Fixed Point Theory in Ordered Sets and Applications From Differentia
This monograph provides a unified and comprehensive treatment of an order-theoretic fixed point theory in partially ordered sets and its various useful interactions with topological structures. It begins with a discussion of some simple examples of the or
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Siegfried Carl Seppo Heikkilä
Fixed Point Theory in Ordered Sets and Applications From Differential and Integral Equations to Game Theory
Siegfried Carl Martin-Luther-Universität Halle-Wittenberg Institut für Mathematik Halle Germany [email protected]
Seppo Heikkilä Department of Mathematical Sciences University of Oulu Oulu Finland [email protected]
ISBN 978-1-4419-7584-3 e-ISBN 978-1-4419-7585-0 DOI 10.1007/978-1-4419-7585-0 Springer New York Dordrecht Heidelberg London Mathematics Subject Classification Codes (2010): 06Axx, 06Bxx, 03F60, 28B05, 34Axx, 34Bxx, 34Gxx, 34Kxx, 35B51, 35J87, 35K86, 45N05, 46G12, 47H04, 47H10, 47J20, 49J40, 91A10, 91B16, 91B50, 58D25 © Springer Science+Business Media, LLC 2011 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, USA), 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 in gratitude and high esteem to
Professor V. Lakshmikantham
Preface
Fixed point theory is one of the most powerful and fruitful tools of modern mathematics and may be considered a core subject of nonlinear analysis. In recent years a number of excellent monographs and surveys by distinguished authors about fixed point theory have appeared such as, e.g., [2, 4, 7, 25, 31, 32, 100, 101, 103, 104, 108, 155, 196]. Most of the books mentioned above deal with fixed point theory related to continuous mappings in topological or even metric spaces (work of Poincar´e, Brouwer, Lefschetz–Hopf, Leray–Schauder) and all its modern extensions. This book focuses on an order-theoretic fixed point theory and its applications to a wide range of diverse fields such as, e.g., (multi-valued) nonlocal and/or discontinuous partial differential equations of elliptic and parabolic type, differential equations and integral equations with discontinuous nonlinearities in general vector-valued normed spaces of non-absolutely integrable functions containing the standard Bochner integrable functions as special case, and mathematical economics and game theory. In all these topics we are faced with the central problem of handling the loss of continuity of mappings and/or missing appropriate geometric and topological structure of their underlying domain of definition. For example, it is noteworthy that, in particular, for proving the existence of certain optimal strategies in game theory,
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