Table Of ContentGrundlehren der mathematischen Wissenschaften 263
A Series of Comprehensive Studies in Mathematics
M.A. Krasnosel'skfi
P.P. Zabrelko
Geometrical Methods
of Nonlinear Analysis
Springer-Verlag Berlin Heidelberg NewYork Tokyo
Grundlehren der
mathematischen Wissenschaften 263
A Series of Comprehensive Studies in Mathematics
Editors
M. Artin S.S. Chern J.M. Frohlich A. Grothendieck
E. Heinz H. Hironaka F. Hirzebruch L. Hormander
S. Mac Lane W. Magnus C.C. Moore J.K. Moser
M. Nagata W. Schmidt D.S. Scott J. Tits
B.L. van der Waerden M. Waldschmidt S. Watanabe
Managing Editors
M. Berger B. Eckmann S.R.S. Varadhan
M.A. Krasnosel'skiI P.P. Zabreiko
Geometrical Methods
of Nonlinear Analysis
Translated from the Russian
by Christian C. Fenske
Springer-Verlag
Berlin Heidelberg NewYork Tokyo
1984
Mark Aleksandrovic Krasnosel'skiI
Department of Mathematics
Voronez State University
Vo ronez, USSR
Petr Petrovic Zabre'fko
Department of Mathematics
Belorussian State University
Minsk, USSR
Christian C. Fenske
Mathematisches Institut
der Justus-Liebig-Universitat
ArndtstraBe 2
D-6300 GieBen
Title of the Russian original edition:
Geometriceskie metody nelineinogo analiza
Publisher Nauka, Moscow 1975
AMS Subject Classification (1980): 47H15, 54H25, 55M25, 58C30
ISBN 3-540-12945-6
Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-12945-6
Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data
Krasnosel'skil, M.A. (Mark Aleksandrovich), 1920-
Geometrical methods of nonlinear analysis.
(Grundlehren der mathematischen Wissenschaften; 263)
Bibliography: p. I. Operator equations, Nonlinear - Numerical solutions.
I. Zabreiko, P. P. (Petr Petrovich), 1939-
11. Title. III. Series.
QA329.8.K713 1984 515.7'248 83-16986
ISBN 0-387-12945-6 (U.S.)
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Preface to the English Edition
Geometrical (in particular, topological) methods in nonlinear analysis were
originally invented by Banach, Birkhoff, Kellogg, Schauder, Leray, and others
in existence proofs. Since about the fifties, these methods turned out to be
essentially the sole approach to a variety of new problems: the investigation
of iteration processes and other procedures in numerical analysis, in bifur
cation problems and branching of solutions, estimates on the number of
solutions and criteria for the existence of nonzero solutions, the analysis of
the structure of the solution set, etc. These methods have been widely
applied to the theory of forced vibrations and auto-oscillations, to various
problems in the theory of elasticity and fluid mechanics, to control theory,
theoretical physics, and various parts of mathematics.
At present, nonlinear analysis along with its geometrical, topological,
analytical, variational, and other methods is developing tremendously
thanks to research work in many countries. Totally new ideas have been
advanced, difficult problems have been solved, and new applications have
been indicated. To enumerate the publications of the last few years one
would need dozens of pages. On the other hand, many problems of non
linear analysis are still far from a solution (problems arising from the
internal development of mathematics and, in particular, problems arising in
the process of interpreting new problems in the natural sciences).
We hope that the English edition of our book will contribute to the
further propagation of the ideas of nonlinear analysis.
Finally, we take the opportunity to express our sincere thanks to Pro
~essor Fenske for his efforts in translating the book and eliminating several
maccuracies.
The authors
Preface
This book is intended for mathematicians and mechanical engineers in
terested in the qualitative behaviour of operator equations (existence theo
rems, estimates on the number of solutions, nontrivial solutions, connec
tivity of the solution set, approximation schemes for the construction of
solutions, dependence of solutions on parameters, etc.). We use methods
which are derived from geometric ideas: rotation of vector fields, fixed point
theory, concavity or monotonicity of nonlinearities, etc. Theorems and prin
ciples will be illustrated by applying them for instance to the theory of
nonlinear vibrations, boundary value problems, nonlinear integral equations,
nonlinear mechanical problems, and critical values of functionals.
In physical applications, one is led to investigate not only single opera
tor equations; one is rather interested in general classes of operators which
- in a natural way - permit certain modifications of the equation, e.g.,
continuous deformation, or perturbation of constituents of the equation
(integral kernels, right hand sides of differential equations, boundary con
ditions, or parameters). As a rule, the only mathematical theories that are of
practical interest are those stable under small perturbations of equations.
