Constructive Existence of Solutions of Multi-point Boundary Value Problem for Hilfer Fractional Differential Equation at Resonance

In this paper, we study the constructive existence of solutions of multi-point boundary value problem for Hilfer fractional differential equation at resonance. The constructive existence of solutions is very important in practice, because it gives approximation method to solve equation as well as the existence. First, we obtain reversibility of some fractional integral operator which is useful for description and get analytic expressions of the inverse operator with Mittag-Leffler function. Next, we have the solution expressions of initial value problem and boundary-value problem for linear Hilfer fractional differential equations. And we get the sufficient conditions for the existence and nonexistence of solutions of multi-point value problem for nonlinear Hilfer fractional differential equation at resonance and establish the approximation method to solve it by the method of upper and lower solutions. Finally, we illustrate our results with an example.