Characterizing fixed points

Abstract

A set of sufficient conditions which guarantee the existence of a point x⋆ such
that f(x⋆) = x⋆ is called a "fixed point theorem". Possibly hundreds of such theorems exist in the literature. About twenty of them are explicitly mentioned in Wikipedia where they are named after well-known mathematicians and economists. Fixed point theorems are among most useful ones in applied mathematics, especially in economics and game theory. Particularly important theorem in these areas is Kakutani's fixed point theorem which ensures existence of fixed point for point-to-set mappings, e.g., [2, 3, 4]. John Nash developed and applied Kakutani's ideas to prove the existence of (what became known as)"Nash equilibrium" for finite games with mixed strategies for any
number of players. This work earned him a Nobel Prize in Economics that he shared with two mathematicians. Nash's life was dramatized in the movie "Beautiful Mind" in 2001. In this paper, we approach the system f(x) = x differently. Instead of studying existence of its solutions our objective is to determine conditions which are both necessary and sufficient that an arbitrary point x⋆ is a fixed point, i.e., that it satises f(x⋆) = x⋆. The existence
of solutions for continuous function f of the single variable is easy to establish using the Intermediate Value Theorem of Calculus. However, characterizing fixed points x⋆, i.e., providing answers to the question of finding both necessary and sufficient conditions for an arbitrary given x⋆ to satisfy f(x⋆) = x⋆, is not simple even for functions of the single variable. It is possible that constructive answers do not exist. Our objective is to find them.
Our work may require some less familiar tools. One of these might be the "quadratic envelope characterization of zero-derivative point" recalled in the next section. The results are taken from the author's current Research project "Studying the Essence of Fixed Points". They are believed to be original. The author has received several feedbacks on the preliminary report and on parts of the project which can be seen on Internet [9].