Unique Solution of a Coupled Fractional Differential System Involving Integral Boundary Conditions from Economic Model

Abstract

We study the existence and uniqueness of the positive solution for the fractional differential system involving the Riemann-Stieltjes integral boundary conditions $-{\mathcal{D}}_t^{\alpha }x(t)=f(t,y(t))$, $-{\mathcal{D}}_t^{\beta }y(t)=g(t,x(t))$, $t\in (\mathrm{0,1})$, $x(\mathrm{0})=y(\mathrm{0})=\mathrm{0}$, $x(\mathrm{1})={\int }_{\mathrm{0}}^{\mathrm{1}}‍x(s)dA(s)$, and $y(\mathrm{1})={\int }_{\mathrm{0}}^{\mathrm{1}}‍y(s)dB(s)$, where , $\beta \le \mathrm{2}$, and ${\mathcal{D}}_t^{\alpha }$ and ${\mathcal{D}}_t^{\beta }$ are the standard Riemann-Liouville derivatives, $A$ and $B$ are functions of bounded variation, and ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{‍}{\mathcal{D}}_t^{\beta }x(s)dA(s)$ and ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{‍}{\mathcal{D}}_t^{\beta }y(s)dB(s)$ denote the Riemann-Stieltjes integral. Our results are based on a generalized fixed point theorem for weakly contractive mappings in partially ordered sets.