Abstract
In this paper, we study the existence of positive solutions to a boundary value problem for fractional type compartment model
\(\begin{cases}(D_{0^{+}}^{\upsilon}\xi(z))^{\prime}+aD_{0^{+}}^{\beta}(D_{0^{+}}^{\upsilon}\xi(z))+\Upsilon(z,\xi(z))=0,\quad z\in(0,1),\\ D_{0^{+}}^{\upsilon}\xi(0)-bD_{0^{+}}^{\upsilon}\xi(1)=0,\quad\xi(0)=0,\quad D_{0^{+}}^{\sigma}\xi(1)=\sum_{i=1}^{m}\mu_{i}D_{0^{+}}^{\sigma}\xi(\eta_{i}),\end{cases}\)
where \(m\geq 1\) is integer, \(\mu_{i}>0\) , \(0<\eta_{i}<1\) for \(0\leq i\leq{m}\) , \(\sum_{i=1}^{m}\mu_{i}\eta_{i}^{\upsilon-\sigma-1}<1\) , \(a,b>0\) , \(0<\beta<1\) , \(1<\upsilon\leq 2\) , \(0<\sigma<1\) , \(\upsilon-\sigma-1\geq 0\) , \(D_{0^{+}}^{\upsilon}\) is the Riemann–Liouville fractional derivative operator of order \(\upsilon\) , \(\Upsilon\in C([0,1]\times[0,+\infty),[0,+\infty))\) . By using some fixed point theorems, the existence of positive solutions to the given problem are obtained, which are well-illustrated with the aid of examples.
