We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{��,��}_��}$, which is a convex combination of the left Caputo fractional derivative of order $��$ and the right Caputo fractional derivative of order $��$. The fractional variational problems under our consideration are formulated in terms of ${{^CD}^{��,��}_��}$. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

This is a preprint of a paper whose final and definite form has been published in: Fract. Calc. Appl. Anal., Vol. 14, No 4 (2011), pp. 523--537; DOI: 10.2478/s13540-011-0032-6