Canada a drabek p fonda a (eds ) handbook of differential equations ordinary differential equations vol 3

The con-tribution by De Coster, Obersnel and Omari deals with qualitative properties of solutions varia-of two kinds varia-of scalar differential equations: first order ODEs, and second

H ANDBOOK

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

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This is the third volume in a series devoted to self contained and up-to-date surveys in thetheory of ordinary differential equations, written by leading researchers in the area Allcontributors have made an additional effort to achieve readability for mathematicians andscientists 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 theeditors hope that it will become very useful for research, learning and teaching We expressour deepest gratitude to all contributors to this volume for their clearly written and elegantarticles

This volume consists of seven chapters covering a variety of problems in ordinary ential equations Both, pure mathematical research and real word applications are reflectedpretty well by the contributions to this volume They are presented in alphabetical orderaccording to the name of the first author The paper by Andres provides a comprehensivesurvey on topological methods based on topological index, Lefschetz and Nielsen num-bers Both single and multivalued cases are investigated Ordinary differential equationsare studied both on finite and infinite dimensions, and also on compact and noncompactintervals There are derived existence and multiplicity results Topological structures ofsolution sets are investigated as well The paper by Bonheure and Sanchez is dedicated

differ-to show how variational methods have been used in the last 20 years differ-to prove existence

of heteroclinic orbits for second and fourth order differential equations having a tional structure It is divided in 2 parts: the first one deals with second order equations andsystems, while the second one describes recent results on fourth order equations The con-tribution by De Coster, Obersnel and Omari deals with qualitative properties of solutions

varia-of two kinds varia-of scalar differential equations: first order ODEs, and second order parabolicPDEs Their setting is very general, so that neither uniqueness for the initial value prob-lems nor comparison principles are guaranteed They particularly concentrate on periodicsolutions, their localization and possible stability The paper by Han is dedicated to thetheory 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’s16th 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 delayedfunctional 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 empha-size on particular models and on the emerging theory from the dynamical systems point

v

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, theauthor 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 in-volve 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 non-linear singular boundary value problems for ordinary differential equations on the compactinterval The nonlinearities in differential equations may be singular both in the time andspace variables Location of all singular points need not be known

With this volume we end our contribution as editors of the Handbook of DifferentialEquations We thank the staff at Elsevier for efficient collaboration during the last threeyears

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)

vii

1 Topological principles for ordinary differential equations 1

J Andres

2 Heteroclinic orbits for some classes of second and fourth order differential

D Bonheure and L Sanchez

3 A qualitative analysis, via lower and upper solutions, of first order periodic lutionary equations with lack of uniqueness 203

evo-C De Coster, F Obersnel and P Omari

4 Bifurcation theory of limit cycles of planar systems 341

M Han

5 Functional differential equations with state-dependent delays: Theory and

F Hartung, T Krisztin, H.-O Walther and J Wu

6 Global solution branches and exact multiplicity of solutions for two point

Contents of Volume 1

1 A survey of recent results for initial and boundary value problems singular in the

R.P Agarwal and D O’Regan

2 The lower and upper solutions method for boundary value problems 69

C De Coster and P Habets

3 Half-linear differential equations 161

O Došlý

4 Radial solutions of quasilinear elliptic differential equations 359

J Jacobsen and K Schmitt

5 Integrability of polynomial differential systems 437

Contents of Volume 2

1 Optimal control of ordinary differential equations 1

V Barbu and C Lefter

2 Hamiltonian systems: periodic and homoclinic solutions by variational methods 77

T Bartsch and A Szulkin

3 Differential equations on closed sets 147

O Cârj˘a and I.I Vrabie

4 Monotone dynamical systems 239

M.W Hirsch and H Smith

5 Planar periodic systems of population dynamics 359

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: andres@inf.upol.cz

Contents

1 Introduction 3

2 Preliminaries 8

2.1 Elements of ANR-spaces 8

2.2 Elements of multivalued maps 10

2.3 Some further preliminaries 13

3 Applied fixed point principles 15

3.1 Lefschetz fixed point theorems 15

3.2 Nielsen fixed point theorems 18

3.3 Fixed point index theorems 23

4 General methods for solvability of boundary value problems 29

4.1 Continuation principles to boundary value problems 29

4.2 Topological structure of solution sets 45

4.3 Poincaré’s operator approach 59

5 Existence results 61

5.1 Existence of bounded solutions 61

5.2 Solvability of boundary value problems with linear conditions 70

5.3 Existence of periodic and anti-periodic solutions 73

6 Multiplicity results 76

6.1 Several solutions of initial value problems 76

6.2 Several periodic and bounded solutions 79

6.3 Several anti-periodic solutions 93

7 Remarks and comments 95

7.1 Remarks and comments to general methods 95

7.2 Remarks and comments to existence results 96

7.3 Remarks and comments to multiplicity results 97

Acknowledgements 97

References 97

* 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

1

1 Introduction

The classical courses of ordinary differential equations (ODEs) start either with the Peanoexistence theorem (see, e.g., [54]) or with the Picard–Lindelöf existence and uniquenesstheorem (see, e.g., [71]), both related to the Cauchy (initial value) problems



