Here's a hint. At a fixed and for small enough , we have . Now let tend to...
Howdy all, got stuck on this one.
Let F be a continuous function and suppose |F(x,y) - F(x,z)| <= k(x)|y-z| on the strip 0<x<a. Show that if the improper integral of k(x)dx from 0 to a is finite, then y' = F(x,y) has at most one solution satisfying y(0) = 0.
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There are no restrictions on k so far as I can tell, it's possible that the supremum of k on [0,a] is infinity. If the supremum is finite, F is Lipschitz and the conclusion follows immediately. Am I headed down the right road here? Thanks in advance for any help.