Handbook of differential equations ordinary differential equations volume 3

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H ANDBOOK
OF D IFFERENTIAL E QUATIONS
O RDINARY D IFFERENTIAL E QUATIONS
VOLUME III


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H ANDBOOK
OF D IFFERENTIAL E QUATIONS
O RDINARY D IFFERENTIAL
E QUATIONS
VOLUME III
Edited by

A. CAÑADA
Department of Mathematical Analysis, Faculty of Sciences,
University of Granada, Granada, Spain

P. DRÁBEK
Department of Mathematics, Faculty of Applied Sciences,
University of West Bohemia, Pilsen, Czech Republic

A. FONDA
Department of Mathematical Sciences, Faculty of Sciences,
University of Trieste, Trieste, Italy

Amsterdam • Boston • Heidelberg • London • New York • Oxford
Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo


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First edition 2006
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Preface
This is the third volume in a series devoted to self contained and up-to-date surveys in the
theory of ordinary differential equations, written by leading researchers in the area. All
contributors have made an additional effort to achieve readability for mathematicians and
scientists from other related fields, in order to make the chapters of the volume accessible
to a wide audience. These ideas faithfully reflect the spirit of this multi-volume and the
editors hope that it will become very useful for research, learning and teaching. We express
our deepest gratitude to all contributors to this volume for their clearly written and elegant
articles.
This volume consists of seven chapters covering a variety of problems in ordinary differential equations. Both, pure mathematical research and real word applications are reflected
pretty well by the contributions to this volume. They are presented in alphabetical order
according to the name of the first author. The paper by Andres provides a comprehensive
survey on topological methods based on topological index, Lefschetz and Nielsen numbers. Both single and multivalued cases are investigated. Ordinary differential equations
are studied both on finite and infinite dimensions, and also on compact and noncompact
intervals. There are derived existence and multiplicity results. Topological structures of
solution sets are investigated as well. The paper by Bonheure and Sanchez is dedicated

to show how variational methods have been used in the last 20 years to prove existence
of heteroclinic orbits for second and fourth order differential equations having a variational structure. It is divided in 2 parts: the first one deals with second order equations and
systems, while the second one describes recent results on fourth order equations. The contribution by De Coster, Obersnel and Omari deals with qualitative properties of solutions
of two kinds of scalar differential equations: first order ODEs, and second order parabolic
PDEs. Their setting is very general, so that neither uniqueness for the initial value problems nor comparison principles are guaranteed. They particularly concentrate on periodic
solutions, their localization and possible stability. The paper by Han is dedicated to the
theory of limit cycles of planar differential systems and their bifurcations. It is structured
in three main parts: general properties of limit cycles, Hopf bifurcations and perturbations
of Hamiltonian systems. Many results are closely related to the second part of Hilbert’s
16th problem which concerns with the number and location of limit cycles of a planar
polynomial vector field of degree n posed in 1901 by Hilbert. The survey by Hartung,
Krisztin, Walther and Wu reports about the more recent work on state-dependent delayed
functional differential equations. These equations appear in a natural way in the modelling
of evolution processes in very different fields: physics, automatic control, neural networks,
infectious diseases, population growth, cell biology, epidemiology, etc. The authors emphasize on particular models and on the emerging theory from the dynamical systems point
v


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vi

Preface

of view. The paper by Korman is devoted to two point nonlinear boundary value problems
depending on a parameter λ. The main question is the precise number of solutions of the
problem and how these solutions change with the parameter. To study the problem, the
author uses bifurcation theory based on the implicit function theorem (in Banach spaces)
and on a well known theorem by Crandall and Rabinowitz. Other topics he discusses involve pitchfork bifurcation and symmetry breaking, sign changing solutions, etc. Finally,
the paper by Rach˚unková, Stanˇek and Tvrdý is a survey on the solvability of various nonlinear singular boundary value problems for ordinary differential equations on the compact

interval. The nonlinearities in differential equations may be singular both in the time and
space variables. Location of all singular points need not be known.
With this volume we end our contribution as editors of the Handbook of Differential
Equations. We thank the staff at Elsevier for efficient collaboration during the last three
years.


