Some results for two classes of two-point local fractional proportional boundary value problems

Abstract

In this paper, we consider two classes of boundary value problems in the frame of local proportional fractional derivatives. For both of these classes, we obtain the associated Green?s functions and discuss their properties. Using these properties, we go about the uniqueness of the solutions. In addition, we establish Lyapunov-type and Hartman-Wintner-type inequalities and build sharp estimated for the unique solutions of the considered equations.