Eigenvalue of Fractional Differential Equations withp-Laplacian Operator

We investigate the existence of positive solutions for the fractional order eigenvalue problem with p-Laplacian operator -tβ(φp(tαx))(t)=λf(t,x(t)),  t∈(0,1),  x(0)=0,  tαx(0)=0,  tγx(1)=∑j=1m-2ajtγx(ξj), where tβ,  tα,  tγ are the standard Riemann-Liouville derivatives and p-Laplacian operator is defined as φp(s)=|s|p-2s,  p>1.f:(0,1)×(0,+∞)→[0,+∞) is continuous and f can be singular at t=0,1 and x=0. By constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of fractional differential equation is established