Can someone please explain to me how the assumption of Lipschitz continuity ensures uniqueness of a solution in a IVP,
Thanks
If we have an IVP $\displaystyle dx/dt=f(x)$; $\displaystyle x(t_0)=x_0$ and f is Lipschitz, we can get successive approximation to the solution using the operator $\displaystyle T(x)=x_0+\int_{t_0}^t f(x(u))\,du$. Then $\displaystyle T(x_0)$, $\displaystyle T(T(x_0))$, ... 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).