✍内容

Consider the differential equation ${\displaystyle \dot{\mathit{x}}=\mathit{v}(t,x)}$. Define the Picard mapping to be the mapping $A$ that takes the function ${\displaystyle \phi :t\mapsto \mathit{x}}$ to the function ${\displaystyle A\phi :t\mapsto \mathit{x}}$, where

$$(A\phi )(t)={\mathit{x}}_0+{\int }_{t_0}^t\mathit{v}(\tau ,\phi (\tau ))d\tau$$

And the followings are equivalent:

• ${\displaystyle \phi }$ is a solution with the initial condition ${\displaystyle \phi (t_0)=x_0}$
• ${\displaystyle \phi =A\phi }$

To prove the convergence of the successive approximations we shall construct a complete metric space in which the Picard mapping is a contraction. We begin by recalling some facts from analysis.