Positive solutions of infinite coupled system of fractional differential equations in the sequence space of weighted means

We first discuss the existence of solutions of the infinite system of $(n-1,n)$-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations

$$\{\begin{array}{ll}{D}_{0_{+}}^{\alpha }{u}_i(\rho )+\eta f_i(\rho ,v(\rho ))=0,& \rho \in (0,1),\\ D_{0_{+}}^{\alpha }{v}_i(\rho )+\eta g_i(\rho ,u(\rho ))=0,& \rho \in (0,1),\\ u_i^{(j)}(0)=v_i^{(j)}(0)=0,& 0\le j\le n-2,\\ u_i(1)=\zeta {\int }_0^1{u}_i(\vartheta )d\vartheta ,v_i(1)=\zeta {\int }_0^1{v}_i(\vartheta )d\vartheta ,& i\in \mathbb{N},\end{array}$$

in the sequence space of weighted means $c_0(W_1,W_2,\mathrm{\Delta })$, where $n\ge 3$, $\alpha \in (n-1,n]$, $\eta ,\zeta$ are real numbers, $0<\eta <\alpha ,$ $D_{0_{+}}^{\alpha }$ is the Riemann-Liouville's fractional derivative, and $f_i,g_i,$ $i=1,2,\dots$, are semipositone and continuous. Our approach to the study of solvability is to use the technique of measure of noncompactness. Then, we find an interval of $\eta$ such that for each $\eta$ lying in this interval, the system of $(n-1,n)$-type semipositone BVPs has a positive solution. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.