Existence of a weak solution for fractional Euler-Lagrange equations

TL;DR

This paper proves the existence of weak solutions for fractional Euler-Lagrange equations using a variational approach, addressing equations involving Riemann-Liouville fractional derivatives of order between 0 and 1.

Contribution

The paper introduces a general variational theorem that guarantees the existence of weak solutions for a class of fractional Euler-Lagrange equations involving Riemann-Liouville derivatives.

Findings

Established a general existence theorem for weak solutions.

Applied variational methods to fractional differential equations.

Extended classical calculus of variations to fractional derivatives.

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