Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations

In this paper, we study the existence of positive solutions for the singular nonlinear fractional differential equation boundary value problem D₀₊^α u(t) + f(t,u(t)) = 0, 0 <; t <; 1, u(0) = u(1) = u´(0) = 0, where 2 <; α ≤ 3 is a real number, D₀₊^α is the Riemann-Liouville fractional derivative, and f : (0,1] × [0, + ∞) → [0, + ∞) is continuous, lim_t→0+ f(t, ·) = + ∞ (i.e., f is singular at t = 0). Our analysis rely on nonlinear alternative of Leray-Schauder type and Krasnosel´skii fixed point theorem on a cone. As an application, an example is presented to illustrate the main results.