Existence of positive solutions for the singular fractional differential equations

In this paper, we consider the following two-point fractional boundary value problem. We provide sufficient conditions for the existence of multiple positive solutions for the following boundary value problems that the nonlinear terms contain i-order derivative cD0+α u(t) + f(t, u(t), u'(t),⋯, ui(t)) = 0, 0 < t < 1, u(0) = u'(0) = ⋯ = ui-1(0) = ui+1 (0) = ⋯ u(n-1)(0) = 0, ui(1) = 0, where n-1<α≤n is a real number, n is natural number and n≥2, α-i>1, i×N and 0≤i≤n-1. cD0+α is the standard Caputo derivative. f(t,x 0,x 1,.,x i ) may be singular at t=0. © 2013 Korean Society for Computational and Applied Mathematics.