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Math Help - Uniqueness by Lipschitz

  1. #1
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    Uniqueness by Lipschitz

    Can someone please explain to me how the assumption of Lipschitz continuity ensures uniqueness of a solution in a IVP,

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  2. #2
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    Re: Uniqueness by Lipschitz

    If we have an IVP dx/dt=f(x); x(t_0)=x_0 and f is Lipschitz, we can get successive approximation to the solution using the operator T(x)=x_0+\int_{t_0}^t f(x(u))\,du. Then T(x_0), T(T(x_0)), ... converges to the solution. This is because the fact that f is Lipschitz guarantees that T is a contraction, and the existence and uniqueness of the fixpoint of T (which is also the solution to the IVP) follows from the Banach fixpoint theorem.

    For a complete proof, see these notes (PDF).
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