Studies on Sturm-Liouville boundary value problems for multi-term fractional differential equations

The Sturm-Liouville boundary value problem of the multi-order fractional differential equation { D α 0+ [p(t)D β 0+ u(t)] + q(t)f(t, u(t)) = 0, t ∈ (0, 1), a limt→0 t 1−βu(t) − b limt→0 t 1−α p(t)D β 0+ u(t) = 0, c limt→1 u(t) + d limt→1 p(t)D β 0+ u(t) = 0 is studied. Results on the existence of solutions are established. The analysis relies on a weighted function space and a fixed point theorem. An example is given to illustrate the efficiency of the main theorems.