Re: Contraction Mapping Problem
- From: quasi <quasi@xxxxxxxx>
- Date: Tue, 08 Jan 2008 19:17:20 -0500
On Tue, 8 Jan 2008 15:05:51 -0800 (PST), maud.july@xxxxxxxxx wrote:
I also see that if a solution takes the fixed point of the cosine
function, then it is constant thereafter, by the uniquness of a
solution.
On Jan 8, 4:48 pm, maud.j...@xxxxxxxxx wrote:
Sorry if this posts twice, did't seem to go through the first time.
I understand that any two solutions, u, taking the value b will look
indentical (although maybe shifted) Thus the lim t->infty u(t) will be
the same for any two solutions taking the same value. However I still
don't see (1) why a limit exists and (2) how to see that there is one
solution function u that takes any two prescibed values.
Please don't top post.
As a guideline, only show the relevant parts of the prior message, and
then either repy at the bottom, or else intersperse your reply (or
parts of it) after appropriate sections of the prior message.
On Jan 8, 4:17 pm, quasi <qu...@xxxxxxxx> wrote:
On Tue, 8 Jan 2008 12:27:36 -0800 (PST), maud.j...@xxxxxxxxx wrote:
Once again, I'm studying for a PhD general and am working through old
problems. Thanks to everyone who has helped me before. Here's another
question I'm stuck on:
Consider the first order ode: u'+u=cos(u) posed with initial-value
u(0)=c
(a) Use the contraction mapping principle to show there is exactly one
solution for a given c. and (b) prove that there is a unique number z
such that u(t) tends to z as t-> infty regardless of u(0)=c.
Using the contraction mapping theorem (and the mean value theorem, (a)
isn't too bad as we can solve the equation for a given durration of
time ind. of c. And then adjust the constant c and extend the
solution. However I'm stuck on part b and don't know how to get
started? any hints or help would be appreciated!
Hints for (b):
(1) Find a constant solution, call it z.
(2) Consider 3 cases, u(0) < z, u(0) = z, u(0) > z.
Additional hints:
For t1 >= 0,
if u(t1) < z, u is increasing at t = t1
if u(t1) > z, u is decreasing at t = t1.
if u(t1) = z, u(t) = z for all t.
Deduce that for any value of u(0), u(t) must approach a limit, as t
approaches infinity.
Show that such a limit must be equal to z.
quasi
.
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