Positive solutions of singular Caputo fractional differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C3[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.

► We study positive solutions of integral boundary value problems with fractional order.
► Necessary and sufficient conditions of super-linear and or sub-linear are given.
► The tools are the fixed point theorems on cones.
► Nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.