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

CHAPTER 3. APPROXIMATE SOLUTIONS 51
3.3 The Fundamental Inequality
We can prove that the Epsilon-approximate solution converges to an exact solution
of the differential equation, giving us a proof of the fundamental existence theorem.
This proof will be presented in the following section. Here we will present a basic
inequality that is often used in the theory of differential equations.
Lemma 3.1. Suppose that ∂f/∂y is bounded for all (t, y) ∈ D. Then f(t, y) is
Lipshitz on D with Lipshitz Constant
K = sup
∂f
∣ ∂y ∣ (3.66)
Proof. By the Mean Value theorem there exists some number c between between y 1
and y 2 such that
∂f(t, c)
f(t, y 2 ) − f(t, y 1 ) = (y 2 − y 1 ) (3.67)
∂y
Since ∣ ∣∣∣ ∂f(t, c)
∂y ∣ ≤ K (3.68)
we have
and thus the function is Lipshitz in y.
|f(t, y 2 ) − f(t, y 1 )| ≤ K|y 2 − y 1 | (3.69)
Lemma 3.2. Suppose D is not convex, but can be embedded in a convex domain
D ′ . Let δ > 0 be the shortest distance between the boundaries of D and D ′ . Suppose
that f is continuous and bounded by M in D, and that ∂f/∂y exists and is bounded
by N in D ′ then f(t, y) is Lipshitz in y in D with Lipshitz Constant
(
K = max N, 2M )
(3.70)
δ
Proof. Let P = (t, y 1 ), Q = (t, y 2 ) ∈ D.
Then if |y 1 − y 2 | > δ,
∣ f(t, y 1 ) − f(t, y 2 ) ∣∣∣
∣
≤ |f(t, y 1)| + |f(t, y 2 )|
= 2M y 1 − y 2 min |y 1 − y 2 | δ
thus
(3.71)
|f(t, y 1 ) − f(t, y 2 )| ≤ 2M δ |f(t, y 1) − f(t, y 2 )| (3.72)
(
≤ max N, 2M )
|f(t, y 1 ) − f(t, y 2 )| (3.73)
δ
≤ K|f(t, y 1 ) − f(t, y 2 )| (3.74)
Alternatively, if |y 1 −y 2 | ≤ δ, the line segment PQ still lies entirely in D ′ . But since
D ′ is convex, we can apply lemma 3.1 to conclude that N is a Lipshitz constant.
Hence the Lemma is proved.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge