# Calculus of variations and Bernstein's theorem

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.