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 , , $u(\mathrm{0})=v(\mathrm{0})=\mathrm{0},\hspace{0.17em}\hspace{0.17em}{a}_{\mathrm{1}}{D}_{{\mathrm{0}}^{+}}^{\beta }u(\mathrm{1})=b_{\mathrm{1}}{D}_{{\mathrm{0}}^{+}}^{\beta }v(\xi )$, $a_{\mathrm{2}}{D}_{{\mathrm{0}}^{+}}^{\beta }v(\mathrm{1})=b_{\mathrm{2}}{D}_{{\mathrm{0}}^{+}}^{\beta }u(\eta ),\hspace{0.17em}\hspace{0.17em}\eta ,\xi \in (\mathrm{0,1}),$ 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.