Elementary fixed point theorems This book provides a primary resource in basic fixed point theorems due to Banach Brouwer Schauder and Tarski and their applications. Key topics covered include Sharkovsky’s theorem on periodic points Thron’s results on the convergence of certain real iterates Shield’s common fixed theorem for a commuting family of analytic functions and Bergweiler’s existence theorem on fixed points of the composition of certain meromorphic functions with transcendental entire functions. Generalizations of Tarski’s theorem by Merrifield and Stein and Abian’s proof of the equivalence of Bourbaki Zermelo fixed point theorem and the Axiom of Choice are described in the setting of posets. A detailed treatment of Ward’s theory of partially ordered topological spaces culminates in Sherrer fixed point theorem. It elaborates Manka’s proof of the fixed point property of arcwise connected hereditarily unicoherent continua based on the connection he observed between set theory and fixed point theory via a certain partial order. Contraction principle is provided with two proofs: one due to Palais and the other due to Barranga. Applications of the contraction principle include the proofs of algebraic Weierstrass preparation theorem a Cauchy Kowalevsky theorem for partial differential equations and the central limit theorem. It also provides a proof of the converse of the contraction principle due to Jachymski a proof of fixed point theorem for continuous generalized contractions a proof of Browder Gohde Kirk fixed point theorem a proof of Stallings generalization of Brouwers theorem examine Caristis fixed point theorem and highlights Kakutanis theorems on common fixed points and their applications Chapter 1. Prerequisites Chapter 2. Fixed Points of Some Real and Complex Functions Chapter 3. Fixed Points and Order Chapter 4. Partially Ordered Topological Spaces and Fixed Points Chapter 5. Contraction Principle Chapter 6. Applications of the Contraction Principle Chapter 7. Caristi’s ﬁxed point theorem Chapter 8. Contractive and Nonexpansive Mappings Chapter 9. Geometric Aspects of Banach Spaces and Nonexpansive Mappings Chapter 10. Brouwer’s Fixed Point Theorem Chapter 11. Schauder’s Fixed Point Theorem and Allied Theorems Chapter 12. Basic Analytic Degree Theory af a Mapping