Therefore, one should try first to investigate those properties of an equation
which are preserved under small perturbations. After this is done, one may
use the information obtained to solve the given concrete question. This is a
well-known procedure in mathematics. It is especially useful when the
quantities in question may be effectively computed or estimated and, if it is
possible, to formulate, in terms of these quantities, problems that are suf
ficiently interesting. In fact, it is just this situation that has developed in the
theory of large classes of nonlinear operator equations. This enables us to
adopt a common approach to many problems which at first glance look
completely different.
The methods we deal with in the present book originated long ago. They
date back to Kronecker, Poincare, Brouwer, and Hopf who developed the
topological theory of continuous mappings in finite dimensional spaces. A
second stream of ideas originated from the investigations of Birkhoff and
Kellogg, Banach, Cacciopoli, Schauder, and A.N. Tihonov on fixed point
theory. A major contribution was supplied by the work of A.M. Ljapunov,
E. Schmidt, A.I. Nekrasov, P.S. Uryson, Hammerstein, and Golomb on the
theory of various classes of integral equations as well as by L.A. Ljusternik's
work on variational problems. The contributions by N.N. Bogoljubov and
N.M. Krylov, M.V. Keldys, G.I. Petrov, and A.I. Lur'e to projection meth
ods for approximate solutions of equations particularly influenced the de-
Preface VII
velopment of methods in nonlinear operator equations. These methods are
fundamental to many schemes of construction of topological invariants for
mappings in infinite dimensional spaces. In the course of constructing these
methods, ideas from L.V. Kantorovic's approximation theory proved very
useful. The last fifteen to twenty years, finally, brought novel approaches
and solutions to new problems.
In our time, qualitative geometric methods for the investigation of non
linear operator equations are forming a well established theory with many
applications to integral equations, various boundary value problems, and
problems from physics and mechanics.
In the first chapter we study vector fields in finite dimensional spaces.
Here, the notion of rotation of a vector field on the boundary of a domain
turns out to be an important concept. A detailed discussion of this concept
can be found in many textbooks on topology. In our context, the main
emphasis will be on theorems and methods allowing us to effectively com
pute or estimate the rotation of concrete vector fields. In order to under
stand the first chapter, the reader is not assumed to have any special
knowledge of topological facts. This is because in the main constructions,
we shall use only three main properties of rotation which are easy to
formulate.
Rotation for various classes of vector fields in function spaces is dis
cussed in Chapters 2 and 4. The most detailed description of the theory of
completely continuous vector fields has originally been given by Leray and
Schauder. We shall also consider various other classes of fields involving
Fredholm operators or operators which are positive, condensing, or multi
valued or which have other special properties. The main emphasis will
always be on the problem of how to actually compute or estimate the
rotation.
To compute or estimate rotations of vector fields, one uses theorems on
relatedness or invariance of rotation. Chapter 3 is devoted to this topic.
Theorems on relatedness or invariance of rotation contain formulae relating
rotations of different vector fields (on the boundaries of the respective
domains) describing one and the same problem in different function spaces.
The idea underlying the theorem on relatedness may, for instance, be
illustrated by problems on forced vibrations in systems described by or
dinary differential equations. The initial values of periodic solutions are then
singular points of the vector field of translation (by the period) along
trajectories of the vector field. These fields are defined in finite dimensional
phase space. Although the fields are not explicitly known they can be
analyzed if one uses well established methods from the qualitative theory of
differential equations. On the other hand, the existence problem for periodic
solutions can be transformed into a problem in nonlinear integral equa
tions; periodic solutions then correspond to singular points of a vector field
in some function space with a completely continuous integral operator given
by an explicit formula. In the problem on forced vibrations, the theorem on
relatedness provides a formula relating the rotation of the vector field of
VIII Preface
translation in phase space with the rotation of the field of the corresponding
integral operator on the boundary of an appropriate domain in a function
space. The theorems on relatedness and invariance of the rotation allow us
to apply general principles developed for abstract operator equations to
concrete problems.
In Chapter 5, we develop geometric methods for existence proofs. Select
ing the material for this chapter was a particularly difficult task. In doing
so, the authors have restricted themselves to theorems which are manifestly
important for applications to concrete problems and to results which are
strikingly surprising and beautiful. Many important principles which are
dealt with in detailed expository articles, in monographs, or textbooks are
just mentioned.
