Topological Methods in Nonlinear

Functional Analysis

AMERICAn MATHEMATICAL SOCIETY u 0 L u m E 21

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COnTEMPORARY MATHEMATICS

Volume 21

Topological Methods in Nonlinear

Functional Analysis

S. P. Singh1 S. Thomaiar1

and B. Watson1 Editors

AMERICAn MATHEMATICAL SOCIETY PI'OUIBCI • RhOde ISland

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 21-26,1982

1980 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 0271-4132; 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), 1937- . II. Thomaier, S. Ill. Watson, B., 1946- IV. Series: Contemporary mathematics (American Mathematical Society); v. 21. OA321.5.T66 1983 515.7 83-11824 ISBN 0-8218-5023-7

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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.

Introduction Contractors and Fixed Points

M. ALTMAN

CONTENTS

vi 1

The Degree of Mapping, and its Generalizations • 15 F.E. BROWDER

Multiple Fixed Points of Compact Maps on Wedgelike ANRS in Banach Spaces . . • . . . . • • • • • • . . • • • • . • . ••• 41 R.F. BROWN

The Nielsen Numbers on Surfaces • • • • • • • . . • . • • • • • • • 59 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

115

• 125

Case • . • • • • • • . • • • . . • • • . • • • • . . • • . • 137 R. GUZZARDI

On Fixed Points of Nonexpansive Mappings • . . • . . . . . . . • 147 D.S. JAGGI

Large Oscillations of Forced Nonlinear Differential Equations. • 151 M. MARTELLI

Fixed Points and Sequences of Iterates in Locally Convex Spaces • 159 S.A. NAIMPALLY, K.L. SINGH & J.H.W. WHITFIELD

Fixed Point Theorems and Jung Constant in Banach Spaces • • • 167 P.L. PAPINI

Some Results on Multiple Positive Fixed Points of Multivalued Condensing Maps • • • • • • • • • • • • • • • • • . • • • 171 W.V. PETRYSHYN

Some Problems and Results in Fixed Point Theory • 179 S. REICH

Contractive Definitions Revisited • . • • • • • • 189 B.E. RHOADES

Fixed Points, Antipodal Points and Coincidences of n-Acyclic Multifunctions • . . • • • • . • • • • • • • • 207 H. SCHIRMER

A Coincidence Theorem for Topological Vector Spaces 213 V.M. SEHGAL, S.P. SINGH & B. WATSON

Some Random Fixed Point Theorems V.M. SEHGAL & C. WATERS

v

215

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, 1982.

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 quasi-nonexpansive 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