Remerciement

Ce n'est pas une chose simple de résumer ici, en quelques lignes, combien ont pu compter pour moi les personnes qui ont participé de près ou de loin à ce doctorat, que ce soit en travaillant avec moi, ou bien en partageant des instants de vie. Je tiens avant tout à exprimer ma reconnaissance et mes remerciements particuliers à mon directeur, le professeur Khaled Latrach, pour ses excellents conseils, son soutien constant, ses encouragements, ses qualités scientifiques et notamment sa patience dans la direction de ce travail. Je tiens à lui exprimer ma profonde gratitude pour m'avoir accepté en thèse. Je remercie Mohamed AZIZ-TAOUDI et Jésus GARCIA-FALSET d'avoir eu l'obligeance de rapporter sur cette thèse malgré leurs nombreuses occupations. Vos remarques précieuses m'ont sans doute permis d'améliorer et préciser mon propos. Je voudrais également remercier Véronique BAGLAND d'avoir accepté de participer à mon jury. Je ne saurais oublier de remercier mon ami et frère Abdul-majeed pour sa coopération permanente et son appui inconditionnel dans une ambiance de fraternité pour la réussite de ce projet. J'adresse un merci particulier à tonton Ahemed Mohamed ainsi qu'à sa merveilleuse femme Saada Ahemed, à mes amis Bouh, Abdi-chakour pour les moments passés ensemble les dimanches, qui me remplissaient de joie et d'énergie pour le reste de la semaine. Je remercie aussi à mes collègues de l'université de Djibouti et en particulier à mes collègues et amis de la faculté des sciences. Parmi eux Liban Ismael, Abdirazack Mohamed, Yahyeh lakios, Liban Ohtman, Kosad, Thomas, Yonis... Mon coeur bat pour ma famille et belle famille, étendue et proche, Un immense merci à mes parents sans qui rien n'aurait été possible. En particulier à ma mère, qui a oeuvré pour ma réussite, de par son amour, son soutien, tous les sacrifices consentis et ses précieux conseils, pour toute son assistance et sa présence dans ma vie, reçois à travers ce travail aussi modeste soit-il, l'expression de mes sentiments et de mon éternelle gratitude. J'exprime toute ma gratitude du font du coeur à ma belle-mère pour son amour infini pour moi et mes enfants, soutiens constant, son empressement à faire passer nos besoins avant les siens et son soin le plus attentif m'ont donné le luxe et l'assurance nécessaires pour m'aventurier dans le monde de la recherche en les sachant en sécurité et entourée de tendresse. A mon brillant frère Mahamoud Yassin et adorable soeur Linda Yassin qui ont su m'encourager. Je tiens également à remercier mes belles soeurs, Deka, Mariamouna, Fozia et en particulier à Dr Djama et Farhan pour leurs soutiens et confiances à mon égard. Je ne peux finir sans remercier et dédier cette thèse à ma merveilleuse femme Neima Elmi et la mère la plus tendre et la plus attentionnée, a été une compagnon de vie rêvé. Notre histoire n'est pas terminée, et j'attends avec impatience les nombreuses aventures à venir. L'amour un détail qui reste très sensible, celui qui change le parcours de notre vie, on ressent des choses étranges, jamais vue auparavant, des sentiments profondes nous envahissent, il construit un lien très fort entre-deux personnes pour les réunir sous les mêmes ambitions et fonder leur propre foyer. Ton sourire, tes encouragements, tes échanges ma permis te tenir jusqu'au bout. Merci pour tous, merci d'avoir supporté mon absence et d'élever avec amour nos deux adorables enfants Hikma et Yassin.

Chapter 1

Introduction générale

Depuis les années 1950, la théorie de points fixes est une branche très active des mathématiques et elle est au coeur de l'analyse puisqu'elle fournit un outil puissant pour établir l'existence de solutions à de nombreux problèmes non linéaires posés en théorie des jeux, économie, mécanique, biologie, etc... En fait, dans de nombreux problèmes stationnaires concrets, en général, on transforme le problème en un problème de point fixe et on cherche les solutions comme étant les points fixes du problème en utilisant des hypothèses sur les opérateurs intérvenants dans le problème ou sur la structure de l'espace de Banach. Les théorèmes du points fixes trouvent leurs origines au XIX siécle dans les travaux des grands mathématiciens. Citons à titre d'exemple : Picard, Banach, Brouwer, Schauder, etc... Dans ce projet de thèse, on présente quelques résultats portant sur les théorèmes de points fixes pour une classe d'applications univoques et multivoques, ensuite on donne des applications pour des problèmes intervenant en neutronique et d'autres portant sur des équations intégrales de type Volterra. Ce travail est composé d'une introduction, cinq chapitres et d'une liste de références bibliographies.

• Dans le chapitre 2, nous rappelons les principaux concepts, définitions et résultats qui sont requis tout au long des chapitres suivants. Tout d'abord, on rappelle la définition des mesures de non compacités et, en particulier celle de Kuratowskii et celle de Hausdorff. On introduit aussi la mesure de non faible compacité de De Blasi, que nous utiliserons dans les chapitres 4-6. On donne les définitions des opérateurs contractants ou k-contractants, non expansifs, pseudo-contractifs, contractions non linéaires et les opérateurs ww-compacts et ws-compacts. Nous définissons également des opérateurs de superposition. Enfin, ce chapitre est clôturé en introduisant la définition d'opérateur multivoque, et quelques résultats de points fixes qui sont nécessaires dans la suite.

• Dans le chapitre 3, On établit des théorèmes de points fixes pour des opérateurs condensants dénombrables par rapport à une mesure de non compacité. La théorie des l'opérateurs condensants a commencé en 1967 avec le travail de Sadovskii [START_REF] Sadovski | A fixed point principle[END_REF]. Soient K un sous-ensemble borné, convexe fermé de l'epace de Banach X et f : K → K un opérateur univoque. Si f est continu et condensant par rapport à la mesure de non compacité µ c'est à dire pour tout sous-ensemble borné C de K, µ( f (C)) ≤ µ(C), alors f admet un point fixe. Le résultat de Sadovskii a été amélioré en utilisant différentes idées: Daher [START_REF] Daher | On a fixed point principle of Sadovskii[END_REF] a montré qu'il est valide pour les opérateurs condensants dénombrables par rapport à une mesure de non compacité. Motivé par l'article de H. Monch [START_REF] Mönch | Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces[END_REF], In-Soon Kim [START_REF] Kim | Fixed points of countably condensing mappings and its application to nonlinear eigenvalue problems[END_REF] obtient un théorème de type Larey-Schauder pour des opérateurs dénombrablement condensants univoques. Dans [START_REF] Himmelberg | Fixed point theorems for condensing multifunction[END_REF], Himmelberg, Porter et Van Vleck ont également étendu la définition des opérateurs condensants aux applications multivoques et ont obtenu une nouvelle version du théorème de Sadovskii. Un résultat analogue au théorème du point fixe de Daher a été obtenu par Agarwal et O'Regan [START_REF] Agarwal | A generalization of the Krasnoselskii-Petryshyn compression and expansion theorem: an essential map approach[END_REF] (pour les multivoques semi-continues supérieures) dans un espace de Banach séparable. Pour plus d'information sur les opérateurs dénombrablement condensants multivoques, on renvoie le lecteur aux références [START_REF] Cardinali | Rubbioni Countably condensing multimaps and fixed points[END_REF][START_REF] Dhage | A fixed point theorem for multi-valued mappings in ordered Banach spaces with applications I[END_REF]. Nos résultats pour les opérateurs condensant dénombrable présentés dans ce chapitre concernent précisement des théorèmes de points fixes de type Altman, de type Leray-Schauder, de type Krasnoselskii et de type Krasnoselskii-Shaefer (le cas du domaine non borné).

• Le Chapitre 4, est consacré au problèmes de points fixes pour les opérateurs ws-compact et ww-compact (voir Définition 2.1.13). En 2006, K. Latrach, M.A. Taoudi et A. Zeghal [START_REF] Latrach | Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations[END_REF], ont introduit les opérateurs ws-compact et ww-compact et ont obtenu des nouveaux théorèmes de points fixes. En fait, l'intérêt des opérateurs ws-compacts réside dans le fait qu'ils permettent d'obtenir des théorèmes de points fixes de type Schauder, Darbo, Sadovskii, Krasnosel'skii ou encore de type Schaefer (dans des domaines non bornés) pour les opérateurs transformant un sous-ensemble faiblement compact dans lui-même sans que ceux-ci soient faiblement continus. Observant que, sur un espace de Banach, tout opérateur de Dunford-Pettis (voir Définition 2.1.14) est ws-compact et que tout opérateur faiblement compact sur un espace de Banach vérifient la propriété de Dunford-Pettis (voir Définition 2.1.14) est un opérateur ws-compact. L'objet de ce chapitre est de développer des théorèmes de points fixes de type Schauder et Krasnosel'skii pour des problèmes faisant intervenir deux ou trois opérateurs tel que un est ww-compacts, l'autre wscompacts et le dernier est contractif.

• Le chapitre 5 est dédié aux opérateurs multivoques. Soient X, Y deux espaces normés. Un opérateur multivoque T de X vers Y est une correspondance qui associe à tout élément x ∈ X un sous ensemble T x de Y (on désigne généralement un operateur multivoque par T : X → P(Y) où P(Y) est l'ensemble des parties de Y). Un opérateur multivoque T peut être traité comme un opérateur univoque de X dans P(Y) [46]. Nadler [START_REF] Sam | Multi-valued contraction mappings[END_REF] a commencé le développement de la théorie du point fixe des opérateurs multivoques contractants pour la distance de Pompieu-Hausdorff et il a prouvé l'existence des points fixes dans les espaces métriques complets, en généralisant ainsi le principe de contraction de Banach. Ensuite, il y a eu de nombreux travaux concernant l'existence de points fixes pour les opérateurs multivoques. Des résultats de type Schauder [START_REF] Himmelberg | Fixed points of compact multifunctions[END_REF] et de type Krasnosel'skii [START_REF] Al-Thagafi | Krasnosel'skii-type fixed-set results[END_REF] (voir aussi [START_REF] Basoc | A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps[END_REF][START_REF] Biondini | Existence of solutions for a nonlinear integral equation via a hybrid fixed point theorem[END_REF][START_REF] Cardinali | Monch sets and fixed point theorems for multimaps in locally convex topological vector spaces[END_REF][START_REF] Dhage | Multivalued operators and fixed-point theorems in Banach algebras[END_REF][START_REF] Dhage | Mutli-valued mappings and fixed points II[END_REF][START_REF] Graef | Multivalued versions of a Krasnosel'skii-type fixed point theorem[END_REF][START_REF] Ok | Fixed set theorems of Krasnoselskii type[END_REF][START_REF] Reich | Fixed points of condensing functions[END_REF][START_REF] Somyot | Fixed point theorems of Krasnosel'skii type for the sum of two multivalued mappings in Banach spaces[END_REF]). Ce travail est motivé par un résultat de Himmelberg [START_REF] Himmelberg | Fixed points of compact multifunctions[END_REF] cf. Théorème 2.3.1 qui affirme que, si M est un sous-ensemble convexe fermé non vide d'un espace de Banach X et F : M → P cl,cv (M) est un opérateur multivoque semicontinu supérieurement et F(M) est relativement compact, alors il existe x ∈ M tel que x ∈ F(x). En utilisant ce résultat et en prenant des selections ws-compact et des perturbations ww-compact (voir Définition 2.1.13), on montre de nouveaux théorèmes de points fixes de type Krasnosel'skii pour la somme F + B où F est un opérateur multivoque et B est une perturbation (selon le cas, elle pourra être supposée univoque ou multivoque).

• Dans le chapitre 6, on va présenter quelques applications de nos résultats.

La première section de ce chapitre est consacrée à deux équations intégrales de type Volterra de la forme: La deuxième section, est dédiée à l'existence de solution de l'équation de transport des neutrons rétardés de la forme:

ϕ(t) = H(t) +
                     v.∇ x f 0 (x, v) + σ(x, v, f 0 (x, v)) + λ f 0 (x, v) = + R N κ 0 (x, v, v )Θ 0 (x, v , f 0 (x, v ))dµ(v ) + d i=1 λ i β i (x, v) f i (x, v) λ i f i (x, v) = R N κ i (x, v, v )Θ i (x, v , f 0 (x, v ))dµ(v ), 1 ≤ i ≤ d (1.0.3) où (x, v) ∈ D × R N .
Les conditions aux limites sont modélisées par l'équation

f - 0 = H( f + 0 ). (1.0.4)
Après quelques transformations, le problème (1.0.3)-(1.0.4) se réduit au problème de point fixe suivant

f 0 = F(λ) f 0 + B(λ) f 0 , f 0 |Γ -= H( f 0 |Γ + ).
(1.0.5)

Pour montrer que l'équation (1.0.3)-(1.0.4) admet des solutions, il suffit de montrer que le problème de point fixe (1.0.5) admet des solutions. Pour cela on vérifie que (1.0.5) satisfait aux hypothèses du Corollaire 4.2.1.

Dans la dernière section, on étudie l'existence de solution de l'équation de transport avec des opérateurs de collisions élastiques et inélastiques de la forme :

v. ∂ϕ ∂x (x, v) + σ(x, v, ϕ(x, v)) + λϕ(x, v) = V k c (x, v, v ) f (x, v , ϕ(x, v ))dµ(v ) + l j=1 S N-1 k j d (
x, ρ j , ω, ω )ϕ(x, ρ j ω )dω

+ S N-1
k e (x, ρ, ω, ω )g(x, ρ, ω , ϕ(x, ρ, ω ))dω (1.0.6) où (x, v) ∈ Ω × V, (V est appelé l'espace des vitesses admissibles et Ω est un ensemble ouvert et borné de R N (N ≥ 3) muni de la mesure de Lebesgue dx). Les conditions aux limites sont modélisées par

ϕ | Γ-= 0, (1.0.7)
Notre but dans cette section est de résoudre le problème stationnaire (1.0.6)-(1.0.7) dans L p -espace (1 ≤ p < ∞), la stratégie consiste à transformer le problème (1.0.6)-(1.0.7) au problème de point fixe suivant

ψ = A(ψ) + C(ψ)
où A, et C sont des opérateurs non linéaires. Pour résoudre le dernier problème, on utilise les arguments de compacité établis dans [START_REF] Boumhamdi | Existence results for a nonlinear transport equation with unbounded admissible velocities space[END_REF][START_REF] Latrach | Existence results for a nonlinear transport equation in bounded geometry on L 1 -spaces[END_REF], et les théorèmes du points fixes de Krasnosels'kii (voir, [START_REF] Abdallah | Some fixed point theorems in Banach space and aplication to a transport equation with delayed neutrons[END_REF]Corollary 3.8]).

General introduction

The theory of fixed points, since 1950 is a very active branch of mathematics and is at the heart of nonlinear analysis since it provides powerful tool to establish existence of solutions for many nonlinear problems arising in mechanic, economics, biology etc.. In fact, in many concrete problems, we seek solutions as fixed point of the problem using hypotheses on the mappings involved in the problem or on the structure of the Banach space.

In this thesis project we present some fixed point results for single-valued as well as multivalued mappings. It is divided into four chapters.

• In Chapter 2, we recall the main concepts, definitions and results which are required throughout Chapters 3-6. First, we recall the definition of measures of non-compactness and measures of weak noncompactness. As examples we recall the measure of noncompactness of Kuratowskii and that of Hausdorff. The measure of weak noncompactness of De Blasi, which we will use in Chapters 3-6, is also recalled. We give the definitions of the contraction or k-contractive, nonexpansive, pseudocontractive, accretive and ψ-contraction operators, ws-compact and ww-compact operators.

The definition of superposition operators and some of their properties are given. Further, the concept of multivalued mapping is introduced. Throughout this chapter, some fixed point results required in the sequel are stated.

• In Chapter 3, we discuss fixed point results for countably condensing mappings with respect to the measure of noncompactness. The theory of condensing operator began in 1967 with the paper of Sadovskii [START_REF] Sadovski | A fixed point principle[END_REF]. Let K be a nonempty closed, bounded and convex subset of a Banach space X and let f : K → K be a single valued maps. If f is continuous and condensing with respect to a measure of noncompactness µ, that is for all bounded subset C of K, µ( f (C)) < µ(C), then f has a fixed point. Sadovskii's result was improved in the literature using different ideas: Daher [START_REF] Daher | On a fixed point principle of Sadovskii[END_REF] showed that it is still true for countably condensing maps with respect to a measure of noncompactness; motivated by H. Mönch paper [START_REF] Mönch | Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces[END_REF], In-Soon Kim [START_REF] Kim | Fixed points of countably condensing mappings and its application to nonlinear eigenvalue problems[END_REF] derives a Leray-Schauder's type fixed point theorem for countably condensing mappings and applies it to discuss an eigenvalue problem. In [START_REF] Himmelberg | Fixed point theorems for condensing multifunction[END_REF], Himmelberg, Porter and Van Vleck extended the definition of condensing maps to multimaps and derived a multivalued version of Sadovskii's theorem; a multivalued analogue of Daher's fixed point theorem was obtained by Agarwal and O'Regan [START_REF] Agarwal | A generalization of the Krasnoselskii-Petryshyn compression and expansion theorem: an essential map approach[END_REF] (for upper semi-continuous multimaps) in a separable Banach space. For further results about countably condensing multimaps, we refer the reader to papers [START_REF] Cardinali | Rubbioni Countably condensing multimaps and fixed points[END_REF][START_REF] Dhage | A fixed point theorem for multi-valued mappings in ordered Banach spaces with applications I[END_REF] and the references therein.

Here, beside a result of Altma's type, our analysis is focussed on results of Leray-Shauder's type, Krasnoselskii's type and Krasnoselskii-Shaefer's type (the case of unbounded domain).

• Chapter 4, is devoted to fixed point problem involving ws-compact and ww-compact (see Definition 2.1.13). In 2006, K. Latrach, M.A. Taoudi and A. Zeghal [START_REF] Latrach | Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations[END_REF], introduced ws-compact and ww-compact operators and obtained new fixed point theorems. The interest of the class of ws-compact operators lies in the fact that they make possible to obtain fixed point theorems of types Schauder, Darbo, Sadovskii, Krasnosel'skii or even Schaefer type (in unbounded domains)

for operators transforming each weakly compact subsets into itself. Here we derive new fixed points results of Schauder's type and others of Krasnosel'skii 's for fixed point problems involving ws-compact and ww-compact mappings. Observing that, on a Banach space, every Dunford-Pettis operator (see Definition 2.1.14) is ws-compact and that every weakly compact operator on a Banach space with the Dunford-Pettis property (see Definition 2.1.14) is ws-compact operaors, we extend our results to these two classes of operators.

