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’s 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.
