Nothing due May 11. The last homework was going to be a little project in
doing a piece of mathematics but it is too late given the fact
that there will be a final exam. To test yourself and see if you have
gained in understanding from the course, try to write all this out as
clearly as you can after you figure it out.
Read Rudin Chapter 9, ``The contraction principle'' on
your own. Let
be continuous and Lipschitz
continuous in the second variable, meaning that there is a constant
Suppose that for some
is a differentiable function, with
the differential equation
- Show that is continuously differentiable on
and that it
satisfies the integral equation
- Conversely show that if
continuous function which satisfies (4) then it is
differentiable and satisfies (3).
- Fix some real number
is closed with respect to the supremum metric.
and as above show that the map
defines a map from into and that this map satisfies
where is the constant in (2).
- Conclude that for sufficiently small (depending only on and
but not there is a unique differentiable function on
- No big hints for the last part! Show that, given
and under the same conditions as above there is a unque differentiable
function on satisfying the intial value problem