Can someone please explain to me how the assumption of Lipschitz continuity ensures uniqueness of a solution in a IVP,
If we have an IVP ; and f is Lipschitz, we can get successive approximation to the solution using the operator . Then , , ... 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).