• Chapter 5 is dedicated to multivalued mappings, that is, mappings from X into P λ (Y) (the set of all subsets of Y) where X and Y are two normed spaces. In [START_REF] Sam | Multi-valued contraction mappings[END_REF], Nadler began the development of fixed point theory for contracting multivalued maps using the Pompieu-Hausdorff distance which is defined on P cl,bd (X) (the set of all closed bounded subsets of Y). He proved existence fixed points for this class of mappings. Later, several works dedicated to fixed point theory for multivalued mappings were published. Recently, results involving the weak topology and the measure of weak noncompactness were derived (see, for example, [START_REF] Al-Thagafi | Krasnosel'skii-type fixed-set results[END_REF][START_REF] Basoc | A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps[END_REF][START_REF] Biondini | Existence of solutions for a nonlinear integral equation via a hybrid fixed point theorem[END_REF][START_REF] Cardinali | Monch sets and fixed point theorems for multimaps in locally convex topological vector spaces[END_REF][START_REF] Dhage | Multivalued operators and fixed-point theorems in Banach algebras[END_REF][START_REF] Dhage | Mutli-valued mappings and fixed points II[END_REF][START_REF] Graef | Multivalued versions of a Krasnosel'skii-type fixed point theorem[END_REF][START_REF] Ok | Fixed set theorems of Krasnoselskii type[END_REF][START_REF] Reich | Fixed points of condensing functions[END_REF][START_REF] Somyot | Fixed point theorems of Krasnosel'skii type for the sum of two multivalued mappings in Banach spaces[END_REF] and the reference therein). This work is motivated by a Himmelberg's result [START_REF] Himmelberg | Fixed points of compact multifunctions[END_REF] cf. Theorem 2.3.1 which asserts that, if M is a nonempty closed, convex subset of a Banach space X and F : M → P cl,cv (M) is a upper semicontinuous multivalued map and F(M) is relatively compact, then there exists x ∈ M such that x ∈ F(x). Using this result together with conditions ws-compact and ww-compact ( see Definition 2.1.13), we prove several new Krasnosel'skii-type fixed set theorems for the sum F + B where F is a multivalued mapping and B is a perturbation. Various kinds of perturbations were considered. The main tools used in this chapter are the measure of weak noncompactness, the concept of condensing multivalued (or single valued) mappings and the concept of ws-compact and ww-compact and Theorem 2.3.1.

• In Chapter 6 we present applications of our results to the study of the existence of solutions to some integro-differential equations.

The first section of this chapter deals with two Volterra integral equations:

ϕ(t) = H(t) + t a f (s, ϕ(s))ds, ∀t ∈ [a, b] (1.0.8)
where

f is a map from [a, b] × X into X and H maps [a, b] into X. ϕ(t) = H(ϕ(t)) + t 0 g(s, ϕ(s))ds t ∈ [0, 1]; (1.0.9)
where g : [0, 1] × X → X, H : X → X are two maps and X is a Banach space. Our objective is discuss existence of the solutions to equation (1.0.8) (resp. (1.0.9)) in the space C([a, b], X) (resp. C([0, 1], X)). The idea of proof consists in transforming these two equations to two fixed point problems and we check that the involved operators satisfy the assumptions of Proposition 3.3.2.

In the second section, we will discuss the existence of solution to the following boundary value problem which models a transport equation with delayed neutrons

                     v.∇ x f 0 (x, v) + σ(x, v, f 0 (x, v)) + λ f 0 (x, v) = + R N κ 0 (x, v, v )Θ 0 (x, v , f 0 (x, v ))dµ(v ) + d i=1 λ i β i (x, v) f i (x, v) λ i f i (x, v) = R N κ i (x, v, v )Θ i (x, v , f 0 (x, v ))dµ(v ), 1 ≤ i ≤ d (1.0.10)
where (x, v) ∈ D × R N . The boundary conditions are modeled by

f - 0 = H( f + 0 ). (1.0.11)
After some transformations, Problem (1.0.10)-(1.0.11) reduces to the following fixed point problem

f 0 = F(λ) f 0 + B(λ) f 0 , f 0 |Γ -= H( f 0 |Γ + ), (1.0.12)
So, to solve Problem (1.0.10)-(1.0.11), it suffices to seek solutions to the fixed point problem (1.0.12). This can be made with the help of Corollary 4.2.1

In the last section, we will study the existence of solution of the transport equation with elastic and inelastic collision operators of the form :

v. ∂ϕ ∂x (x, v) + σ(x, v, ϕ(x, v)) + λϕ(x, v) = V k c (x, v, v ) f (x, v , ϕ(x, v ))dµ(v ) + l j=1 S N-1 k j d (x, ρ j , ω, ω )ϕ(x, ρ j ω )dω + S N-1 k e (x,
ψ = A(ψ) + C(ψ)
where A, and C are nonlinear operators. To solve the last problem we use the compactness arguments established in [START_REF] Boumhamdi | Existence results for a nonlinear transport equation with unbounded admissible velocities space[END_REF][START_REF] Latrach | Existence results for a nonlinear transport equation in bounded geometry on L 1 -spaces[END_REF], and the Krasnoselskii fixed point theorems (see, [START_REF] Abdallah | Some fixed point theorems in Banach space and aplication to a transport equation with delayed neutrons[END_REF]Corollary 3.8]).

7

Chapter 2

Preliminary

In this chapter, we recall some definitions and results required in the next chapters.

Some definitions and properties

Let X be a Banach space endowed with the • . We introduce the following notations:

P(X) = {D ⊂ X : D is a non-empty subsets of X}, B(X) = {D ∈ P(X) such that D is bounded}, C r (X) = {D ∈ P(X) such that D is relatively compact}.
Let K be a subset of X. We denote by B c (K) the set (f) Algebraic semi-additivity:

B c (K) = D ⊂ K such that D
β(B 1 + B 2 ) ≤ β(B 1 ) + β(B 2 ), ∀ B 1 , B 2 ∈ B(X), (g) if (M n ) n∈N
is a decreasing sequence of nonempty and closed sets in B(X) such that

lim n→∞ β(M n ) = 0, then the intersection set M ∞ = ∩ ∞ n=1 M n is nonempty.
The concept of measure of noncompactness was first introduced by Kuratowski in [START_REF] Kuratowski | Sur les espaces complete[END_REF] (Kuratowskii measure of noncompactness). It was defined in the following way:

α : B(X) → R + , B → α(B) = inf r > 0 : B ⊂ ∪ n i=1 D i , diam(D i ) ≤ r .
Another classical example is the Hausdorff measure of noncompactness χ. It was introduced by Goldenstein et al. [START_REF] Goldenstein | Investigations of somme properties of bounded linear operators with their q-norms[END_REF].

It is defined by

χ : B(X) → R + , B → χ(B) = inf r > 0 : B ⊂ ∪ n i=1 B r (x i ), x i ∈ X .
Notation: Throughout this chapter, β(•) will denote a measure of noncompactness in the sense of Definition 2.1.1.

Definition 2.1.2 Let K be a subset of a Banach space X and let β be a measure of noncompactness on X. Let f : K → X be a map.

(a) We say that f satisfies the Darbo condition for bounded countable sets if there exists k > 0 such that

β( f (B)) ≤ kβ(B), ∀B ∈ B c (K); (2.1.1)
(b) f is said to be countably β-k-contractive if it satisfies the condition (2.1.1) for some k ∈ [0, 1);

(c) f is said to be countably β-condensing if β( f (B)) < β(B) for all B ∈ B c (K) with β(B) > 0.

(d) f is said to be countably

1-set-contractive if β( f (B)) ≤ β(B) for all B ∈ B c (K) with β(B) > 0.
It is clear that, every β-k-contractive (resp. β-condensing) map is countably β-k-contractive (resp. countably β-condensing) but the converse is not in general true.

Recall that a map f :

K ⊂ X → X is said to be k-Lipschitzian if f (x) -f (y) ≤ k x -y with k ≥ 0. If k = 1, f is called nonexpansive . Further, if k ∈ [0, 1), then f is called a contraction (or k-contractive).
The following lemma is required in the sequel.

Lemma 2.1.1 Let K be a nonempty subset of a Banach space X and let f : K → X be a k-Lipschitizian map. Then, for each C ∈ B c (K), we have

β( f (C)) ≤ kβ(C).
Proof. It is well known that if f is k-Lipschitizian, then, for each bounded subset C of K, we have β( f (C)) ≤ kβ(C). The result follows from the fact that countably bounded subsets of K are also bounded in K.

2

We recall the definition of φ-contraction mappings.

Definition 2.1.3 Let X be a Banach space. A mapping B : X → X is said to be a Φ-contraction if there exists a continuous function φ : R + → R + such that φ(r) < r for any r > 0 and, for all x, y ∈ X, we have

B(x) -B(y) ≤ φ x -y .
The function φ is called the Φ-function of B.

Moreover, if the function φ is nondecreasing, then B is called a nonlinear contraction .

2 Remark 2.1.1 We note that if φ(r) = αr, α ∈ (0, 1), then B is α-contractive. Hence, any αcontractive map, for some α ∈ (0, 1), is a nonlinear contraction, however, the converse is, in general, not true.

2

Recall that it is well known (cf. [START_REF] Boyd | On nonlinear contractions[END_REF]) that each nonlinear contraction on a Banach space B : X → X possesses a unique fixed point and, moreover, (I -B) is invertible and, for each subset K of X,

(I -B) is a homeomorphism from K onto (I -B)(K).
It is clear that, every β-k-contractive (resp. β-condensing) map is countably β-k-contractive (resp. countably β-condensing) but the converse is, in general, false.

The following lemma is required in the sequel.

Lemma 2.1.2 Let K be a nonempty subset of the Banach space X . If the map g : K ⊂ X → X is a nonlinear φ-contraction, then Ig is a homeomorphism from K onto Ig (K).

Proof. Since g is a nonlinear φ-contraction, there exists a continuous function φ : R + → R + such that φ(r) < r for any r > 0 and, for all x, y ∈ X, we have

g(x) -g(y) ≤ φ x -y .
Let x, y ∈ K, x y we have

(I -g)x -(I -g)y = (x -y) -(g(x) -g(y)) ≥ x -y -g(x) -g(y) ≥ x -y -φ( x -y ) > 0.
Hence, (Ig) is an injective map on K and therefore (Ig) -1 exists from (Ig)(K) to K. To show the continuity of (Ig) -1 suppose that there exists a point x and a sequence (x n ) n∈N of points of K such that (Ig)(x n ) → (Ig)(x) and lim n→+∞ sup k≥n x kx = a. Due to inequality

(I -g)x n -(I -g)x ≥ x n -x -φ( x n -x ),
we see that 0 ≥ a -φ(a). Since φ(a) < a, we deduce that a = 0 and so (Ig) -1 is continuous. 2 Definition 2.1.4 Let X be a Banach space and let f : K ⊂ X → X be a mapping on X .

(a) We say that f is pseudocontractive if, for all x, y ∈ K and, r > 0, one has xy ≤ r( f (y)f (x)) + (1 + r)(xy) .

(b) We say that f is accretive if the inequality

xy + λ( f (x)f (y)) ≥ xy holds for all, λ > 0, and x, y ∈ K.

It is not difficult to check that every nonexpansive map is pseudocontractive. For a proof, let x, y ∈ K and r > 0. We have

r( f (y) -f (x)) + (1 + r)(x -y) ≥ (1 + r) x -y -r f (x) -f (y) ≥ (1 + r) x -y -r x -y ≥ x -y 2
Remark 2.1.2 Note that there exists connection between pseudocontractive operators and accretive operators which we shall use later. It may be asserted as follows: f is pseudocontractive if and only if If is accretive where I is the identity operator.

2

We recall also the following elementary lemmas where the proofs can be found, for example, in [START_REF] Latrach | Introduction à la théorie des points fixes métrique et topologique: avec applications et exercices corrigés[END_REF].

Lemma 2.1.3 Let K be a subset of the Banach space X and let g : K → X be a k-contractive for some k ∈ (0, 1). Then the operator Ig is a homeomorphism from K onto Ig (K) and

I -g -1 (x) -I -g -1 (y) ≤ 1 1 -k x -y , ∀ x, y ∈ I -g (K)
Proof. Let x, y ∈ X, we have

(I -g)(x) -(I -g)(y) ≥ x -y -g(x) -g(y) ≥ (1 -k) x -y .
This shows that the map (Ig) is injective and therefore invertible with inverse (Ig) -1 : (Ig)(K) → K. The continuity of (Ig) -1 results from the previous inequality. Let u, v ∈ (Ig)(K).

By substituting in the previous inequality (Ig) -1 (u) by x and (Ig) -1 (v) by y, we obtain

(I -g) -1 (u) -(I -g) -1 (v) ≤ 1 1 -k u -v . 2
Definition 2.1.5 Let K be a subset of a Banach space X. The mapping f : K → X is said to be expansive, if there exists a constant h > 1 such that, for all x, y ∈ K we have f (x)f (y) ≥ h xy .

Lemma 2.1.4 Let K be a subset of the Banach space X and let g : K → X be a continuous expansive mapping with constant h > 1. Then the operator Ig is a homeomorphism from K onto Ig (K) and

I -g -1 (x) -I -g -1 (y) ≤ 1 h -1 x -y , ∀ x, y ∈ I -g (K).
Proof. We note that (Ig) is a continuous mapping. Let x, y ∈ X, we have

(I -g)(x) -(I -g)(y) = (g(x) -g(y)) -(x -y) ≥ (h -1) x -y . (2.1.2)
Hence, (Ig) is an injective map on K and therefore (Ig) -1 exists from (Ig)(K) to K.Let u, v ∈ (Ig)(K). By substituting in the equation (2.1.2) (Ig) -1 (u) by x and (Ig) -1 (v) by y, we obtain

(I -g) -1 (u) -(I -g) -1 (v) ≤ 1 h -1 u -v . 2
In this note, we give the following definition:

Definition 2.1.6 An operator f : D( f ) ⊂ X → X is said ψ-expansive if there is a function ψ : [0, ∞) → [0, ∞) such that for every x, y ∈ D( f ), the inequality f (x) -f (y) ≥ ψ( x -y ) holds. The function ψ satisfies • ψ(0) = 0;
• ψ(r) > 0 for any r > 0;

• either it is continuous or it is nondecreasing.

Clearly, if we take φ(t) = αt with α > 1, then f is an expansive mapping in the sense of Definition 2.1.5.

We need also the following result. We think that it is probably known but, since we have any reference, we shall give a proof.

Lemma 2.1.5 Let K be a nonempty subset of a Banach space X, β a measure of noncompactness on X and g : K → X a map. If g is a nonlinear φ-contraction, then, for each bounded subset S of K with β(S ) > 0, we have β(g(S )) < β(S ).

Proof. Let ε > 0 be an arbitrary real. Suppose that S ⊆ ∪ n i=1 S i with diam(S i ) ≤ β(S ) + ε. Then we have g(S ) ⊆ ∪ n i=1 g(S i ) = ∪ n i=1 F i where F i := g(S i ). Let y 1 , y 2 ∈ F i where i ∈ 1, 2, • • •, n , there exists x 1 , x 2 ∈ S i such that y 1 = g(x 1 ) and y 2 = g(x 2 ). Since g is a nonlinear φ-contraction, one can write g(x 2 )g(x 1 ) ≤ φ x 2x 1 ≤ φ β(S ) + ε . Hence, we conclude that diam(F i ) ≤ φ β(S ) + ε and therefore β(g(S )) ≤ φ β(S ) + ε < β(S ) + ε because g is a nonlinear φ-contraction. Since ε is an arbitrary real number, we infer that β(g(S )) < β(S ).

2 Definition 2.1.7 Let X be a normed topological vector space and let K be a subset of X.

• We say that K is absorbing if, for every x ∈ X, there exists t ≥ 0 such that x ∈ tK.

• We say that K is balanced if λK ⊂ K for every λ with |λ| ≤ 1. Now we can define the Minkowski's Functionals. Definition 2.1.8 (Minkowski's functional) Let K be a convex absorbing subset of a normed space X. We define the Minkowski functional of K, µ K (•), by µ K (x) = inf t > 0 : x ∈ t K .

Note that for each x ∈ X, the set of real η > 0 such that x ∈ ηK is not empty (because K is absorbing), so the definition of µ K (•) has a meaning. Lemma 2.1.6 [START_REF] Takác | A spectral mapping theorem for the exponential function in linear transport theory[END_REF] Let K be an open convex absorbing subset of a normed space X, θ ∈ K. Then the Minkowski functional µ K (•) of K is nonnegative and continuous on X. Moreover,

(a) µ K (λx) = λµ K (x), ∀λ ≥ 0; (b) x ∈ K if and only if 0 ≤ µ K (x) < 1; (c) x ∈ ∂K if and only if µ K (x) = 1; (d) x K if and only if µ K (x) > 1.
Definition 2.1.9 Let K be a subset of X and f : K → X. We say that f has a fixed point in K if there exists x ∈ K such that f (x) = x. The set of all fixed points of f is denoted by Fix( f ).

We now recall the following result due to Daher [START_REF] Daher | On a fixed point principle of Sadovskii[END_REF]. It is an extension of Sadovskii's fixed point theorem to countably β-condensing maps .

Theorem 2.1.1 (Daher) Let K be a nonempty, bounded, closed and convex subset of a Banach space X and let β be a measure of noncompactness on X. If f : K → K is a continuous countably β-condensing map, then Fix( f ) ∅.

We recall also the following result (cf. [START_REF] Xiang | A class of expansive-type Krasnosel'skii fixed point theorems[END_REF]).

Theorem 2.1.2 Let K be a closed subset of X and let f : K → X be an expansive mapping. If f (K) ⊃ K, then Fix( f ) ∅ .

We will introducing the definition of semi-closed mappings.

Definition 2.1.10 Let K be a nonempty subset of a Banach space X. A map f : K → X is said to be semi-closed if for every closed subset F of K, the set (If )(F) is closed. Lemma 2.1.7 Let K be nonempty closed, bounded and convex subset of a Banach space X, and let f : K → K be a continuous, countably 1-set-contractive map. If f is semi-closed and f (K) is bounded, then Fix( f ) ∅.

Proof. Let z ∈ K and define the maps f n by f n := t n f +(1-t n )z where n ∈ N and (t n ) n∈N a sequence of (0, 1) such that t n → 1 as n → +∞. The convexity of K implies that f n maps K into itself. Let C ∈ B c (K). Then we have

β( f n (C)) ≤ β t n f (C) + (1 -t n )z ≤ t n β( f (C)) ≤ t n β(C). If β(C) 0, then β( f n (C)) < β(C) and therefore f n is countably β-condensing. Hence, according to Theorem 2.1.1, f n has a fixed point x n ∈ K, that is x n = t n f (x n ) + (1 -t n )z. Note that, ∀x ∈ K, we have f (x) -f n (x) = f (x) -t n f (x) + (1 -t n )z = |1 -t n | f (x) -z .
Moreover, the boundedness of f (K) implies f (x)f n (x) → 0 as n → ∞. Now using the equality above we get (

I -f )(x n ) = ( f n (x n ) -f )(x n ) = |(1 -t n )| z -f (x n ) → 0 as n → ∞.
Next, using the fact that f is semi-closed and K is closed, we infer that 0

∈ (I -f )(K). Hence, Fix( f ) ∅.
Let X be a Banach space. Let us first introduce the following two subsets W r (X) = {D ∈ B(X) such that D is relatively weakly compact}, and

W(X) = {D ∈ B(X) such that D is weakly compact}.
In the sequel, we denote by B r the closed ball in X centred at 0 with radius r.

Definition 2.1.11 A map µ : B(X) → [0, +∞[ is said to be a measure of weak noncompactness on X if it satisfies the following conditions:

(a) The family ker µ For the axiomatic definition of the measure of weak noncompactness we refer, for example, to [START_REF] Banaś | On measures of weak noncompactness[END_REF].

