Fractional boundary value problems with singularities in space variables

We are concerned with the existence of solutions for the singular fractional boundary value problem ^cD^αu = f (t,u), u(0) + u(1) = 0, u ′ (0) = 0, where α ∈ (1, 2), f ∈ C([0, 1] × (ℝ \ {0})) and lim_x→0 f (t,x)=∞ for all t ∈ [0, 1]. Here, ^cD is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0, 1), that is, they "pass through" the singularity of f inside of (0, 1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.