Elementary Fixed Point Theorems

This resource explores key fixed-point theorems and their applications, including contributions from notable mathematicians. Elementary Fixed Point Theorems is essential for understanding these concepts.

This book serves as a comprehensive resource on fundamental fixed-point theorems attributed to prominent mathematicians such as Banach, Brouwer, Schauder, and Tarski, alongside their various applications. It delves into significant topics, including Sharkovsky’s theorem concerning periodic points and Thron’s findings on the convergence of specific real iterates. Additionally, it explores Shield’s common fixed theorem for a commuting family of analytic functions and Bergweiler’s existence theorem related to fixed points of the composition of certain meromorphic functions with transcendental entire functions.

Elementary Fixed Point Theorems also discusses the generalizations of Tarski’s theorem by Merrifield and Stein, as well as Abian’s proof linking the Bourbaki–Zermelo fixed-point theorem to the Axiom of Choice within the context of posets. Furthermore, it provides an in-depth examination of Ward’s theory on partially ordered topological spaces, culminating in Sherrer's fixed-point theorem. The book elaborates on Manka’s proof regarding the fixed-point property of arcwise connected hereditarily unicoherent continua, highlighting the relationship between set theory and fixed-point theory through a specific partial order.

The contraction principle is presented with two distinct proofs: one by Palais and the other by Barranga. Applications of this principle are extensive, including proofs of the algebraic Weierstrass preparation theorem, a Cauchy–Kowalevsky theorem for partial differential equations, and the central limit theorem. The book also includes a proof of the converse of the contraction principle by Jachymski, alongside various fixed-point theorems and their implications, making it an essential read for those interested in this mathematical field.