Editing Picard–Lindelöf theorem (section)

== Proof sketch ==
A standard proof relies on transforming the differential equation into an integral equation, then applying the [[Banach fixed-point theorem]] to prove the existence and uniqueness of solutions.

Integrating both sides of the differential equation <math display="inline">y'(t)=f(t,y(t))</math> shows that any solution to the differential equation must also satisfy the [[integral equation]]

:<math>y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds.</math>

Given the hypotheses that <math>f</math> is continuous in <math>t</math> and Lipschitz continuous in <math>y</math>, this integral operator is a [[Contraction (operator theory)|contraction]]{{why|date=February 2025}} and so the Banach fixed-point theorem proves that a solution can be obtained by [[fixed-point iteration]] of successive approximations. In this context, this fixed-point iteration method is known as [[Picard iteration]].

Set
:<math>\varphi_0(t)=y_0</math>
and
:<math>\varphi_{k+1}(t)=y_0+\int_{t_0}^t f(s,\varphi_k(s))\,ds.</math>

It follows from the Banach fixed-point theorem that the sequence of "Picard iterates" <math display="inline">\varphi_k</math> is [[Limit of a sequence|convergent]] and that its limit is a solution to the original initial value problem:

:<math>\lim_{k\to \infty} \varphi_k(t) = y(t)</math>.

Since the Banach fixed-point theorem states that the fixed-point is unique, the solution found through this iteration is the unique solution to the differential equation given an initial value.