Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions

Abstract

In this paper, we consider a four-point coupled boundary value problem for systems of the nonlinear semipositone fractional differential equation \begin{gather*}\left\{ \begin{array}{ll} \mathbf{D}_{0+}^\alpha u+\lambda f(t,u,v)=0,\quad 00,\\ \mathbf{D}_{0+}^\alpha v+\lambda g(t,u,v)=0,\\ u^{(i)}(0)=v^{(i)}(0)=0, 0\leq i\leq n-2,\\ u(1)=av(\xi), v(1)=bu(\eta), \xi,\eta\in(0,1) \end{array}\right.\end{gather*} where $\lambda$ is a parameter, $a, b, \xi,\eta$ satisfy $\xi,\eta\in(0,1)$, $0<ab\xi\eta<1$, $\alpha \in(n-1, n]$ is a real number and $n\geq 3$, and $\mathbf{D}_{0+}^\alpha$ is the Riemann-Liouville's fractional derivative, and $f,g$ are continuous and semipositone. We derive an interval on $\lambda$ such that for any $\lambda$ lying in this interval, the semipositone boundary value problem has multiple positive solutions.

Keywords

• riemann-liouville's fractional derivative
• semipositone fractional differential equation
• four-point coupled boundary value problem
• positive solution
• fixed-point theorem