˙x = f (t, x),

where f ∈ C([0, τ] × R n ,Rn ), and

f (t, x) − f (t, y)  L |x − y|, for all t ∈ [0, τ] and x, y ∈ R n , (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 T τ:Rn→ Rn, at the

time τ > 0, along the trajectories of ˙x = f (t, x), defined as follows:

T τ (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]×R n , for which problem (1.1) with (0, x0) ∈ U is not uniquely

solv-able, is meager, i.e a set of the first Baire category In other words, the generic continuousCauchy problems (1.1) are solvable in a unique way Therefore, no wonder that the first ex-ample of nonuniqueness was constructed only in 1925 by M.A Lavrentev (cf [71] and, formore information, see, e.g., [1]) The same is certainly also true for Carathéodory ODEs,

because the notion of a classical (C1-) 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 Riemannintegral

On the other hand, H Kneser [80] proved in 1923 that the sets of solutions to continuousCauchy problems (1.1) are, at every time, continua (i.e compact and connected) Thisresult was later improved by M Hukuhara [75] who proved that the solution set itself is

a continuum in C( [0, τ], R n ) N Aronszajn [41] specified in 1942 that these continua are

R δ -sets (see Definition 2.3 below), and as a subsequence, multivalued operators T τ 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 T τ admits a fixed point, say ˆx ∈ R n,i.e ˆx ∈ T τ ( ˆx), then ˆx determines a τ -periodic solution of ˙x = f (t, x), and vice versa This

is one of stimulations why to study the fixed point theory for multivalued mappings inorder to obtain periodic solutions of nonuniquely solvable ODEs Since the regularity of

(multivalued) Poincaré’s operator T τis the same (see Theorem 4.17 below) for differentialinclusions˙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 suchdifferential inclusions with this respect Moreover, initial value problems for differentialinclusions are, unlike ODEs, typically nonuniquely solvable (cf [42]) by which Poincaré’soperators are multivalued

In this context, an interesting phenomenon occurs with respect to the Sharkovskii cyclecoexistence 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 2n · · ·  22 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

m -periodic points of T τ and (subharmonic) mτ -periodic solutions of (1.4) Nevertheless,

the analogy of classical Sharkovskii’s theorem does not hold for subharmonics of (1.4) Infact, 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 T τ in (1.3) is admissible (see Theorem 4.17below) which inR means (cf Definition 2.5 below) that T τ is upper semicontinuous (cf.Definition 2.4 below) and the sets of values consist either of single points or of compactintervals In a series of our papers [16,29,36], we developed a version of the Sharkovskiicycle coexistence theorem which applies to (1.4) as follows:

THEOREM1.1 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

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.

REMARK1.1 As pointed out, Theorem 1.1 holds only in the lack of uniqueness; wise, it is empty On the other hand, the right-hand side of the given (multivalued) ODEcan be a (multivalued upper) Carathéodory mapping with nonempty, convex and compactvalues (see Definition 2.10 below)

other-REMARK 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

fol-lows from T Matsuoka’s results in [87–89] that three (harmonic) τ -periodic solutions of

the planar (i.e inR2) 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 inpapers [87–89]

THEOREM 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 B3and Z is its center ( for definitions, see, e.g.,

[22, Chapter III.9]) Then there exist infinitely many (subharmonic) kτ -periodic solutions

For n > 2, statements like Theorem 1.1 or Theorem 1.2 appear only rarely Nevertheless,

if f = (f1, f2, , f n )has a special triangular structure, i.e

f i (x) = f i (x1, , x n ) = f i (x1, , x i ), i = 1, , n, (1.5)then Theorem 1.1 can be extended to hold inRn(see [35])

THEOREM 1.3 Under assumption (1.5), the conclusion of Theorem 1.1 remains valid

inRn

REMARK 1.5 Similarly to Theorem 1.1, Theorem 1.3 holds only in the lack of ness Without the special triangular structure (1.5), there is practically no chance to obtain

unique-an unique-analogy to Theorem 1.1, for n 2

There is also another motivation for the investigation of multivalued ODEs, i.e

differ-ential 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] × R n× Rn→ Rn and u ∈ U are control parameters such that u(t) ∈ R n,

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

and satisfies|f (t, x)|  α + β|x|, for all (t, x) ∈ [0, τ] × R n, with some nonnegative

con-stants α, β Thus, by a Filippov solution of ˙x = f (t, x), it is so understood a Carathéodory

