Minimization Problems for Functionals Depending on Generalized Proportional Fractional Derivatives

In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with and without boundary conditions, and with additional restrictions like isoperimetric and holonomic. Herglotz's variational problem and when in presence of time delays are also considered.