# Picard’s theorem

###### Theorem 1 (Picard’s theorem [KF]).

Let $E$ be an open subset of $\mathbb{R}^{2}$ and a continuous function $f(x,y)$ defined as $f\colon E\to\mathbb{R}$. If $(x_{0},y_{0})\in E$ and $f$ satisfies the Lipschitz condition in the variable $y$ in $E$:

 $|f(x,y)-f(x,y_{1})|\leq M|y-y_{1}|$

where $M$ is a constant. Then the ordinary differential equation defined as

 $\frac{dy}{dx}=f(x,y)$

with the initial condition

 $y(x_{0})=y_{0}$

has a unique solution $y(x)$ on some interval $|x-x_{0}|\leq\delta$.

The above theorem is also named the Picard-Lindelöf theorem and can be generalized to a system of first order ordinary differential equations

###### Theorem 2 (generalization of Picard’s theorem [KF]).

Let $E$ be an open subset of $\mathbb{R}^{n+1}$ and a continuous function $f(x,y_{1},\ldots,y_{n})$ defined as $f=(f_{1},\ldots,f_{n})\colon E\to\mathbb{R}^{n}$. If $(t_{0},y_{10},\ldots,y_{n0})\in E$ and $f$ satisfies the Lipschitz condition in the variable $y_{1},\ldots,y_{n}$ in $E$:

 $|f_{i}(x,y_{1},\dots,y_{n})-f_{i}(x,y_{1}^{\prime}\ldots,y_{n}^{\prime})|\leq M% \max_{1\leq j\leq n}|y_{j}-y_{j}^{\prime}|$

where $M$ is a constant. Then the system of ordinary differential equation defined as

 $\displaystyle\frac{dy_{1}}{dx}$ $\displaystyle=f_{1}(x,y_{1},\ldots,y_{n})$ $\displaystyle\vdots$ $\displaystyle\frac{dy_{n}}{dx}$ $\displaystyle=f_{n}(x,y_{1},\ldots,y_{n})$

with the initial condition

 $y_{1}(x_{0})=y_{10},\ldots,y_{n}(x_{0})=y_{n0}$

has a unique solution

 $y_{1}(x),\ldots,y_{n}(x)$

on some interval $|x-x_{0}|\leq\delta$.