Positive Solutions for a Class of Third-Order Three-Point Boundary Value Problem

We investigate the problem of existence of positive solutions for the nonlinear third-order three-point boundary value problem u'''(t) + lambda a(t)f(u(t)) = 0, 0 < t < 1, u(0) = u'(0) = 0, u ''(1) =alpha u ''(eta), where lambda is a positive parameter, alpha is an element of (0, 1), eta is an element of (0, 1), f : (0,infinity) -> (0,infinity), a : (0, 1) -> (0,infinity) are continuous. Using a specially constructed cone, the fixed point index theorems and Leray-Schauder degree, this work shows the existence and multiplicities of positive solutions for the nonlinear third-order boundary value problem. Some examples are given to demonstrate the main results.