Topological Methods in Nonlinear Functional Analysis

Transcription

1 Topological Methods in Nonlinear Functional Analysis AMERICAn MATHEMATICAL SOCIETY u u m 0 L E 21

2 COnTEMPORARY MATHEMATICS Titles in this Series VOLUME 1 VOLUME 2 VOLUME 3 Markov random fields and their applications Ross Kindermann and J. Laurie Snell Proceedings of the conference on integration, topology, and geometry in linear spaces William H. Graves, Editor The closed graph and P-closad graph properties in general topology T. R. Hamlett and L. L. Herrington VOLUME 4 VOLUME 5 VOLUME 6 VOLUME 7 VOLUME 8 VOLUME I Problems of elastic stability and vibrations Vadim Komkov, Editor Rational constructions of modules for simple Lie algebras George B. Seligman Umbral calculus and Hopf algebras Robert Morris, Editor Complex contour integral representation of cardinal spline functions Walter Schempp Ordered fields and real algebraic geometry D. W. Dubois and T. Recio, Editors Papers in algebra, analysis and statistics R. Lidl, Editor VOLUME 10 Operator algebras and K-thaory Ronald G. Douglas and Claude Schochet, Editors VOLUME 11 Plane ellipticity and related problems Robert P. Gilbert, Editor VOWME 12 Symposium on algebraic topology in honor of Jose Adam Samuel Gitler, Editor

3 Titles in this series VOLUME 13 Algebraists homage: Papers in ring theory and related topics Edited by S. A. Amitsur, D. J. Saltman and G. B. Seligman VOLUME 14 Lactures an Nielsen fixed paint theory Boju Jiang VOWME 15 Advanced analytic number theory. Part 1: Ramification theoretic methods Carlos J. Moreno VOLUME 16 Complex representations of GL(2 1 K) far finite fields K llya Piatetski-Shapiro VOLUME 17 Nonlinear partial differential equations Joel A. Smaller, Editor VOLUME 18 Fixed paints and nanexpansive mappings Robert C. Sine, Editor VOLUME 18 Proceedings of the Northwestern homotopy theory conference Haynes R. Miller and Stewart B. Priddy, Editors VOLUME 20 Law dimensional topology Samuel J. Lomonaco, Jr., Editor VOLUME 21 Topological methods in nonlinear functional analysis Edited by S. P. Singh, S. Thomaier, and B. Watson VOLUME 22 Factorizations of bn ± 1 1 b = up to high powers John Brillhart, D. H. Lehmer, J. L.Selfrldge, Bryant Tuckerman, and S. S. Wagstaff, Jr. VOLUME 23 Chapter I of Ramanujan e eecond nateboaklnfinite sariaa identitiaa 1 tranefal"llllltia, and evaluations Bruce C. Berndt and Padmini T. Joshi

4 COnTEMPORARY MATHEMATICS Volume 21 Topological Methods in Nonlinear Functional Analysis S. P. Singh 1 S. Thomaiar 1 and B. Watson 1 Editors AMERICAn MATHEMATICAL SOCIETY PI'OUIBCI RhOde ISland

5 PROCEEDINGS OF THE SPECIAL SESSION ON FIXED POINT THEORY AND APPLICATIONS 86TH SUMMER MEETING OF THE AMERICAN MATHEMATICAL SOCIETY HELD AT THE UNIVERSITY OF TORONTO TORONTO, CANADA AUGUST 21-26, Mathematics Subject Classification. Primary 47Hxx, 54H25. Library of Congress Cataloging in Publication Data Main entry under title: Topological methods in nonlinear functional analysis. (Contemporary mathematics, ISSN ; v. 21) "Proceedings of the special session on fixed point theory and applications, 86th summer meeting of the American Mathematical Society, held at the University of Toronto, Toronto, Canada"-T. p. verso. Includes bibliographies. 1. Nonlinear functional analysis-congresses. 2. Fixed point theory-congresses. I. Singh, S. P. (Sankatha Prasad), II. Thomaier, S. Ill. Watson, B., IV. Series: Contemporary mathematics (American Mathematical Society); v. 21. OA321.5.T ISBN Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island The appearance of the code on the first page of an article in this volume indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to Copyright Clearance Center, Inc. 21 Congress Street, Salem, Massachusetts This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotion purposes, for creating new collective works or for resale. Copyright 1983 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government This volume was printed directly from author prepared copy.

