Positive solutions of second-order three-point boundary value problems with sign-changing coefficients

In this article, we investigate the boundary-value problem {x ''(t) + h(t)f(x(t)) = 0, t is an element of [0, 1], x(0) = beta x'(0), x(1) = x(eta), where beta >= 0, eta is an element of (0, 1), f is an element of C([0, infinity), [0, infinity)) is nondecreasing, and importantly h changes sign on [0, 1]. By the Guo-Krasnosel'skii fixed-point theorem in a cone, the existence of positive solutions is obtained via a special cone in terms of superlinear or sublinear behavior of f.