1
$\begingroup$

I am reading Gelfand's and Fomin's book Calculus of Variations.

There are this one theorem and there is no proof for it. I haven't yet tried to proof it myself.


If the functions $F$, $F_{y}$ and $F_{y'}$ are continuous at every finite point $(x,y)$ for any finite $y'$, and if a constant $k>0$ and functions $$ \alpha=\alpha (x,y)\geq 0\quad \beta=\beta (x,y)\geq0 $$ (which are bounded in every finite region of the plane) can be found such that $$ F_{y}(x,y,y')>k,\quad|F(x,y,y')|\leq\alpha y'^{2}+\beta, $$ then one and only one integral curve of equation $y''=F(x,y,y')$ passes through any two points $(a,A)$ and $(b,B)$ with different abscissas $(a\neq b)$.


This theorem is on the page 16 in the book that I mentioned above. It is about the existence and uniqueness of the solutions "in the large" of an equation of the form $y''=F(x,y,y')$.

I am very pleased if I could get a some proof about this theorem.

$\endgroup$

1 Answer 1

1
+50
$\begingroup$

The book gives the paper of Bernstein as a reference, indicating that this might not be a completely trivial proof... Anyway, if you can not access the original paper, here is a more recent one with a proof of a slightly more general result: http://projecteuclid.org/euclid.pjm/1102810437

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.