Disclamer: This is a student prepared note based on the lecure of Monday September 21st.
Def.
is called Lipschitz if
(a Lipschitz constant of f) such that
.
Note that any function that is Lipschitz is uniformly continuous, and that if a function f and its derivative are both continous on a compact set then f is Lipschitz.
Thm. Existence and Uniqueness Theorem for ODEs
Let
be continuous and uniformly Lipschitz relative to y. Then the equation
with
has a unique solution
where
where M is a bound of f on
.
Let
and let
.
Claim 1:
is well-defined. More precisely,
is continuous and
,
where b is as referred to above.
Proof of Claim 1:
The statement is trivially true for
. Assume the claim is true for
.
is continuous, being the integral of a continuous function.
Claim 2: For
,
.
Proof of Claim 2: