Summary: We discuss the existence and uniqueness of a solution to the non-local problem for a fractional differential equation \[ \begin{gathered} D^\alpha_{0^+} u(t)= f(t, u(t)),\qquad\text{a.e. in }(0,1),\\ I^{1-\alpha}_{0^+} u(0)=\beta I^{1-\alpha}_{c^+}u(\xi),\end{gathered} \] using the contraction principle and a continuation method.