Fractional Euler–Lagrange Equations Under Periodic and Antiperiodic Boundary Conditions

In this work, we derive necessary optimality conditions for a class of fractional variational problems involving Caputo-type derivatives. We consider functionals defined on appropriate spaces of absolutely continuous functions and study both periodic and antiperiodic boundary conditions, treated in a unified framework. The analysis covers the cases 0<α<1 and 1<α<2, leading to fractional Euler–Lagrange equations supplemented by suitable transversality conditions. We further extend the results to problems with integral constraints and holonomic constraints, as well as to a fractional Herglotz variational principle.