Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following second-order three-point boundary value problem is studied: x ''(t) + f(t, x(t)) = 0, 0 < t < 1, x '(0) = 0, x(1) + delta x '(eta) = 0, where eta is an element of (0, 1), delta is an element of [0, infinity), f is an element of C([0, 1] x [0, infinity), [0, infinity)). Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.