Coincidence Degree, and Nonlinear Differential Equations

266 Pages·1977·4.65 MB·English

by  Robert E. Gaines, Jean L. Mawhin (auth.)

Lecture Notes ni
Mathematics
Edited yb .A Dold dna .B Eckmann
568
Robert .E Gaines
Jean .L Mawhin
Coincidence Degree, and
Nonlinear Differential Equations
Springer-Verlag
Berlin.Heidelberg • New York 7791
Authors
Robert E. Gaines
Colorado State University
Department of Mathematics
Fort Collins
Colorado 80523/USA
Jean L. Mawhin
Universit@ Catholique de Louvain
Institut Math@matique
B-1348 Louvain-la-Neuve/Belgium
Library of Congress Cataloging in Publication Data
Gaines, Robert E 1941-
Coincidence degree, and non-
linear differential equations.
(Lecture notes in mathematics ; 568)
Includes bibliographical references and index.
i. Differential equations~ Nonlinear. 2. Bound-
ary value problems. o5 Coincidence theory (Mathe-
matics) I. Mawhin~ J., joint author. II. Title.
III. Series: Lecture notes in mathematics (Berlin)
568.
QA3.L28 no. 568 [QA372] 510'.8s [515'.35] 76-58459
AMS Subject Classifications (1970): 34 B15, 34 K10, 35J 65, 47 H ,51 55C20
ISBN 3-540-08067-8 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-08067-8 Springer-Verlag New York • Heidelberg - Berlin
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TABLE OF CONTENTS
I. Introduction
II. Alternative problems : an historical perspective
III. Coincidence degree for perturbations of Fredholm
mappings 10
IV. A generalized continuation theorem and existence
theorems for Lx = Nx 26
V. Two-point boundary value problems : nonlinearities
without special structure 36
VI. Approximation of solutions - The projection method 104
VII. Quasibounded perturbations of Fredholm mappings 134
VIII. Boundary value problems for some semilinear elliptic
partial differential equations 151
IX. Periodic solutions of ordinary differential equations
with quasibo~nded nonlinearities and of functional
differential equations 166
X. Coincidence index, multiplicity and bifurcation
theory 189
XI. Coincidence degree for k-set contractive perturbations
of linear Fredholm mappings 209
XII. Nonlinear perturbations of Fredholm mappings of
nonzero index 922
References 242
Index 261
.I INTRODUCTION
This work has its origin in lectures given by J. Mawhin in 1974 at the
University of Brasilia and by R.E. Gaines in 1975 at the University of
Louvain. Those lectures respectively covered chapters II to IV, VII to IX
and chapters V-VI. Chapters X to XII have been added to include more
recent material.
The emphasis of the work is on the use of topological degree techniques
in studying alternative problems , i.e. problems which can be written as
operator equations of the form
(1.1) ~ = ~x
in a suitable abstract space, with L linear and non-invertible. As shown
in chapter II many techniques have been developed to handle equations of
the form (1.1) and research in this field is still very active.
A principal aim of these lecture notes is to show that by proving once
and for all, for coupled mappings (L,N) satisfying certain conditions,
a number of properties quite similar to those of Leray-Schauder degree,
one is able to study many problems of type (1.1) in an unified way.
This is the so-called coincidence degree theory which is described in chapters
III and leads in chapter IV to general useful coincidence theorems for L
and N, i.e. existence theorems for (1.1) and in particular to a continuation
theorem of Leray-Schauder type.
The applicability of those theorems, as in any degree theory, depends
upon the obtention of a priori bounds for the solutions of the equation.
Chapter V consider the problem of a priori bounds in the case of boundary
value problems, includin~ period~g solutions, for ordinary differential
equations. The emphasis is placed on the determination of a priori estimates
through the geometric properties of the vector field defined by the dif-
ferential equation. The covered material, which includes lower and upper
solutions, differential inequalities, Nagumo conditions, Gustafson-Schmitt-
Bebernes convexity conditions, Krasnosel'skii's guiding functions, ... is
generally classical but the approach is original in several places. In
chapter VI we consider the problem of the approximation of the solutions of
(1.1) by Galerkin-type methods when existence has been proved by the
techniques of Chapter IV. The speciTic case of linear boundary value
problems for nonlinear ordinary differential equations is treated in detail.
