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We study ordinary differential equations of the type $u^{(n)}(t)=f(u(t))$with initial conditions $u(0) = u'(0) =... = u^{(m-1)}(0) = 0 $ and $u^{(m)}(0)\neq 0$ where $m \geq n$, no additional assumption is made on $f$. We establishsome uniqueness results and show that $f$ is always H\"older continuous.

Uniqueness Theorems for Ordinary Differential Equations with H\"older Continuity