Existence of a weak solution for fractional Euler-Lagrange equations

Abstract

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

This is a preprint of a paper whose final and definite form is published in Journal of Mathematical Analysis and Applications