Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions

In this paper, we investigate the four-point coupled boundary value problem of nonlinear semipositone Hadamard fractional differential equations

$$\begin{aligned} D^\alpha u(t)\,+\,\lambda f(t,u(t),v(t))\!=\!0, \quad D^\beta v(t)\,+\,\lambda g(t,u(t),v(t))\!=\!0, \quad t\!\in \!(1,e),\quad \!\!\lambda \!>\!0,\\ u^{(j)}(1)\!=\!v^{(j)}(1)\!=\!0,\quad 0\!\le \! j\le n-2, \quad u(e)\!=\!av(\xi ),\quad v(e)\!=\!bu(\eta ), \quad \xi ,\eta \in (1,e), \end{aligned}$$

where \(\lambda ,a,b\) are three parameters with \(0<ab(\log \eta )^{\alpha -1}(\log \xi )^{\beta -1}<1\), \(\alpha ,\beta \in (n-1,n]\) are two real numbers and \(n\ge 3\), \(D^\alpha , D^\beta \) are the Hadamard fractional derivative of fractional order, and \(f,g\) are continuous and may be singular at \(t=0\) and \(t=1\). We firstly give the corresponding Green’s function for the boundary value problem and some of its properties. Moreover, by applying Guo-Krasnoselskii fixed point theorems, we derive an interval of \(\lambda \) such that any \(\lambda \) lying in this interval, the singular boundary value problem has at least one positive solution. As applications, two interesting examples are presented to illustrate the main results.

     

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