:= M ∈ B(X) : µ(M) = 0 is nonempty and ker µ is contained in W r (X). (b) Monotonicity: M 1 ⊂ M 2 ⇒ µ(M 1 ) ≤ µ(M 2 ), for all M 1 , M 2 ∈ B(X) (c) Invariance under passage to the closed convex hull: ∀M ∈ B(X), µ(co(M)) = µ(M) where co denotes the closed convex hull of M. (d) µ(M 1 + M 2 ) ≤ µ(M 1 ) + µ(M 2 ). for all M 1 , M 2 ∈ B(X) (e) Homogeneity µ(αM) = |α| µ(M) for any real number α and M ∈ B(X) (f) Generalized Cantor intersection theorem: If (M n ) n≥1 is a sequence of nonempty, weakly closed subsets of X with M 1 bounded and M 1 ⊇ M 2 ⊇ . . . ⊇ M n ⊇ . . . such that lim n→∞ µ(M n ) = 0, then ∞ n=1 M n ∅ and µ( ∞ n=1 M n ) = 0.
We recall that the first example of measure of weak noncompactness was introduced by De Blasi in 1977 (cf. [START_REF] De Blasi | On a property of the unit sphere in a Banach space[END_REF]). The De Blasi measure of weak noncompactness ω(•) is defined by

ω : B(X) → [0, +∞[, A -→ ω(A) with ω(A) = inf t > 0 : there exists K ∈ W(X) such that A ⊆ K + tB 1
where B 1 denotes the closed unit ball of X. It satisfies all axioms listed in Definition 2.1.11 and more. For the proofs of the properties of ω(•) we refer to [START_REF] De Blasi | On a property of the unit sphere in a Banach space[END_REF][START_REF] Emmanuele | Measure of weak noncompactness and fixed point theorems[END_REF] or [48, p. 225].

Notation : In the remainder of this report ω(•) will always denote the measure of weak noncompactness of De Blasi.

Definition 2.1.12 Let X be a Banach space and B : X → X a single-valued mapping. We say that B is weakly sequentially continuous if it is sequentially continuous for the weak topology σ(X, X * ) of X and B is said to be weakly continuous if it is continuous for the weak topology σ(X, X * ) of X.

For the definition and properties of the weak topology σ(X, X * ) on a normed space X we refer the reader to [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF] or [55, p.124] We recall the following definition introduced in [51, p. 260]:

Definition 2.1.13 Let X be a Banach space and let f : D( f ) ⊂ X → X be a continuous mapping.

(i) We say that f is ws-compact if for each weakly convergent sequence

(x n ) n∈N of D( f ) the sequence ( f (x n ))
n∈N has a strongly convergent subsequence.

(ii) We say that f is ww-compact if for each weakly convergent sequence (x n ) n∈N of D( f ), the sequence ( f (x n )) n∈N has a weakly convergent subsequence.

Remark 2.1.3 For the properties ws-compact or ww-compact we refer, for example, to [START_REF] Garcia-Falset | Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness[END_REF][START_REF] Latrach | Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations[END_REF][START_REF] Latrach | Existence results for a nonlinear transport equation in bounded geometry on L 1 -spaces[END_REF] or [48, pp. 221-222]. We note that with the help of Eberlein-Smulian (theorem [31, p. 430]), ww-compact operators map weakly compact sets into weakly compact sets. Next since bounded linear operators are weakly continuous, so each bounded linear operator is ww-compact.

2

We now recall the following two theorems concerning these two classes of operators established established in [START_REF] Latrach | Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations[END_REF].

Theorem 2.1.3 Let K be a nonempty closed convex subset of a Banach space X. Assume that f : K → K is a ws-compact mapping. If f (K) is relatively weakly compact, then Fix( f ) ∅.

Theorem 2.1.4 Let K be a nonempty closed bounded convex subset of a Banach space X. Suppose that f : K → K and g : X → K such that:

(a) f is ws-compact and f (K) is relatively weakly compact, (b) g is ww-compact and k-contractive for some k ∈ (0, 1),

(c) f (K) + g(K) ⊂ K.
Then there exists x ∈ K such that f (x)

+ g(x) = x.
Let X, Y be two Banach spaces and T ∈ L(X, Y), where L(X, Y) denotes the set of all bounded linear operators from X into Y.

Definition 2.1.14 Let X and Y be two Banach spaces and T ∈ L(X, Y).

(a) We recall that T is called weakly compact if it carries norm bounded subset of X to relatively weakly compact subset of Y.

(b) We say that T is a Dunford-Pettis operator if T maps weakly compact subsets of X into norm compact sets of Y.

2 Theorem 2.1.5 (Krein-Smulian) Let X be a Banach space and K a weakly compact subset of X (that is, K is compact when X is endowed with the weak topology). Then the closed convex hull of K in X is weakly compact.

For a proof of Krein-Smulian's theorem, we refer, for example, to [31, p. 434].

We recall the following lemma established in [35, pp. 3438-3439].

Lemma 2.1.8 Let X be a Banach space and let B : X → X be a ww-compact mapping. If B is a Φ-contraction with Φ-function φ, then

ω(B(S )) ≤ φ ω(S ) , for all S ∈ B(X).
Note that if B is a k-contractive and ww-compact maps, then according to Remark 2.1.1, B is a Φ-contraction with Φ-function φ where, for each x ∈ X, φ(x) = kx. Hence, the estimate in Lemma 2.1.8 may be written simply as ω(B(S )) ≤ k ω(S ), for all S ∈ B(X).

(2.1.

3)

It should be noticed that Equation (2.1.3) is in fact valid for all k-Lipschitzian and ww-compact maps (cf. [START_REF] Agarwal | Browder-Krasnoleskii-type fixed point theorems in Banach spaces[END_REF]).

Remark 2.1.4 It should be noticed that each bounded linear operator from a normed space X into itself is ww-compact (use Remark 2.1.3). Morever, it is T -Lipschitzian where T is norm of T (in the operator topology), so according to (2.1.3), we have

ω T (M) ≤ T ω(M), for all M ∈ B(X). (2.1.4) 2
In Lemma 2.1.8, if we replace the hypothesis f is ww-compact by f is weakly sequentially continuous, the result remains valid. This motivated by the following Lemma.

Lemma 2.1.9 Let X be a Banach space and let B : X → X be a weakly sequentially continuous mapping. If B is a Φ-contraction with Φ-function φ, then ω(B(S )) ≤ φ ω(S ) , for all S ∈ B(X).

Proof. Let S ∈ B(X) and r > ω(S ). There exists r 0 ∈ [0, r) and an element K ∈ W(X) such that S ⊂ K + B r 0 . Since f is a Φ-contraction with Φ-function φ, we have

f (S ) ⊂ f (K) + B φ(r 0 ) ⊂ f (K) w + B φ(r 0 )
where f (K) w denote the closure of f (K) in the weak topology. Since f is weakly sequentially continuous, using Eberlin-Smulian's theorem (see, for example [31, Theorem 1, p.430]), we infer that f w is weakly compact. So ω( f (S )) ≤ φ(r 0 ) < φ(r).

Now, using the continuity of φ and letting r goes to ω(S ), we get

ω( f (S )) ≤ φ(ω(S )).

Superposition operators

Now we shall recall some facts concerning superposition operators required below. Let Ω be a subset of R N . A function g :

Ω × C -→ C is said to be a Carathéodory function on Ω × C if        t -→ g(t, u) is measurable on Ω for all u ∈ C, u -→ g(t, u) is continuous on C for almost all t ∈ Ω.
If g is a Carathéodory function, we can define the operator N g on the set of functions ψ : Ω -→ C by

(N g ψ)(z) := g(z, ψ(z))
for every z ∈ Ω. The operator N g is called the Nemytskii operator generated by g. In L p -spaces the Nemytskii operator has been extensively investigated (see [START_REF] Appell | The superposition operator in function spaces -A survey[END_REF][START_REF] Chow | Methods of bifurcation theory[END_REF][START_REF] Krasnoselskii | Integral operators in spaces of summable functions[END_REF] and the references therein). However, we recall the following result due to Krasnosel'skii which states a basic fact for the theory of these operators on L p . Lemma 2.2.1 Assume that g satisfies the Carathéodory conditions. If the operator N g acts from L p into L p , then N g is continuous and maps bounded sets into bounded sets. Moreover, there is a constant η > 0 and a function h(.) ∈ L + p (the positive cone of L p ) such that |g(x, y)| ≤ h(x) + η y a.e. in x, for all y ∈ R.

Proof. For the proof we refer, for example, to [24, p. 35] or [START_REF] Krasnoselskii | Integral operators in spaces of summable functions[END_REF].

Multivalued maps

Now we introduce notations and definitions which are required in the paper. Let X be Banach space and define the sets

P cv (X) = {M ⊂ X : M is nonempty and convex}, P cl,cv (X) = {M ⊂ X : M is nonempty, convex and closed}.
Before going further we introduce the following definitions which are required below.

Definition 2.3.1 Let (X, d) and (Y, d) be two metric spaces and let F : X → P cl,cv (Y) be a multivalued map. We say that F is upper semicontinuous if, for every open set U of Y, the set

F -1 (U) is open in X, where F -1 (U) = x ∈ X : F(x) ⊂ U .
Definition 2.3.2 Let X and Y be two metric spaces and F :

X → P cl,cv (Y) be a multivalued map. A single valued map f : X → Y is called a selection of F if for every x ∈ X, f (x) ∈ F(x).
Definition 2.3.3 Let M be a nonempty closed, convex subset of a Hausdorff locally convex linear topological space X, µ(•) a measure of weak noncompactness on X and F : M → P(X) a multivalued mapping. We say that F is condensing with respect to µ(•) (or µ-condensing) if We recall the following lemma established in [START_REF] Agarwal | Browder-Krasnoleskii-type fixed point theorems in Banach spaces[END_REF].

Lemma 2.3.1 Let X be a Banach spaces and let B :

X → X be a ww-compact map. If B is k-lipschitzian with k ∈ R, then ω(B(S )) ≤ k ω(S ), for all S ∈ B(X).
We need also the following lemma.

Lemma 2.3.2 Let X be a Banach space and let B : X → X be a k-contractive map for some k ∈ [0, 1). If B is a ww-compact, then (I -B) -1 is a ww-compact.

Proof. We know that I -B is invertible and (I -B) -1 : X → X is continuous. Let (x n ) n∈N be a sequence of points belonging to X which converges weakly to some x ∈ X. Thus the set N = x n : n ∈ N is relatively weakly compact and therefore ω(N) = 0. Moreover, it follows from the equality (I -B)

-1 = I + B(I -B) -1 that ω (I -B) -1 N ≤ ω N + ω B(I -B) -1 N . Next, the use of Lemma 2.1.8 implies that ω (I -B) -1 N ≤ kω (I -B) -1 N which is a contradiction.
Thus ω (I -B) -1 N = 0 which prove that (I -B) -1 N is weakly compact. Hence, the sequence (I -B) -1 x n ) n∈N has a weakly convergent subsequence which ends the proof.

M → P(M) be a multivalued mapping. A point x ∈ M is called a fixed point of B if x ∈ B(M).
We close this section by recalling the following result owing to Himmelberg [START_REF] Himmelberg | Fixed points of compact multifunctions[END_REF] which will play a crucial role in our further analysis.

Theorem 2.3.1 Let X be a Hausdorff locally convex linear topological space and let M ∈ P cl,cv (X). Let F : M → P cl,cv (M) be a upper semicontinuous multivalued mapping such that F(M) is relatively compact. Then there exists x ∈ M such that x ∈ F(x).

Chapter 3

Fixed point results for countably condensing mappings

This section is devoted to fixed points results for countably condensing mappings. It is organised as follows. In Section 3.1, we give a fixed point of Altman's type. In Section 3.2, we present some fixed point results of Leray-Schauder's type. Section 3.3 deals with results of Krasnosel'skii's type where various kinds of perturbations were considered. In Section 3.4, results of Krasnosel'skii-Schaefer's type were discussed.

A fixed point result of Altman's type

Theorem 3.1.1 Let K be a bounded open absorbing convex subset of a Banach space X, and let f : K → X be a countably β-condensing map.

If ∀x ∈ ∂K, f (x) -x 2 ≥ f (x) 2 -x 2 , then Fix( f ) ∅. Proof. Define the function r(•) on X by r(x) = (max{1, µ K (x)}) -1 x where , µ K (x) is the Minkowski functional of K. Let λ = (max{1, , µ K (x)}) -1 . It follows from Lemma 2.1.6 that λ ∈ (0, 1]. One can check easily that r : X → K is a retraction (i.e. r is continuous, r(x) = x, ∀x ∈ K). Moreover, we have r • f (K) ⊂ K. We claim that the map r(•) is a countably 1-set-contraction. Indeed, since λ ∈ (0, 1], for C ∈ B c (K), we have r(C) = λC ⊂ co{C, θ} and then β(r(C)) ≤ β(co{C, θ}) = β(C) which proves our claim. Moreover, for any C ∈ B c (K) with β(C) > 0, we have β(r • f (C)) ≤ β( f (C)) < β(C). Hence r • f is countably β-condensing. Applying Theorem 2.
1.1 we deduce that there exists

x 1 ∈ K such that x 1 = (r • f )(x 1 ) ∈ K. • If f (x 1 ) ∈ K, then r f (x 1 ) = f (x 1
) and then f (x 1 ) = x 1 .

• If f (x 1 ) K, then, by Lemma 2.1.6, µ K ( f (x 1 )) ≥ 1 and therefore

x 1 = r( f (x 1 )) = (µ K ( f (x 1 ))) -1 f (x 1 ). Hence µ K (x 1 ) = µ K (r( f (x 1 ))) = µ K f (x 1 ) µ K ( f (x 1 )) = µ K ( f (x 1 )) µ K ( f (x 1 )) = 1
which proves that x 1 ∈ ∂K (use Lemma 2.1.6).

Let us now suppose that

x 1 ∈ ∂K. If f (x 1 ) ∈ K, then x 1 = r • f (x 1 ) = f (x 1
). If f (x 1 ) K, then, by Lemma 2.1.6, we have µ K ( f (x 1 )) > 1 and thus

x 1 = r • f (x 1 ) = (µ K ( f (x 1 ))) -1 f (x 1 ). Letting k = µ K ( f (x 1 )
), one obtains f (x 1 ) = kx 1 , and consequently,

f (x 1 ) -x 1 2 = (k -1) 2 x 1 2 ≥ f (x 1 ) 2 -x 1 2 = (k 2 -1) x 1 2 .
This implies that (k -1) 2 ≥ (k 2 -1), and therefore k ≤ 1. This contradicts the fact that k = µ K ( f (x 1 )) > 1. So, we have necessarily that µ K ( f (x 1 )) ≤ 1 which prove that x 1 is a fixed point of f .

2

An immediate consequence of Theorem 3.1.1 we have:

Corollary 3.1.1 Let K be a bounded open absorbing convex subset of a Banach space X, and let f : K → X be a countably β-condensing map. Assume that one of following condition is satisfied:

(a) f (x) ≤ x , ∀x ∈ ∂K (Roth's condition type ); (b) x -f (x) ≥ f (x)
, ∀x ∈ ∂K (Petryshyn's condition type);

(c) for Hilbert space X, f (x), x ≤ x 2 , ∀x ∈ ∂K (Krasnosel'skii's condition type).

Then Fix( f ) ∅ in K.

Fixed point results of Leray-Schauder's type

We start this section by establishing a result of Schaefer type for countably β-condensing operators .

Proposition 3.2.1 Let X be a Banach space and let f : X → X be a map. If f is continuous, countably β-condensing, then (a) either Fix( f ) ∅ or, (b) the set x ∈ X : x = λ f (x), for some λ ∈ (0, 1) is unbounded.

Proof. Let τ > 0 and let ζ τ be the radial retraction mapping from X into B τ (the closed ball with center θ and radius τ). One can define the mapping J τ :

B τ → B τ by J τ (x) = ζ τ f (x) .
It is clear that J τ is continuous. Hence for each x ∈ B τ , we have two possibilities:

• If f (x) ≤ τ, then ζ τ f (x) = f (x) ∈ f B τ ⊂ co f (B τ ) ∪ {0} . • If f (x) > τ, alors ζ τ f (x) = τ f (x) f (x) + 1 - τ f (x) 0 is an element of co f (B τ ) ∪ {0} .
Hence, in all cases we have J τ B τ ⊂ co f (B τ ) ∪ {0} , and so, for any countably subset C of B τ , we have J τ C ⊂ co f (C) ∪ {0} . Next, using the properties if β(•) and the fact that f is countably β-condensing we get

β J τ (C) ≤ β f (C) ∪ {0} < β(C).
This shows that J τ is countably β-condensing. Now the use of Theorem 2.1.1 guarantees that there exists z ∈ B τ such that

J τ (z) = z = ζ τ f (z) . Note that, if f (z) ≤ τ, then we have ζ τ f (z) = f (z) = z , and so z is a fixed point of f . Now if f (z) > τ, then ζ τ f (z) = τ f (z)
f (z) = z, and therefore z is a solution to the equation

x = λ f (x) with λ = τ f (z) ∈ (0, 1).
Hence, either, for some real τ > 0, we obtain a solution of f (x) = x, or for each τ > 0, we obtain an eigenvector of norm τ for some eigenvalue in (0, 1). In the second case the set of such eigenvectors is unbounded. We note that, since K is closed in X, the boundaries of U in X and in K are the same.

Proof. Suppose (b) is false and f has no fixed point on ∂U. Then z λ f (z) + (1 -λ)p for all z ∈ ∂U and λ ∈ (0, 1). Let

Λ := {x ∈ U : x = λ f (x) + (1 -λ)p for some λ ∈ [0, 1]}.
Note that Λ ∅ is nonempty since p ∈ Λ and, by the continuity of f , it is closed. Notice Λ ∩ ∂U = ∅. By Uryshon's theorem (see, for example, Theorem 1.1.1 in [START_REF] Latrach | Introduction à la théorie des points fixes métrique et topologique: avec applications et exercices corrigés[END_REF]), there exists a continuous mapping ζ : U → [0, 1] separating Λ and ∂U, i.e., ζ(Λ) = 1 and ζ(∂U) = 0. Define the function η :

K → K by η(x) :=        ζ(x) f (x) + (1 -ζ(x))p if x ∈ U p if x ∈ K\U.
Note that η is continuous. Further, for any subset C ∈ B c (K) with β(C) > 0, the use of the inclusion

η(C) ⊂ co( f (C ∩ U) ∪ { p}) implies that β(η(C)) ≤ β( f (C ∩ U)). Hence, if β(C ∩ U) = 0, then C ∩ U is relatively compact, so f (C ∩ U) is relatively compact too, and therefore β η(C) ≤ β( f (C ∩ U)) = 0 < β(C). Assume now that β(C ∩ U) 0, then β( f (C ∩ U)) < β(C ∩ U) ≤ β(C).
Thus in both cases we have β(η(C)) < β(C) which prove that η is countably β-condensing. It follows from Theorem 2.1.1 that there exists

x ∈ K such that η(x) = x. Since p ∈ U, the point x ∈ U, hence, x = ζ(x) f (x) + (1 -ζ(x))p. Because ζ(x) ∈ [0, 1], one sees that x ∈ Λ and therefore ζ(x) = 1 that is x = f (x).
2

If we take p = 0, then we obtain the following corollary. Proof. Suppose that (b) is false and f has no fixed point on ∂U. Define the maps f n , n ∈ N, by f n = t n f where (t n ) n∈N is a sequence of (0, 1) such that t n → 1 as n → +∞. The convexity of K together with the fact that θ ∈ K implies that f n maps U into K. Because t n ∈ (0, 1), the set f n (U) is bounded. Assume that λ n f n (y n ) = y n for some y n ∈ ∂U and some λ n ∈ (0, 1). Therefore we have y n = λ n t n f (y n ), which contradicts our assumption because λ n t n ∈ (0, 1). Since f is countably 1-set-contractive, f n is countably β-condensing. It follows from Theorem 2.1.1 that f n has a fixed point say x n ∈ U. This yields

x n -f (x n ) = t n f (x n ) -f (x n ) = |t n -1| f (x n ) .
Using the fact that f (U) is bounded, we deduce that x nf (x n ) → 0 as n → +∞. Since f is semi-closed and K is closed, we conclude that 0 ∈ (If )(K), which ends the proof.

