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,, L s '" L " Dill! ' ' U: 1 uc ~ .. ~· " ., I, ) ti! I ~ " ,, ~ ff 8 I I • ,,• II Ill p. ~ 'l•l .~ jl Mathematical Analysis I " ' '' Translations of Mathematical Monographs Volume 12 QUALITATIVE METHODS . In MATHEMATICAL ANALYSIS by L. E. El' sgol' c AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1964 KAqECTBEHHhlE METO~hl B MATEMATMqECKOM AHAJH13E JI. 3. 3JibcrOJibU rocyAapCTBeHHOe M3AaTeHbCTBO Texa11Ko-TeopeT11qecKo.i1 Jl11TepaTypb1 MocKsa 1955 Translated from the Russian by A. A. Brown and J.M. Danskin Publication aided by grant NSF-GN 57 from the NATIONAL SCIENCE FOUNDATION Text composed on Photon, partly subsidized by NSF Grant G21913 Library of Congress Card Number 64-16170 Copyright © 1964 by the American Mathematical Society All rights reserved. No portion of this book may be reproduced without the written permission of the publisher. Printed in the United States of America Table of Contents FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER I. QUALITATIVE METHODS IN EXTREMAL PROBLEMS . . . . . . . 1 1. Fundamental method of estimation of the number of critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1· 2. Estimate of the number of analytically distinct critical points. . 5 3. Estimate of the number of geometrically distinct critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4. Changes in the topological properties of level surfaces . . . . . 31 5. Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6. The minimun-maximum principle and its generalization . . . . 38 7. Some generalizations in finite-dimensional space . . . . . . . . . 45 8. Generalization to the infinite case . . . . . . . . . . . . . . . . . . 51 CHAPTER II. QUALITATIVE METHODS IN THE THEORY OF FUNCTIONS OF COMPLEX VARIABLES . . . . . . . . . . . . . . . . . . . . 55 1. Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . 55 2. Interdependences among the zeros, critical points, and poles of a meromorphic function. . . . . . . . . . . . . . . . . . . . . . . 58 3. Functions of several complex variables . . . . . . . . . . . . . . . 62 CHAPTER III. THE FIXED POINT METHOD . . . . . . . . . . . . . . . . . . 66 1. Theorems on fixed points . . . . . . . . . . . . . . . . . . . . . . . 66 2. Some applications of fixed point theorems . . . . . . . . . . . . . 73 3. Theorems on fixed points using invariants of category type . . 78 CHAPTER IV. QUALITATIVE METHODS IN THE THEORY OF DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . 81 1. Estimates of the number of stationary points . . . . . . . . . . . 81 2. Dependence of the solutions on small coefficients of the highest derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3. Some asymptotic properties of the solutions of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4. Dynamical systems with integral invariants . . . . . . . . . . . 115 iii TABLE OF CONTENTS iv 5. Stability of the solutions of differential equations . . . . . . . 120 6. Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 CHAPTER V. DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS • • • • • • • • • •