## International Journal of Differential Equations

### Positive Solutions for a Coupled System of Nonlinear Semipositone Fractional Boundary Value Problems

#### Abstract

In this paper, we consider a four-point coupled boundary value problem for system of the nonlinear semipositone fractional differential equation ${D}_{{\mathrm{0}}^{+}}^{\alpha }u(t)+\lambda f(t,u(t),v(t))=\mathrm{0}, \mathrm{0}, ${D}_{{\mathrm{0}}^{+}}^{\alpha }v(t)+\mu g(t,u(t),v(t))=\mathrm{0}, \mathrm{0}, $u(\mathrm{0})=v(\mathrm{0})=\mathrm{0}, {a}_{\mathrm{1}}{D}_{{\mathrm{0}}^{+}}^{\beta }u(\mathrm{1})={b}_{\mathrm{1}}{D}_{{\mathrm{0}}^{+}}^{\beta }v(\xi )$, ${a}_{\mathrm{2}}{D}_{{\mathrm{0}}^{+}}^{\beta }v(\mathrm{1})={b}_{\mathrm{2}}{D}_{{\mathrm{0}}^{+}}^{\beta }u(\eta ), \eta ,\xi \in (\mathrm{0,1}),$ where the coefficients ${a}_{i},{b}_{i},i=\mathrm{1,2}$ are real positive constants, $\alpha \in (\mathrm{1,2}],\beta \in (\mathrm{0,1}],$${D}_{{\mathrm{0}}^{+}}^{\alpha }$, ${D}_{{\mathrm{0}}^{+}}^{\beta }$ are the standard Riemann-Liouville derivatives. Values of the parameters $\lambda$ and $\mu$ are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.

#### Article information

Source
Int. J. Differ. Equ., Volume 2019 (2019), Article ID 2893857, 9 pages.

Dates
Revised: 22 December 2018
Accepted: 31 December 2018
First available in Project Euclid: 15 March 2019