MHB Prove Local Uniqueness of DE Solutions on Interval

[onie mti]

if a function ls locally lip then considering this diff eq x'(t)= f(x(t) where now x and y are solutions of the DE on some interval J
and x(s)=y(s) for some s in J. then how can I prove that there exists a positive number delta such that x=y on (s-delta, s+delta)∩ J

 

[Evgeny.Makarov]

The proof of the Picard–Lindelöf theorem converts the original IVP
\begin{align*}
x'(t)&=f(x(t))\\
x(t_0)&=x_0
\end{align*}
into an integral equation
\[
x(t)=x_0+\int_{t_0}^tf(x(s))\,ds.\qquad{(*)}
\]
Define an operator $P(x)(t)=x_0+\int_{t_0}^tf(x(s))\,ds$, so (*) becomes
\[
x(t)=P(x).
\]
Thus, $x(t)$ is a fixpoint of $P$ iff $x(t)$ is a solution to the original IVP. The proof shows that there exists a $\delta$ such that $P$ is a contraction on $C[t_0-\delta,t_0+\delta]$ and thus has a unique fixpoint. Therefore, the solution to the IVP is also unique.