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List of Contributors
Andres, J., Palacký University, Olomouc-Hejˇcín, Czech Republic (Ch. 1)
Bonheure, D., Université Catholique de Louvain, Louvain-La-Neuve, Belgium (Ch. 2)
De Coster, C., Université du Littoral-Côte d’Opale, Calais Cédex, France (Ch. 3)
Han, M., Shanghai Normal University, Shanghai, China (Ch. 4)
Hartung, F., University of Veszprém, Veszprém, Hungary (Ch. 5)
Korman, P., University of Cincinnati, Cincinnati, OH, USA (Ch. 6)
Krisztin, T., University of Szeged, Szeged, Hungary (Ch. 5)
Obersnel, F., Università degli Studi di Trieste, Trieste, Italy (Ch. 3)
Omari, P., Università degli Studi di Trieste, Trieste, Italy (Ch. 3)
Rach˚unková, I., Palacký University, Olomouc, Czech Republic (Ch. 7)
Sanchez, L., Universidade de Lisboa, Lisboa, Portugal (Ch. 2)
Stanˇek, S., Palacký University, Olomouc, Czech Republic (Ch. 7)
Tvrdý, M., Mathematical Institute, Academy of Sciences of the Czech Republic, Praha,
Czech Republic (Ch. 7)
Walther, H.-O., Universität Gießen, Gießen, Germany (Ch. 5)
Wu, J., York University, Toronto, Canada (Ch. 5)

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Contents
Preface
List of Contributors
Contents of Volume 1
Contents of Volume 2

v
vii
xi
xiii

1. Topological principles for ordinary differential equations
J. Andres
2. Heteroclinic orbits for some classes of second and fourth order differential equations
D. Bonheure and L. Sanchez
3. A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness
C. De Coster, F. Obersnel and P. Omari
4. Bifurcation theory of limit cycles of planar systems
M. Han
5. Functional differential equations with state-dependent delays: Theory and applications
F. Hartung, T. Krisztin, H.-O. Walther and J. Wu
6. Global solution branches and exact multiplicity of solutions for two point boundary value problems
P. Korman

7. Singularities and Laplacians in boundary value problems for nonlinear ordinary
differential equations
I. Rach˚unková, S. Stanˇek and M. Tvrdý
Author index
Subject index

1

103

203
341

435

547

607

725
735

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Contents of Volume 1
Preface
List of Contributors

v
vii

1. A survey of recent results for initial and boundary value problems singular in the
dependent variable
R.P. Agarwal and D. O’Regan
2. The lower and upper solutions method for boundary value problems
C. De Coster and P. Habets
3. Half-linear differential equations
O. Došlý
4. Radial solutions of quasilinear elliptic differential equations
J. Jacobsen and K. Schmitt
5. Integrability of polynomial differential systems
J. Llibre
6. Global results for the forced pendulum equation
J. Mawhin
7. Wa˙zewski method and Conley index
R. Srzednicki
Author index
Subject index

1
69
161

359
437
533
591

685
693

xi


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Contents of Volume 2
Preface
List of Contributors
Contents of Volume 1

v
vii
xi

1. Optimal control of ordinary differential equations
V. Barbu and C. Lefter
2. Hamiltonian systems: periodic and homoclinic solutions by variational methods

T. Bartsch and A. Szulkin
3. Differential equations on closed sets
O. Cârj˘a and I.I. Vrabie
4. Monotone dynamical systems
M.W. Hirsch and H. Smith
5. Planar periodic systems of population dynamics
J. López-Gómez
6. Nonlocal initial and boundary value problems: a survey
S.K. Ntouyas
Author index
Subject index

1
77
147
239
359
461

559
565

xiii


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CHAPTER 1

Topological Principles for Ordinary Differential
Equations
Jan Andres∗
Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40,
779 00 Olomouc-Hejˇcín, Czech Republic
E-mail:

Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Elements of ANR-spaces . . . . . . . . . . . . . . . . . . . . .
2.2. Elements of multivalued maps . . . . . . . . . . . . . . . . . .
2.3. Some further preliminaries . . . . . . . . . . . . . . . . . . . .
3. Applied fixed point principles . . . . . . . . . . . . . . . . . . . . .
3.1. Lefschetz fixed point theorems . . . . . . . . . . . . . . . . . .
3.2. Nielsen fixed point theorems . . . . . . . . . . . . . . . . . . .
3.3. Fixed point index theorems . . . . . . . . . . . . . . . . . . . .
4. General methods for solvability of boundary value problems . . . .
4.1. Continuation principles to boundary value problems . . . . . .
4.2. Topological structure of solution sets . . . . . . . . . . . . . .
4.3. Poincaré’s operator approach . . . . . . . . . . . . . . . . . . .
5. Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Existence of bounded solutions . . . . . . . . . . . . . . . . .
5.2. Solvability of boundary value problems with linear conditions
5.3. Existence of periodic and anti-periodic solutions . . . . . . . .
6. Multiplicity results . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1. Several solutions of initial value problems . . . . . . . . . . .
6.2. Several periodic and bounded solutions . . . . . . . . . . . . .
6.3. Several anti-periodic solutions . . . . . . . . . . . . . . . . . .
7. Remarks and comments . . . . . . . . . . . . . . . . . . . . . . . .
7.1. Remarks and comments to general methods . . . . . . . . . . .
7.2. Remarks and comments to existence results . . . . . . . . . . .
7.3. Remarks and comments to multiplicity results . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* Supported by the Council of Czech Government (MSM 6198959214).

HANDBOOK OF DIFFERENTIAL EQUATIONS
Ordinary Differential Equations, volume 3
Edited by A. Cañada, P. Drábek and A. Fonda
© 2006 Elsevier B.V. All rights reserved
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Topological principles for ordinary differential equations

3

1. Introduction
The classical courses of ordinary differential equations (ODEs) start either with the Peano
existence theorem (see, e.g., [54]) or with the Picard–Lindelöf existence and uniqueness
theorem (see, e.g., [71]), both related to the Cauchy (initial value) problems
x˙ = f (t, x),
x(0) = x0 ,

(1.1)

where f ∈ C([0, τ ] × Rn , Rn ), and
f (t, x) − f (t, y)

L|x − y|,

for all t ∈ [0, τ ] and x, y ∈ Rn ,

(1.2)

in the latter case.
In fact, if f satisfies the Lipschitz condition (1.2), then “uniqueness implies existence”
even for boundary value problems with linear conditions that are “close” to x(0) = x0 ,
as observed in [53]. Moreover, uniqueness implies in general (i.e. not necessarily, under
(1.2)) continuous dependence of solutions on initial values (see, e.g., [54, Theorem 4.1 in
Chapter 4.2]), and subsequently the Poincaré translation operator Tτ : Rn → Rn , at the
time τ > 0, along the trajectories of x˙ = f (t, x), defined as follows:
Tτ (x0 ) := x(τ ) | x(.) is a solution of (1.1) ,

(1.3)

is a homeomorphism (cf. [54, Theorem 4.4 in Chapter 4.2]).
Hence, besides the existence, uniqueness is also a very important problem. W. Orlicz
[92] showed in 1932 that the set of continuous functions f : U → Rn , where U is an open
subset relative to [0, τ ]×Rn , for which problem (1.1) with (0, x0 ) ∈ U is not uniquely solvable, is meager, i.e. a set of the first Baire category. In other words, the generic continuous
Cauchy problems (1.1) are solvable in a unique way. Therefore, no wonder that the first example of nonuniqueness was constructed only in 1925 by M.A. Lavrentev (cf. [71] and, for
more information, see, e.g., [1]). The same is certainly also true for Carathéodory ODEs,
because the notion of a classical (C 1 -) solution can be just replaced by the Carathéodory
solution, i.e. absolutely continuous functions satisfying (1.1), almost everywhere (a.e.).
The change is related to the application of the Lebesgue integral, instead of the Riemann
integral.
On the other hand, H. Kneser [80] proved in 1923 that the sets of solutions to continuous
Cauchy problems (1.1) are, at every time, continua (i.e. compact and connected). This
result was later improved by M. Hukuhara [75] who proved that the solution set itself is
a continuum in C([0, τ ], Rn ). N. Aronszajn [41] specified in 1942 that these continua are

