Geodesic
Existence of a solution of a difference scheme for one variational problem
Zapiski Nauchnykh Seminarov POMI, Computational methods and automatic programming, Tome 58 (1976), pp. 40-47

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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. 
@article{ZNSL_1976_58_a4,
     author = {Z. A. Vlasova and O. I. Nikolaev},
     title = {Existence of a solution of a difference scheme for one variational problem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {40--47},
     publisher = {mathdoc},
     volume = {58},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_58_a4/}
}
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Z. A. Vlasova; O. I. Nikolaev. Existence of a solution of a difference scheme for one variational problem. Zapiski Nauchnykh Seminarov POMI, Computational methods and automatic programming, Tome 58 (1976), pp. 40-47. http://geodesic.mathdoc.fr/item/ZNSL_1976_58_a4/