Lecture Notes in Differential Equations - Bruce E. Shapiro

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Theorem 26.5. Under the same conditions as theorem 26.4 except that
the condition of equation 26.7 is replaced with the following condition:
f(t) is Lipshitz with Lipshitz constant K < 1. Then fixed point iteration
converges.
Proof. Lipshitz gives equation 26.16. The rest of the the proof follows as
before.
The Lipshitz condition can be generalized to apply to functions on a vector
space.
Definition 26.6. Lipshitz Condition on a Vector Space. Let V be a
vector space and let t ∈ R. Then f(t, y) is Lipshitz if there exists a real
constant K such that
for all vectors y, z ∈ V.
|f(t, y) − f(t, z)| ≤ K|y, z| (26.21)
Definition 26.7. Let V be a normed vector space, S ⊂ V. A contraction
is any mapping T : S ↦→ V such that
‖T y − T z‖ ≤ K‖y − z‖ (26.22)
where 0 < K < 1, holds for all y, z ∈ S. We will call the number K
the contraction constant. Observe that a contraction is analogous to a
Lipshitz condition on operators with K < 1.
We will need the following two results from analysis:
1. A Cauchy Sequence is a sequence y 0 , y 1 , . . . of vectors in V such
that ‖v m − v n ‖ → 0 as n, m → ∞.
2. Complete Vector Field. If every Cauchy Sequence converges to an
element of V, then we call V complete.
The following lemma plays the same role for contractions that Lemma (12.6)
did for functions.
Lemma 26.8. Let T be a contraction on a complete normed vector space
V with contraction constant K. Then for any y ∈ V
‖T n y − y‖ ≤ 1 − Kn
‖T y − y‖ (26.23)
1 − K