Positive solutions of a third-order three-point BVP with sign-changing Green's function (Q1718301)

Summary: We are concerned with the following third-order three-point boundary value problem: \(u^{\prime \prime \prime} \left(t\right) = f \left(t, u \left(t\right)\right), t \in \left[0, 1\right], u'\left(0\right) = u \left(1\right) = 0\) and \(u^{\prime \prime} \left(\eta\right) - \alpha u'\left(1\right) = 0,\) where \(\alpha \in \left[0, 1\right)\) and \(\eta \in \left[(14 + \alpha) /(24 - 3 \alpha), 1\right)\). Although the corresponding Green's function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on \(f\) by using the two-fixed-point theorem due to Avery and Henderson.