From: israel@math.ubc.ca (Robert Israel) Subject: Re: non-solutions to ODE Date: 5 Feb 2001 01:50:39 GMT Newsgroups: sci.math Summary: Peano's local existence theorem for solutions to ODEs In article <3A7D87CB.50CDF030@math.missouri.edu>, Stephen Montgomery-Smith wrote: >OK, a unique solution y(t) (a<=t<=b) to the diff equ >dy/dt = f(t,y) >y(a) = alpha >can be guaranteed if f(t,y) is continuous and Lipschitz in >y for a<=t<=b. So I am interested in counterexamples when >f(t,y) is not Lipschitz in y. >But I remember seeing an example of a function f(t,y) where >there did not exist a solution for a<=t<=a+e for any e>0. >Can anyone refresh my memory? No. Peano's local existence theorem holds as long as f(t,y) is continuous in a - epsilon <= t <= a + epsilon, alpha - epsilon <= y <= alpha + epsilon. See e.g. Birkhoff & Rota, Ordinary Differential Equations, Sec. V.13. In your case you only have a <= t <= a+epsilon, but you can define f(t,y)=f(2a-t,y) for a-epsilon <= t <= a to make it continuous in a-epsilon <= t <= a+epsilon. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: israel@math.ubc.ca (Robert Israel) Subject: Re: non-solutions to ODE Date: 5 Feb 2001 17:05:34 GMT Newsgroups: sci.math In article <3A7E1F84.A66E0AB1@math.missouri.edu>, Stephen Montgomery-Smith wrote: >I know I should go to the library and read the book, but could >you give me a quick idea of how Peano's theorem is proved? The proof in Birkhoff and Rota is due to Tonelli. You show that the iteration y_{n+1}(t) = c + int_a^t F(s,y_n(s)) ds has a fixed point, using the Arzela-Ascoli Theorem to get a limit point of the sequence y_n. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2