Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives

[摘要] In this paper we study the existence of unique positive solutions for the following coupled system: $$\begin{aligned} \textstyle\begin{cases} D_{0^{+}}^{\alpha }x(\tau )+f_{1}(\tau ,x(\tau ),D_{0^{+}}^{\eta }x( \tau ))+g_{1}(\tau ,y(\tau ))=0, \\ D_{0^{+}}^{\beta }y(\tau )+f_{2}(\tau ,y(\tau ),D_{0^{+}}^{\gamma }y( \tau ))+g_{2}(\tau ,x(\tau ))=0, \\ \tau \in (0,1),\qquad n-13$ and $1\leq \gamma \leq \xi \leq n-2$, $1\leq \eta \leq \zeta \leq n-2$, $f_{1},f_{2}:[0,1]\times \mathbb{R^{+}}\times \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, $g_{1},g_{2}:[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ and $k_{1},k_{2}:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ are continuous functions, $D_{0^{+}}^{\alpha }$ and $D_{0^{+}}^{\beta }$ stand for the Riemann–Liouville derivatives. An illustrative example is given to show the effectiveness of theoretical results.

[学科分类] 航空航天科学

[关键词] Fractional differential equation;Mixed monotone operator;Normal cone;Coupled system