Fractional boundary value problems with singularities in space variables

We are concerned with the existence of solutions for the singular fractional boundary value problem $^{c}kern-1pt D^{alpha}u = f(t,u)$ , u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(??{0})) and lim _x→0 f(t,x)=∞ for all t∈[0,1]. Here, $^{c}kern-1pt D$ 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.