Lecture Notes in Mathematics

Transcription

1 i i Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich 104 George H. Pimbley, Jr. University of California Los Alamos Scientific Laboratory, Los Alamos, New Mexico Eigenfunction Branches of Nonlinear Operators, and their Bifurcations $ Springer-Verlag Berlin-Heidelberg- New York 1969

2 Work performed under the auspices of the U. S. Atomic Energy Commission All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer by Springer.Verlag Berlin" Heidelberg 1969 Library of Congress Catalog Card Number Printed in Germany. Title No. 3710

3 -1 - TABLE OF CONT~VgS Introduction... I. An Example The Extension of Branches of Solutions for Nonlinear Equations in Ban~ch Sl~ces Development of Branches of Solutions for Nonlinear Equations near an Exceptional Point. Bifurcation Theory Solution of the Bifurcation Equation in the Case n = i; Bifurcation at the Origin.... The Eigenvalue Problem; Hammerstein Operators; Sublinear and Superllnesr Operators; Oscillation Kernels On the Extension of Branches of Eigenfunctions; Conditions Preventing Secondary BifUrcation of Brancke~... ~ Extension of Branches of Eigenfumctions of Hammerstein Operators ii O 8. The Example of Section l, Reconsidered A Two-Point Boundary Value Problem... i0. S,mm~y; Collection of Hypotheses; Unsettled Questions.. Bibliogral~hy.... Additional References... Appendix: Another Bifurcation Method; the Example of Section i~ Reconsidered Again if4 Zl6 120

4 -2- INTRODUCTION The series of lectures on nonlinear operators covered by these lecture notes was given at the Battelle Memorial Institute Advanced Studies Center in Geneva, Switzerland during the period June 27 - August 5, 1968, at the invitation of Dr. Norman W. Bazley of the Battelle Research Center in Geneva. The material is taken from the results of approximately seven years of work on the ~rt of the author at the Los Alamos Scientific Laboratory of the University of California, Los Alamos, New Mexico. Much of this material had previously been published in the open literature (see the Bibliography). This effort was generated by the need for a nonlinear theory observed in connection with actual problems in physics at Los Alamos. In deriving nonlinear theory, abstract formulation is perhaps a desired end; but in the newer Im~rts of the theory, as with secondary bii%xrcation in these notes, progress seems to be made more easily with concrete assumptions, as with our preoccul~tion with HA~merstein operators with oscillation kernels. The entire lecture series had to do with the eigenvalue problem kx = T(x), where T(x) is a bounded nonlinear operator. Other authors, with a view to applications in nonlinear differential equations with appropriate use of Sobolev Sl~Ces to render the operators bounded, have preferred to study eigenvalue problems of the form (LI+NI)U = k(l2+n2)u, where LI, L 2 are linear and NI, N 2 are nonlinear. Such is the case with M. S. Berger [ref. 4]. In these notes we had the less ambitious goal of understanding nonlinear integral equations, whence we concentrated on the

5 - 3 - simpler form AX = T(x}. In the more abstract part of the notes, X is any Banach space with real scalars. When we take up Hammerstein operators, the space is the set C(O,l} of continuous real functions xes} defined on the interval [0,1] with norm!1xll = sup Ix(s}l. When inner products are in evidence and when O S s ~ l variational principles are used to study eigenvalues, it will be clear that the space L 2 (0,1} of functions square integrable on the interval [0,1] is in the background, and we are using the setwise embedding The author wishes to thank Dr. Norman W. Bazley and Dr. Bruno Zwahlen of the Battelle Research Center, Geneva, for the opportunity to visit and present these lectures. He wishes to thank them also for many helpful suggestions. Thanks for good suggestions is also owed to Prof. Melvyn S. Berger, University of Minnesota; Prof. David W. Dean of Ohio state University, Prof. William G. Faris of Cornell University, Prof. Karl E. Gustafson of the University of Colorado, Prof. K. Kirchgassner of Friburg University, and Prof. Joachim Weidmann of the University of Munich. Appreciation is accorded also to Mrs. Margaret A. Gore of Los Alamos whose skill in typing a technical manuscript is amply demonstrated herein, and to Mrs. Barbara C. Powell for the various drawings. The attention of the reader is directed to the summary of results in Section 10.