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: $${\displaystyle \frac{\mathrm{\partial }L}{\mathrm{\partial }x}}(u,D_{-}^{\alpha }u,t)+D_{+}^{\alpha }({\displaystyle \frac{\mathrm{\partial }L}{\mathrm{\partial }y}}(u,D_{-}^{\alpha }u,t))=0$$ on a real interval $[a,b]$ and where $D_{-}^{\alpha }$ and $D_{+}^{\alpha }$ are the fractional derivatives of Riemann-Liouville of order $0<\alpha <1$.