# Picard–Lindelöf theorem

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In mathematics, the **Picard–Lindelöf theorem** or **Picard's existence theorem** on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem

has exactly one solution if *f* is Lipschitz continuous in , continuous in as long as
stays bounded.

A simple proof of existence of the solution is successive approximation: (also called Picard iteration)

Set

and

It can then be shown rather easily, by using the Banach fixed point theorem, that the sequence of the (called the Picard iterates) is convergent and that the limit is a solution to the problem.

An application of Grönwall's lemma to , where and are two solutions, shows that , thus proving the uniqueness.