The Nonexistence of Positive Solutions for A Coupled System of Non-separated Boundary Value Problems

Abstract

In this paper, we consider the non-separated boundary value problem for system of nonlinear Riemann–Liouville fractional differential equations

$$\begin{aligned} \begin{aligned} D_{0^+}^{\alpha }x(t)+\lambda f(t, x(t), y(t))=0,~~0<t<1,\\ D_{0^+}^{\alpha }y(t)+\mu g(t, x(t), y(t))=0, ~~0<t<1, \end{aligned} \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} \begin{aligned} x(0)=y(0)=0,~~ u_1D_{0^+}^{\beta }x(1)=v_1D_{0^+}^{\beta }y(\xi ),\\ u_2D_{0^+}^{\beta }y(1)=v_2D_{0^+}^{\beta }x(\eta ),~~\eta ,\xi \in (0,1), \end{aligned} \end{aligned}$$

where the coefficients $u_{i},v_{i},i=1,2$ are real positive constants, we give sufficient conditions on $\lambda , \mu , f$ and g such that the system has no positive solutions. An example is given to demonstrate the main result.