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

TL;DR

This paper investigates the uniqueness of solutions to certain higher-order ordinary differential equations with initial conditions, demonstrating that the function involved must be H"older continuous without extra assumptions.

Contribution

It establishes new uniqueness theorems for ODEs with minimal assumptions on the function, showing that the function must be H"older continuous.

Findings

Uniqueness results for specific classes of ODEs.

Proof that the function f is always H"older continuous.

No additional assumptions are needed on f for the results.

Abstract

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 establish some uniqueness results and show that $f$ is always H\"older continuous.