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Topology and Approximate Fixed Points

  • Book
  • © 2022

Overview

Authors:
  1. Afif Ben Amar

    1. Faculty of Sciences, Mathematics University of Sfax, Sfax, Tunisia

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    You can also search for this author in PubMed   Google Scholar

  2. Donal O'Regan

    1. School of Mathematical and Statistical Sciences, National University of Ireland, Galway, Ireland

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Explores in detail approximate fixed point theory in different classes of topological spaces
  • Covers weak topologies and their properties, in particular sequential properties of the weak topology
  • Provides appropriate background to those seeking to advance in topology and fixed point theory

Part of the book series: Developments in Mathematics (DEVM, volume 71)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Basic Concepts

    • Afif Ben Amar, Donal O’Regan
    Pages 1-97

  3. Almost Fixed Points

    • Afif Ben Amar, Donal O’Regan
    Pages 99-144

  4. Approximate Fixed Points in Ultrametric Spaces

    • Afif Ben Amar, Donal O’Regan
    Pages 145-153

  5. Synthetic Approaches to Problems of Fixed Points

    • Afif Ben Amar, Donal O’Regan
    Pages 155-185

  6. Approximate Fixed Points in Topological Vector Spaces

    • Afif Ben Amar, Donal O’Regan
    Pages 187-237

  7. Back Matter

    Pages 239-251
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Keywords

About this book

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 areweak 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 pointtheory.



Reviews

“The book is very well written, with accurate and detailed arguments in the proofs and inspiring interpretations of the statements of the results. This book is a valuable resource for graduate students who wish to learn about advanced topology and fixed point theory but it will also be useful to researchers in the area.” (Jarosław Górnicki, zbMATH 1492.54018, 2022)

Authors and Affiliations

  • Faculty of Sciences, Mathematics University of Sfax, Sfax, Tunisia

    Afif Ben Amar

  • School of Mathematical and Statistical Sciences, National University of Ireland, Galway, Ireland

    Donal O'Regan

About the authors

Afif Ben Amar is a Professor in the Department of Mathematics, Faculty of Sciences, at the University of Sfax, Tunisia. His research interests lie in operator theory, fixed point theory, nonlinear spectral theory, partial differential equations, integral equations, and applications of mathematics to natural sciences. He co-authored (with Donal O’Regan) the book “Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications”, published by Springer.

Donal O’Regan is a Professor in the School of Mathematics, Statistics, and Applied Mathematics at the National University of Ireland, Galway. His research interests include differential equations, nonlinear analysis, and fixed point theory. He has authored several books, including the Springer titles “Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications (with Afif Ben Amar) and “An Introduction to Ordinary Differential Equations” (with Ravi P. Agarwal).


Bibliographic Information

  • Book Title: Topology and Approximate Fixed Points

  • Authors: Afif Ben Amar, Donal O'Regan

  • Series Title: Developments in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-92204-7

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Hardcover ISBN: 978-3-030-92203-0Published: 26 January 2022

  • Softcover ISBN: 978-3-030-92206-1Published: 27 January 2023

  • eBook ISBN: 978-3-030-92204-7Published: 25 January 2022

  • Series ISSN: 1389-2177

  • Series E-ISSN: 2197-795X

  • Edition Number: 1

  • Number of Pages: XIII, 251

  • Topics: Functional Analysis, Operator Theory, Topology

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