Fixed point results for sums of mappings

We start our discussion by etablishing the following elementary results of Krasnosel'skii type.

Proposition 3.3.1 Let K be a nonempty convex closed and bounded subset of a Banach space X, and let f : K → K and g : K → K be two continuous maps. Assume that f and g satisfy (a) f (K) is relatively compact;

(b) g is countably β-condensing;

(c) f (K) + g(K) ⊆ K.
Then Fix( f + g) ∅.

Proof. Let C ∈ B c (K). Using that fact that f (C) is relatively compact, we get β((

f + g)(C)) ≤ β( f (C)) + β(g(C)) ≤ β(g(C)) < β(C).
Hence the operator f + g is countably β-condensing. The use of Theorem 2.1.1 concludes the proof.

2 Proposition 3.3.2 Let K be a nonempty convex closed and bounded subset of a Banach space X, and let f : K → K and g : X → X be two continuous maps. Assume that f and g satisfy (a) f is countably β-η-contractive for some η ∈ (0, 1/2);

(b) g is (1 -η)-contractive; (c) (x = g(x) + f (y), y ∈ K) =⇒ x ∈ K.
Then Fix( f + g) ∅.

Proof. It is well known that by conditions (b) and (c) the map (Ig) -1 f is well defined and maps K into itself. Using Lemmas 2.1.1 and 2.1.3, we get

β (I -g) -1 f )(C) ≤ η 1 -η β(C) for all C ∈ B c (K).
Since η ∈ (0, 1/2), we have η 1 -η < 1. Hence, (Ig) -1 f is countably β-condensing. Now, applying Theorem 2.1.1, we conclude that there exists z ∈ K such that z = (Ig) -1 f (z) which ends the proof.

2 Theorem 3.3.1 Let K be a nonempty convex closed and bounded subset of a Banach space X, and let f : K → X and g : X → X be two continuous maps. Assume that f and g satisfy (a) g is pseudocontractive and countably β-k-contractive for some k ∈ [0, 1);

(b) (I -g) is ψ-expansive, (c) f is countably β-s-contractive for some s ∈ (0, 1 -k), (d) x = g(x) + f (y), y ∈ K =⇒ x ∈ K.
Then Fix( f + g) ∅.

Proof.

(1) We show that the operator Ig : X → (Ig)(X) is bijective. Consider x, y ∈ X with x y. Since (Ig) is ψ-expansive, we have

(I -g)(x) -(I -g)(y) ≥ ψ( x -y ).
Thus (Ig) is one-to-one, so Ig : X → (Ig)(X) is bijective. Arguing as in the point (b) of the proof of Theorem 3.3 in [START_REF] Garcia-Falset | Krasnosel'skii-type fixed point theorems for weakly sequentially continuous mappings[END_REF] one sees that (Ig) -1 f (K) ⊂ K and so f (K) ⊂ (Ig)(K).

(

) We claim that (I -g) -1 f is continuous. Indeed, it is clear that (I -g) -1 f : K → K ⇐⇒ (I -g) -1 : f (K) → K ⇐⇒ (I -g) -1 : (I -g)(K) → K. 2
Let show that (Ig) -1 : (Ig)(K) → K is continuous. We first assume that ψ is continuous. Let (x n ) n∈N be a sequence of points of (Ig)(K) such that (x n ) n∈N converges to a point x 0 ∈ (Ig)(K) and set y n = (Ig) -1 x n and y 0 = (Ig) -1 x 0 . So we have (Ig)y n = x n and (Ig)y 0 = x 0 . Because (Ig) is ψ-expansive, we get

ψ( y n -y 0 ) ≤ (I -g)(y n ) -(I -g)(y 0 ) = x n -x 0 .
Since ψ is continuous with ψ(0) = 0 and ψ(r) > 0, for all r > 0, we get (I-g) -1 (x n ) → (I-g) -1 (x 0 ), which proves the continuity of (Ig) -1 .

We suppose now that ψ is increasing. Let (x n ) n∈N be a sequence of points of (Ig)(K) such that (x n ) n∈N converges to some point x 0 ∈ (Ig)(K) and set y n = (Ig) -1 x n and y 0 = (Ig) -1 x 0 .

It is assumed that the sequence (y n ) n∈N does not converge to y 0 . Hence, there exists > 0 and a subsequence (x n k ) k∈N from (x n ) n∈N such that, for all k ∈ N, we have

(I -g) -1 (x n k ) -(I -g) -1 (x 0 ) > .
We put δ := ψ( ) > 0. Because (x n ) n∈N converges to x 0 , we have

x n k -x < δ for k large enough. So δ = ψ( ) ≤ ψ( (I -g) -1 (x n k ) -(I -g) -1 (x 0 ) ) ≤ x n k -x < δ,
which is absurd. Hence the operator (Ig) -1 is continuous and so is (Ig) -1 f .

(3) We shall show now that (Ig) -1 f is countably β-condensing. Using the equation

(I -g) -1 f = f + g(I -g) -1 f (3.3.1)
we get, for all C ∈ B c (K)

β((I -g) -1 f (C)) ≤ β( f (C)) + β(g(I -g) -1 f (C)) ≤ sβ(C) + kβ((I -g) -1 f (C))
and therefore

β((I -g) -1 f (C)) ≤ s 1 -k β(C).
This proves that (Ig) -1 f : K → K is countably β-condensing.

To conclude, it suffices to apply Theorem 2.1.1 to the map (Ig) -1 f . 2 Corollary 3.3.1 Let K be a nonempty convex closed and bounded subset of a Banach space X, and let f : K → X and g : X → X be two continuous maps. Assume that f and g satisfy (a) g is pseudocontractive and, for all B ∈ B(X), g(B) is a relatively compact;

(b) (I -g) is ψ-expansive; (c) f is countably β-condensing; (d) (x = g(x) + f (y), y ∈ K) =⇒ x ∈ K.
Then Fix( f + g) ∅.

Proof. Arguing as in the proof of Theorem 3. 2 Theorem 3.3.2 Let K be a nonempty convex closed and bounded subset of a Banach space X and let f : K → X and g : X → X be continuous maps. Assume that f and g satisfy (a) f (K) is relatively compact;

(b) g is pseudocontractive and countably 1-set-contractive;

(c) g |K is nonexpansive;

(d) if (x n ) n∈N is a sequence of element of K such that (I -g)(x n ) → y, then (x n ) n∈N has a convergent subsequence;
(e) if λ ∈ (0, 1) and x = λg(x) + f (y), for some y ∈ K, then x ∈ K.

Before establishing this result, we first recall the following facts in connection with accretive maps.

To this end, we denote by [•, •] s the directional derivative of the function x → x , that is, [x, y] s = lim λ 0

x + λyx λ , ∀x, y ∈ X.

A characterization of [•,

•] s is given by (see, for example, [START_REF] Barroso | Krasnoslskii's fixed point theorem for weakly continuous maps[END_REF][START_REF] Ph | Evolution equations governed by accretive operators[END_REF])

[x, y] s = max y, x * : x * ∈ J(x) , ∀x, y ∈ X,
where J(•) denotes the duality mapping from X into 2 X * defined by

J(x) = x * ∈ X * such that x * , x = x and x * ≤ 1 .
It is well known that an operator f is accretive if and only if, for any x, y ∈ D(A), there exists

x * ∈ J(x -y) such that f (x) -f (y), x * ≥ 0.
For more information, we refer the reader, for example, to the books [START_REF] Barroso | Krasnoslskii's fixed point theorem for weakly continuous maps[END_REF][START_REF] Ph | Evolution equations governed by accretive operators[END_REF] and the references there in.

Proof of Theorem 3.2. It is clear that λg is continuous and β-λ-contractive for any λ ∈ (0, 1).

We claim that, for any λ ∈ (0, 1), the operator λg is pseudocontractive. Indeed, let λ ∈ (0, 1). Since g is pseudocontractive, by Remark 2.1.2, the operator Ig is accretive; so, as it was remembered before the proof, for every x, y ∈ X, there exists j(xy) ∈ J(xy) such that (Ig)x -(Ig)y, j(xy) ≥ 0.

Hence, one can write

(I -λg)(x) -(I -λg)(y), x -y s ≥ (I -λg)(x) -(I -λg)(y), j(x -y) = λx -λg(x) -λy + λg(y), j(x -y) + (1 -λ)(x -y), j(x -y) = λ (I -g)(x) -(I -g)(y), j(x -y) +(1 -λ) λx -y, j(x -y) ≥ (1 -λ) x -y 2 ≥ 0.
Hence, the operator λg is pseudocontractive. On the other hand, the same inequality yields that

(x -λg(x)) -(y -λg(y)) x -y ≥ x -λg(x) -(y -λg(y)), x -y s ≥ (1 -λ) x -y 2 ≥ 0. Consequently, (x -λg(x)) -(y -λg(y)) ≥ ψ( x -y ),
where ψ(t) = (1 -λ)t. This proves that the map I -λg is ψ-expansive. The above arguments show that f and λg satisfy the conditions of Theorem 3.3.1. Hence, for each λ ∈ (0, 1), there exists x λ ∈ K such that

x λ = λg(x λ ) + f (x λ ).
Let (λ n ) n∈N be a sequence of (0, 1) such that λ n → 1 and let (x n ) n∈N be a sequence in K such that

x n = λ n g(x n ) + f (x n ), ∀n ≥ 0. (3.3.2)
Using the fact that f (K) is relatively compact and passing eventually to a subsequence, we may assume that ( f (x n )) n∈N converges to some y ∈ K. Accordingly

f (x n ) = x n -λ n g(x n ) = (I -λ n g)(x n ) → y.
Since K is bounded and g |K is nonexpansive, we infer that the sequence g(x n ) n∈N is bounded. Consequently,

(x n -g(x n )) -(x n -λ n g(x n )) = (1 -λ n ) g(x n ) → 0 as n → +∞,
and therefore

x ng(x n ) → y.

In view of (d) there exists a subsequence (x n k ) k∈N of (x n ) n∈N which converges to some z ∈ K. The use of (3.3.2) and the fact that f and g are continuous, imply that f (z) + g(z) = z.

2 Theorem 3.3.3 Let K be a nonempty convex closed and bounded subset of a Banach space X. Suppose that f : K → X and g : X → X are two maps such that (a) f is continuous, countably β-k-contractive for some k ∈ [0, 1), and f (K) is relatively compact;

(b) g is a nonlinear φ-contraction with φ(r) < (1k)r for r > 0;

(c) x = g(x) + f (y), y ∈ K =⇒ x ∈ K.
Proof. By Lemma 2.1.2, I-g has a continuous inverse on X, and then J := (I-g) -1 f is well defined on K. By rotinuous calculations, one sees that J is continuous and Then Fix( f + g) ∅ in K.

J(K) ⊂ K. Let C ∈ B c (K)
Proof. For each x ∈ K, by (c), we see that there exists y ∈ H such that yg(y) = f (x). By Lemma 2.1.4, we have y

= (G • f (x)) ∈ H with G(x) := (I -g) -1 (x). It follows from Lemma 2.1.4, (b) and (a) that G • f : K → H is k k -1 -contractive because k h -1 < 1 and therefore from Lemma 2.1.1 it is countably β-condensing.
In what follows, we shall check that the condition (c) of Corollary 3.1.1 is satisfied. Indeed for x ∈ ∂K, the use of the equality ρ( f (x)) -

x 2 = ρ( f (x)) 2 + x 2 -2 ρ( f (x)),
x together with the hypothesis (d) we derive the estimate

ρ( f (x)) 2 -x 2 -ρ( f (x)) -x 2 = 2[(x, ρ( f (x)) -x 2 ] ≤ 0.
This shows that the condition (c) of Corollary 3.1.1 is satisfied. This ends the proof.

Fixed point results of Krasnoselskii-Schaefer's type

Let X be a Banach space. We say that a map f from X into itself is completely continuous if it maps each bounded set into a relatively compact one .

Theorem 3.4.1 Let X be a Banach space and let f, g : X → X be two continuous mappings. If f and g satisfy the conditions (a) g is pseudocontractive and β-k-contractive, for some k ∈ (0, 1);

(b) Ig is ψ-expansive where ψ is either strictly increasing or lim r→∞ ψ(r) = ∞;

(c) f is countably β-s-contractive for some s ∈ (0, 1k).

Then (i) either Fix( f + g) ∅, or (ii) the set x ∈ X such that x = λg x λ + λ f (x) for some λ ∈ (0, 1) is unbounded.

Proof. It is easy to check that x ∈ X is a solution of the equation x = f (x) + g(x) if and only if x is a fixed point of the map (Ig) -1 f , whenever it is well defined. According to the proof of Theorem 3.2 in [START_REF] Garcia-Falset | Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness[END_REF] and Theorem 3.3 in [START_REF] Garcia-Falset | Krasnosel'skii-type fixed point theorems for weakly sequentially continuous mappings[END_REF], we can prove that (Ig) -1 f : X → X is well defined and continuous. By Proposition 3.2.1 we only need to show that (Ig) -1 f is countably β-condensing. Let C ∈ B c (X) such that β(C) > 0. We first show that (Ig) -1 f (C) is bounded. Let x, y ∈ (Ig) -1 f (C). Hence, there exist z 1 , z 2 ∈ C such that x = (Ig) -1 f (z 1 ), y = (Ig) -1 f (z 2 ). Then, xg(x) = f (z 1 ), yg(y) = f (z 2 ). Since Ig is ψ-expansive, we can write

ψ( x -y ) ≤ x -g(x) -(y -g(y)) = f (z 1 ) -f (z 2 ) ≤ diam ( f (C)) < +∞.
If (Ig) -1 f (C) is not a bounded set, then there exist x n , y n ∈ (Ig) -1 f (C) such that x ny n → +∞. Hence,

ψ( x n -y n ) ≤ diam ( f (C)). If ψ is such that lim r→∞ ψ(r) = ∞, then necessarily diam ( f (C)) = +∞, which is a contradiction.
Else, if ψ is strictly increasing, then ψ has an inverse on [0, +∞), which is strictly increasing as well. Then

x n -y n ≤ ψ -1 ( diam ( f (C))) < +∞,
which gives another contradiction, Hence, in any case, (Ig) -1 f (C) is bounded.

Next using equation (3.3.1), one sees that

β((I -g) -1 f (C)) ≤ β( f (C)) + β(g(I -g) -1 f (C)) ≤ sβ(C) + kβ((I -g) -1 f (C)) ≤ s 1 -k β(C) < β(C).
which shows that (I-g) -1 f is countably β-condensing. Now the use of Proposition 3.2.1 completes the proof.

2 Theorem 3.4.2 Let X be a Banach space and let f, g : X → X be two continuous mappings. If f and g satisfy the conditions (a) f is completely continuous;

(b) g is pseudocontractive and countably β-condensing;

(c) Ig is ψ-expansive where ψ is either strictly increasing or lim r→∞ ψ(r) = ∞.

Then (i) either Fix( f + g) ∅, or (ii) the set x ∈ X such that x = λg x λ + λ f (x) for some λ ∈ (0, 1) is unbounded.

Proof. In view of the proof of a Theorem 3.4.1, the map (Ig) -1 f : X → X is well defined, and continuous. Let C ∈ B c (X) be such that β(C) > 0. Using equality (3.3.1), we get

β((I -g) -1 f (C)) ≤ β( f (C)) + β(g(I -g) -1 f (C)) = β(g(I -g) -1 f (C)) < β((I -g) -1 f (C)),
which is a contradiction. So (I-g) -1 f maps bounded sets into relatively compact ones, so (I-g) -1 f is completely continuous on X. Now the use of Theorem 3.5.2 in [START_REF] Latrach | Introduction à la théorie des points fixes métrique et topologique: avec applications et exercices corrigés[END_REF] completes the proof.

2 Theorem 3.4.3 Let X be a Banach space. Suppose that f, g : X → X are two continuous mappings satisfying the conditions (a) f is countably β-k-contractive, for some k ∈ [0, 1);

(b) g is a nonlinear φ-contraction;

(c) φ satisfies, for each r > 0, φ(r) < (1k)r and lim r→∞ [r -φ(r)] = +∞.

Then (i) either Fix( f + g) ∅, or (ii) the set x ∈ X such that x = λg x λ + λ f (x) for some λ ∈ (0, 1) is unbounded.

Arguing as in the proofs of the previous theorems, one sees that J := (I -G) -1 f is well defined and continuous. Let us prove that J := (Ig) -1 f maps bounded sets into a bounded sets. Let A be a bounded set and let x, y ∈ (Ig) -1 f (A). Hence, there exist

z 1 , z 2 ∈ A such that x = (I -g) -1 f (z 1 ), y = (I -g) -1 f (z 2 ) or again x -g(x) = f (z 1 ), y -g(y) = f (z 2 )
. By assumption (b) and the boundedness of f (A), we have

(x -y) -(g(x) -g(y)) = f (z 1 ) -f (z 2 ) x -y -g(x) -g(y) ≤ f (z 1 ) -f (z 2 ) x -y -φ( x -y ) ≤ f (z 1 ) -f (z 2 ) ≤ diam ( f (A)) < +∞. If J(A) = (I -g) -1 f (A) is unbounded, then there exist x n , y n ∈ (I -g) -1 f (A) such that x n -y n → +∞.
Using assumption (c) we infer that

x n -y n -φ( x n -y n ) → +∞. This is a contradiction with diam ( f (A)) < +∞. Hence J(A) is bounded. Let C ∈ B c (

Fixed point theorems of Schauder's type

In this section we shall present some fixed point theorems involving two or three mappings.

Theorem 4.1.1 Let X be a Banach space and let K be a nonempty closed convex subset of X. Let f : X → K and g : K → X be two continuous maps and set F = f • g. Assume that (a) F(K) is relatively weakly compact, (b) f satisfies ws-compact, (c) g satisfies ww-compact.

Then Fix(F) ∅.

Proof. Let C = co(F(K)). Since K is a closed convex subset of X satisfying F(K) ⊆ K, then C ⊆ K and therefore F(C) ⊆ F(K) ⊆ co(F(K)) = C
. This shows that F maps C into itself. Since F(K) is relatively weakly compact, by Krein-Smulian theorem [31, p. 434], C is weakly compact too.

Let (x n ) n∈N be a sequence in C. Using the fact that C is weakly compact and g satisfies wwcompact, we infer that there exists a subsequence (x n k ) k∈N of (x n ) n∈N such that (g(x n k )) k∈N converges weakly in C. Next, using the fact that f is satisfies ws-compact, we conclude that there exists a subsequence (x n k j ) j∈N of (x n k ) k∈N such that the sequence (F(x n k j )) j∈N converges strongly in C. This shows that F satisfies condition ws-compact. Because F(C) is relatively weakly compact, the use of Theorem 2.1.3 concludes the proof. Then Fix(F) ∅.

Proof. We note that according to Remark 4.1.2, the operator f satisfies condition ws-compact, so the result follows from Theorem 4.1.1.

2

Remark 4.1.3 A Banach space X has the Dunford-Pettis property if every linear weakly compact operator defined on X takes weakly compact sets into norm compact sets. It is clear that if X is a Banach space with the Dunford-Pettis property, then every linear weakly compact operator on X is a Dunford-Pettis operator.

2

Making use of this observation we get the following result.

