Fractional tempered variational calculus

Abstract

In this paper, we derive sufficient conditions ensuring the existence of a weak solution $u$ for a tempered fractional Euler-Lagrange equations $$\frac{\partial L}{\partial x}(u,{}^C{\mathbb{D}}_{a^{+}}^{\alpha ,\sigma }u,t)+{\mathbb{D}}_{b^{-}}^{\alpha ,\sigma }\left(\frac{\partial L}{\partial y}(u,{}^C{\mathbb{D}}_{a^{+}}^{\alpha ,\sigma }u,t)\right)=0$$ on a real interval $[a,b]$ and ${}^C{\mathbb{D}}_{a^{+}}^{\alpha ,\sigma },{\mathbb{D}}_{b^{-}}^{\alpha ,\sigma }$ are the left and right Caputo and Riemann-Liouville tempered fractional derivatives respectively of order $\alpha$. Furthermore, we study a fractional tempered version of Noether theorem and we provide a very explicit expression of a constant of motion in terms of symmetry group and Lagrangian for fractional problems of calculus of variations. Finally we study a mountain pass type solution of the cited problem.