Existence of a unique positive solution for a singular fractional boundary value problem

Abstract

In the present work, we discuss the existence of a unique positive solution of a boundary value problem for a nonlinear fractional order equation with singularity. Precisely, order of equation D-0+(alpha) u(t) = f(t, u(t)) belongs to (3, 4] and f has a singularity at t = 0 and as a boundary conditions we use u(0) = u(1) = u'(0) - u'(1) - 0. Using a fixed point theorem, we prove the existence of unique positive solution of the considered problem.