# Bernstein's Theorem on Unique Extremal

## Theorem

Let $F, F_y, F_{ y' }$ be functions.

Let $F, F_y, F_{ y' }$ be continuous at every point $\left ( { x, y } \right )$ for all finite $y'$.

Suppose a constant $k > 0$ and functions $\alpha = \alpha \left ( { x, y } \right ) \ge 0$, $\beta = \beta \left ( { x, y } \right ) \ge 0$ bounded in every finite region of the plane exist such that

$F_y \left ( { x, y, y' } \right ) > k$
$\left \vert { F \left ( { x, y, y' } \right ) } \right \vert \le \alpha y'^2 + \beta$

Then one and only one integral curve of equation $y'' = F \left ( { x, y, y' } \right )$ passes through any two points $\left ( { a, A } \right )$ and $\left ( { b, B } \right)$ such that $a \ne b$.

## Source of Name

This entry was named for Sergei Natanovich Bernstein.