Fixed Point Theory for Lipschitzian-type Mappings with Applications
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-contai
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Topological Fixed Point Theory and Its Applications
Fixed Point Theory for Lipschitzian-type Mappings
Fixed Point Theory for Lipschitzian-type Mappings with Applications Ravi P. Agarwal Donal O’Regan D.R. Sahu
Ravi P. Agarwal
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 agarwal@fit.edu
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 is to present many of the basic techniques and results of this theory. Of course, not all aspects of this theory could be included in this exposition. The book contains eight chapters. The first chapter is devoted to some of the basic results of nonlinear functional analysis. The final section in this chapter deals with the classic results of fixed point theory. Our goal is to study nonlinear problems in Banach spaces. We remark here that it is hard to study these without the geometric properties of Banach spaces. As a result in Chapter 2, we discuss elements of convexity and smoothness of Banach spaces and properties of duality mappings. This chapter also includes many int
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