Contraction maps and equivalent linearization

This study is primarily concerned with the question: If the method of equivalent linearization indicates the existence of a periodic solution, is there actually a periodic solution near the approximation of equivalent linearization? To answer this question, we use a modification of the contraction mapping fixed point theorem. We discuss applications to differential equations and difference-differential equations (with forcing functions). Also, we show that our use of contraction maps is not applicable (without modification) to autonomous systems because the mapping evaluated in the neighborhood of a periodic solution to an autonomous system is not a contraction in a space of periodic functions.