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.
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