Left-definite fractional Hamiltonian systems: Titchmarsh-Weyl theory

Hamiltonian systems are useful when formally symmetric boundary value problems generated by ordinary derivatives are considered. However, if the ordinary derivatives are changed by non-integer-order (fractional) derivatives, it is not easy to investigate the corresponding problems. In this paper, we introduce a systematic approach to dealing with fractional boundary value problems by constructing a fractional Hamiltonian system. In particular, we consider a left-definite system, and we construct nested-circles theory (Weyl theory) for this system of equations. Using the Titchmarsh-Weyl function, we prove that at least $r$-solutions of the $2r$-dimensional system of equations should be Dirichlet-integrable on a given interval.