Rδ -sets (see Definition 2.3 below), and as a subsequence, multivalued operators Tτ in (1.3)
become admissible in the sense of L. Górniewicz (see Definition 2.5 below).
Obviously if, for f (t, x) ≡ f (t + τ, x), operator Tτ admits a fixed point, say xˆ ∈ Rn ,
ˆ then xˆ determines a τ -periodic solution of x˙ = f (t, x), and vice versa. This
i.e. xˆ ∈ Tτ (x),
is one of stimulations why to study the fixed point theory for multivalued mappings in
order to obtain periodic solutions of nonuniquely solvable ODEs. Since the regularity of


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4

J. Andres

(multivalued) Poincaré’s operator Tτ is the same (see Theorem 4.17 below) for differential
inclusions x˙ ∈ F (t, x), where F is an upper Carathéodory mapping with nonempty, convex
and compact values (see Definition 2.10 below), it is reasonable to study directly such
differential inclusions with this respect. Moreover, initial value problems for differential
inclusions are, unlike ODEs, typically nonuniquely solvable (cf. [42]) by which Poincaré’s
operators are multivalued.
In this context, an interesting phenomenon occurs with respect to the Sharkovskii cycle
coexistence theorem [95]. This theorem is based on a new ordering of the positive integers,
namely
3 5 7 · · · 2 · 3 2 · 5 2 · 7 · · · 22 · 3 22 · 5 22 · 7 · · ·
2n · 3 2n · 5 2n · 7 · · · 2n+1 · 3 2n+1 · 5 2n+1 · 7 · · ·
2n+1

· · · 22

2n

2 1,

saying that if a continuous function g : R → R has a point of period m with m k (in the
above Sharkovskii ordering), then it has also a point of period k.
By a period, we mean the least period, i.e. a point a ∈ R is a periodic point of period m
if g m (a) = a and g j (a) = a, for 0 < j < m.
Now, consider the scalar ODE
x˙ = f (t, x),

f (t, x) ≡ f (t + τ, x),

(1.4)

where f : [0, τ ] × R → R is a continuous function.
Since
Tτm = Tτ ◦ · · · ◦ Tτ = Tmτ
m times

holds for the Poincaré translation operator Tτ along the trajectories of Eq. (1.4), defined in
(1.3), there is (in the case of uniqueness) an apparent one-to-one correspondence between
m-periodic points of Tτ and (subharmonic) mτ -periodic solutions of (1.4). Nevertheless,
the analogy of classical Sharkovskii’s theorem does not hold for subharmonics of (1.4). In
fact, we only obtain an empty statement, because every bounded solution of (1.4) is, under
the uniqueness assumption, either τ -periodic or asymptotically τ -periodic (see, e.g., [94,
pp. 120–122]).
This handicap is due to the assumed uniqueness condition. On the other hand, in the
lack of uniqueness, the multivalued operator Tτ in (1.3) is admissible (see Theorem 4.17
below) which in R means (cf. Definition 2.5 below) that Tτ is upper semicontinuous (cf.

Definition 2.4 below) and the sets of values consist either of single points or of compact
intervals. In a series of our papers [16,29,36], we developed a version of the Sharkovskii
cycle coexistence theorem which applies to (1.4) as follows:
T HEOREM 1.1. If Eq. (1.4) has an mτ -periodic solution, then it also admits a kτ -periodic
solution, for every k m, with at most two exceptions, where k m means that k is less


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Topological principles for ordinary differential equations

5

Fig. 1. Braid σ .

