In this paper,we mainly establish existence and uniqueness of positive solution for the Hadamard-type fractional two-point boundary value problem (\mathcal{D}^{α}_{1^{+}}x(t))’‘+λ^2 \mathcal{D}^{α}_{1^{+}}x(t)+f(t,x(t),-\mathcal{D}^{α}_{1^{+}}x(t))=0, t∈ (1,e), x(1)=x’(1)=x’(e)=0 , \mathcal{D}^{α+1}_{1^{+}}x(1)=\mathcal{D}^{α+1}_{1^{+}}x(e)=0, by using the fixed point theorems. In addition, we also study the Ulam-Hyers-Rassias stability of the related problem. On the other hand, when f(t,u,v) is singular at u=0 and v=0, we study the existence and uniqueness of its solution. Finally, some examples are included to show the applicability of our results.