Variational problems of Herglotz type with complex order fractional derivatives and less regular Lagrangian

We derive optimality conditions for variational problems of Herglotz type whose Lagrangian depends on fractional derivatives of both real and complex order, and resolve the case of subdomain when the lower bounds of variational integral and fractional derivatives differ. Moreover, we consider a problem of the Herglotz type that corresponds to the case when the Lagrangian depends on the fractional derivative of the action and give an example of the problem that corresponds to the oscillator with a memory. Since our assumptions on the Lagrangian are weaker than in the classical theory, we analyze generalized Euler–Lagrange equations by the use of weak derivatives and the appropriate technics of distribution theory. Such an example is discussed in detail.