Predictor homotopy analysis method: Two points second order boundary value problems

The purpose of the present paper is to give a proof for a systematic method namely, predictor homotopy analysis method (PHAM) to predict the multiplicity of the solutions of two points nonlinear second order boundary value problems of type u″=f(x,u,u′). We present briefly the method first, and then some theorems are given to clarify this issue that, how one can predict the existence of the multiple solutions and obtain them simultaneously as well.