than m in the above Sharkovskii ordering of positive integers. In particular, if m = 2k , for
all k ∈ N, then infinitely many (subharmonic) periodic solutions of (1.4) coexist.
R EMARK 1.1. As pointed out, Theorem 1.1 holds only in the lack of uniqueness; otherwise, it is empty. On the other hand, the right-hand side of the given (multivalued) ODE
can be a (multivalued upper) Carathéodory mapping with nonempty, convex and compact
values (see Definition 2.10 below).
R EMARK 1.2. Although, e.g., a 3τ -periodic solution of (1.4) implies, for every k ∈ N,
with a possible exception for k = 2 or k = 4, 6, the existence of a kτ -periodic solution of
(1.4), it is very difficult to prove that such a solution exists. Observe that a 3τ -periodic
solution of (1.4) implies the existence of at least two more 3τ -periodic solutions of (1.4).
The Sharkovskii phenomenon is essentially one-dimensional. On the other hand, it follows from T. Matsuoka’s results in [87–89] that three (harmonic) τ -periodic solutions of
the planar (i.e. in R2 ) system (1.4) imply “generically” the coexistence of infinitely many
(subharmonic) kτ -periodic solutions of (1.4), k ∈ N. “Genericity” is this time understood
in terms of the Artin braid group theory, i.e. with the exception of certain simplest braids,
representing the three given harmonics.
The following theorem was presented in [8], on the basis of T. Matsuoka’s results in

papers [87–89].
T HEOREM 1.2. Assume that a uniqueness condition is satisfied for planar system (1.4).
Let three (harmonic) τ -periodic solutions of (1.4) exist whose graphs are not conjugated
to the braid σ m in B3 /Z, for any integer m ∈ N, where σ is shown in Fig. 1, B3 /Z denotes
the factor group of the Artin braid group B3 and Z is its center ( for definitions, see, e.g.,
[22, Chapter III.9]). Then there exist infinitely many (subharmonic) kτ -periodic solutions
of (1.4), k ∈ N.
R EMARK 1.3. In the absence of uniqueness, there occur serious obstructions, but Theorem 1.2 still seems to hold in many situations; for more details see [8].
R EMARK 1.4. The application of the Nielsen theory considered in Section 3.2 below
might determine the desired three harmonic solutions of (1.4). More precisely, it is more
realistic to detect two harmonics by means of the related Nielsen number (see again Section 3.2 below), and the third one by means of the related fixed point index (see Section 3.3
below).


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6

J. Andres

For n > 2, statements like Theorem 1.1 or Theorem 1.2 appear only rarely. Nevertheless,
if f = (f1 , f2 , . . . , fn ) has a special triangular structure, i.e.
fi (x) = fi (x1 , . . . , xn ) = fi (x1 , . . . , xi ),

i = 1, . . . , n,

(1.5)

then Theorem 1.1 can be extended to hold in Rn (see [35]).
T HEOREM 1.3. Under assumption (1.5), the conclusion of Theorem 1.1 remains valid

in Rn .
R EMARK 1.5. Similarly to Theorem 1.1, Theorem 1.3 holds only in the lack of uniqueness. Without the special triangular structure (1.5), there is practically no chance to obtain
an analogy to Theorem 1.1, for n 2.
There is also another motivation for the investigation of multivalued ODEs, i.e. differential inclusions, because of the strict connection with
(i) optimal control problems for ODEs,
(ii) Filippov solutions of discontinuous ODEs,
(iii) implicit ODEs, etc.
ad (i): Consider a control problem for
x˙ = f (t, x, u),

u ∈ U,

(1.6)

where f : [0, τ ] × Rn × Rn → Rn and u ∈ U are control parameters such that u(t) ∈ Rn ,
for all t ∈ [0, τ ]. In order to solve a control problem for (1.6), we can define a multivalued
map F (t, x) := {f (t, x, u)}u∈U . The solutions of (1.6) are those of
x˙ ∈ F (t, x),

(1.7)

and the same is true for a given control problem. For more details, see, e.g., [27,79].
ad (ii): If function f is discontinuous in x, then Carathéodory theory cannot be applied
for solving, e.g., (1.1). Making, however, the Filippov regularization of f , namely
conv f Oδ (t, x) \ r ,

