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

34 CHAPTER 2. SUCCESSIVE APPROXIMATIONS
for all i. Hence
Since K > 0,
Thus p i → p as i → ∞.
|p i − p| ≤ K 2 |p i−2 − p| ≤ K 3 |p i−2 − p| ≤ · · · ≤ K i |p 0 − p| (2.36)
0 ≤ lim
i→∞
|p i − p| ≤ |p 0 − p| lim
i→∞
K i = 0 (2.37)
Theorem 2.9. Under the same conditions as theorem 2.8 except that the condition
of equation 2.25 is replaced with the following condition: f(t) is Lipshitz with
Lipshitz constant K < 1. Then fixed point iteration converges.
Proof. Lipshitz gives equation 2.34. The rest of the the proof follows as before.
2.3 Fixed Point Iteration in a Normed Vector Space
Definition 2.2 (Vector Space). A vector space V is a set that is closed under two
operations that we call addition and scalar multiplication such that the following
properties hold:
Closure For all vectors u, v ∈ V, and for all a ∈ R,
Commutivity of Vector Addition For all u, v ∈ V,
u + v ∈ V (2.38)
av ∈ V (2.39)
u + v = v + u (2.40)
Associativity of Vector Addition For all u, v, w ∈ V,
u + (v + w) = (u + v) + w (2.41)
Identity for Addition There is some element 0 ∈ V such that for all v ∈ V
0 + v = v + 0 = v (2.42)
Inverse for Addition For each v ∈ V there is a vector −v ∈ V such that
v + (−v) = (−v) + v = 0 (2.43)
Associativity of Scalar multiplication For all v ∈ V and for all a, b ∈ R,
a(bv) = (ab)v (2.44)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007