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

Abstract

In this paper, we investigate the existence and uniqueness of solutions to the coupled system of nonlinear fractional differential equations $$\left \{ \begin{array}{@{}l} -D^{\nu_{1}}_{0^{+}}y_{1}(t)=\lambda_{1}a_{1}(t)f(y_{1}(t),y_{2}(t)), -D^{\nu_{2}}_{0^{+}}y_{2}(t)=\lambda_{2}a_{2}(t)g(y_{1}(t),y_{2}(t)), \end{array} \right . $$ where $D^{\nu}_{0^{+}}$ is the standard Riemann-Liouville fractional derivative of order ν, $t\in(0,1)$ , $\nu_{1}, \nu_{2} \in(n-1,n]$ for $n>3$ and $n \in\mathbf{N} $ , and $\lambda_{1}, \lambda_{2} > 0$ , with the multi-point boundary value conditions: $y^{(i)}_{1}(0)=0=y^{(i)}_{2}(0)$ , $0 \leq i \leq n-2$ ; $D^{\beta}_{0^{+}}y_{1}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{1}(\xi_{i})$ ; $D^{\beta}_{0^{+}}y_{2}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{2}(\xi_{i})$ , where $n-2 < \beta< n-1$ , $0 < \xi_{1} < \xi_{2} < \cdots< \xi_{m-2} <1$ , $b_{i} \geq0$ ( $i=1,2,\ldots,m-2$ ) with $\rho_{1}: = \sum^{m-2}_{i=1} b_{i}\xi^{\nu_{1}-\beta-1}_{i}<1$ , and $\rho_{2}: =\sum^{m-2}_{i=1} b_{i}\xi^{\nu_{2}-\beta-1}_{i}<1$ . Our analysis relies on the Banach contraction principle and Krasnoselskii’s fixed point theorem.