Positive solutions for a fourth-order three-point BVP with sign-changing Green's function

This paper is concerned with the following fourth-order three-point boundary value problem \begin{equation*} \begin{cases} u^{(4)}(t) = f\big(t,u(t)\big), ~t\in{[0,1]},\\ u'(0) = u''(0) = u'''(\eta) = u(1) = 0, \end{cases} \end{equation*} where $ \eta\in\big[\frac{1}{3}, 1\big)$. In spite of sign-changing Green's function, for arbitrary positive integer $n\ (\geq 2)$, we still obtain the existence of at least $n-1$ decreasing positive solutions to the above problem by imposing some suitable conditions on $f$. The main tool used is the fixed point index theory