| Summary: | Abstract In this study, we discuss the existence of positive solutions for the system of m-singular sum fractional q-differential equations Dqαixi+gi(t,x1,…,xm,Dqγ1x1,…,Dqγmxm)+hi(t,x1,…,xm,Dqγ1x1,…,Dqγmxm)=0 $$ \begin{gathered} D_{q}^{\alpha_{i}} x_{i} + g_{i} \bigl(t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma _{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr) \\ \quad{} +h_{i} \bigl(t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr)=0 \end{gathered} $$ with boundary conditions xi(0)=xi′(1)=0 $x_{i}(0) = x_{i}' (1) = 0$ and xi(k)(t)=0 $x_{i}^{(k)}(t) = 0$ whenever t=0 $t=0$, here 2≤k≤n−1 $2\leq k \leq n-1$, where n=[αi]+1 $n= [\alpha_{i}]+ 1$, αi≥2 $\alpha_{i} \geq2$, γi∈(0,1) $\gamma_{i} \in(0,1)$, Dqα $D_{q}^{\alpha}$ is the Caputo fractional q-derivative of order α, here q∈(0,1) $q \in(0,1)$, function gi $g_{i}$ is of Carathéodory type, hi $h_{i}$ satisfy the Lipschitz condition and gi(t,x1,…,x2m) $g_{i} (t , x_{1}, \ldots, x_{2m})$ is singular at t=0 $t=0$, for 1≤i≤m $1 \leq i \leq m$. By means of Krasnoselskii’s fixed point theorem, the Arzelà-Ascoli theorem, Lebesgue dominated theorem and some norms, the existence of positive solutions is obtained. Also, we give an example to illustrate the primary effects.
|
|---|