positive solutions for a nonlocal fractional differential equation

in this paper, we study the following singular boundary value problem of a nonlocal

fractional differential equation

{d_α⁰₊u(t) + q(t)f (t, u(t)) = 0, 0 < t < 1, n − 1 < α ≤ n,

u(0) = u′(0) = · · · = u⁽ⁿ⁻²⁾(0) = 0, u(1) =

∫ ¹₀u(s)da(s),

where α ≥ 2, d_α⁰₊ is the standard riemann–liouville derivative,∫ ¹₀ u(s)da(s) is given by riemann–stieltjes integral with a signed measure, q may be singular at t = 0 and/or t = 1, f (t, x) may also have singularity at x = 0. the existence and multiplicity of positive solutions are obtained by means of the fixed point index theory in cones.