6 Introduction Contractors and Fixed Points M. ALTMAN CONTENTS The Degree of Mapping, and its Generalizations F.E. BROWDER 15 Multiple Fixed Points of Compact Maps on Wedgelike ANRS in Banach Spaces R.F. BROWN The Nielsen Numbers on Surfaces E.R. FADELL & S. HUSSEIN! A Good Class of Eventually Condensing Maps. 99 G. FOURNIER Iteration Process for Nonexpansive Mappings.... K. GOEBEL & W.A. KIRK The Best Approximation of Bivariate Functions by Separable Functions M. von GOLITSCHEK & E.W. CHENEY Positive Solutions of Operator Equations in the Nondifferentiable Case R. GUZZARDI 137 On Fixed Points of Nonexpansive Mappings D.S. JAGGI 147 Large Oscillations of Forced Nonlinear Differential Equations. M. MARTELLI 151 Fixed Points and Sequences of Iterates in Locally Convex Spaces S.A. NAIMPALLY, K.L. SINGH & J.H.W. WHITFIELD 159 Fixed Point Theorems and Jung Constant in Banach Spaces P.L. PAPINI 167 Some Results on Multiple Positive Fixed Points of Multivalued Condensing Maps. W.V. PETRYSHYN 171 Some Problems and Results in Fixed Point Theory S. REICH 179 Contractive Definitions Revisited. B.E. RHOADES 189 Fixed Points, Antipodal Points and Coincidences of n-acyclic Multifunctions... H. SCHIRMER 207 A Coincidence Theorem for Topological Vector Spaces V.M. SEHGAL, S.P. SINGH & B. WATSON 213 Some Random Fixed Point Theorems V.M. SEHGAL & C. WATERS 215 v vi 1

7 INTRODUCTION This volume contains the Proceedings of the Special Session on FIXED POINT THEORY AND APPLICATIONS held during the 86th Summer Meeting of the American Mathematical Society, which took place at the University of Toronto during August 21-26, The theory of contractors and contractor directions is developed and used to obtain existence theory under rather weak conditions. Theorems on the existence of fixed points of nonexpansive mappings and the convergence of the sequence of iterates of nonexpansive and quasinonexpansive mappings are given. Degree of mapping and its generalizations are given in detail. A class of eventually condensing mappings is studied and multivalued condensing mappings with multiple fixed potnts are also given. Topological fixed points including the study of the Nielsen number of a selfmap on a compact surface, extensions of a well-known result of Krasnosebskii's Compression of a Cone Theorem, are given. Also, fixed points, antipodal points, coincidences of multifunctions are discussed. Several results with applications in the field of partial differential equations are given. Application of fixed point theory in the area of Approximation Theory is also illustrated. As organizers of the Special Session we wish to thank Memorial University of Newfoundland for support and, in particular, the Department of Mathematics and Statistics, the Vice-President (Academic), Dr. E.R.W. Neale, and the Dean of the School of Graduate Studies, Dr. F.A. Aldrich. We also want to thank all those who helped before and after the conference. Our special thanks go to all participants whose contributions made this Special Session a success. As editors of the Proceedings, we wish to express our appreciation to all contributors for their help and kind cooperation. We also thank Miss Yvonne MacNeil for the typing of the manuscripts for offset printing. Finally, we thank Dr. R. James Milgram for including our Proceedings in the Contemporary Mathematics series of the American Mathematical Society and to the Editorial Staff of the Society for their cooperation. St. John's, Newfoundland April, 1983 vi S.P. Singh S. Thomeier B. Watson

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