Metric Fixed Point Theory

Abstract

The aim of this chapter is to give a brief history of metric fixed point theory. In this section, we discuss the pioneer metric fixed point theorem that was given by Banach [56]. This outstanding result is known as the contraction mapping principle or the Banach contraction mapping principle. The main advantage of Banach’s metric fixed point theorem is the following property: This theorem not only guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces but also indicates how to construct an iterative sequence that provides the desired fixed points. It is worth mentioning that this famous fixed point theorem was formulated in his thesis in 1920 and published in 1922 in the setting of normed linear spaces (not metric spaces). As we mentioned in the previous section, Banach Contraction Mapping Principle is not the first fixed point theorem in the literature but the most interesting fixed point theorem in the context of metric fixed point theory. Indeed, Brouwer gave the first result, which only guarantees the existence of the fixed point. Unfortunately, Brouwer’s fixed point theorem does not explain how to get the guaranteed fixed point and how to ensure the uniqueness of this mentioned fixed point. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.