Existence of positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator

In this paper, we investigate the existence of positive solutions of fractional differential equations with p-Laplacian operator: D₀₊^γ(φ_p(D₀₊^αu(t)))=f(t,u(t)), 0<;t<;1,u(0) = u´(0)=u´(1)=0, D₀₊^αu(0) = D₀₊^αu´(1) = 0, 1<;γ≤2, 2<;α≤3, φ_p(s)=|s|^p-2 s, p >; 1, where (φ_p)^-1= φ_q, 1/p + 1/q= 1 and D₀₊^α, D₀₊^γ is the standard Riemann-Liouville differentiation. It is valuable to point out that the nonlinearity f can be singular at ^{t = 0, 1} or ^{u= 0}. By the use of fixed point theorem on cone and the upper and lower solutions method, the existence of positive solutions is obtained.