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 − α u , t ) + D + α ( ∂ L ∂ y ( u , D − α u , 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 .