Bernstein's Theorem on Unique Extremal

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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 $.


Proof


Source of Name

This entry was named for Sergei Natanovich Bernstein.


Sources