F (t, x) :=

(1.8)

δ>0 r⊂[0,τ ]×Rn
μ(r)=0

where μ(r) denotes the Lebesgue measure of the set r ⊂ Rn and
Oδ (y) := z ∈ [0, τ ] × Rn | |y − z| < δ ,
multivalued F is well known (see [60]) to be again upper Carathéodory with nonempty,
convex and compact values (cf. Definition 2.10 below), provided only f is measurable
and satisfies |f (t, x)| α + β|x|, for all (t, x) ∈ [0, τ ] × Rn , with some nonnegative constants α, β. Thus, by a Filippov solution of x˙ = f (t, x), it is so understood a Carathéodory


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Topological principles for ordinary differential equations

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solution of (1.7), where F is defined in (1.8). As an example from physics, dry friction
problems (see, e.g., [84,91]) can be solved in this way.
ad (iii): Let us consider the implicit differential equation
x˙ = f (t, x, x),
˙

(1.9)

where f : [0, τ ] × Rn × Rn → Rn is a compact (continuous) map and the solutions are
understood in the sense of Carathéodory. We can associate with (1.9) the following two
differential inclusions:
x˙ ∈ F1 (t, x)

(1.10)

x˙ ∈ F2 (t, x),

(1.11)

and

where F1 (t, x) := Fix(f (t, x, ·)), i.e. the (nonempty, see [22, p. 560]) fixed point set of
f (t, x, ·) w.r.t. the last variable, and F2 ⊂ F1 is a (multivalued) lower semicontinuous (see
Definition 2.4 below) selection of F1 . The sufficient condition for the existence of such a
selection F2 reads (see, e.g., [22, Chapter III.11, pp. 558–559]):
dim Fix f (t, x, ·) = 0,

for all (t, x) ∈ [0, τ ] × Rn ,

(1.12)

where dim denotes the topological (covering) dimension.
Denoting by S(f ), S(F1 ), S(F2 ) the sets of all solutions of initial value problems to
(1.9), (1.10), (1.11), respectively, one can prove (see [22, p. 560]) that, under (1.12),
S(f ) = S(F1 ) ⊂ S(F2 ) = ∅. For more details, see [19] (cf. [22, Chapter III.11]).
Although there are several monographs devoted to multivalued ODEs (see, e.g., [22,42,
45,58,61,74,79,91,96,97]), topological principles were presented mainly for single-valued
ODEs (besides [22,45,58] and [61] for differential inclusions, see, e.g., [62,64,65,82,83,
90]). Hence our main object will be topological principles for (multivalued) ODEs; whence
the title. We will consider without special distinguishing differential equations as well as
inclusions; both in Euclidean and Banach spaces. All solutions of problems under our consideration (even in Banach spaces) will be understood at least in the sense of Carathéodory.
Thus, in view of the indicated relationship with problems (i)–(iii), many obtained results
can be also employed for solving optimal control problems, problems for systems with
variable structure, implicit boundary value problems, etc.

The reader exclusively interested in single-valued ODEs can simply read “continuous”,
instead of “upper semicontinuous” or “lower semicontinuous”, and replace the inclusion
symbol ∈ by the equality =, in the given differential inclusions. If, in the single-valued
case, the situation simplifies dramatically or if the obtained results can be significantly
improved, then the appropriate remarks are still supplied.
We wished to prepare an as much as possible self-contained text. Nevertheless, the reader
should be at least familiar with the elements of nonlinear analysis, in particular of fixed
point theory, in order to understand the degree arguments, or so. Otherwise, we recommend the monographs [69] (in the single-valued case) and [22] (in the multivalued case).


