Existence of Positive Solutions for Nonlinear Third-Order Boundary Value Problem

We are concerned with the existence of positive solutions for the nonlinear third-order three-point boundary value problem u"'(t) + lambda(g)(t) f(u(t)) = 0, 0 < t < 1, u(0) = alpha u(eta), u'(0) = u"(1) = 0, where 0 < eta < 1, 0 < alpha < 1, lambda is a positive parameter, g : (0, 1) -> [0, infinity), and f : [0, infinity) -> [0, infinity) is continuous. We construct Green's function for the associated linear boundary value problem and obtain some useful properties of Green's function. Finally, by using fixed-point index theorem in cones, we establish the existence results of positive solutions for the boundary value problem an example illustrates the application of the results obtained.