The main subject of this book is the use of the Chain Generating Recursion Principle in introducing iteration methods which lead to the proof of fixed point theorems on partially ordered sets (posets). These fixed point theorems are applied in numerous cases of subtle problems, like existence and comparison results for operator equations and inclusions, partial differential equations and inclusions, ordinary differential and integral equations in Banach spaces. The Chain Generating Recursion Principle is actually a way to derive well-ordered chains in posets by using well-ordered sets in them. Moreover, these fixed point results, which are contained in the Chapter 2 of the book, are useful in the proof of the existence and the study of the properties of Nash equilibria for normal-form games. They are also applicable in the study of the existence of winning strategies for (ordered) pursuit and evasion games. Note that pursuit and evasion games rely in general on monotone parts of (possibly transfinite) sequences.