Leray-Schauder Type Alternatives, Complemantarity Problems and Variational Inequalities
Complementarity theory, a relatively new domain in applied mathematics, has deep connections with several aspects of fundamental mathematics and also has many applications in optimization, economics and engineering. The study of variational inequalities i
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Nonconvex Optimization and Its Applications VOLUME 87 Managing Editor: Panos Pardalos University of Florida, U.S.A.
Advisory Board: J. R. Birge University of Chicago, U.S.A. Ding-Zhu Du University of Minnesota, U.S.A. C. A. Floudas Princeton University, U.S.A. J. Mockus Lithuanian Academy of Sciences, Lithuania H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A. G. Stavroulakis Technical University Braunschweig, Germany H.Tuy National Centre for Natural Science and Technology, Vietnam
LERAY-SCHAUDER TYPE ALTERNATIVES, COMPLEMENTARITY PROBLEMS AND VARIATIONAL INEQUALITIES
By GEORGE ISAC Royal Military College of Canada, Kingston, Ontario, Canada
Springer
Library of Congress Control Number: 2006921731 ISBN-10: 0-387-32898-X
e-ISBN: 0-387-32900-5
ISBN-13: 978-0-387-32898-0
Printed on acid-free paper.
AMS Subject Classifications: 90C33, 47J20, 47J25, 90C30 © 2006 Springer Science+Business Media, Inc. 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, Inc., 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 in the United States of America. 987654321 springer.com
To my family
The motion of Truth is cyclical, The way of Truth is pliant The Works of Lao Zi Truth and Nature
CONTENTS PREFACE 1 PRELIMINARY NOTIONS 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Topological spaces. Some fundamental notions Metric spaces Some classes of topological vector spaces Compactness and compact operators Measures of noncompactness and condensing operators Topological degrees Zero-epi mappings Convex cones Projection operators
xi 1 1 4 6 13 14 19 29 35 40
2 COMPLEMENTARITY PROBLEMS AND VARIATIONAL INEQUALITIES 49 2.1 2.2 2.3
Complementarity problems Variational inequalities Complementarity problems, variational inequalities, equivalences and equations
49 59 62
3 L E R A Y - S C H A U D E R ALTERNATIVES
71
3.1
The Leray-Schauder alternative by topological degree
72
3.2
The Leray-Schauder alternative by the fixed point theory
74
3.3 3.4
The Leray-Schauder alternative by the topological transversality theory 76 Some classes of mappings and Leray-Schauder type alternatives... 81
3.5
An implicit Leray-Schauder alternative
90
3.6
Leray-Schauder type alternatives for set-valued mappings
95
4 THE ORIGIN OF THE NOTION OF EXCEPTIONAL FAMILY OF ELEMENTS 4.1
109
Exceptional family of elements, topological degree and nonlinear complementarity problems in R" 109
viii
LeraySchauder
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