Pata V. Fixed Point Theorems and Applications

Springer, 2019. — 171 p. — (UNITEXT 116). — ISBN: 978-3-030-19669-1.This book addresses fixed point theory, a fascinating and far-reaching field with applications in several areas of mathematics. The content is divided into two main parts. The first, which is more theoretical, develops the main abstract theorems on the existence and uniqueness of fixed points of maps. In turn, the second part focuses on applications, covering a large variety of significant results ranging from ordinary differential equations in Banach spaces, to partial differential equations, operator theory, functional analysis, measure theory, and game theory. A final section containing 50 problems, many of which include helpful hints, rounds out the coverage. Intended for Master’s and PhD students in Mathematics or, more generally, mathematically oriented subjects, the book is designed to be largely self-contained, although some mathematical background is needed: readers should be familiar with measure theory, Banach and Hilbert spaces, locally convex topological vector spaces and, in general, with linear functional analysis.The Banach Contraction Principle
The Boyd-Wong Theorem
Further Extensions of the Contraction Principle
Weak Contractions
Contractions of ε-Type
Sequences of Maps and Fixed Points
Fixed Points of Non-expansive Maps
The Riesz Mean Ergodic Theorem
The Brouwer Fixed Point Theorem
The Schauder-Tychonoff Fixed Point Theorem
Further Consequences of the Schauder-Tychonoff Theorem
The Markov-Kakutani Theorem
The Kakutani-Ky Fan Theorem
The Implicit Function Theorem
Location of Zeros
Ordinary Differential Equations in Banach Spaces
The Lax-Milgram Lemma
An Abstract Elliptic Problem
Semilinear Evolution Equations
An Abstract Parabolic Problem
The Invariant Subspace Problem
Measure Preserving Maps on Compact Hausdorff Spaces
Invariant Means on Semigroups
Haar Measures
Game Theory
Problems