A new approach on fractional variational problems and Euler-Lagrange equations

In this paper we generalize fractional variational problems in [a,b]. We allow for the possibility that functions in the space of solution for the optimization problem can blow up at boundary points. The appropriate fractional derivative spaces are introduced and a compact embedding theorem demonstrated. We prove the existence of minimizers for the variational problems which satisfy the Euler-Lagrange equations with Riemann-Liouville boundary conditions. Our method is based on the fractional calculus of variations. An example is given to illustrate the results.