Corollary 4.1.2 Let X be a Banach space with the Dunford-Pettis property and let K be a nonempty closed convex subset of X. Let f : X → K and g : K → X be two continuous maps and set F = f • g. Assume that (a) f is a linear weakly compact operator, (b) g satisfies ww-compact and g(K) is bounded.

Then Fix(F) ∅.

Proof. Since X is a Banach space with Dunford-Pettis property and f is a weakly compact linear operator, according to Remark 4.1.3, f is a Dunford-Pettis operator. To complete the proof, it suffices to show that F(K) is relatively weakly compact. This follows from the fact that g(K) is bounded and hypothesis (a). Now the use of Corollary 4.1.1 ends the proof. Then Fix(F + B) ∅.

Proof. Since B is a nonlinear contraction with Φ-function φ, it is known that (I-B) is a homeomorphism from K onto (I -B)(K). Let y be a point in K and define the map K x → B(x) + F(y). It is clear that it is a nonlinear contraction with Φ-function φ. Hence, according to Boyd-Wong's theorem (see, for example, [START_REF] Boyd | On nonlinear contractions[END_REF] or [48, Theorem 2.7.2, p. 52]), the equation z = B(z)+F(y) has a unique solution x ∈ X. Using hypothesis (e) we infer that x ∈ K and therefore

x = (I -B) -1 F(y) ∈ K. Hence, (I -B) -1 F(K) ⊂ K. (4.2.1)
We define a sequence (K n ) n∈N of subsets of K by

K 0 = K and K n+1 = co (I -B) -1 F(K n ) .
It is clear that (K n ) n∈N is a sequence of nonempty closed convex subsets of K. Moreover, the inclusion (4.2.1) shows that (K n ) n∈N is decreasing (in the sense of the inclusion). Furthermore, elementary calculations show that

(I -B) -1 F = F + B(I -B) -1 F. (4.2.2)
Hence, using equation (4.2.2) we get

(I -B) -1 F(K n ) ⊂ F(K n ) + B co((I -B) -1 F(K n )) ⊂ F(K n ) + B(K n+1 ). Since (K n ) n∈N is decreasing, we get (I -B) -1 F(K n ) ⊂ F(K n ) + B(K n ). Thus, assumption (b) yields µ(K n+1 ) ≤ µ F(K n ) + B(K n ) ≤ γµ(K n ).
By induction we get µ(K n+1 ) ≤ γ n µ(K) and therefore lim n→+∞ µ(K n ) = 0 because γ ∈ (0, 1). Hence, it follows from the generalized Cantor intersection theorem that

K ∞ = n≥0 K n is a nonempty convex weakly compact subset of K. Moreover, we have (I -B) -1 F(K ∞ ) ⊂ K ∞ and consequently the set (I -B) -1 F(K ∞ ) is relatively weakly compact.
Let (x n ) n∈N be a sequence in K ∞ , so it has a weakly convergent subsequence which we denote again (x n ) n∈N . Using hypothesis (c), one sees that there exists a subsequence denoted by (x n k ) k∈N such that (g(x n k )) k∈N is weakly convergent. Next, using the fact that f satisfies ws-compact, we conclude that there exists a subsequence (x n k j ) j∈N of (x n k ) k∈N such that ( f (g(x n k j ))) j∈N converges strongly in K ∞ . Because (I -B) -1 is continuous, the sequence (I -B) -1 f (g(x n k j )) j∈N converges strongly in K ∞ . Hence, the map (I -B) -1 F satisfies condition ws-compact. So, invoking the fact that (I -B) -1 F(K ∞ ) is relatively weakly compact and Theorem 2.1.3, we conclude that Fix (I -B) -1 F ∅, which concludes the proof. Then Fix(F + B) ∅.

Note that if X is Banach space with the Dunford-Pettis property, then, according to Remark 4.1.3, every weakly compact linear operator on X is a Dunford-Pettis operator. This yields the following result.

Corollary 4.2.2 Let X be a Banach space with the Dunford-Pettis property, K a nonempty closed bounded convex subset of X and µ(•) a measure of weak noncompactness on X. Let f : X → K, g : K → X and B : K → X be continuous maps and set F = f • g. Suppose that (a) f is a weakly compact linear operator, (b) there exists γ ∈ (0, 1) such that µ F(S ) + B(S ) ≤ γµ(S ) for all S ⊂ K with µ(S ) > 0, (c) g satisfies ww-compact and g(K) is bounded, (d) B is a nonlinear contraction with Φ-function φ,

(e) x = B(x) + F(y), y ∈ K =⇒ x ∈ K. Then Fix(F + B) ∅. Remark 4.
B is k-contractive for some k ∈ (0, 1) and satisfies ww-compact,

(e) x = B(x) + F(y), y ∈ K =⇒ x ∈ K. Then Fix(F + B) ∅. Proof. Because B is k-contractive, we know that (I -B) is a homeomorphism from K onto (I - B)(K).
Let y be a point in K and define the map

K x → B(x) + F(y).
It is clear that it is k-contractive, hence by the Banach contraction principle, the equation z = B(z) + F(y) has a unique solution x ∈ X. Using hypothesis (e), we infer that x ∈ K and therefore x = (I -B) -1 F(y) ∈ K. Hence (I -B) -1 F(K) ⊂ K. We define a sequence (K n ) n∈N of subsets of K by

K 0 = K and K n+1 = co (I -B) -1 F(K n ) .
Thus, (K n ) n∈N is a decreasing sequence of nonempty closed convex subsets of K. Further, using equation (4.2.2), we see that

(I -B) -1 F(K n ) ⊂ F(K n ) + B co (I -B) -1 F(K n ) ⊂ F(K n ) + B(K n ).
The properties of the measure of weak noncompactness ω(•), assumptions (b) and (d) and Eq.

(2.1.3) yield that

ω(K n+1 ) ≤ ω F(K n )) + ω(B(K n ) ≤ kω(K n ).
Hence, for all n ∈ N, we have ω(K n ) ≤ k n ω(K) and therefore lim n→+∞ ω(K n ) = 0. By the generalized

Cantor intersection theorem we conclude that K ∞ = n≥0 K n is a nonempty convex weakly compact subset of K and

(I -B) -1 F(K ∞ ) ⊂ K ∞ . This implies that (I -B) -1 F(K ∞ ) is a relatively weakly compact subset of X.
To complete the proof, it suffices to argue as in the last paragraph of the proof of Theorem 4.2.1.

2

We recall now the following result established in [START_REF] Garcia-Falset | Existence of fixed points for the sum of two operators[END_REF].

Lemma 4.2.1 Let X be a Banach space and let K be a nonempty closed bounded and convex subset of X. If B : K → K is a nonexpansive mapping such that I -B is ψ-expansive, then B has a unique fixed point in K.

Theorem 4.2.3 Let X be a Banach space and let K be a nonempty closed bounded convex subset of X. Let f : K → K, g : K → K and B : K → X be continuous maps and set F = f • g. Suppose that (a) f satisfies ws-compact and f (K) is relatively weakly compact, (b) g satisfies ww-compact, (c) B is nonexpansive and ω-condensing,

(d) I -B is ψ-expansive, (e) x = B(x) + F(y), y ∈ K =⇒ x ∈ K.
Then Fix(F + B) ∅.

Proof. Note that according to the proof of Theorem 3.2 in [START_REF] Garcia-Falset | Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness[END_REF], under assumptions (c) and (d) the operator (I -B) is invertible, (I -B) -1 is continuous and its domain contains the range of F. Since f and g are continuous, the operator (I -B) -1 F : K → K is continuous and maps K into itself.

We claim that (I -B) -1 F satisfies condition ws-compact. Indeed, if (x n ) n∈N is a weakly convergent sequence of points of K, then there exists a subsequence (x n k ) k∈N of (x n ) n∈N such that (g(x n k )) k∈N converges in the weak topology because g satisfies ww-compact. Next, using the fact that f satisfies ws-compact we infer that there exists a subsequence (g(x n k j )) j∈N of (g(x n k )) k∈N such that f g((x n k j )) j∈N converges in the strong topology. Next, using the continuity of the operator (I-B) -1 we conclude that the sequence (I-B) -1 f g((x n k j ))

j∈N converges in the strong topology.

Consequently, the operator (I -B) -1 F satisfies condition ws-compact.

We claim that the subset (I -B) -1 F(K) is relatively weakly compact. To see this, we first note that (I -B) -1 F(K) is bounded and (I -B) -1 F(K) ⊆ K. Moreover, since g(K) ⊂ K, we conclude that F(K) is relatively weakly compact (use assumption (a)). If (I -B) -1 F(K) is not relatively weakly compact, then using (4.2.2), the fact that B is ω-condensing and the properties of ω we get

ω (I -B) -1 F(K) = ω F + B (I -B) -1 F (K) ≤ ω(F(K)) + ω(B (I -B) -1 F(K)) = ω(B(I -B) -1 F(K)) < ω((I -B) -1 F(K))
which is a contradiction. This yields that ω (I -B) -1 F(K) = 0 and therefore (I -B) -1 F(K) is relatively weakly compact. To complete the proof it suffices to apply Theorem 2.1.3 to the map

(I -B) -1 • f • g = (I -B) -1 F. 2 Theorem 4.2.4
Let X be a Banach space and let K be a nonempty closed bounded convex subset of X. Let f : K → K, g : K → K and B : K → X be continuous maps and set F = f • g. Suppose that (a) f satisfies ws-compact,

(b) g satisfies ww-compact, (c) B is pseudocontractive and I -B is ψ-expansive, (d) ω F(S ) + B(S ) < ω(S ) for all S ⊂ K such that ω(S ) > 0, (e) x = B(x) + F(y), y ∈ K =⇒ x ∈ K.
Then Fix(F + B) ∅.

Proof. As in the proofs of the above theorems, it can be seen easily that x ∈ K is a solution for the equation x = F(x) + B(x) if, and only if, x is a fixed point for the operator (I -B) -1 F whenever it is well-defined. According to the proof of Theorem 3.3 in [START_REF] Garcia-Falset | Krasnosel'skii-type fixed point theorems for weakly sequentially continuous mappings[END_REF], we can prove that (I -B) -1 F is well defined.

The map (I -B) -1 F satisfies ws-compact. Indeed, let (x n ) n∈N be a weakly convergent sequence in K. Since g satisfies condition ww-compact, then there exists a subsequence (x n k ) k∈N of (x n ) n∈N such that (g(x n k )) k∈N is weakly convergent. Because f satisfies ws-compact, there exists a subsequence (x n k j ) j∈N of (x n k ) k∈N such that f (g(x n k j )) k∈N converges in the strong topology. This shows that F satisfies ws-compact. Next, using the continuity of (I -B) -1 we infer that (I -B) -1 F satisfies the condition ws-compact.

Next, we prove that there exists a subset C of M such that C is weakly compact and the map (

I -B) -1 F : C → C is continuous.
Given an element ζ ∈ K, we define the set

Λ := A ⊂ K : A is closed convex, ζ ∈ A and (I -B) -1 F(A) ⊂ A ,
and put M :

= A∈Λ A. Let C := co (I -B) -1 F(M ∪ {ζ} . Because ζ ∈ M and (I -B) -1 F(M) ⊂ M, it follows that C ⊂ M. This implies that (I -B) -1 F(C) ⊂ (I -B) -1 F(M) ⊂ C. (4.2.3)
Moreover, since ζ ∈ C, we obtain that C ∈ Λ. This yields that C = M. If C is not relatively weakly compact, then the use of (4.2.2), (4.2.3) and the properties of ω gives

ω(C) = ω co (I -B) -1 F(M ∪ {ζ} = ω (I -B) -1 F(M) = ω (I -B) -1 F(C) = ω (F + B (I -B) -1 F)(C) ≤ ω(F(C) + B(C)) < ω(C).
This implies ω(C) = 0. Hence C is relatively weakly compact. Because (I -B) -1 F(C) ⊂ C, we infer that (I -B) -1 F(C) is relatively weakly compact too. To complete the proof it suffices to apply Theorem 2.1.3 to the map (

I -B) -1 F : C → C. B p ) -1 k=p-1 k=0
B k and I -B is an homeomorphism from M onto (I -B) -1 M. (cf. [START_REF] Nashed | Some variants of a fixed point theorem of Krasnosel'skii and applications to nonlinear integral equations[END_REF] or [48, p. 155]).

2

We note that one of the main tools in the proof of Theorem 5.1.1 is the fact (I -B) -1 is weakly continuous. If we suppose that B is a weakly continuous k-contractive mapping B for some k ∈ [0, 1), as it is shown in the next lemma, the map (I -B) -1 is also weakly continuous.

Theorem 5.1.2 Let M be a nonempty closed, convex subset of a Banach space X. Let F : M → P cl,cv (M) be a upper semicontinuous multivalued mapping and B : X → X a weakly sequentially continuous single valued mapping satisfying ww-compact. Suppose that (a) all selections of F are ws-compact,

(b) F(M) is relatively weakly compact, (c) B is a k-contractive map, for some k ∈ [0, 1), (d) for each x ∈ M, (I -B) -1 F(x) ∈ P cv (M), (e)
F(M) + B(M) ⊂ M.
Then there exists y ∈ M such that y ∈ F(y) + B(y).

Remark 5.1.2 Note that in Theorem 5.1.1, since (I -B) -1 is a linear homeomorphism operator, it maps each closed convex set F(x), x ∈ M, onto a closed convex subset of M. In Theorem 5.1.2, even if, for all x ∈ M, (I -B) -1 F(x) is closed, it is not necessary convex. So the hypothesis (d) was added to guarantee the convexity of (I -B) -1 F(x) for all x ∈ M.

2

Proof. It is clear that according to assumption (c) and Lemma 2.3.3, the operator I -B is invertible and its inverse (I -B) -1 is a weakly sequentially continuous mapping.

Note that, for all x ∈ M, F(x) ∈ P cl,cv (M), and therefore by assumption (d) we conclude that (I -B) -1 F(x) ∈ P cl,cv (M). Hence, we can define the mapping N :

M → P cl,cv (M) by x → N(x) = (I -B) -1 F(x).
Let x ∈ M and z ∈ (I -B) -1 F(x). It is clear that there exists y ∈ F(x) such that z = (I -B) -1 (y) and therefore z = B(z) + y ⊆ B(z) + F(x). Using the hypothesis (e) one sees that z ∈ M. Hence, N(M) ⊂ M. On the other hand, because F(M) is relatively weakly compact and (I -B) -1 is weakly sequentially continuous on F(M), hence the use of Theorem 1.1 in [START_REF] O'regan | Fixed point theory of Mönch type for weakly sequentially upper semicontinuous maps[END_REF] shows that (I -B) -1 is weakly continuous on F(M). This yields that N(M) is relatively weakly compact.

Next, set K = co(N(M)). It follows from Krein-Ŝmulian's theorem 2.1.5 that the set K is a convex weakly compact subset of X. Moreover, because M ∈ P cl,cv (X), we have K ⊂ M and therefore N(K) ⊆ N(M) ⊂ co(N(M)) = K. This proves that N(K) ⊆ K and then N(K) is relatively weakly compact. Note also that N is is upper semicontinuous because (I -B) -1 is continuous and F is upper semicontinuous.

To complete the proof we have just to show that N(K) is relatively compact and to apply Theorem 2.3.1. The proof of the compactness of N(K) uses condition ws-compact and is similar to the last part of the proof of Theorem 5.1.1.

Condensing perturbations

In this section we establish some fixed point results in the spirit of Theorems 5.1.1 and 5.1.2 for condensing perturbations with respect to a measure of weak noncompactness. Then there exists x ∈ M such that x ∈ F(x) + B(x).

Proof. Let K be a bounded subset of M such that ω(K) 0. Since F(M) is relatively weakly compact, we have

ω((F + B)(K)) ≤ ω(F(K)) + ω(B(K)) = ω(B(K)).
(5.2.1)

According to Lemma 5.2.1, the operator B is ω-condensing. Consequently, Equation (5.2.1) may be written in the form

ω((F + B)(K)) < ω(K).
This shows that the mapping F + B is ω-condensing.

Let k 0 ∈ M and consider the set Π defined by

Π := {A ⊂ M such that (F + B)(A) ⊆ A, k 0 ∈ A and A is closed, convex}.
As in the proof of Theorem 5.2.1, we show that the set

C := A∈Π A = co (F + B)(C) ∪ {k 0 } (5.2.2)
belongs to Π and consequently

(F + B)(C) ⊂ C. (5.2.3)
The set C is weakly compact. Indeed, using the properties of ω and the fact that F + B is ωcondensing, we can write

ω(C) = ω co (B + F)(C) ∪ {k 0 } = ω (B + F)(C) < ω(C)
which is a contradiction. Hence ω(C) = 0 and so C is weakly compact.

Since B is a nonlinear contraction, it is well known that (I-B) is continuous, invertible and (I-B) -1 is continuous. Moreover, the use of the hypothesis (d) shows that (I -B) -1 F(x) ∈ P cl,cv (C) (here we use the fact the preimage of each closed subset of C by (I -B) -1 is closed). So, we can define the map N : C → P cl,cv (C) by

x → N(x) = (I -B) -1 F(x).
Since F is upper semicontinuous and (I -B) -1 is continuous, we conclude that N is upper semicontinuous. On the other hand, because F(C) is relatively weakly compact and the map (I -B) -1 is weakly continuous, we conclude that the set N(C) is relatively weakly compact.

We claim that N(C) is compact. To see this, let (x n ) n∈N be a sequence of points in C. Since C is weakly compact, there exists a subsequence (x n k ) k∈N such that x n k x as n → +∞ (x ∈ C because C is weakly closed). Let (y n k ) k∈N be a sequence in N(K) such that, for each k ∈ N, y n k ∈ N(x n k ), that is y n k ∈ (I -B) -1 F(x n k ). Hence, there exists a selection f of F such that y n k = (I -B) -1 f (x n k ), for each k ∈ N. Since f satisfies condition ws-compact, we infer that the sequence f (x n k ) k≥0 has a strongly convergent subsequence in K, say f (x n k j )) j≥0 . Moreover, since the operator (I -B) -1 is continuous, the sequence (I -B) -1 f (x n k j ) j≥0 converges strongly in C. This proves that the sequence N(x n k ) k∈N has a strongly convergente sequence. Hence N(C) is relatively compact.

         D ⊂ M, D = co {x 0 } ∪ B(D) and D = C with C ⊂ D countable =⇒ D is compact, (f) F(M) + B(M) ⊂ M.
Then there exists y ∈ M such that y ∈ F(y) + B(y). Thus, using the continuity of T ϕ n (•) and T ϕ(•) (see (6.1.2)), for > 0 (taken in the last step), there exists δ > 0 such that, for all t 1 , t 2 ∈ [a, b] with |t 2t 1 | < δ, we have

T ϕ n (t 1 ) -T ϕ n (t 2 ) X ≤ /3,
T ϕ n (t) -T ϕ(t) X ≤ T ϕ n (t) -T ϕ n (s) X + T ϕ n (s) -T ϕ(s) X + T ϕ(s) -T ϕ(t) X ≤
and therefore T ϕ -T ϕ n 0 ≤ ε. This proves the continuity of T . Now, we show that T is countably β-condensing. Indeed, let D be a countably bounded subset of U, we have

β(T (D)(t)) = β H(t) + t a f (s, ϕ(s))ds : ϕ ∈ D ≤ β t a f (s, ϕ(s))ds : ϕ ∈ D ≤ β (t -a)conv({ f (s, ϕ(s)) : ϕ ∈ D}) ≤ (t -a)β(conv({ f ([a, t] × D)})) = (t -a)β({ f ([a, t] × D)}).
Because U is bounded and equicontinuous, D is also bounded and equicontinuous. Using condition (c) together with Lemma 6.1.1, we get

β(T (D)(t)) ≤ ζ(b -a)β(D) < β(D) with β(D) > 0.
Hence T is countably β-condensing.

