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Math Help - Generalized Lipschitz Condition

  1. #1
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    Generalized Lipschitz Condition

    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.
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  2. #2
    Super Member Rebesques's Avatar
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    Here's a hint. At a fixed x and for small enough h>0, we have \int_{x}^{x+h}|F(s,y)-F(s,z)|ds\leq|y-z|\int_x^{x+h}k(s)ds\leq |y-z|\int_0^ak(s)ds=:C|y-z|. Now let h tend to...
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