In this article, we state and prove the Cauchy Lipschitz theorem for the existence and uniqueness of solutions to nonlinear ordinary differential equations. The key proof of this theorem is the Banach-Picard fixed point theorem. We give some applications of this theorem.

Local and maximal solutions to nonlinear Cauchy problems

Throughout this section, $I$ is an interval of $\mathbb{R}$ and $\Omega$ is an open set of $\mathbb{R}^d$. In addition, let $(t_0,x_0)\in \Omega$ et $f:I\times \Omega\to\mathbb{R}^n$ be a continuous function.

We look for additional conditions on $f$ so as the following Cauchy problem \begin{align*}\tag{Eq} u(t_0)=x_0,\quad \dot{u}(t)=f(t,u(t)),\quad t\in I,\end{align*}admits a “kind” of solutions.

By a solution to $({\rm Eq})$ we mean a couple $(J,u),$ where $J\subset I$ is an interval such that $t_0\in J,$ and $u:J\to \Omega$ is a $C^1$ function that satisfies $({\rm Eq})$.

On the other hand, we define an order in the set of all solutions to the Cauchy problem $({\rm Eq})$. In fact, we say that a solution $(J_2,u_2)$ extends another solution $(J_1,u_1),$ of (Eq) if $J_1\subset J_2$ and $u_2(t)=u_1(t)$ for any $t\in J_1$.

A maximum solution of $({\rm Eq})$ is a solution that does not admit an extension to another solution.

Remark: Niote that $(J,u)$ is a solution of $({\rm Eq})$ if and only if it satisfies the following integral equation \begin{align*}\tag{IE} u(t)=x_0+\int^t_{t_0}f(s,u(s))ds,\quad\forall t\in J.\end{align*}

A particular version of the Cauchy Lipschitz theorem

In this section, let $\alpha>0,\;r>0$ and $(t_0,x_0)\in I\times \Omega$ such that \begin{align*} \tag{H1}Q:=[t_0-\alpha,t_0+\alpha]\times \overline{B}(x_0,r)\subset I\times \Omega\end{align*}\begin{align*} \tag{H2} f:Q\to \mathbb{R}^d\; \text{is continuous, and }\;M=\sup_Q\|f\|<\infty.\end{align*}\begin{align*} \tag{H3} \exists C>0, \forall (t,x),(t,y)\in Q, \quad \|f(t,x)-f(t,y)\|\le C|x-y|.\end{align*}

Theorem: Under the condition $(H1)$ to $(H3)$, the Cauchy problem $({\rm Eq})$ admits a unique solution $(J,u)$ such that \begin{align*} J=[t_0-T,t_0+T]\quad\text{with}\quad T:=\min\left\{alpha,\frac{r}{M}\right\}.\end{align*}\begin{align*} (s,u(s))\in Q,\quad \forall s\in J.\end{align*}

Proof: We shall use Banach-Picard fixed point theorem. The latter said that if $E$ is a Banach space and $\Phi: E\to E$ is a contraction, that is there exists $\gamma\in (0,1)$ such that $\|\Phi(x)-\Phi(y)\|\le \gamma \|x-y\|$ for any $x,y\in E;$ then there exists a unique $u\in E$ such that $\Phi(u)=u$. In this case, we also have $\Phi^n(u)=u$ for any $n\in \mathbb{N}$, where $\Phi^n=\Phi\circ\Phi\circ\cdots\Phi$. Conversely, il there exists $m\in \mathbb{N}$ and a unique $u\in E$ such that $\Phi^m(u)=u$, then $u$ is a fixed point for $\Phi,$ that is $\Phi(u)=u$.

Now come back to the proof of the theorem. For $u\in E:=\mathcal{C}(J, \overline{B}(x_0,r) )$, we define for any $t\in J,$ \begin{align*}\left(\Phi(u)\right)(t)=x_0+\int^t_{t_0}f(s,u(s))ds.\end{align*}Then $\Phi:E\to E$. By recurrence we show that for any $t\in J$ and $n\in \mathbb{N}$ we have \begin{align*} \|\Phi^n(v)(t)- \Phi^n(w)(t) \|\le \frac{C^n}{n!}|t-t_0|^n \|u-v\|_\infty.\end{align*} For a large $n,$ we have \begin{align*}\gamma:= \frac{C^n}{n!} \left(\frac{r}{M}\right)^n < 1.\end{align*} Then $\Phi^n$ is a contraction. Thus there exists a unique $u\in E= \mathcal{C}(J, \overline{B}(x_0,r) )$ such that $\Phi(u)=u$. This means that $u:J\to \overline{B}(x_0,r) $ such that \begin{align*} u(t)=x_0+\int^t_{t_0}f(s,u(s))ds,\qquad \forall t\in J.\end{align*}This ends the proof.

You may also consult the concept of maximal solutions in detail.