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] × R n× Rn→ Rn is a compact (continuous) map and the solutions areunderstood in the sense of Carathéodory We can associate with (1.9) the following twodifferential inclusions:

˙x ∈ F1(t, x) (1.10)and

˙x ∈ F2(t, x), (1.11)

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⊂ F1is a (multivalued) lower semicontinuous (see

Definition 2.4 below) selection of F1 The sufficient condition for the existence of such a

selection F2reads (see, e.g., [22, Chapter III.11, pp 558–559]):

dim Fix

f (t, x, ·) = 0, for all (t, x) ∈ [0, τ] × R n , (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-valuedODEs (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; whencethe title We will consider without special distinguishing differential equations as well asinclusions; both in Euclidean and Banach spaces All solutions of problems under our con-sideration (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 resultscan be also employed for solving optimal control problems, problems for systems withvariable 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 inclusionsymbol∈ 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 significantlyimproved, then the appropriate remarks are still supplied

We wished to prepare an as much as possible self-contained text Nevertheless, the readershould be at least familiar with the elements of nonlinear analysis, in particular of fixedpoint theory, in order to understand the degree arguments, or so Otherwise, we recom-mend the monographs [69] (in the single-valued case) and [22] (in the multivalued case)

Furthermore, one is also expected to know several classical results and notions from thestandard courses of ODEs, functional analysis and the theory of integration like the Gron-wall inequality, the Arzelà–Ascoli lemma, the Mazur Theorem, the Bochner integral, etc.

We will study mainly existence and multiplicity of bounded, periodic and anti-periodicsolutions 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 rathertraditional in this framework, it might not be so when the maps, representing, e.g., prob-lems on noncompact intervals, operate in nonnormable Fréchet spaces This is due to theunpleasant locally convex topology possessing bounded subsets with an empty interior Wehad therefore to develop with my colleagues our own fixed point index theory The applica-tion 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 Interna-tional 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 Beforepresenting general methods for solvability of boundary value problems in Section 4, wetherefore make a sketch of the applied fixed point principles in Section 3 Hence besidesSection 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.

DEFINITION2.1 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.

PROPOSITION2.1

(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.

(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, X2are closed ANRs such that X1∩ X2is an ANR, then X1∪ X2is 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.

DEFINITION2.2 A (metrizable) space X is an absolute retract (AR) if, for each able) Y and every closed A ⊂ Y , each continuous mapping f : A → X is extendable over Y

(metriz-PROPOSITION2.2

(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, X2are closed ARs such that X1∩ X2is an AR, then X1∪ X2is 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

con-tractible in U ) and, as follows from Proposition 2.2(i) that every AR X is concon-tractible (i.e.

if idX : X → X is homotopic to a constant map).

DEFINITION 2.3 X is called an R δ -set if, there exists a decreasing sequence {X n} of

compact, contractible sets X n such that X= {X n | n = 1, 2, }.

Although contractible spaces need not be ARs, X is an R δ-set if and only if it is an

intersection of a decreasing sequence of compacts ARs Moreover, every R δ -set is acyclic

w.r.t any continuous theory of homology (e.g., the ˇCech homology), i.e homologicallyequivalent 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 ⊂ R δ⊂ compact + acyclic,

and all the above inclusions are proper

For more details, see [47] (cf also [22,67,69])

2.2 Elements of multivalued maps

In what follows, by a multivalued map ϕ : X  Y , i.e ϕ : X → 2 Y\{0}, we mean the one

with at least nonempty, closed values

DEFINITION2.4 A map ϕ : X  Y is said to be upper semicontinuous (u.s.c.) if, for every

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

con-tinuous Moreover, the compact-valued map ϕ : X  Y is continuous if and only if it

is continuous, i.e continuous w.r.t the metric d in X and the metric d H in{B ⊂ Y | B is nonempty and bounded}, where d H (A, B) := inf{ε > 0 | A ⊂

Hausdorff-O ε (B) and B ⊂ O ε (A) } and O ε (B) := {x ∈ X | ∃y ∈ B: d(x, y) < ε} Every u.s.c map

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

DEFINITION2.5 Assume that we have a diagram X p q

⇒ X is a continuous Vietoris map, namely

= 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

the ˇCech homology functor with compact carriers and coefficients in the fieldQ of

rationals,

→ 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.

−→ Y be two admissible maps, i.e.

