  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.