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8

J. Andres

Furthermore, one is also expected to know several classical results and notions from the
standard courses of ODEs, functional analysis and the theory of integration like the Gronwall inequality, the Arzelà–Ascoli lemma, the Mazur Theorem, the Bochner integral, etc.
We will study mainly existence and multiplicity of bounded, periodic and anti-periodic
solutions of (multivalued) ODEs. Since our approach consists in the application of the fixed
point principles, these solutions will be either determined by, (e.g., τ -periodic solutions
x(t) by the initial values x(0) via (1.3)) or directly identified (e.g., solutions of initial value
problems (1.1)) with fixed points of the associated (Cauchy, Hammerstein, etc.) operators.
Although the usage of the relative degree (i.e. the fixed point index) arguments is rather
traditional in this framework, it might not be so when the maps, representing, e.g., problems on noncompact intervals, operate in nonnormable Fréchet spaces. This is due to the
unpleasant locally convex topology possessing bounded subsets with an empty interior. We
had therefore to develop with my colleagues our own fixed point index theory. The application of the Nielsen theory, for obtaining multiplicity criteria, is very delicate and quite rare,
and the related problem is named after Jean Leray who posed it in 1950, at the first International Congress of Mathematics held after World War II in Cambridge, Mass. We had also
to develop a new multivalued Nielsen theory suitable for applications in this field. Before
presenting general methods for solvability of boundary value problems in Section 4, we

therefore make a sketch of the applied fixed point principles in Section 3. Hence besides
Section 4, the main results are contained in Section 5 (Existence results) and Section 6
(Multiplicity results). The reference sources to our results and their comparison with those
of other authors are finally commented in Section 7 (Remarks and comments).

2. Preliminaries
2.1. Elements of ANR-spaces
In the entire text, all topological spaces will be metric and, in particular, all topological
vector spaces will be at least Fréchet. Let us recall that by a Fréchet space, we understand
a complete (metrizable) locally convex space. Its topology can be generated by a countable
family of seminorms. If it is normable, then it becomes Banach.
D EFINITION 2.1. A (metrizable) space X is an absolute neighbourhood retract (ANR) if,
for each (metrizable) Y and every closed A ⊂ Y , each continuous mapping f : A → X is
extendable over some neighbourhood of A.
P ROPOSITION 2.1.
(i) If X is an ANR, then any open subset of X is an ANR and any neighbourhood
retract of X is an ANR.
(ii) X is an ANR if and only if it is a neighbourhood retract of every (metrizable) space
in which it is embedded as a closed subset.
(iii) X is an ANR if and only if it is a neighbourhood retract of some normed linear
space, i.e. if and only if it is a retract of some open subset of a normed space.


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Topological principles for ordinary differential equations

9

(iv) If X is a retract of an open subset of a convex set in a Fréchet space, then it is an

ANR.
(v) If X1 , X2 are closed ANRs such that X1 ∩ X2 is an ANR, then X1 ∪ X2 is an ANR.
(vi) Any finite union of closed convex sets in a Fréchet space is an ANR.
(vii) If each x ∈ X admits a neighbourhood that is an ANR, then X is an ANR.
D EFINITION 2.2. A (metrizable) space X is an absolute retract (AR) if, for each (metrizable) Y and every closed A ⊂ Y , each continuous mapping f : A → X is extendable over Y .
P ROPOSITION 2.2.
(i) X is an AR if and only if it is a contractible (i.e. homotopically equivalent to a one
point space) ANR.
(ii) X is an AR if and only if it is a retract of every (metrizable) space in which it is
embedded as a closed subset.
(iii) If X is an AR and A is a retract of X, then A is an AR.
(iv) If X is homeomorphic to Y and X is an AR, then so is Y .
(v) X is an AR if and only if it is a retract of some normed space.
(vi) If X is a retract of a convex subset of a Fréchet space, then it is an AR.
(vii) If X1 , X2 are closed ARs such that X1 ∩ X2 is an AR, then X1 ∪ X2 is an AR.
Furthermore, it is well known that every ANR X is locally contractible (i.e. for each
x ∈ X and a neighbourhood U of x, there exists a neighbourhood V of x that is contractible in U ) and, as follows from Proposition 2.2(i) that every AR X is contractible (i.e.
if idX : X → X is homotopic to a constant map).
D EFINITION 2.3. X is called an Rδ -set if, there exists a decreasing sequence {Xn } of
compact, contractible sets Xn such that X = {Xn | n = 1, 2, . . .}.
Although contractible spaces need not be ARs, X is an Rδ -set if and only if it is an
intersection of a decreasing sequence of compacts ARs. Moreover, every Rδ -set is acyclic
ˇ
w.r.t. any continuous theory of homology (e.g., the Cech
homology), i.e. homologically
equivalent to a one point space, and so it is in particular nonempty, compact and connected.
The following hierarchies hold for metric spaces:
contractible ⊂ acyclic
∪
convex ⊂ AR ⊂ ANR,

