Existence of a weak solution for fractional Euler-Lagrange equations

We derive sufficient conditions ensuring the existence of a weak solution u for fractional Euler-Lagrange equations of the type: (ELα)∂L∂x(u,D−αu,t)+D+α(∂L∂y(u,D−αu,t))=0, on a real interval [a,b] and where D−α and D+α are the fractional derivatives of Riemann-Liouville of order 0<α<1.