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 Sharkovskys theorem on periodic points, Throns results on the convergence of certain real iterates, Shields common fixed theorem for a commuting family of analytic functions and Bergweilers existence theorem on fixed points of the composition of certain meromorphic functions with transcendental entire functions. Generalizations of Tarskis theorem by Merrifield and Stein and Abians proof of the equivalence of BourbakiZermelo fixed-point theorem and the Axiom of Choice are described in the setting of posets. A detailed treatment of Wards theory of partially ordered topological spaces culminates in Sherrer fixed-point theorem. It elaborates Mankas 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 viaa 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 CauchyKowalevsky 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 BrowderGohdeKirk fixed point theorem, a proof of Stalling's generalization of Brouwer's theorem, examine Caristi's fixed point theorem, and highlights Kakutani's theorems on common fixed points and their applications.