In this paper, by using a recent fixed point theorem, we discuss a class of m-point boundary value problems of fractional differential equations on an infinite interval $$\begin{aligned} \left\{ \begin{array}{l} D^\alpha _{0^+}u(t)+\lambda {a(t)f(t, u(t))}=0,~t\in (0,+\infty ),\\ u(0)=u'(0)=0,~D^{\alpha -1}_{0^+}{u(+\infty )} =\sum \limits _{i=1}^{m-2}\beta _iu(\xi _i), \end{array}\right. \end{aligned}$$ where $$2<\alpha <3$$ , $$D_{0^+}^\alpha $$ is the usual Riemann–Liouville fractional derivative, $$\lambda $$ is a positive parameter, $$a:[0,+\infty )\rightarrow [0,+\infty )$$ and $$f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )$$ are continuous, $$0<\xi _1<\xi _2<\cdots<\xi _{m-2}<+\infty $$ , $$\beta _i\ge 0$$ , $$i=1,2,\ldots ,m-2,$$ and $$0<\sum \nolimits _{i=1}^{m-2}\beta _{i} \xi _i^{\alpha -1}<\Gamma (\alpha )$$ . It is shown that, for any given parameter $$\lambda >0$$ , the above problem has a unique positive solution $$u_\lambda ^*$$ in a special set $$K_h$$ , here $$h(t)=t^{\alpha -1},~t\in [0,+\infty )$$ . Further, we give some good properties of positive solutions which depend on the parameter $$\lambda >0$$ , namely, the positive solution $$u_\lambda ^*$$ is strictly increasing, continuous in $$\lambda $$ ; and $$\lim \nolimits _{\lambda \rightarrow 0^+}\Vert u_\lambda ^*\Vert =0$$ , $$\lim \nolimits _{\lambda \rightarrow +\infty }\Vert u_\lambda ^*\Vert =+\infty $$ . In the end, a simple example is given.