  In this PhD thesis we introduce a generalized fractional calculus of
variations. We consider variational problems containing generalized fractional
integrals and derivatives, and study them using standard (indirect) and direct
methods. In particular, we obtain necessary optimality conditions of
Euler-Lagrange type for the fundamental and isoperimetric problems, natural
boundary conditions, and Noether theorems. Existence of solutions is shown
under Tonelli type conditions. Moreover, we apply our results to prove
existence of eigenvalues, and corresponding orthogonal eigenfunctions, to
fractional Sturm-Liouville problems.
