Abstract
We study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem
D(0+)(alpha)u(t) = f(t, u(t), u(t)), 0 < t < 1,
u(0) = u(1) = u'(0) = u'(1) = 0,
where 3 < alpha <= 4 is a real number, D-0+(alpha) is the Riemann-Liouville fractional derivative and f : (0, 1] x [0,+infinity) x [0,+infinity) -> [0,+infinity) is continuous, lim(t -> 0+) f(t, ., .) = +infinity (f is singular at t = 0). Our approach is based on a coupled fixed point theorem on ordered metric spaces. An example is given to illustrate our main result.