ϕ0= q0◦ p−10 and ϕ1= q1◦ p1−1 We say that ϕ0is admissibly homotopic to ϕ1(written

ϕ0∼ ϕ1or (p0, q0) ∼ (p1, q1) ) if there exists an admissible map X × [0, 1] p 0

Thus, admissible maps are always u.s.c with nonempty, compact and connected ues Moreover, their class is closed w.r.t finite compositions, i.e a finite composition ofadmissible maps is also admissible In fact, a map is admissible if and only if it is a fi-

val-nite composition of acyclic maps with compact values, i.e u.s.c maps with acyclic and

compact values

The class of admissible maps so contains u.s.c maps with convex and compact

val-ues, u.s.c maps with contractible and compact valval-ues, R δ -maps (i.e u.s.c maps with

R δ-values), acyclic maps with compact values and their compositions

The class of compact admissible maps ϕ : X  Y , i.e ϕ(X) is compact, will be denoted

byK(X, Y ), or simply by K(X), provided ϕ is a self-map (an endomorphism) If the

ad-missible homotopy in Definition 2.6 is still compact, then we say that ϕ0∈ K(X, Y ) and

ϕ1∈ K(X, Y ) are compactly admissibly homotopic.

Another important class of admissible maps are condensing admissible maps denoted by

C(X, Y ) For this, we need to recall the notion of a measure of noncompactness (MNC).

Let E be a Fréchet space endowed with a countable family of seminorms . s , s ∈ S

(S is the index set), generating the locally convex topology Denoting by B = B(E) the set

of nonempty, bounded subsets of E, we can give

DEFINITION 2.7 The family of functions α = {α s}s ∈S:B → [0, ∞) S , where α s (B):=

inf{δ > 0 | B ∈ B admits a finite covering by the sets of diam s  δ}, s ∈ S, for B ∈ B,

is called the Kuratowski measure of noncompactness and the family of functions γ =

{γ s}s ∈S:B → [0, ∞) S , where γ s (B) := inf{δ > 0 | B ∈ B has a finite ε s-net}, s ∈ S, for

B ∈ B, is called the Hausdorff measure of noncompactness.

These MNC are related as follows:

γ (B)  α(B)  2γ (B), i.e γ s (B)  α s (B)  2γ s (B), for each s ∈ S.

Moreover, they satisfy the following properties:

PROPOSITION2.3 Assume that B, B1, B2∈ B Then we have (component-wise):

where μ denotes either α or γ

DEFINITION2.8 A bounded mapping ϕ : E ⊃ U  E, i.e ϕ(B) ∈ B, for B  B ⊂ U, is

said to be μ-condensing (shortly, condensing) if μ(ϕ(B)) < μ(B), whenever B  B ⊂ U

and μ(B) > 0, or equivalently, if μ(ϕ(B))  μ(B) implies μ(B) = 0, whenever B  B ⊂

U , where μ = {μ s}s ∈S:B → [0, ∞) Sis a family of functions satisfying at least conditions

(μ1) –(μ5) Analogously, a bounded mapping ϕ : E ⊃ U  E is said to be a k-set

contrac-tion w.r.t μ = {μ s}s ∈S:B → [0, ∞) S satisfying at least conditions (μ1) –(μ5)(shortly, a

k-contraction or a set-contraction) if μ(ϕ(B))  kμ(B), for some k ∈ [0, 1), whenever

B  B ⊂ U.

Obviously, any set-contraction is condensing and both α-condensing and γ -condensing maps are μ-condensing Furthermore, compact maps or contractions with compact values (in vector spaces, also their sum) are well known to be (α, γ )-set-contractions, and so (α, γ )-condensing.

Besides semicontinuous maps, measurable and semi-Carathéodory maps will be also of

importance Hence, assume that Y is a separable metric space and ( , U, ν) is a

measure ν on U A typical example is when is a bounded domain in R n, equipped withthe Lebesgue measure

DEFINITION 2.9 A map ϕ :  Y is called strongly measurable if there exists a

se-quence of step multivalued maps ϕ n :  Y such that d H (ϕ n (ω), ϕ(ω))→ 0, for a.a

ω ∈ , as n → ∞ In the single-valued case, one can simply replace multivalued step

maps by single-valued step maps and d H (ϕ n (ω), ϕ(ω))byϕ n (ω) − ϕ(ω).

A map ϕ :  Y is called measurable if {ω ∈ | ϕ(ω) ⊂ V } ∈ U, for each open V ⊂ Y

A map ϕ :  Y is called weakly measurable if {ω ∈ ... < /p>

(i) X is ANR-space, e.g., a retract of an open subset of a convex set in a Fréchet space, < /p> Trang 39

(ii)... sections. < /p> Trang 34

REMARK3. 4 (Important) We have a counter-example in [24]... assume that < /p>

(i) X is a (metric) ANR-space, e.g., a retract of an open subset of a convex set in a< /i> < /p>

Fréchet space, < /p>

(ii) ϕ is a compact (i.e ϕ(X) is compact) composition