In this paper, we study the fourth-order m-point boundary value problem \begin{equation*} \left \{\begin{array}{lcr} u^{(4)}(t)=f(t,u(t)),\ t\in [0,1],\\ u’(0)=u’‘(0)=u(1)=0 , u’‘(1)-\sum^{m-2}_{i=1}\alpha_i u”’(\xi_i)=0, \end{array}\right. \end{equation*} with sign-changing Green’s function. By using some fixed theorems and the properties of Green’s function, we mainly establish the existence and multiplicity of positive solution for the problems under some suitable conditions.

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.