Boundary value problems for nonlinear fractional differential equations with integral and ordinary-fractional flux boundary conditions

In this paper, we consider a new class of boundary value problems of Caputo type fractional differential equations supplemented with classical/nonlocal Riemann-Liouville integral and flux boundary conditions and obtain some existence results for the given problems. The flux boundary condition x′(0) = b cDβx(1) states that the ordinary flux x′(0) at the left-end point of the interval [0, 1] is proportional to a flux cDβx(1) of fractional order β ∈ (0, 1] at the right-end point of the given interval. The coupling of integral and flux boundary conditions introduced in this paper owes to the novelty of the work. We illustrate our results with the aid of examples. Our work not only generalizes some known results but also produces new results for specific values of the parameters involved in the problems at hand.