The authors investigate the existence of multiple positive solutions of fractional differential equations with integral boundary conditions \[ \begin{cases} D^\beta_{0^+} u(t)+a(t)f(t,u(t),u'(t)=0, \quad t\in \mathbb{R}_+,\\ u(0)=u'(0)=0,\\ \lim_{t\to+\infty}D_{0^+}^{\beta-1}u(t)=\int_0^\infty h(s)u'(s)ds+\sum_{i=1}^{\infty}\eta_i D^{\gamma}_{0^+}u(\xi). \end{cases} \] Here, \(D^\alpha\) is the Riemann-Liouville fractional derivative, \(\beta\in(2,3],\)\(0\in[0,\beta-1]\) and \(0\leq\xi_0\leq\xi_2\leq\ldots\leq\xi_k\leq\xi_{k+1}\ldots<\infty,\)\(\eta_k>0,\ k\in \mathbb{N}.\)
Such multiplicity results for the above problem are deduced by constructing the Green’s function related to the linear fractional problem \[ \begin{cases} D^\beta_{0^+} u(t)+a(t)g(t)=0, \quad t\in \mathbb{R}_+,\\ u(0)=u'(0)=0,\\ \lim_{t\to+\infty}D_{0^+}^{\beta-1}u(t)=\int_0^\infty h(s)u'(s)ds+\sum_{i=1}^{\infty}\eta_i D^{\gamma}_{0^+}u(\xi) \end{cases} \] by proving the positivity and continuity of the Green function and using the R. Avery and A. Peterson, fixed point theorem.
An example illustrating the theoretical result is presented at the end of the paper.