Positive solutions for superlinear fractional boundary value problems

We establish the existence, uniqueness, and global behavior of a positive solution for the following superlinear fractional boundary value problem: D(alpha)u (x) = u(x)phi(x, u(x)), x is an element of(0,1), lim(x -> 0)+ D alpha-1U(X) = -a, u(1) = b, where 1 < alpha <= 2, D-alpha is the standard Riemann-Liouville fractional derivative, a, b are nonnegative constants such that a + b > 0 and phi(X, t) is a nonnegative continuous function in (0,1) x [0, infinity) that is required to satisfy some appropriate conditions related to a certain class of functions kappa(alpha). Our approach is based on estimates of the Green's function and on perturbation arguments.