compact + convex ⊂ compact AR ⊂ compact + contractible ⊂ Rδ ⊂ compact + acyclic,
and all the above inclusions are proper.
For more details, see [47] (cf. also [22,67,69]).


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10

J. Andres

2.2. Elements of multivalued maps
In what follows, by a multivalued map ϕ : X
with at least nonempty, closed values.

Y , i.e. ϕ : X → 2Y \{0}, we mean the one

Y is said to be upper semicontinuous (u.s.c.) if, for every
D EFINITION 2.4. A map ϕ : X
open U ⊂ Y , the set {x ∈ X | ϕ(x) ⊂ U } is open in X. It is said to be lower semicontinuous
(l.s.c.) if, for every open U ⊂ Y , the set {x ∈ X | ϕ(x) ∩ U = ∅} is open in X. If it is both
u.s.c. and l.s.c., then it is called continuous.
Obviously, in the single-valued case, if f : X → Y is u.s.c. or l.s.c., then it is continuous. Moreover, the compact-valued map ϕ : X
Y is continuous if and only if it
is Hausdorff-continuous, i.e. continuous w.r.t. the metric d in X and the Hausdorffmetric dH in {B ⊂ Y | B is nonempty and bounded}, where dH (A, B) := inf{ε > 0 | A ⊂
Oε (B) and B ⊂ Oε (A)} and Oε (B) := {x ∈ X | ∃y ∈ B: d(x, y) < ε}. Every u.s.c. map
ϕ :X
Y has a closed graph ϕ , but not vice versa. Nevertheless, if the graph ϕ of a
compact map ϕ : X
Y is closed, then ϕ is u.s.c.

The important role will be played by the following class of admissible maps in the sense
of L. Górniewicz.
p

q

D EFINITION 2.5. Assume that we have a diagram X ⇐
−→ Y ( is a metric space),
where p : ⇒ X is a continuous Vietoris map, namely
(i) p is onto, i.e. p( ) = X,
(ii) p is proper, i.e. p −1 (K) is compact, for every compact K ⊂ X,
(iii) p −1 (x) is acyclic, for every x ∈ X, where acyclicity is understood in the sense of
ˇ
the Cech
homology functor with compact carriers and coefficients in the field Q of
rationals,
and q : → Y is a continuous map. The map ϕ : X
Y is called admissible if it is induced
by ϕ(x) = q(p −1 (x)), for every x ∈ X. We, therefore, identify the admissible map ϕ with
the pair (p, q) called an admissible (selected) pair.
q0

p0

p1

p1

D EFINITION 2.6. Let X ⇐ 0 −→ Y and X ⇐ 1 −→ Y be two admissible maps, i.e.
ϕ0 = q0 ◦ p0−1 and ϕ1 = q1 ◦ p1−1 . We say that ϕ0 is admissibly homotopic to ϕ1 (written

q

p

ϕ0 ∼ ϕ1 or (p0 , q0 ) ∼ (p1 , q1 )) if there exists an admissible map X × [0, 1] ⇐
such that the following diagram is commutative:
pi

−→ Y

qi

X

Y

i
fi

ki

X × [0, 1]

0

q

p

for ki (x) = (x, i), i = 0, 1, and fi : i → is a homeomorphism onto p −1 (X ×i), i = 0, 1,

i.e. k0 p0 = pf0 , q0 = qf0 , k1 p1 = pf1 and q1 = qf1 .