In recent years, the fixed point theory of Lipschitzian-type mappings has rapidly grown into an important field of study in both pure and applied mathematics. It has become one of the most essential tools in nonlinear functional analysis.
This self-contained book provides the first systematic presentation of Lipschitzian-type mappings in metric and Banach spaces. The first chapter covers some basic properties of metric and Banach spaces. Geometric considerations of underlying spaces play a prominent role in developing and understanding the theory. The next two chapters provide background in terms of convexity, smoothness and geometric coefficients of Banach spaces including duality mappings and metric projection mappings. This is followed by results on existence of fixed points, approximation of fixed points by iterative methods and strong convergence theorems. The final chapter explores several applicable problems arising in related fields.
This book can be used as a textbook and as a reference for graduate students, researchers and applied mathematicians working in nonlinear functional analysis, operator theory, approximations by iteration theory, convexity and related geometric topics, and best approximation theory.
Fixed Point Theory for Lipschitzian-type Mappings with Applications
Topological Fixed Point Theory and Its Applications VOLUME 6
Fixed Point Theory for Lipschitzian-type Mappings with Applications by Ravi P. Agarwal Florida Institute of Technology Melbourne, FL, USA
Donal O’Regan National University of Ireland Galway, Ireland
and D.R. Sahu Banaras Hindu University Varanasi, India
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Ravi P. Agarwal Department of Mathematical Sciences Florida Insitute of Technology Melbourne, FL 32901 USA
[email protected]
Donal O’Regan Institute of Mathematics University College Galway National University of Ireland Galway, Ireland
[email protected]
D.R. Sahu Department of Mathematics Faculty of Science Banaras Hindu University Varanasi, India
[email protected]
ISBN 978-0-387-75817-6 e-ISBN 978-0-387-75818-3 DOI 10.1007/978-0-387-75818-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009927662 AMS Subject Classifications (2000): 47H09, 47H10
c 2009 Springer Science+Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to our daughters Sheba Agarwal Lorna Emily O’Regan Gargi Sahu
Preface Over the past few decades, fixed point theory of Lipschitzian and nonLipschitzian mappings has been developed into an important field of study in both pure and applied mathematics. The main purpose of this book i