Note also that T (U) is bounded because

T ϕ 0 ≤ H 0 + (b -a) 1/q µ r p for all ϕ ∈ U It is clear that Π maps B τ into C [0, 1]
, X (argue as in the previous example). The continuity of Π is contained in the proof of the continuity of the map T in the previous example. Now, we show that Π is countably β-ζ-contractive map. Let D be a countably bounded subset of B τ , we have

β(Π(D)(t)) = β t 0 f (s, ϕ(s))ds : ϕ ∈ D ≤ β tconv({ f (s, ϕ(s)) : ϕ ∈ D}) ≤ tβ(conv({ f ([0, t] × D)})) = tβ({ f ([0, t] × D)})
Since D is bounded and equicontinuous (because B τ is equicontinuous and bounded), using condition (b) together with Lemma 6.1.1, we get

β(Π(D)(t)) ≤ ζtβ(D). Hence Π is countably β-ζ-contractive. We show that if ϕ = Πφ + S ϕ, φ ∈ B τ , then ϕ ∈ B τ . Let ϕ ∈ C [0, 1], X and φ ∈ B τ such that ϕ = S ϕ + Πφ. For all t ∈ [0, 1] we have ϕ(t) = H(ϕ(t)) + t 0 f (s, φ(s))ds
and therefore

ϕ(t) ≤ H(ψ(t)) -H(0) + H(0) + t 1 f (s, φ(s)) ds ≤ (1 -ζ) ϕ(t) + H(0) + µ τ p . Hence ϕ 0 ≤ (1 -ζ) ϕ 0 + H(0) + µ τ p .
This writes simply as Let D be a smooth open bounded convex subset of R n and let µ be a positive Radon measure on R N such that µ({0}) = 0.

ζ ϕ 0 ≤ H(0) + µ τ p .
The boundary of the phase space writes as ∂D × R N := Γ -∪ Γ + with

Γ ± = {(x, v) ∈ ∂D × R N , ±v.ν x ≥ 0},
where ν x stands for the outer unit normal vector at x ∈ ∂D.

Definition 6.2.1 Let (x, v) ∈ D × R N . We set t ± (x, v) = sup{t > 0, x ± sv ∈ D, 0 < s < t} = inf{t > 0, x ± tv D} and τ(x, v) := t + (x, v) + t -(x, v) for all (x, v) ∈ D × R N .
2

We now introduce the following functional space

W p = ψ ∈ X p such that v.∇ x f ∈ X p , where
X p := L p (D × R N ; dx ⊗ dµ(v)) (1 ≤ p < ∞).
It is well known (cf. [START_REF] Cessenat | Théorèmes de traces L p pour des espaces de fonctions de la neutronique[END_REF][START_REF] Cessenat | Théorèmes de traces pour des espaces de fonctions de la neutronique[END_REF][START_REF] Beals | Abstract time-dependent transport equations[END_REF] or [START_REF] Greenberg | Boundary value problems in abstract kinetic theory[END_REF]) that any function f in W p possesses traces f ± on Γ ± belonging to L ± p,loc (Γ ± ; |v.ν x |dγ x dµ(v)), dγ x being the Lebesgue measure on ∂D. In applications, suitable L p -spaces for the traces are

L ± p := L p (Γ ± ; |v.ν x |dγ x dµ(v)).
So, we define the set

W p = f ∈ W p ; f -∈ L - p .
It is well known that if f ∈ W p and f -∈ L - p , then f + ∈ L + p and vice versa [START_REF] Cessenat | Théorèmes de traces L p pour des espaces de fonctions de la neutronique[END_REF][START_REF] Cessenat | Théorèmes de traces pour des espaces de fonctions de la neutronique[END_REF][START_REF] Greenberg | Boundary value problems in abstract kinetic theory[END_REF]. More precisely we have the identity

W p = f ∈ W p ; f -∈ L - p = f ∈ W p ; f + ∈ L + p .
The boundary conditions may be written abstractly as an operator H relating the incoming and the outgoing fluxes, that is

H( f + ) = f -, D(H) = L + p and H(L + p ) ⊆ L - p .
Let H ∈ L(L + p , L - p ) be a positive boundary operator (the positivity is taken in the lattice sense, that is H transforms the positive cone of L + p into the positive cone of L - p ). Define the free streaming operator

T H        T H : D(T H ) ⊆ X p -→ X p , T H f (x, v) = -v.∇ x f (x, v) with domain D(T H ) = f ∈ W p such that f -= H( f + ) .
Let λ ∈ C and let g be a given function of X p . We consider the following boundary value problem

v.∇ x f (x, v) + λ f (x, v) = g(x, v) f -= H( f + ) (6.2.3)
where the unknown f is to be found in D(T H ). For Reλ > 0, the solution of equation (6.2.3) can be given formally by

f (x, v) = f (x -t -(x, v)v, v)e -λt -(x,v) + t -(x,v) 0 e -λs g(x -sv, v)ds. (6.2.4)
Moreover, for (x, v) ∈ Γ + , the equation (6.2.4) becomes

f + = f -e -λτ(x,v) + τ(x,v) 0 e -λs g(x -sv, v)ds. ( 6

.2.5)

To allow the abstract formulation of (6.2.4) and (6.2.5), let us define the following operators depending on the parameter λ:

M λ : L - p → L + p , u → M λ u := ue -λτ(x,v) ; B λ : L - p → X p , u → M λ u := ue -λt -(x,v) ;          G λ : X p → L + p , g → G λ g := τ(x,v) 0 e -λs g(x -sv, v)ds; and            C λ : X p → X p , g → C λ g := t -(x,v) 0
e -λs g(xsv, v)ds;

For p = 1, to show the dissipativity of the operator A, it is sufficient to show that Re Au, sign(u) ≤ 0 for all u ∈ D(A) where

sign(ψ)(x) =              1 when ψ(x) > 0, 0 when ψ(x) = 0, -1 when ψ(x) < 0.
The following observation is required below: For any ψ ∈ D(A) we have

ψ 1 sign(ψ), ψ = Ω ψ 1 sign(ψ)(x) ψ(x)dx = ψ 1 Ω sign(ψ)(x) ψ(x)dx = ψ 1 Ω |ψ(x)| dx = ψ 2 1 .
Hence, for any ψ ∈ D(A) we have

ψ 2 1 = ψ 1 sign(ψ), ψ .
Remark 6.2.3 It is well known that. For all boundary operator H such that ||H|| ≤ 1, the free streaming operator T H is dissipative, that is for all ψ ∈ D(T H ), we have T H ψ, |ψ| p-2 ψ ≤ 0. Lemma 6.2.1 Let 1 ≤ p < +∞ and assume that H ≤ 1. Then, for any λ ∈ C 0 , we have

(λ -T H ) -1 ≤ 1 Reλ .
Proof. For p ∈ (1 + ∞), we refer to Lemma 2.2 in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF]. Consider now the case p = 1. We first check that T H is dissipative. We have

T H ψ, sign(ψ) = - D×R N v∇ x ψ(x, v)sign(ψ)(x, v)dxdµ(v) = - D×R N v∇ x (|ψ(x, v)|)dxdµ(v) = - ∂D×R N |ψ| vν x dγ(x) dµ(v) = ψ -L 1,--ψ + L 1,+ = Hψ + L 1,--ψ + L 1,+ .
Hence, for every ψ ∈ D(T H ),

T H ψ, ψ X 1 sign(ψ) = ψ X 1 T H ψ, sign(ψ) ≤ 0 Case p = 1 Definition 6.2.4
Let X and Y be two normed spaces. A set W of L(X, Y) is said to be collectively weakly compact if, and only if, the set W(B 1 ) = {U(x), U ∈ W, x ∈ B 1 } is relatively weakly compact in Y.

2 Now, recall regular collision operators in X 1 [START_REF] Lods | On linear kinetic equations involving unbounded cross-sections: Math[END_REF][START_REF] Takác | A spectral mapping theorem for the exponential function in linear transport theory[END_REF]. Definition 6.2.5 We say that K is a regular collision operator on X 1 if, for almost all x ∈ D, the operator

φ ∈ L 1 (R N ; dµ(v)) -→ R N κ(x, v, v )φ(v )dµ(v ) ∈ L 1 (R N ; dµ(v))
is weakly compact on L 1 (R N ; dµ(v)) and the family of such operators on L 1 (R N ; dµ(v)) indexed by x ∈ D is collectively weakly compact.

2

We now recall the following result established in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF][START_REF] Mokhtar-Kharroubi | Mathematical topics in neutron transport theory[END_REF] which will play a crucial role. Proposition 6.2.2 Let K be a regular collision operator in X p with nonnegative kernel and assume that the conditions (A2) and (A3) are satisfied. Then, for any λ ∈ C 0 , (λ -T H ) -1 K is a Dunford-Pettis operator on X 1 .

Proof. see Lemma 3.3 in [START_REF] Boumhamdi | Existence results for a nonlinear transport equation with unbounded admissible velocities space[END_REF].

Let B(X 1 ) denote the collection of all nonempty bounded subsets of X 1 and let W(X 1 ) be the subset of B(X 1 ) consisting of all relatively weakly compact subsets of X 1 . Inspired by measures of weak noncompactness introduced in the works [START_REF] Boumhamdi | Existence results for a nonlinear transport equation with unbounded admissible velocities space[END_REF] and [START_REF] Latrach | Existence results for a class of nonlinear singular transport equations in bounded spatial domains[END_REF], we define the map

ζ : B(X 1 ) → [0, +∞) by ζ(M) = ζ 1 (M) + ζ 2 (M),
where

ζ 1 (M) = lim sup ε→0        sup ψ∈M E |ψ(x, v)|dxdµ(v), |E| ≤ ε        , (6.2.10)
with |E| denotes the measure of E ⊂ D × R n with respect to dxdµ(v) and

ζ 2 (M) = lim m→+∞        sup ψ∈M D |v|≥m |ψ(x, v)|dxdµ(v)        . ( 6

.2.11)

Arguing as in the proof of Proposition 2.8 in [START_REF] Boumhamdi | Existence results for a nonlinear transport equation with unbounded admissible velocities space[END_REF] or Proposition 2 in [START_REF] Latrach | Existence results for a class of nonlinear singular transport equations in bounded spatial domains[END_REF] we establish that ζ(•) is a regular measure of weak noncompactness on X 1 . In fact, it satisfies axioms of Definition 2.1.11 and more. To avoid repetitions we will not recall them. and therefore

F(λ)(B p r ) + B(λ)(B p r ) ⊂ B p r .
Next, we show that, for appropriate value of λ ∈ C 0 , the operator B(λ) is a contraction on X p . Indeed, let ψ 1 , ψ 2 ∈ X p and let λ ∈ C 0 . Using Lemma 6.2.1 and (A5)-(A6), we obtain

B(λ)ψ 1 -B(λ)ψ 2 X p ≤ 1 Reλ         σ 0 ∞ + d i=1 |λ i | β i ∞ K i ρ i ∞         ψ 1 -ψ 2 X p .
Let τ 2 be a nonnegative real number such that

k := 1 τ 2 ( σ 0 ∞ + d i=1 |λ i | β i ∞ K i ρ i ∞ ) < 1.
Hence, for all λ ∈ C τ 2 , B(λ) is k-contractive.

From now, we must treat separately the cases p ∈ (1, +∞) and p = 1.

(a) Let p ∈ (1, +∞). We show that the operator F(λ) is compact. Since K 0 is a regular collision operator, then according to Proposition 6.2.1 (a) (λ-T H ) -1 K 0 is compact. Therefore, by Lemma 2.2.1 and assumption (A4), we infer that N Θ 0 is continuous and consequently F(λ) is compact on X p . Set θ(r) = max(τ 1 , τ 2 ). Then, for any λ ∈ C θ(r) , the operators B(λ) and F(λ) satisfy the conditions of the classical Krasnosel'skii fixed point theorem. So, the fixed point Problem (6. We shall show that the operators F(λ) and B(λ) appearing in Eq. (6.2.15) satisfy the hypotheses of Corollary 4.2.1. To do so, we first observe that, according to Proposition 6.2.2, (λ -T H ) -1 K 0 is a Dunford-Pettis operator. Hence the condition (a) of Corollary 4.2.1 is satisfied.

Let S ⊂ B 1 r . Since the maps Θ i , i = 0, •, •, •, d, satisfy the condition (A4), by Lemma 2.2.1, there exist η i > 0 and h i (.) ∈ X + 1 (the positive cone of X 1 ) such that

|Θ i (x, v, f (x, v))| ≤ h i (x, v) + η i | f (x, v)|.
for all f ∈ S and for almost all (x, v) ∈ D × R N . Accordingly,

E |(N Θ i f )(x, v)|dxdµ(v) ≤ E h i (x, v)dxdµ(v) +η i E | f (x, v)|dxdµ(v), (6.
≤ (λ -T H ) -1         K 0 ζ 1 (N Θ 0 S ) + ζ 1 (N -σ S ) + d i=1 Λ i K i ζ 1 (N Θ i S )         ≤ 1 Reλ         η 0 K 0 + η σ + d i=1 η i β i ∞ K i         ζ 1 (S ).
Now, let τ 3 be a nonnegative real number such that

γ := 1 τ 3 (η 0 K 0 + η σ + d i=1 η i |λ i | β i ∞ K i ) < 1.
Hence, for all λ ∈ C τ 3 and for all subset S of B 1 r , we have ζ 1 F(λ)(S ) + B(λ)(S ) ≤ γζ 1 (S ).

Similarly, by using (6.2.17 Next, since Θ 0 satisfies (A4), it follows from Lemma 3.2 in [START_REF] Boumhamdi | Existence results for a nonlinear transport equation with unbounded admissible velocities space[END_REF] that N Θ 0 satisfies condition wwcompact.

Set θ(r) = max(τ 1 , τ 2 , τ 3 ). The steps above show that, for all λ ∈ C θ(r) , the assumptions of Corollary 4.2.1 are satisfied and so the fixed point problem (6.2.14) has a solution f 0 ∈ B 1 r . Therefore, the boundary value problem (6.2.1)-(6.2.2) has at least one solution ( f 0 , ..., f d ) with f 0 ∈ B 1 r .

By approximating, we assume that the functions f and g are constants ( f (•) ∈ L ∞ (S N-1 ) and g(•) ∈ L ∞ (S N-1 ). In such case, it suffices to consider the case K e (λ -T ) -1 and (λ -T ) -1 K e map L s (D × I × S N-1 ) into itself for all s ∈ [1, ∞] so that (by interpolation) it suffices to give a proof for p = 2. We consider first K e (λ -T ) -1 . Let M g be the averaging operator

M g : ϕ ∈ L 2 D × I × S N-1 → S N-1
ϕ(x, ρ, ω )dω ∈ L 2 (D).

It suffices to show that

M g (λ -T ) -1 : L 2 D × I × S N-1 → L 2 (D) is compact.
This amounts to

M g : D(T ) = ϕ ∈ L 2 D × V ; v. ∂ϕ ∂x ∈ L 2 D × V , ϕ | Γ-= 0 → L 2 (D).
We denote

W 2 (R n x × R + × S N-1 ) = ϕ ∈ L 2 R n x × R + × S N-1 such that v. ∂ϕ ∂x ∈ L 2 R n x × R + × S N-1 .
Furthermore, since D is convex, it follows from the lemma of extention that there exists a continuous extension operator (see [START_REF] Golse | Regularity of the moment of the solution of a neutron transport equation[END_REF]) such that E : W 2 (D × R + × S N-1 ) → W 2 (R n x × R + × S N-1 ). Let O ⊂ W 2 (D × R + × S N-1 ) be bounded. Then E(O) ⊂ W 2 (R n x × R + × S N-1 ). Applying Theorem 3.1 in [START_REF] Mokhtar-Kharroubi | Mathematical topics in neutron transport theory[END_REF], we conclude that the set {Eϕ |D , ϕ ∈ O} is relatively compact in L 2 (D). And consequently K * e ((λ -T ) -1 ) * is compact. Now using the Schauder theorem [31, Theorem 2, p. 485], we conclude that (λ-T ) -1 K e is compact by duality. Next, we shall check that, for suitable λ in C 0 , the operator G λ is a contraction on L p D × V . To see this, let ψ 1 , ψ 2 ∈ L p D × V and let λ ∈ C 0 . Using (A5) and the estimate of Eq. (6.3.4), one sees that

G λ (ψ 1 ) -G λ (ψ 2 ) L p D×V ≤ K d Reλ ψ 1 -ψ 2 L p D×V + 1 Reλ N -σ (ψ 1 ) -N -σ (ψ 2 ) L p D×V ≤ K d + ρ ∞ Reλ ψ 1 -ψ 2 L p D×V .
Let θ 2 be a nonnegative real number such that

α := K d + ρ ∞ θ 2 < 1.
Hence, for any λ such that Reλ > θ 2 , the operator G λ is α-contractive.

The rest of the proof is divided into two steps following 1 < p < ∞ or p = 1.

a) Let 1 < p < ∞. We first observe that the operator F λ is compact. Indeed, since K c and K e are a regular operators, according to Lemma 6.3.1 and Proposition 6.3.1, we infer that the operators (λ -T ) -1 K c and (λ -T ) -1 K e are compact. Further, using (A10) together with Lemma 2.2.1 one sees that N f and N g are continuous and consequently F λ is compact. Next, set θ r = max(θ 1 , θ 2 ), the operators F λ and G λ satisfy the conditions of the classical Krasnosel'skii fixed point theorem and therefore Problem (6. The operator G λ = (λ -T ) -1 N -σ + (λ -T ) -1 K d is ww-compact. Indeed, since (λ -T ) -1 and (λ -T ) -1 K d are linear oerator, according to Remark 2.1.4, they are ww-compact. Moreover, by Lemma 6.3.3, we know that N -σ is ww-compact. It is not dificult to check that the composition and the sum of ww-compact operator are also ww-compact. So, we conclude that G λ is ww-compact.

We claim that the operator F λ is ws-compact. Let (x n ) n∈N be a weakly convergent sequence in L 1 D × V and set U = {x 0 , x 2 , • • • }. It is clear that U is relativelly compact in L 1 D × V . Using Lemma 6.3.3 and the fact that the (λ -T ) -1 K c and (λ -T ) -1 K e are Dunford-Pettis operators we conclude that the sets (λ -T ) -1 K c N f (U) and (λ -T ) -1 K e N g (U) are relatively compacts and therefore the set (λ -T ) -1 K c N f (U) + (λ -T ) -1 K e N g (U) is relatively compact. This proves that F λ is ws-compact.

Quelques théorèmes de points fixes pour des applications univoques et multivoques et applications

Résumé On présente dans cette thèse quelques théorèmes de points fixes pour des applications univoques et multivoques dans les espaces de Banach. Dans la première, des théorèmes de points fixes de type Altman, Sadovskii, Leray-Schauder et Krasnosel'skii ont été établis pour les applications dénombrablement condensantes. Ensuite des résultats de points fixes faisant intervenir deux ou trois applications ont été présentés. Ces résultats reposent sur les concepts d'applications wscompactes et ww-compactes. Pour les applications multivoques on a établi de nombreux résultats de points fixes de type Krasnosel'skii en considérant divers types de perturbations (applications univoques et mutivoques). Dans la dernière partie, on a présenté des applications de ces résultats aux équations non linéaires intégrales de Volterra et à deux modèles d'équations de transport neutronique stationnaires.

