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 {(c)D(0+)(alpha)u(t) + f(t, u(t), u '(t), ...,u((i)) (t)) = 0, 0 < t < 1, u(0) = u '(0) = center dot center dot center dot = u((i-1)) (0) = u((i+1)) (0) = center dot center dot center dot = u((n-1)) (0) = 0, u((i))(1) = 0, where n - 1 < alpha <= n is a real number, n is natural number and n >= 2, alpha - i > 1, i is an element of N and 0 <= i <= n - 1. D-c(0+)alpha is the standard Caputo derivative. f (t, x(0), x(1),..., x(i)) may be singular at t = 0.