## Abstract and Applied Analysis

### 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 $-{\mathcalbf{D}}_{t}^{\alpha }x(t)=f(t,y(t))$, $-\mathrm{ }{\mathcalbf{D}}_{t}^{\beta }y(t)=g(t,x(t))$, $t\in (\mathrm{0,1})$, $x(\mathrm{0})=y(\mathrm{0})=\mathrm{0}$, $x(\mathrm{1})={\int }_{\mathrm{0}}^{\mathrm{1}}‍x(s)dA(s)$, and $y(\mathrm{1})={\int }_{\mathrm{0}}^{\mathrm{1}}‍y(s)dB(s)$, where $\mathrm{1}<\alpha$, $\beta \le \mathrm{2}$, and ${\mathcalbf{D}}_{t}^{\alpha }$ and ${\mathcalbf{D}}_{t}^{\beta }$ are the standard Riemann-Liouville derivatives, $A$ and $B$ are functions of bounded variation, and ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{‍}{\mathcalbf{D}}_{t}^{\beta }x(s)dA(s)$ and ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{‍}{\mathcalbf{D}}_{t}^{\beta }y(s)dB(s)$ denote the Riemann-Stieltjes integral. Our results are based on a generalized fixed point theorem for weakly contractive mappings in partially ordered sets.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 615707, 6 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393443521

Digital Object Identifier
doi:10.1155/2013/615707

Mathematical Reviews number (MathSciNet)
MR3090294

#### Citation

Li, Rui; Zhang, Haoqian; Tao, Hao. Unique Solution of a Coupled Fractional Differential System Involving Integral Boundary Conditions from Economic Model. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 615707, 6 pages. doi:10.1155/2013/615707. https://projecteuclid.org/euclid.aaa/1393443521

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