Mots-clés: Théorèmes de point fixe, mesure de non-compacité et de non faible compacité, applications dénombrablement condensantes, applications multivoques semi-continues supérieurement, opérateurs de Dunford-Pettis, équations intégrales non linéaires de Volterra, équations stationnaires de transport neutronique.

Some fixed point theorems for single and multivalued mapping and applications Abstract In this thesis, we present some fixed point theorems for single and multivalued mappings in Banach spaces. In the first part, fixed point theorems of the Altman, Sadovskii, Leray-Schauder and Krasnosel'skii type have been established for countably condensing maps. Further, some fixed point results involving two or three maps were presented. These results are based on the concepts of ws-compact and ww-compact maps. For multivalued mapping, many fixed point results of Krasnosel'skii's type have been established for various kind of perturbations (singlevalued and multivalued maps). In the last part of this work, we present applications of our results to solve nonlinear integral Volterra equations and two models of stationary neutron transport equations.

Keywords

  , ϕ(s))ds, ∀t ∈ [a, b] (1.0.1) où f est une application de [a, b] × X dans X et H une application [a, b] dans X. Quant à la seconde, elle est de la formeϕ(t) = H(ϕ(t)) + t 0 g(s, ϕ(s))ds t ∈ [0, 1]; (1.0.2) où g : [0, 1] × X → X, H : X → X sontdeux applications et X est un espace de Banach. Notre objectif est d'étudier l'existence des solutions de l'équation (1.0.1) (resp. (1.0.2)) dans l'espace C([a, b], X) (resp. C([0, 1], X)). L'idée consiste à transformer l'équation (1.0.1) (resp, (1.0.2)) sous la forme d'un problème de point fixe de type Krasnosel'skii et ensuite vérifier si toutes les hypothèses de la Proposition 3.3.2 sont satisfaites. Lorsque c'est le cas, on obtiendrait des solutions de l'équation (1.0.1) (resp,(1.0.2)).

  ρ, ω, ω )g(x, ρ, ω , ϕ(x, ρ, ω ))dω (1.0.13) where (x, v) ∈ D × V, (V is called the space of admissible velocities and Ω is an open and bounded set of R N (N ≥ 3) endowed with the Lebesgue measure dx). The boundary conditions are modeled by ϕ | Γ-= 0. (1.0.14) The aim of this section is to solve the stationary Problem (1.0.13)-(1.0.14) in L p -espace (1 ≤ p < ∞). The strategy consists in transforming problem (1.0.13)-(1.0.14) into the following fixed point problem

  (a) Regularity: β(B) = 0 if and only if B ∈ C r (X). (b) Invariant under closure: β(B) = β(B), ∀B ∈ B(X).

( c )

 c Iinvariant under passage to the convex hull: β(co(B)) = β(B), ∀B ∈ B(X).

(

  d) Monotonicity: for any B 1 , B 2 ∈ B(X) we have B 1 ⊂ B 2 =⇒ β(B 1 ) ≤ β(B 2 ).

(

  e) Homogeneity: β(λB) = |λ| β(B) for any real λ and B ∈ B(X).

2

  The family ker µ given in first assertion is called the kernel of the measure of weak noncompactness µ(•). It should be noticed that the containment M ⊆ M w ⊆ co(M) together with the item (3) of Definition 2.1.11 implies that µ(M w ) = µ(M) where M w stands for the weak closure of M.

2

 2

  (a) F(M) is bounded, (b) µ(F(D)) < µ(D), for all bounded subset D of M with µ(D) > 0.

2 Theorem 3 . 2 . 1

 2321 Let K be a nonempty convex closed and bounded subset of a Banach space X, and let U ⊂ K be an open subset of K with p ∈ U. Let f : U → K be a continuous countably β-condensing map and assume that f (U) is bounded. Then (a) either Fix( f ) ∅, or (b) there exist u ∈ ∂U and λ ∈ (0, 1) such that u = λ f (u) + (1 -λ)p, here ∂U denotes the boundary of U in X.

Corollary 3 . 2 . 1

 321 Let K be a nonempty convex closed and bounded subset of a Banach space X, U ⊂ K an open subset of K with θ ∈ U. Let f : U → K be a continuous countably β-condensing. If f (U) is bounded, then (a) either Fix( f ) ∅, or (b) there exist u ∈ ∂U and λ ∈ (0, 1) such that u = λ f (u). Proposition 3.2.2 Let K be a nonempty convex closed and bounded subset of a Banach space X, U an open subset of K such that θ ∈ U and f : U → K a continuous, countably 1-set-contractive map. If f (U) is bounded and f is semi-closed, then (a) either Fix( f ) ∅, or (b) there exist u ∈ ∂U and λ ∈ (0, 1) such that u = λ f (u).

3 . 1

 31 one sees that the operator ζ : K → K defined by ζ(x) = (Ig) -1 f (x) is well defined and continuous. Using (3.3.1) one sees that ζ = f + gζ, so, for any subset C ∈ B c (K), we have β(ζ(C)) ≤ β( f (C)) + β(g(ζ(C))) < β(C) (use the hypothesis (b) and the fact that g maps bounded sets into relatively compact ones). This shows that ζ is countably β-condensing. The result is now follows from Theorem 2.1.1.

2 Theorem 3 . 3 . 4

 2334 be such that β(C) > 0. Using equation (3.3.1) together with the hypotheses (a), (b) and Lemma 2.1.5, we get β(J(C)) ≤ β( f (C)) + β(gJ(C)) ≤ kβ(C) + φ(β(J(C))). (3.3.3) If k = 0, inequality (3.3.3) becomes β(J(C)) ≤ φ(β(J(C))), and then β(J(C)) = 0 (use the fact that g is a nonlinear φ-contraction). Otherwise, the use of the fact, for r > 0, φ(r) < (1k)r, inequality (3.3.3) becomes β(J(C)) < kβ(C) + (1k)β(J(C)), and therefore β(J(C)) < β(C).In both cases, J is countably β-condensing. Now the use of Theorem 2.1.1 achieves the proof. Let K be a bounded open absorbing convex set of a Hilbert space H, and let f : K → H and g : H → H be two mappings such that (a) f is countably β-k-contractive, for some k ∈ (0, 1);(b) g is expansive with constant h > k + 1;(c) f (K) ⊂ (Ig)(X);(d) θ ∈ K and x, G( f (x) ≤ x 2 for each x ∈ ∂K, where G = (Ig) -1 .

2

 2

  X) with β(C) > 0. By equation (3.3.1), we have J(C) ⊂ f (C) + g(J(C)). Since f is countably β-k-contractive and g satisfies β(g(J(C))) ≤ φ(β(J(C))) we obtainβ(J(C)) ≤ β( f (C)) + β(gJ(C)) ≤ kβ(C) + φ(β(J(C))).(3.4.1)If k = 0, inequality (3.4.1) becomes β(J(C)) ≤ φ(β(J(C))) and therefore β(J(C)) = 0. Otherwise, by recalling the assumption that φ(r) < (1k)r for r > 0, inequality (3.4.1) becomes β(J(C)) < β(C). In both cases, J is countably β-condensing. Now, applying Proposition 3.2.1 to J we obtain the desired result.

2 35 4

 24 ChapterSome fixed point theorems for ws-compact and ww-compact operators This chapter is dedicated to fixed point results for Schauder's type and Krasnoselkii's type for ws-compact and ww-compact operators. As a consequence we derive similar results for Dunford-Pettis operators and weakly compact operators on spaces with the Dunford-Pettis Property because these two classes of operators are ws-compact operators.

2 Remark 4 . 1 . 1 2 Remark 4 . 1 . 2 2 Corollary 4 . 1 . 1

 241124122411 Note that, the identity operator I of the space X belongs to L(X), so it satisfies ww-compact (cf. Remark 2.1.3). Hence, if we take g = I in Theorem 4.1.1, then we recapture Theorem 2.1.3. By definition, linear Dunford-Pettis operators map weakly compact sets into norm compact sets, so they satisfy condition ws-compact. Let X be a Banach space and let K be a nonempty closed convex subset of X. Let f : X → K and g : K → X be two continuous maps and set F = f • g. Assume that (a) F(K) is relatively weakly compact, (b) f is a linear Dunford-Pettis operator, (c) g satisfies ww-compact.

2 4. 2

 22 Fixed point theorems of Krasnosel'skii's type Theorem 4.2.1 Let X be a Banach space, K a nonempty closed bounded convex subset of X and µ(•) a measure of weak noncompactness on X. Let f : X → K, g : K → X and B : K → X be continuous maps and set F = f • g. Suppose that (a) f satisfies ws-compact, (b) there exists γ ∈ (0, 1) such that µ F(S ) + B(S ) ≤ γµ(S ) for all S ⊂ K with µ(S ) > 0, (c) g satisfies ww-compact, (d) B is a nonlinear contraction with Φ-function φ, (e) x = B(x) + F(y), y ∈ K =⇒ x ∈ K.

2

  As an immediate consequence of Theorem 4.2.1 and Remark 4.1.2 we have: Corollary 4.2.1 Let X be a Banach space, K a nonempty closed bounded convex subset of X and µ(•) a measure of weak noncompactness on X. Let f : X → K, g : K → X and B : K → X be continuous maps and set F = f • g. Suppose that (a) f is a Dunford-Pettis operator, (b) there exists γ ∈ (0, 1) such that µ F(S ) + B(S ) ≤ γµ(S ) for all S ⊂ K with µ(S ) > 0, (c) g satisfies ww-compact, (d) B is a nonlinear contraction with Φ-function φ, (e) x = B(x) + F(y), y ∈ K =⇒ x ∈ K.

2 Theorem 4 . 2 . 2

 2422 Let X be a Banach space, K a nonempty closed bounded convex subset of X. Let f : K → K, g : K → K and B : K → X be continuous maps and set F = f • g. Suppose that (a) f satisfies ws-compact, (b) F(K) is relatively weakly compact, (c) g satisfies ww-compact,

2

 2

Theorem 5 . 2 . 1

 521 Let M be a nonempty closed,bounded and convex subset of a Banach space X and µ(•) a measure of weak non-compactness on X. Let F : M → P cl,cv (M) and B : X → P cl,cv (M) be two upper semicontinuous multivalued mappings. Suppose that (a) all selections of F and B are ws-compact,(b) F(M) is relatively weakly compact, (c) B is µ-condensing, (d) F(M) + B(M) ⊂ M.Then there exists y ∈ M such that y ∈ F(y) + B(y).Proof. Let ζ ∈ M and set G := F + B. We define the family Λ of subsets of M byΛ := D ⊆ M : D is closed convex, ζ ∈ D and G : D → P cl,cv (D) . co(G(K) ∪ {ζ}). It is clear that K is a closed, convex subset of M, ζ ∈ K and, for each D ∈ Λ, we have G(K) ⊆ G(D) ⊆ D, hence G(K) ⊆ D∈Λ D = K which proves that K ∈ Λ. Since G(K) ∪ {ζ} ⊆ K, the use of the relation K * = co(G(K) ∪ {ζ}) ⊆ co(K) = K yields that K * ⊆ K. Moreover, because G(K * ) ⊆ G(K) ⊆ K * ,we infer that K * ∈ Λ and then K ⊆ K * . This proves that K = K * . (a) all selections of F are ws-compact, (b) F(M) is relatively weakly compact, (c) B is a nonlinear contraction with Φ-function φ, (d) for each x ∈ M, (I -B) -1 F(x) ⊂ P cv (M) .

  (e) F(M) + B(M) ⊂ M.

  Now applying Theorem 2.3.1, one sees that there existsζ ∈ K ⊆ M such that ζ ∈ (I -B) -1 F(ζ) or equivalently ζ ∈ F(ζ) + B(ζ).

2

  Let us now introduce the following class of mappings introduced in[START_REF] Liu | Schaefer type theorem and periodic solutions of evolution equations[END_REF]. Definition 5.2.1 Let X be a normed space. A mapping B : D(B) ⊆ X → X is said to be a separate contraction if there exist two functions φ, ψ : R + → R + satisfying (a) ψ(0) = 0, ψ is strictly increasing, (b) B(x) -B(y) ≤ φ( xy ) for all x, y ∈ D(B), (c) ψ(r) ≤ r -φ(r) for all r > 0.Corollary 5.2.1 Let M be a nonempty closed, bounded and convex subset of a Banach space X. Let F : M → P cl,cv (M) be a upper semicontinuous multivalued mapping and let B : M → X be a weakly sequentially continuous single valued mapping. Suppose that (a) all selections of F are ws-compact, (b) F(M) is relatively weakly compact, (c) B is a separate contraction, (d) for each x ∈ M, (I -B) -1 F(x) ⊂ P cv (M) .

Theorem 5 . 3 . 1

 531 Let M be a nonempty closed, convex subset of a Banach space X. Let F : M → P cl,cv (M) and B : M → P cl,cv (M) be two upper semicontinuous multivalued mappings. Suppose that (a) all selections of F satisfy ws-compact, (b) F(M) is relatively weakly compact, (c) For each x ∈ M, F(x) ∩ B(x) ∅, (d) B maps compact sets into relatively compact sets, (e) there exists x 0 ∈ M such that

Proof.

  Let x 0 ∈ M and define the iterative sequence of sets (D n ) n∈N by D 0 = {x 0 }, D n = co {x 0 } ∪ B(D n-1 ) for n = 1, 2, • • • . and D = ∞ n=0 D n . Using the hypothesis (b) and by a standard argument (cf. the proof of Theorem 3.1 in [20]), we infer that D = co {x 0 } ∪ B(D) and D is compact. Since B(D) ⊂ D we infer that B(D) is relatively compact. Next, set O = F(M) ∩ B(D). It is clear that O is nonempty (use hypothesis (c)). Since B(D) is relatively compact, we deduce that O is also relatively compact. Thus O is compact and therefore, by hypothesis (d), B(O) is relatively compact. On the other hand, because O ⊂ M, we have F(O) ⊂ F(M) and the use of hypothesis (b) shows that F(O) is relatively weakly compact. We claim that F(O) is relatively compact. To see this, it suffices to use the assumption (a) and to argue as at the end of the proof of Theorem 5.1.1 or Theorem 5.2.1.According to the steps above, the set (F + B)(O) is relatively compact. Now applying Theorem 2.3.1, we conclude that there exists z ∈ O (and then z ∈ M) such that z ∈ (F + B)(z) which ends the proof.We shall apply Theorem Daher. First we prove that T :U → C [a, b], X is continuous. Let (ϕ n ) n∈N be a sequence of U such that ϕ n → ϕ in C [a, b], X . We have to show that T ϕ n → T ϕ in C [a, b], X . We know that ϕ n 0 ≤ M, ϕ 0 ≤ M, and there exists µM ∈ L p [a, b] such that f (s, ϕ n (s)) X ≤ µ M (s) and f (s, ϕ(s)) X ≤ µ M (s),for almost all s ∈ [a, b]. The Lebesgue dominated convergence theorem implies T ϕ n (s) → T ϕ(s) point-wise on [a, b]. This means ∀t, ∀ > 0 ∃N such that T ϕ n (t) -T ϕ(t) ≤ /3.

  By condition (b), we get ϕ 0 ≤ 1 ζ τ ζ = τ and so ϕ belongs to B τ . Moreover, we know from the hypothesis (d) that S is (1 -ζ)-contractive. Finally, applying Proposition 3.3.2 one sees that there exists φ ∈ B τ such that Πφ + S φ = φ.

  2.14) has a solution f 0 ∈ B p r and therefore the boundary value problem (6.2.1)-(6.2.2) has at least one solution ( f 0 , ..., f d ) with f 0 ∈ B p r . (b) Consider now the case p = 1.

  Θ i f )(x, v)|dxdµ(v) ≤ D |v|≥m h i (x, v)dxdµ(v) +η i D |v|≥m | f (x, v)|dxdµ(v),(6.2.17)for all measurable subset E of D × R N , m > 0 and 0 ≤ i ≤ d. This together with the estimate (2.1.4) implies thatζ 1 (F(λ)(S ) + B(λ)(S )) ≤ ζ 1 (F(λ)(S )) + ζ 1 (B(λ)(S ))

), we obtain ζ 2 F

 2 (λ)(S ) + B(λ)(S ) ≤ γζ 2 (S ). Hence the last two estimates yield ζ F(λ)(S ) + B(λ)(S ) ≤ γζ(S ), for all subset S ⊂ B 1 r .

2 Case p = 1 F

 21 In this subsection we are concerned with the transport operator T + K e on the spacesL 1 D × I × S N-1 := L 1 (D × I × S N-1 ; ρ N-1 dxdρdω)The elastic collision operator K e on L 1 D × I × S N-1 defined by and therefore

  3.1)-(6.3.2) has at least a solution ψ in B p r for all λ such that Reλ > θ r . b) For p = 1.

2

  It is not difficult to check that the result of Lemma 2.3.2 remains valid if we replace the hypothesis "B is a ww-compact map" by "B is weakly sequentially continuous map". That is Lemma 2.3.3 Let X be a Banach space and let B : X → X be a k-contractive map for some k ∈ [0, 1). If B is a weakly sequentially continuous map, then (I -B) -1 is a weakly sequentially continuous map.

The proof of this lemma is similar to that of Lemma 2.3.2 it suffices to replace in the proof Lemma 2.1.8 by Lemma 2.1.9. Definition 2.3.4 Let M be a subset of a Banach space X and let B :

  2.1 Note that according to Remark 2.1.2, if B is a contractive mapping, then Theorem 4.2.1 and Corollaries 4.2.1 and 4.2.2 hold true.

  ) together with the fact that [a, b] is compact we conclude that for all t ∈ [a, b] and for all s ∈ [a, b] such that |t -s| < δ, for n large enough, we have

	for all n ∈ N	(6.1.3)
	and
	T ϕ(t 1 ) -T ϕ(t 2 ) X ≤ /3.	(6.1.4)
	Now using (6.1.3), (6.1.4

:

  Fixed point theorem, measure of noncompactness and weak noncompactness, condensing mappings, countably condensing mappings, upper semicontinuous multivalued mappings, Dunford-Pettis operators, Nonlinear integral equations of Volterra, stationary neutron transport equations.

	Index
	equation	nonlinear contraction, 11
	of transport with delayed neutrons, 58	pseudocontractive, 12
	of transport with elastic and inelastic colli-	semi-closed, 15
	sion , 70	upper semicontinuous, 19
	of Volterra integral, 53, 56	weakly compact, 18
		weakly continuous, 17
	function	weakly sequentially continuous, 17
	Carathéodory, 19
	selection, 20	set
	functional	absorbing, 14
	Minkowski functional, 14	balanced, 14
	measure	theorem
	of noncompactness, 9	of Daher, 14
	of noncompactness of Hausdorff, 10	Altman's type, 21
	of noncompactness of Kuratowski, 10	Himmelberg, 20
	of weak noncompactness, 16	Krasnosel'skii's type, 25, 37
	of weak noncompactness of De Blasi, 16	Krasnosl'skii-Schaefer's type, 31
		Krein-Smulian, 18
	operator	Leray-Schauder's type, 22
	Φ-contraction, 11	Schauder's type, 35
	ψ-expansive, 13
	k-Lipschitzian, 10
	k-contractive, 10
	ws-compact, 17
	ww-compact, 17, 18
	accretive, 12
	countably 1-set-contractive, 10, 15
	countably β-k-contractive, 10
	countably β-condensing, 10
	Dunford-Pettis, 18
	expansive, 13
	Nemytskii operator, 19
	nonexpansive, 10, 12
	88

Chapter 5 Some fixed points results for perturbed multivalued mappings

This chapter is devoted to some fixed point results for perturbed multivalued mapping. It is organised as follows: In Section 5.1 deals with contractive perturbations while Section 5.2 is concerned with condensing perturbations. In Section 5.3 we give two fixed set results using the concept of Mönch-sets.

