From: "Volker W. Elling" <volker.elling@post.rwth-aachen.de>
Subject: [Fwd: Existence/uniqueness for ODE]
Date: Fri, 03 Dec 1999 14:00:58 +0100
Newsgroups: sci.math
Keywords: proofs for solutions of initial value ODE problems

Hello,

since initial value problems
  x(0) = x0
  d/dt x(t) = f(x(t))
seem to be the starting point of all analysis of
mathematical physics problems, I am interested in
simple or interesting sufficient or necessary
conditions to f and x0 for the existence/uniqueness of a solution x(t)
for a small time (i.e. for 0 <= t < T with arbitrarily small T>0).
x might have values in an infinite-dimensional Banach space rather
than R^n only.

I know
++ Peanos proof for existence based on continuity and (in the
   Banach-valued case) compactness close to the initial value
++ Picard-Lindeloefs proof for existence and uniqueness based
   on Lipschitz continuity close to the initial value.

However, I suspect Peanos conditions are not necessary for existence
and Picard-Lindeloefs conditions are not necessary for existence
or uniqueness.
In addition, it would be interesting to have a _characterization_
(i.e. a necessary _and_ sufficient condition) at least in the finite-
dimensional case.
I have read the word "Peano funnel" in these contexts, but no
explanation.

Any ideas?
==============================================================================

From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Unsolvable diff. equations?
Date: 27 Oct 1999 23:50:43 GMT
Newsgroups: sci.math
Keywords: Picard's method shows all analytic differential equations solvable

In article <3816D482.E7EB4423@br.homeshopping.com.br>,
 Bruno Barberi Gnecco <brunobg@br.homeshopping.com.br> writes:
 
> 	Are there any unsolvable differential equations in the complex domain?

Unsolvable in what sense?  Do you mean something like this?

Let F(z,w) be an analytic function of two variables z and w (for (z,w) in
some open set U in C^2), and consider the initial value problem w' = F(z,w), 
w(z0) = w0, where (z0,w0) is in U.
Then there is an analytic function f(z) defined on a neighbourhood D of z0
that solves this, i.e. f(z0) = w0 and f'(z) = F(z,f(z)) for z in D.
Thus in a certain sense every analytic differential equation is solvable.

IIRC this can be proved using Picard's method, with a proof quite similar
to the real case.

Of course there are unsolvable equations with non-analytic F, e.g. 
w' = conjugate(z).

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2