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

64 CHAPTER 3. APPROXIMATE SOLUTIONS
This proves Weierstrass’ Approximation theorem for a piecewise linear function.
Since you can approximate any continuous function arbitrarily closely by a piecewise
linear function this proves the theorem for any continuous function.
3.7 Peano Existence Theorem
Our proof of the existence of a solution to the general IVP requires a Lipshitz
Condition be placed upon f(t, y). It turns out that this condition is not necessary.
Equicontinuity
Definition 3.2. Let f k (t 1 , t 2 , . . . ) be a sequence of functions. Then we say that the
set of functions {f k } is Uniformly Bounded if there exists some number M ∈ R
such that for all t 1 , t 2 , ...., k,
|f k (t 1 , t 2 , . . . )| ≤ M (3.186)
In particular, the constant M is the same for every function in the set.
Definition 3.3. A set S of functions f(t) is equicontinuous over an interval [a, b]
if for any ɛ > 0 there exists a δ > 0 such that whenever t, ˜t ∈ [a, b],
for functions f ∈ S.
|t − ˜t| ≤ δ =⇒ |f(t 1 ) − f(t 2 )| ≤ ɛ (3.187)
Lemma 3.3. Suppose that S is an infinite set of uniformly bounded, equicontinuous
functions on [a, b]. The S contains a sequence that converges uniformly on [a, b].
=⇒
Theorem 3.10 (Peano). Let D = [t 0 − a, t 0 + a] × [y 0 − b] × [y 0 + b]; Suppose
that f(t, y) ∈ C(D) be bounded by M. Then there is a solution to the initial value
problem
for |t − t 0 | < h = min(a, b/M).
y ′ = f(t, y) (3.188)
y(t 0 ) = y 0 (3.189)
Proof. Let ɛ n → 0 be a sequence of numbers, and P n (t, y) be a sequence of polynomials
such that
on
|P n (t, y) − f(t, y)| ≤ ɛ n (3.190)
t 0 − h ≤t ≤ t 0 + h (3.191)
y 0 − Mh ≤y ≤ y 0 + Mh (3.192)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007