Special emphasis will be given to fixed point principles for mappings
leaving invariant an appropriate set - the most elementary examples being
the Banach contraction theorem and the Schauder-Tihonov theorem.
Among other things, we study equations involving nonexpansive, condens
ing, limit-compact, set-valued, and discontinuous operators. The main tool
will be the principle of nonvanishing rotation: if the rotation of a vector
field on the boundary of an open set is different from zero, then the field
must have at least one singular point. We show how to apply this principle
along with the principles of relatedness and invariance of rotation to bound
ary value problems and integral equations. We explain the importance of
one-sided estimates in existence theorems: here, important methods are
based on Schaefer's elegant (though scarcely known) contributions to non
linear integral equations. Special criteria for the solvability of operator
equations are provided by so-called covering principles, the importance of
which has been emphasized by S.I. Pohofaev. We also develop methods for
the investigation of equations with dissipative operators (including appli
cations to the theory of Levinson-Pliss dissipative systems of ordinary
differential equations).
In Chapter 6 we give estimates for the number of solutions for equations
with nonlinear operators, and we obtain various principles for the existence
of nontrivial solutions. We emphasize that it is the nontrivial solutions
which are of particular importance in many nonlinear problems (e.g., the
theory of waves, auto-oscillations, or forms of loss of stability in elastic
systems). The material in this chapter uses four main methods. The first one
is the theory of rotation for vector fields which is used to obtain estimates
for the number of solutions for equations. The second method is the theory
of partially ordered spaces. We apply it to nonlinear equations with a
dominant nonlinearity - the nonlinearities we have in mind are step func
tions. (The main result will be that, if there is a sequence of domains such
that the nonlinearity alternates between faster and slower growth on these
domains, then there are as many solutions as there are domains.) Here, we
also explain the importance of concavity and convexity of nonlinearities.
The third method applies to various nonlocal theorems on the solution of
parameter-dependent equations. The fourth method involves connectivity
Preface IX
properties of the solution set. A paradigmatic application is the following
theorem: if a mixed boundary value problem for a quasilinear parabolic
equation has two different solutions, then there must be a continuum of
solutions.
Chapter 7 is a short one. Here, we study approximation methods for the
solution of nonlinear operator equations. We establish a connection be
tween convergence of the method of successive approximations and the
topological index of the solution. Many approximation methods (e.g., the
method of Galerkin-Petrov, the least square method, or the Ljapunov
Cesari method) are based on exact solutions of the approximating equa
tions; we show that the solutions of the approximating equations converge
to the exact solution of the original equation if the rotation of the corre
sponding vector field is different from zero. This is an important fact: it
allows us to translate conditions in existence theorems into conditions
which entail the applicability of various approximation methods. We de
scribe an algorithm that produces a posteriori estimates, and which is
applicable to problems where the linearized equation is only poorly re
flected in the approximating equations. There is a close connection between
convergence of the method of successive approximations and stability of
equilibria for dynamical systems. In Chapter 7, we discuss the following
problem: consider a system possessing an isolated equilibrium state which is
Ljapunov-stable. Is it possible to continuously deform the system so that it
reaches an equilibrium state which is asymptotically stable? We shall see
that in general the answer is no.
In the last chapter, we discuss various problems that are connected with
small perturbations of operator equations. The classical implicit function
theorem and the Ljapunov-Schmidt procedure for bifurcation equations are
just briefly mentioned, as these topics are dealt with in many textbooks. We
discuss a connection between the bifurcation theory for equations with
many scalar parameters and the homotopy classification of mappings on a
sphere into a sphere of lower dimension. The concept of "functionalizing
the parameter" will allow us to study perturbations of equations with
continua of solutions by applying the theory of rotation for vector fields. In
order to give an application, we consider periodic solutions of differential
equations with a retarded argument. We also study geometric methods for
the investigation of bifurcation points for the parameter. These methods
require little information on the equations (in many cases, the linearized
equation is sufficient). Finally, we analyze the stability of critical values of
functionals.
The reader is assumed to have a basic knowledge of functional analysis
and topology.
The main part of the book (almost all subsections in §§ 1-8, 10, 11, 15-
26, 31-36, and some items in §42) have been written jointly by both
~uthors, and the remaining part is due to M.A. Krasnosel'skil In compos
IIlg several items and subsections (and in some cases even entire sections) we
have been supported by N.A. Bobylev, Ju.G. Borisovic, V.M. Gerstein, V.S.
