Lyapunov inequalities of nested fractional boundary value problems and applications

In this paper, we study certain classes of nested fractional boundary value problems including both of the Riemann–Liouville and Caputo fractional derivatives. In addition, since we will use the signed-power operators ${\varphi }_{\nu }z≔|z{|}^{\nu -1}z,\nu \in (0,\infty )$ in the governing equations, so our desired boundary value problems possess half-linear nature. Our investigation theoretically reaches so called Lyapunov inequalities of the considered nested fractional boundary value problems, while in viewpoint of applicability using the obtained Lyapunov inequalities we establish some qualitative behavior criteria for nested fractional boundary value problems such as a disconjugacy criterion that will also be used to establish nonexistence results, upper bound estimation for maximum number of zeros of the nontrivial solutions and distance between consecutive zeros of the oscillatory solutions. Also, considering corresponding nested fractional eigenvalue problems we find spreading interval of the eigenvalues.