In this paper, we broaden the investigation of fractional variational problems of Herglotz type. Specifically, we consider cases where the Lagrangian function depends on distributed-order fractional derivatives with arbitrary smooth kernels, the endpoint conditions, and a real parameter. The existing literature does not provide the means to analyze this category of generalized variational problems. The inclusion of boundary conditions and an arbitrary parameter in the Lagrangian is not an arbitrary extension; on the contrary, it is a crucial formulation applicable to numerous problems, including those in physics and economics.

Work supported by Portuguese funds through the CIDMA and the Portuguese Foundation for Science and Technology, within project UIDB/04106/2020.

published