Existence of a weak solution for fractional Euler-Lagrange equations

In this paper, we state with a variational method a general theorem providing the existence of a weak solution $u$ for fractional Euler-Lagrange equations of the type: $$ \dfrac{\partial L}{\partial x} (u,D^\alpha_- u,t) + D^\alpha_+ (\dfrac{\partial L}{\partial y} (u,D^\alpha_- u,t)) = 0 $$ on a real interval $[a,b]$ and where $D^\alpha_-$ and $D^\alpha_+$ are the fractional derivatives of Riemann-Liouville of order $0 < \alpha < 1$.