
Suppose that a second fixed point q ∈ [a, b], q ≠ p also exists, so that
Hence
q = f(q) (2.27)
|f(p) − f(q)| = |p − q| (2.28)
By the mean value theorem (theorem 2.3) there is some number c between p and q
such that
f ′ f(p) − f(q)
(c) = (2.29)
p − q
Taking absolute values,
and thence
f(p) − f(q)
∣ p − q ∣ = |f ′ (c)| ≤ K < 1 (2.30)
|f(p) − f(q)| < |p − q| (2.31)
which is a contradiction. Thus our assumption that a second, different fixed point
exists must be incorrect. Hence the fixed point is unique.
Theorem 2.8 (Convergence of fixed point iteration). Under the same conditions
as theorem 2.7 then fixed point iteration converges.
Proof. We know from theorem 2.1 that a fixed point exists, and from theorem 2.7
that the fixed point p exists. Pick any starting point p 0 ∈ [a, b], and generate the
sequence of fixed point iterations
We need to prove that p i → p as i → ∞.
p i+1 = f(p i ), i = 1, 2, . . . (2.32)
Since f maps onto a subset of itself, every point p i in the sequence is clearly in ⇐=
the interval [a, b]. Further, since p itself is a fixed point, ⇐=
|p i − p| = |p i − f(p)| = |f(p i−1 ) − f(p)| (2.33)
If for any value of i we have p i = p then we have reached the fixed point and the
theorem is proved. So we assume that p i ≠ p for all i. Then by the mean value
theorem, for each value of i there exists a number c i between p i−1 and p such that
Since f(p) = p and f(p i−1 ) = p i ,
|f(p i−1 ) − f(p)| = |f ′ (c i )||p i−1 − p| ≤ K|p i−1 − p| (2.34)
|p i − p| ≤ K|p i−1 − p| (2.35)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge