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Lipschitz Continuity and Uniqueness

  1. Sep 7, 2005 #1
    Dear all,

    If a differential equation is Lipschitz continuous, then the solution is unique. But what about the implication in the other direction? I know that uniqueness does not imply Lipschitz continuity. But is there a counterexample? A differential equation that is not L-continuous, still the solution is unique?
    I would really like to know the answer to this question.
    Thank you all for your help,

  2. jcsd
  3. Sep 9, 2005 #2
    Now I have found an answer to my question:

    http://archive.numdam.org/ARCHIVE/A..._1996_6_5_4_577_0/ AFST_1996_6_5_4_577_0.pdf

    or, rather, Bernis and Qwang have found an answer. However, my differential equation is only of first degree, like

    \dot K(t) = F(K(t))

    One example would be F(K) = K^\alpha with 0 < \alpha < 1, then you get multiple solutions if you start from K = 0. But that's not surprising, because the differential equation is not Lipschitz continuous in K = 0. My question now would be: Is it possible to have a unique solution even if F were L-continuous in 0?

    Thanks, many greetings,
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