The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives

摘要

In this paper, we study the following singular eigenvalue problem for a higher order fractional differential equation-Dαx(t)=λf(x(t),Dμ1x(t),Dμ2x(t),…,Dμn-1x(t)),00,α-μn-1≤2,α-μ>1, aj∈[0,+∞),0<ξ1<ξ2<⋯<ξp-2<1, 0<∑j=1p-2ajξjα-μ-1<1, Dα is the standard Riemann–Liouville derivative, and f:(0,+∞)n→[0,+∞) is continuous. Firstly, we give the Green function and its properties. Then we established an eigenvalue interval for the existence of positive solutions from Schauder’s fixed point theorem and the upper and lower solutions method. The interesting point of this paper is that f may be singular at xi=0, for i=1,2,…,n.