This book examines in detail approximate fixed point theory in different
classes of topological spaces for general classes of maps. It offers a
comprehensive treatment of the subject that is up-to-date,
self-contained, and rich in methods, for a wide variety of topologies
and maps. Content includes known and recent results in topology (with
proofs), as well as recent results in approximate fixed point theory.
This work starts with a set of basic notions in topological spaces.
Special attention is given to topological vector spaces, locally convex
spaces, Banach spaces, and ultrametric spaces. Sequences and function
spaces-and fundamental properties of their topologies-are also covered.
The reader will find discussions on fundamental principles, namely the
Hahn-Banach theorem on extensions of linear (bounded) functionals; the
Banach open mapping theorem; the Banach-Steinhaus uniform boundedness
principle; and Baire categories, including some applications. Also
included are weak topologies and their properties, in particular the
theorems of Eberlein-Smulian, Goldstine, Kakutani, James and
Grothendieck, reflexive Banach spaces, l_{1}- sequences, Rosenthal's
theorem, sequential properties of the weak topology in a Banach space
and weak* topology of its dual, and the Fréchet-Urysohn property.
The subsequent chapters cover various almost fixed point results,
discussing how to reach or approximate the unique fixed point of a
strictly contractive mapping of a spherically complete ultrametric
space. They also introduce synthetic approaches to fixed point problems
involving regular-global-inf functions. The book finishes with a study
of problems involving approximate fixed point property on an ambient
space with different topologies.
By providing appropriate background and up-to-date research results,
this book can greatly benefit graduate students and mathematicians
seeking to advance in topology and fixed point theory.
