Properties of positive solutions for m-point fractional differential equations on an infinite interval

Abstract

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.