Solvability of Sturm–Liouville boundary value problems for multiple order fractional differential equations

In this paper, definitions of Lα1 function space, ββ-well function space and quasi-Carathéodory function are given, existence results for solutions of a class of Sturm–Liouville boundary value problems of fractional differential equations with multiple order derivatives D0+α[Φ(ρ(t)D0+βx(t))] are established. The analysis relies on a well known fixed point theorem. Some examples are given to illustrate the efficiency of main theorems.