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In this paper, we investigate the existence and uniqueness of solutions to the coupled system of nonlinear fractional differential equations {−D0+ν1y1(t)=λ1a1(t)f(y1(t),y2(t)),−D0+ν2y2(t)=λ2a2(t)g(y1(t),y2(t)),$$\left \{ \begin{array}{@{}l} -D^{\nu_{1}}_{0^{+}}y_{1}(t)=\lambda_{1}a_{1}(t)f(y_{1}(t),y_{2}(t)), \\ -D^{\nu_{2}}_{0^{+}}y_{2}(t)=\lambda_{2}a_{2}(t)g(y_{1}(t),y_{2}(t)), \end{array} \right . $$ where D0+ν$D^{\nu}_{0^{+}}$ is the standard Riemann-Liouville fractional derivative of order ν, t∈(0,1)$t\in(0,1)$, ν1,ν2∈(n−1,n]$\nu_{1}, \nu_{2} \in(n-1,n]$ for n>3$n>3$ and n∈N$n \in\mathbf{N} $, and λ1,λ2>0$\lambda_{1}, \lambda_{2} > 0$, with the multi-point boundary value conditions: y1(i)(0)=0=y2(i)(0)$y^{(i)}_{1}(0)=0=y^{(i)}_{2}(0)$, 0≤i≤n−2$0 \leq i \leq n-2$; D0+βy1(1)=∑i=1m−2biD0+βy1(ξi)$D^{\beta}_{0^{+}}y_{1}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{1}(\xi_{i})$; D0+βy2(1)=∑i=1m−2biD0+βy2(ξi)$D^{\beta}_{0^{+}}y_{2}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{2}(\xi_{i})$, where n−2<β<n−1$n-2 < \beta< n-1$, 0<ξ1<ξ2<⋯<ξm−2<1$0 < \xi_{1} < \xi_{2} < \cdots< \xi_{m-2} <1$, bi≥0$b_{i} \geq0$ (i=1,2,…,m−2$i=1,2,\ldots,m-2$) with ρ1:=∑i=1m−2biξiν1−β−1<1$\rho_{1}: = \sum^{m-2}_{i=1} b_{i}\xi^{\nu_{1}-\beta-1}_{i}<1$, and ρ2:=∑i=1m−2biξiν2−β−1<1$\rho_{2}: =\sum^{m-2}_{i=1} b_{i}\xi^{\nu_{2}-\beta-1}_{i}<1$. Our analysis relies on the Banach contraction principle and Krasnoselskii’s fixed point theorem.

Multi-point boundary value problems for a coupled system of nonlinear fractional differential equations