In this paper, we will apply some fixed-point theorems to discuss the existence of solutions for fractional m-point boundary value problems In addition, we also present Lyapunov’s inequality and Ulam-Hyers stability results for the given m-point boundary value problems.

We mainly consider the existence and stability results of the positive solutions for the fractional differential equation with Hadamard-type by applying fixed point theorems, if the nonlinearity may be continuous or singular. We also construct some examples to show the applicability of the results.

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.