# Uniqueness by Lipschitz

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• August 2nd 2012, 08:05 AM
Mullineux
Uniqueness by Lipschitz
Can someone please explain to me how the assumption of Lipschitz continuity ensures uniqueness of a solution in a IVP,

Thanks
• August 2nd 2012, 08:22 AM
emakarov
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).