In chapter VII we return to abstract equations (1.1) where N ~rows at
most at a linear rate in x. A unified treatment is given of generalized
versions of recent results in this domain due to Cronin, De Figueiredo,
Fabry, Franchetti~ Fu~ik~ Ku~era, NeOns, .... Those results are applied
to semilinear elliptic partial differential equations in chapter VIII
which covers in a systematic way most of the recent contributions in the
line of Landesman and Lazer's pioneering work. In chapter IX we study
periodic solutions for ordinary and functional differential equations
emphasizing nonlinearities with a growth at most linear. However, in the
case of functional differential equations, techniques reminiscent of the
ones of chapter V are also considered.
Chapter X is a short description of the use of coincidence degree in
bifurcation theory. One obtains ~n a more general setting a local
Krasnosel'skii type theory which facilitates corresponding extensions of
the Rabinowitz global results. In chapter XI we briefly describe Hetzer's
extension of coincidence degree to the case of nonlinear perturbations
havin$ k-set contraction pro~ertie~ instead of compactness. Applications
are given to generalizations of results of Kacurovskii, Petryshyn and
Amann. When N is Lipschitzian with a sufficiently small Lipschitz constant,
which is the situation of classical Liapunov-Schmidt method , one then
relates the coincidence degree of L and N to the Brouwer degree of the
mapping associated with the classical bifurcation equation.
In all the above chapters we essentially assume that the Fre~olm
index of L is zero. Chapter XII treats cases where the Fre~holm index
of L is strictly positive in the line of the recent work of Nirenberg,
Rabinowitz and Schechter.
Each of these chapters is followed by bibliographical notes refering
to the original papers and giving sugestions for further reading. All the
references indicated in those bibliographical notes are given in a more
complete setting in the list of references at the end of the volume. This
list moreover contains references to recent papers which are not explicitely
described in this work but which are close in spirit or results. The
chapter(s) to which those papers are related is then indicated.
In preparing this set of notes we have tried, after developing the
general theory, to take our examples more in related papers appearing
in the literature then in our own work. Most of our work is already
written in the spirit of this monograph so that we have found a duplication
unnecessary. In this way we hope to have facilitated the access to this
part of nonlinear f~ctional analysis and nonlinear boundary value
problems. Our expectations will be fulfilled if this work can suggest
further research in the wide area of alternative problems, nonlinear
differential equations and applications to science and engineering.
II. ALTERNATIVE PROBLEMS : AN HISTORICAL PERSPECTIVE
]. The study of differential or integral or other
equations which, written in operator form, are of the type
Lx = Nx ~I. ) I
with L (resp. N) a linear (resp. nonlinear) mapping between
some topological vector spaces X and Z, and with L non inver-
tihle seems to have been initiated by Lyapunov (Zap. Akad. Nauk.
St. Petersbour~ (1906, ~908, 1912, 1914) in his study of integral
equations related to a problem of equilibrium of rotating fluids,
and by E. Schmidt (Math. Ann~_ 6~ (1908) 370-399) in his theoreti-
cal work on nonlinear integral equations. Written in abstract set-
ting their method consists basically to write (11.1) in the equiva-
lent form
Lx = (I - Q)Nx, QNx = 0
(notations are those of Chapterlll for the mappings associated to
L)
or
x - Px = Kp,QNX, QNx = 0 .
Now if we set
Px = y, (I - P)x = z
we obtain the equivalent system
z = Kp, Q~(y+z), Q~(y+z) = 0. (II.2)
Now if some "smallness" and "regularly conditions are assumed
for N, the first equation in (~.2), considered like an equation
in z depending upon the parameter y, will be solved using Banach
fixed point theorem or implicit function theorem to give a solu-
tion z(y) which will depend "regularly" upon y. Hence the solu-
tion of (11.2) will be reduced to the solution of equation (in y)
Q~[y + z(y)] = 0, 3) (IL
usually known as bifurcation or branch in~ or determining equation.
