# Thread: local uniqueness

1. 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

2. The proof of the 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.

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