A Caputo two-point boundary value problem: existence, uniqueness and regularity of a solution

A two-point boundary value problem on the interval $[0,1]$ is considered, where the highest-order derivative is a Caputo fractional derivative of order $2-\delta$ with $0<\delta <1$. A necessary and sufficient condition for existence and uniqueness of a solution $u$ is derived. For this solution the derivative $u'$ is absolutely continuous on $[0,1]$. It is shown that if one assumes more regularity — that $u$ lies in $C^2[0,1]$ — then this places a subtle restriction on the data of the problem.