The Existence of Positive Solutions to A Third-order Three-point Boundary Value Problem with Sign-changing Green's Function

Using the properties of Green's function and the iterative method, the existence of positive solutions to a class of third-order three-point boundary value problems with sign-changing Green's function is studied: \left\ \beginarray*20c \beginarrayl u'''\left( t \right) = f\left( t,u\left( t \right) \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( t \in \left 0,1 \right \right),\\ u\left( 1 \right) = 0,u'\left( 0 \right) = u''\left( 0 \right),\alpha u''\left( \eta \right) + \beta u\left( 0 \right) = 0, \endarray \endarray \right. where f∈C(0, 1×0, ∞), 0, ∞)), α∈0, 1, \frac27α < β < \frac23α, η∈\frac23, 1). The conditions for the existence of positive solutions to the boundary value problem are obtained.