Qualitative methods in mathematical analysis

262 Pages·1964·17.025 MB·English

by  Elsgolc L.E.

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Translations of Mathematical Monographs Volume 12
QUALITATIVE METHODS
.
In
MATHEMATICAL ANALYSIS
by
L. E. El' sgol' c
AMERICAN MATHEMATICAL SOCIETY
PROVIDENCE, RHODE ISLAND
1964
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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 DIFFER-
ENTIAL 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
iv TABLE OF CONTENTS
5. Stability of the solutions of differential equations ....... 120
6. Periodic solutions ........................... 136
CHAPTER V. DIFFERENTIAL EQUATIONS WITH DEVIATING ARGU-
MENTS ••••••••••••••••••••••••••••••• 155
1. Classification of differential equations with deviating argu
ments and establishment of the fundamental boundary-
value problem . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 155
2. The method of successive integrations (method of steps) .. 159
3. Method of successive approximations and the theorem on
existence and uniqueness ...................... 162
4. Integrable types of differential equations with retarded
argument ................................ 163
5. Approximate methods of integration of differential equa-
tions with retarded argument ................... 164
6. Dependence of the solution of a differential equation with
retarded argument on a small coefficient of the leading
derivative ............................... 168
7. Theorems on oscillations of the solution . . . . . . . . . . . . . 172
8. Linear equations ........................... 174
9. Classification of stationary points and estimates of their
number ................................. 179
10. Stability of solutions of differential equations with deviat-
ing arguments ............................. 181
11. Quasilinear equations with retarded argument ......... 200
12. Equations of neutral type. . . . . . . . . . . . . . . . . . . . . . 208
13. Equations with advanced arguments. .............. 211
14. Differential-difference equations involving partial
derivatives. . . . . . . . . . ..................... 213
CHAPTER VI. VARIATIONAL PROBLEMS WITH RETARDED ARGUMENT • 215
1. Statement of the elementary problem .............. 215
2. The fundamental lemmas ...................... 216
3. Fundamental necessity conditions for an extremum ..... 219
4. Further necessity conditions .................... 221
5. Generalization to functionals of more complicated type ... 223
6. Variational problems with moving boundaries ......... 224
7. Conditional extrema . . . . . . . . . . . . . . . . . . . . . . . . . 226
8. Direct methods ............................ 228
9. Estimates of the number of solutions of a variational
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
BIBLIOGRAPHY . . . . . . • • . . • • • • • . . • • • • . . • • • • • • • • • • • • • 230
Foreword
The purpose of the present book "Qualitative methods in mathematical
analysis" is to introduce the reader to the ideas which are characteristic
for qualitative methods and to throw some light on the details of certain
new questions not yet fully developed.
The author, desiring to present to the reader material of a very wide
and diverse character, was obliged to report many results without proof or
with a mere outline of the proof. In these cases he has given references
to the literature.
Chapters I, II and in part Chapter III require from the reader a knowl
edge of the basic elements of topology. Chapters IV, V, and VI, with the
exclusion of one section in each chapter, do not suppose any acquaintance
with topology and do not require that Chapters I, II, and III be studied
first.
Several questions are presented in this book for the first time. Many
other questions have appeared up till now only in scientific papers. Thus
the first more or less systematic exposition of this material will necessarily
contain a number of lacunae. The author nevertheless hopes that in spite
of its deficiencies this book will enable a wide circle of mathematicians to
acquaint themselves with the fundamental ideas and some of the problems
in qualitative methods in analysis.

Introduction
Qualitative methods in mathematics are methods which make it possible
in the absence of a quantitative solution of a mathematical problem to
indicate a number of qualitative properties of the desired solution.
Sometimes the qualitative analysis of a mathematical problem is only
the first step of an investigation, in which one proves the existence of a
solution, estimates the number of solutions, and establishes some peculi
arities of the solutions, thus facilitating in the future their exact or
approximate solution.
However, one not infrequently has to deal with problems in which the
question is from the beginning purely qualitative, and the finding of an
exact or approximate solution of the equations of the problem does not
make it possible to answer the question as posed, and frequently does not
even help in finding the solution of that question.
We present a number of examples of problems which are solved by
qualitative methods:
(1) Given a differential equation, to determine whether its solutions are
stable, whether it has periodic solutions, whether its solutions oscillate,
whether it has singular points, and if so, what are their types.
(2) To estimate the number of solutions of a given problem of finding
extreme values of a function or of a functional, in terms of the properties
of the space on which this function or functional is defined.
(3) Without calculating the roots of the equation f (z) = 0, to indicate
various peculiarities of their distribution, for example to determine
whether the real parts of all the roots of this equation are negative or not.
(4) To estimate the number of singular points of a function f(z) which
is analytic at every point of a region D with the exception of a finite set
of singular points, in terms of the properties of the region D.
(5) To prove the existence of solutions of a given equation and to esti
mate their number.
Particular questions of qualitative analysis have been encountered for a
long time. However, the methods of qualitative analysis have become
widespread only in the twentieth century, and are nowadays applied in ali
branches of mathematics.
Qualitative methods have been developed considerably in the theory
of differential equations, m extremal problems, and in the proof of
existence theorems.
vii