Existence of Positive Solutions for Mixed Compartment Fractional Differential Systems with Multi-Point Fractional Boundary Conditions

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.