The interest of the method is that the spaces in which the left
hand member of (11.3) is defined and take values have been made
"smaller" (often one goes from infinite into finite dimensional
spaces). But, due to the presence 'fo the term z(y), (~.3) is not
usually known in explicit form. This methods is generally known
as Lyapunov-Schmidt method and diversification occurs in the way
of getting and of studying the bifurcation equations. Let us quote
in this line the basic works of Cesari (Atti Accad. Mem. CI. Fis.
Mat.Nat. (6) 11(1940) 633-692), Caccioppoli (Atti Accad. Naz. Lincei
Rend. CI. Sci. Fis. Mat. Natur. 24(1936) 258-268, 416- 421), Shi-
mizu (Math. Japan. I (1948) 36-40) Cronin (Trans. Amer. Math. Soc.
69 (1950) 208-231), Bartle (Trans. Amer. Math. Soc. 75 (1953) 366-
384), Hale (Riv. Mat. Univ. Parma 5 (1954) 281-311) Lewis (Ann. of
Math. 63 (1956) 535-548), Vainberg and Trenogin (Russ. Math. Sur-
veys 17 (1962) 1-60), Antosiewicz (Pacif. J. Math. 17 (1966), 191-
197).
Expositions can be found in the books or lecture notes of Frie-
drichs ("Special Topics in Analysis", Lect. Notes, New York uni-
versity, 1953-54), Nirenberg ("Functional Analysis", Lect. Notes,
New York University, 1960-61), Krasnoselskii et al. ("Topological
Methods in the Theory of Nonlinear Integral Equations", Pergamon,
1963, "Approximate Solutions of Operator Equations", Noordhoff,
1971), Vainberg and Aizengendler ("Progress in Math.", vol. II,
Plenum, 1968), Hale ( Ordlnary Differential Equations", Wiley,
1969; "Applications of Alternative Problems", Brown University
Lect. Notes 71-I, 1971), Mawhin ("Equations non-lin@aires dans
les espaces de Banach", Rapp. S@m. Math. Univ. Louvain ° n 39, 1971).
All the above quoted papers (except the notes by Nirenberg,
Hale and Mawhin) correspond to problems in which the bifurcation
equations are finite-dimensional. The first solved example leading
to infinite-dimensional bifurcation equations seems to be the
problem of periodic solutions of some weakly nonlinear hyperbolic
equations initiated by Cesari ("Nonlinear differential equations
and nonlinear mechanics", 1963, Academic Press, 33-57). An account
and a bibliography about this work can be found in the lectures
notes of Hale quoted above.
2. This problem of perturbed hyperbolic equations has
led Rabinowitz (Comm. Pure Appl. Math. 20(1967), 145-206) more
than fifty years after Lyapunov and Schmidt, to an alternate way
of considering (ll.2)'First solve the second equation in (~2)
considered as an equation in y with z as parameter. Introduce then
the solution y(z), supposed to be sufficiently "regular" in z, in
the first equation of (0.2). Then solve the resulting equation
z = Kp,QN[y(z) + z] (11.4)
using Banach or Schauder fixed point theorem. Usualy, monotone
operator theory is used to solve the first part of the problem
and interesting results using this approach have been obtained
in the theory of periodic solutions of weakly nonlinear equations
by Hall (J. Differential Equ. 7(1970) 509-526; Arch. Rat. Mech.
Anal. 39 (1970), 294-332) ~ de Simon and Torelli (Rend. Sem. Mat.
Univ. Padova 40(1968), 380-401) and the method has been theoreti-
cally improved by Hall (Trans. Am. Math. Soc. 161C1971) 207-218)
and Sova (Comment. Math. Univ. Carolin. 1972). See also the work
of Klingelhofer (Arch. Rat. Mech. Anal. 1970) on some elliptic
equations.
3. A third possible approach for system 61.2) is to try
to solve the two equations simultaneously. A corresponding ite-
rative process has been introduced, for periodic solutions of
ordinary differential equations, by Banfi (Atti Accad. Sci. Torino