Variational Problems Involving a Caputo-Type Fractional Derivative

The aim of this paper is to study certain problems of calculus of variations, that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo--Hadamard fractional derivatives, that are dependent on a real parameter ro. Sufficient and necessary conditions of the first and second order are presented. The cases of integral and holonomic constraints are also considered.