Existence of a solution of a difference scheme for one variational problem

The problem of minimizing the functional
$$ \int_a^b\varphi(x,y,y',y'')\,dx $$
under the conditions
$$ \int_a^b\varphi(x,y,y',y'')\,dx $$
is replaced by the problem of finding the vector $(y_1,y_2,\dots,y_{n-1})$ on which the sum
$$ \sum_{k=0}^nC_k\varphi\biggl(x_k,y_k,\frac{y_{k+1}-y_k}{h},\frac{y_{k+1}-2y_k+y_{k+1}}{h^2}\biggr) $$
takes a minimal value. Under certain conditions on $\varphi$ and $C_k$ it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof.