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# Introduction

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# Introduction

DOI link for Introduction

Introduction book

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## ABSTRACT

Over the past few decades, fixed point theory has been an active area of re-

search with a wide range of applications in several fields. In fact, this theory

constitutes an harmonious mixture of analysis (pure and applied), topology,

and geometry. In particular, it has several important applications in various

fields, such as physics, engineering, game theory, and biology (in which we

are interested). Perhaps, the most well-known result in this theory is Ba-

nach’s contraction principle. More precisely, in 1922, S. Banach formulated

and proved a theorem which focused, under appropriate conditions, on the ex-

istence and uniqueness of a fixed point in a complete metric space (see [149]).

This result leads to several powerful theorems such as inverse map theorem,

Cauchy-Picard theorem for ordinary differential equations among others. In

mathematics, some fixed point theorems in infinite-dimensional spaces gener-

alize a well-known result proved by L. E. J. Brouwer [42] which states that

every continuous map A : B1 −→ B1, where B1 is the closed unit ball in Rn has, at least, a fixed point in B1. These theorems have several applications.

For example, we may refer to the proof of existence theorems for differential

equations. The first result in this field was Schauder’s fixed point theorem,

proved in 1930 by J. Schauder and which asserts that every continuous and

compact mapping from a closed, convex, and bounded subsetM of a Banach space X into M has, at least, a fixed point [147].