Contractive single valued perturbations

In this section we establish a fixed point theorem for the sum of some multivalued mappings and single valued mappings.

Theorem 5.1.1 Let M be a nonempty closed, convex subset of a Banach space X. Let F : M → P cl,cv (M) be a upper semicontinuous multivalued mapping and B : X → X a bounded linear single valued mapping. Suppose that (a) all selections of F are ws-compact, Then there exists y ∈ M such that y ∈ F(y) + B(y).

Proof. It is clear that according to assumption (c), the operator I -B is invertible and the operator (I -B) -1 ∈ L(X). Hence, according to Theorem 3.10 in [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF], (I -B) -1 is weakly continuous.

Define the operator N : M → P cl,cv (M) by x → N(x) = (I -B) -1 F(x).

Since, for each x ∈ M, F(x) ∈ P cl,cv (M) and (I -B) -1 ∈ L(X), the subset N(x) is convex. Futher, using the continuity of I -B and the fact that F(x) is closed, we conclude that N(x) is also closed. Hence, for each x ∈ M, N(x) ∈ P cl,cv (X).

Let x ∈ M and g ∈ (I -B) -1 (F(x)). It is clear that there exists y ∈ F(x) such that g = (I -B) -1 (y) and therefore g = B(g) + y ⊆ B(g) + F(x). Using the hypothesis (d) one sees that g ∈ M. Hence, N(M) ⊂ M. On the other hand, because F(M) is relatively weakly compact and (I -B) -1 is weakly continuous, the set N(M) is relatively weakly compact.

Next, set K = co(N(M)). Using the Krein-Ŝmulian theorem 2.1.5 (ref [31, p. 434]) we infer that K is a weakly compact convex subset of X. Moreover, because M is closed and convex, we have K ⊂ M and consequently we have N(K) ⊆ N(M) ⊂ co(N(M)) = K. This proves that N(K) ⊆ K and therefore N(K) is relatively weakly compact. Because (I -B) -1 is continuous, the operator N is upper semicontinuous.

We claim that N(K) is relatively compact. To see this, let (x n ) n∈N be a sequence of points in K. Since K is weakly compact, there exists a subsequence (x n k ) k∈N such that x n k x (x ∈ K because K is weakly closed), as n → +∞. Let (y n k ) k∈N be a sequence in N(K) such that, for each k ∈ N, y n k ∈ N(x n k ), that is y n k ∈ (I -B) -1 F(x n k ). Hence, there exists a selection f of F such that y n k = (I -B) -1 f (x n k ), for each k ∈ N. Since f satisfies condition ws-compact, we infer that the sequence f (x n k ) k≥0 has a strongly convergent subsequence in K, say f (x n k j )) j≥0 . Moreover, since the operator (I -B) -1 is continuous, the sequence (I -B) -1 f (x n k j ) j≥0 converges strongly in K. This proves that the sequence N(x n k ) k∈N has a strongly convergente sequence. Hence N(K) is relatively compact. Now applying Theorem 2.3.1, one sees that there exists

. This ends the proof. In fact, this result holds true if we suppose that there exists a nonnegative integer p such that B p < 1. In this case the operator I -B is invertible,

B k and I -B is an homeomorphism from M onto (I -B) -1 M. So, the arguments used in the proof above apply.

(b) Note also that Theorem 5.1.1 remains valid if we suppose that, for some nonnegative integer p, B p is a nonlinear contraction. Here again the operator I -B is invertible, (I -B) -1 = (I -We claim that K is relatively weakly compact. If it is not the case, then µ(K) > 0. Using the properties of µ(•) and the fact that B is µ-condensing we can write

which is a contraction. Hence µ(K) = 0 and so K is relatively weakly compact. Next, because G(K) ⊂ K, the set G(K) is relatively weakly compact too.

We claim that G(K) is relatively compact. Indeed, let (x n ) n∈N be a sequence of points in K. Since K is weakly compact, there exists a subsequence (x n k ) k∈N such that x n k x as n → +∞. Let (w n k ) k∈N be a sequence in F(K) such that, for each k ∈ N, w n k ∈ F(x n k ). By hypothesis, there exists a selection f of F such that w n k = f (x n k ) satisfying ws-compact. Also, let (z n k ) k∈N be a sequence in B(K) such that, for each k ∈ N, z n k ∈ B(x n k ). By hypothesis, there exists a selection g of B such that z n k = g(x n k ) satisfying ws-compact. Since f and g satisfy condition ws-compact, we infer that the sequence z n k + w n k k∈N with z n k + w n k = f (x n k ) + g(x n k ), has a strongly convergent subsequence in K. Hence G(K) is relatively compact. Now, since the map G = F + B is upper semicontinuous, applying Theorem 2.3.1, we conclude that there exists z ∈ K ⊆ M such that z ∈ G(z) which ends the proof.

2

In the following we shall give some single-valued perturbations for which the mapping F + B is ω-condensing. To this end we introduce the following lemma which appears in [48, p. 248] in french. For completeness we give a proof. Lemma 5.2.1 Let X be a Banach space and let B : X → X be a weakly sequentially continuous map on X. If B is a nonlinear contraction, then B is ω-condensing.

Proof. Let S ∈ B(X) such that ω(S ) = τ and ε > 0. By definition of ω, there exists W ∈ W such that S ⊆ W + (τ + ε)B 1 . So, for x ∈ S , there exist y ∈ W and z ∈ (τ + ε)B 1 such that x = y + z and then

where φ is the Φ-function of B. Hence, B(S ) ⊆ B(W) + B φ(τ+ε) . On the other hand, since B is weakly sequentially continuous and W is weakly compact, according to Theorem 2.1 in [START_REF] O'regan | Fixed point theory of Mönch type for weakly sequentially upper semicontinuous maps[END_REF], the map B is weakly continuous on W. Hence B(W) is weakly compact, and therefore ω(B(S )) ≤ φ(τ + ε). Since ε is arbitrary, we get ω(B(S )) ≤ φ(τ) < τ = ω(S ). This ends the proof.

2

Theorem 5.2.2 Let M be a nonempty closed, bounded and convex subset of a Banach space X. Let F : M → P cl,cv (M) be a upper semicontinuous multivalued mapping and let B : M → X be a weakly sequentially continuous single valued mapping. Suppose that (e) F(M) + B(M) ⊂ M.

Then there exists x ∈ M such that x ∈ F(x) + B(x).

Proof. This is an immediate consequence of Theorem 5.2.2, it suffices to observe that any separate contraction mapping is a nonlinear contraction.

2

We now introduce the following class of mappings due to Burton [START_REF] Burton | Integral equations, implicit functions, and fixed points[END_REF]. Definition 5.2.2 Let X be a normed space. A mapping B : D(B) ⊆ X → X is said to be a large contraction if, ∀ x, y ∈ D(B) with x y, we have B(x) -B(y) < xy and if, for any ε > 0, there exists a real δ = δ(ε) < 1 such that x, y ∈ X, ε ≤ xy =⇒ B(x) -By) ≤ δ xy . Corollary 5.2.2 Let M be a nonempty closed, bounded and convex subset of a Banach space X. Let F : M → P cl,cv (M) be a upper semicontinuous multivalued mapping and let B : M → X be a weakly sequentially continuous single valued mapping. Suppose that (a) all selections of F are ws-compact, Then there exists x ∈ M such that x ∈ F(x) + B(x).

Proof. Note that according to Lemma 1.1 in [START_REF] Liu | Schaefer type theorem and periodic solutions of evolution equations[END_REF], B is a separate contraction mapping. Hence, the result follows from Corollary 5.2.1.

Countably condensing perturbations

In this section, we present fixed set results for the sum of two upper semicontinuous multivalued mappings using the concept of Mönch-sets. Definition 5.3.1 Let M be a subset of a Banach space X and let µ(•) be a measure of noncompactness on X. Let F : M → P cl,cv (M) be a multivalued mapping. The map F is said to be countably µ-condensing if µ F(M) < ∞ and µ(F(C)) < µ(C) for any countable bounded subset C of M with µ(C) > 0.

Before going further, we first recall the following definition introduced in [START_REF] Cardinali | Monch sets and fixed point theorems for multimaps in locally convex topological vector spaces[END_REF]. Definition 5.3.2 Let M be a convex subset of X and F : M → B(M) be a given map. We say that A ⊂ M is a Mönch-set for F if there exists x 0 ∈ M such that A = co {x 0 } ∪ F(A) and there exists a countable set C ⊂ A with A = C. Corollary 5.3.1 Let M be a nonempty closed, convex bounded subset of a Banach space X and µ(•) a measure of non-compactness on X. Assume that F : M → P cl,cv (M) and B : M → P cl,cv (M) be two upper semicontinuous multivalued mappings. Suppose that the hypotheses (a), (b), (c), (d) and (f) of Theorem 5.3.1 are satisfied. If B is countably µ-condensing, then there exists y ∈ M such that y ∈ F(y) + B(y).

Proof. We prove that B satisfies hypothesis (e) of Theorem 5. We claim that µ(C) = 0. Indeed, the use of (5.3.2), the fact that B is countably µ-condensing and the properties of µ, gives

If µ(N) = 0, then the proof is finished. Otherwise, combining (5.3.1) and (5.3.3) we obtain

which is a contradiction. Hence, µ(C) = 0 which proves our claim. Accordingly, the use of (5. Chapter 6

Applications

In this Section 6.1 we shall apply the results of Chapter 2 to solve two nonlinear Volterra integral equations. In Section 6.2 the results of Chapter 4 were applied to solve a nonlinear transport equations with delayed neutrons while Section 6.3, with the help of Theorem 2.1.4 we discuss existence result for a nonlinear transport equation with partly elastic collisions using.

Volterra integral equation

In this section we shall present two examples. Let us first recall the following result of Ambrosetti's type established in [START_REF] Michell | An existence theorem for weak solutions of differential equation in Banach spaces[END_REF].

Lemma 6.1.1 Let X be a Banach space and let A be a subset of C([0, T ], X) with T > 0. If A is bounded and equicontinuous, then

where β is a measure of noncompactness on X (see Definition

Example 1

We consider the following Voltera problem

where

Here X is a Banach space.

Our objective is to show the the problem (6.1.1) has a solution belonging to the space C [a, b], X .

To this end, we introduce the following hypotheses :

(iii) for each r > 0, there exists

where Proof. Let T be the map defined by :

Note that for any ϕ ∈ C [a, b], X , there exists r > 0 such that ϕ 0 ≤ r and since f is L p -Carathéodory, there exists

where q stands for the conjugate exponent of p.

Finally, applying Theorem Daher, we conclude that T has a fixed point in U, or equivalently, (6.1.1) has a solution in U.

2

Example 2

Now we shall discuss the existence of solutions to the following Volterra integral equation

here f : [0, 1] × X → X, H : X → X are two maps and X is a Banach space.

Our objective is to show the the Problem (6.1.5) has a solution belonging to the space C [0, 1], X .

Let us introduce the following hypotheses :

(iii) for each r > 0, there exists

(b) there exists ζ ∈ (0, 1) such that for any countably bounded subset D of C [0, 1], X and for each t ∈ [0, 1], we have

The real ζ appearing in conditions (b), (c) and (d) is the same and belongs to (0, 1).

A transport equation with delayed neutrons

This section deals with existence of solutions to the following nonlinear boundary value problem which involves a multidimensional transport equation with delayed neutrons in a bounded spatial domain. More precisely, we shall discuss existence of solutions to the nonlinear boundary value problem:

The function f 0 represents the neutrons density, f i (x, v) represents the density of the i-th group delayed neutron emitters and λ i , 1 ≤ i ≤ d} are radioactive decay constants [START_REF] Marti | Mathematical foundations of kinetics in neutron transport theory[END_REF]. The functions σ(•, •, •) and κ i (•, •, •) are called, respectively, the collision frequency and the scattering kernel which are, in general, nonlinear functions of f 0 while

where f - 0 (resp. f + 0 ) is the restriction of f 0 to Γ -(resp. Γ + ), with Γ -(resp. Γ + ) is the incoming (resp. the outgoing ) part of the phase space boundary and H is a linear bounded operator from a suitable space on Γ -to a similar one on Γ + . The well known classical boundary conditions (vacuum boundary conditions, specular reflections, diffuse reflections, periodic and mixed type boundary conditions) are special examples of our framework.

we apply Corollary 4.2.1 to solve the boundary value problem (6.2.1)-(6.2.2). The functional setting of the problem is L p -spaces with p ∈ [1, +∞). We note that to solve Problem (6.2.1)-(6.2.2) in L p -spaces with p ∈ (1, +∞) is not difficult, it uses the compactness results established in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF] and requires the use of the classical Krasnosel'skii's fixed point theorem. However, in L 1space, Problem (6.2.1)-(6.2.2) involves a Dunford-Pettis operator, and its proof requires Corollary 4.2.1.

Basic facts related to the problem

Before going further, we first gather some definitions and results related to the usual neutron transport equation with delayed neutrons required in the sequel. Remark 6.2.1 We note that for λ > 0, the operator M λ , B λ , G λ and C λ are bounded and positive (in the lattice sense).

2

Using these operators and the fact that f must satisfy the boundary condition, Eq. (6.2.5) becomes

The solution of this equation reduces to the invertibility of the operator I -M λ H So, if (I -M λ H) -1 exists, then

On the other hand, Eq (6.2.4) can be written as follows:

Substituting (6.2.6) in this equation we obtain

In the remainder of this section we will use the following notation: For any real number τ, we set

Remark 6.2.2 (see [START_REF] Latrach | Compactness results for transport equations and applications[END_REF]) We note that, for any λ ∈ C 0 , the operator C λ is nothing else but the resolvent of the free streaming operator with absorbing boundary conditions (H = 0) (i.e., C λ = (λ -T 0 ) -1 ).

2

Let X be a Banach space and let X * be its topological dual space. We call the normalized duality map of X the map J : X → 2 X * defined by

Let x ∈ X, by the Hahn-Banach theorem J(x) ∅. A linear operator A with domain and range both in X is called dissipative if for every x ∈ D(A) there is a x * ∈ J(x) such that Re Ax, x * ≤ 0.

Let (Ω, , µ) be a measure space and L p (Ω, dµ) with 1 < p < ∞, J(0) = {0} and for 0 u ∈ L p (Ω, dµ), J(u) is a singleton, that is ,

because H ≤ 1. This proves that T H is dissipative.

Let ψ ∈ D(T H ) and set ϕ = λψ -T H ψ. Using the equality Reλ ψ 2

Reλ which ends the proof.

2

Let us now introduce the following assumptions:

(A1) For p ∈ (1, +∞), the operator H is compact and satisfies

(A2) For p = 1, the operator H is weakly compact and satisfies

For each e ∈ S N-1 , µ{v ∈ R N , v.e = 0} = 0 where S N-1 denotes the unit sphere of R N .

This hypothesis means that the hyperplanes of R N (through the origin) have zero µ-measure.

The physical collision operators K, used in nuclear reactor theory for all types of moderators (gas, liquid or solid), are bounded and most of them are of the form

where k(•, •, •) is a non-negative measurable function (see, for example, [START_REF] Dautray | Analyse mathématique et calcul numérique[END_REF][START_REF] Kaper | Spectral methods in linear transport theory[END_REF][START_REF] Greenberg | Boundary value problems in abstract kinetic theory[END_REF] or [START_REF] Mokhtar-Kharroubi | Mathematical topics in neutron transport theory[END_REF] and the references therein).

Note that K is local with respect to the space variable x ∈ D. So, it may be regarded as an operator valued mapping from D into L(L p (R N ; dµ(v))), that is,

Compactness properties

Let us first recall the definition of collectively compact (resp. collectively weakly compact) operators on Banach spaces.

Case p ∈ (1, ∞) Definition 6.2.2 Let X and Y be two normed spaces. A set C of L(X, Y) is said to be collectively compact if, and only if,

Now we are ready to state the definition of regular collisions operators [START_REF] Mokhtar-Kharroubi | Optimal spectral theory of the linear Boltzmann equation[END_REF] (see also [START_REF] Mokhtar-Kharroubi | Mathematical topics in neutron transport theory[END_REF]).

Definition 6.2.3 Let p ∈ (1, +∞). A collision operator K is said to be regular on X p if:

(a) {K(x) : x ∈ D} is a set of collectively compact operators on L p (R N ; dµ(v)), i.e. K(x)ϕ; x ∈ D, ϕ p ≤ 1 is relatively compact in L p (R N ; dµ(v)).

(b) For every ϕ ∈ L q (R N ; dµ(v)), K (x)ϕ ; x ∈ D, ϕ q ≤ 1 is relatively compact in L q (R N ; dµ(v)).

Here K (x) denotes the dual operator of K(x) and q = p p -1 .

2

We now recall the following result established in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF][START_REF] Boumhamdi | Existence results for a nonlinear transport equation with unbounded admissible velocities space[END_REF][START_REF] Mokhtar-Kharroubi | Mathematical topics in neutron transport theory[END_REF] which will play a crucial role. Proposition 6.2.1 Let K be a regular collision operator in X p with nonnegative kernel and assume that the conditions (A1) and (A3) are satisfied. Then, for any λ ∈ C 0 , the operator (λ -T H ) -1 K is compact on X p with p ∈ (1, +∞),

Proof. Item (a) is Theorem 3.1 in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF].

Existence results

For i ∈ {1, 2, • • •, d}, let Λ i denote the bounded multiplication operators from X p into itself defined by

We also denote by K i , 0 ≤ i ≤ d, the operator defined from X p to X p by

where

Moreover, N g stands for the Nemytskii operator generated by the function g

Now Problem (6.2.1)-( 6.2.2) may be written abstractly as

(6.2.13)

Since, for i = 1, •, •, •, d, the operator Λ i is bounded, we can replace λ i f i (x, ξ) of the first equation of (6.2.13) by K i N Θ i f 0 . Moreover, according to Remark 6.2.2, each λ ∈ C such that λ ∈ C 0 belongs to ρ(T H ). Hence, for such λ, Problem (6.2.1)-(6.2.2) reduces to the following fixed point problem

where (A5) For each r > 0, the function σ(•,

) is a Carathéodory function and N -σ acts from X p into itself, where σ 0 (., .)

Theorem 6.2.1 Let 1 ≤ p < ∞, K 0 be a regular collision operator on X p and assume that (A1)-(A6) hold true. Then, for each r > 0, there exists a real θ(r) > 0 such that, for all λ ∈ C θ(r) , the boundary value problem (6.2.1)-( 6

A transport equation with partly elastic collision operators

The purpose of this section is to discuss existence of solutions to the following boundary value problem

where (x, v) ∈ D × V, (D is an open smooth bounded subset of R N (N ≥ 3) endowed with the Lebesgue measure dx while V stands for the admissible velocities space). The subset V is given by

It is endowed with the Lebesgue measure dv = ρ N-1 dρdω where dω is the Lebesgue measure on the unit sphere S N-1 . The functions σ(•, •, •) and f (•, •, •) are measurable nonlinear of ϕ and the function ϕ(x, v) represents the number density of particles having the position x and the velocity v.

The function σ(•, •, •) is called the collision frequency and the functions

(called classical, elastic and inelastic collision operators respectively).

In our framework, the boundary conditions are modled by

where, Γ -denotes the incoming part of the boundary of the phase space Ω × V and define by

where ν x stands for the outer unit normal vector at x ∈ ∂Ω.

In this model, the collision operator is given by

where K c denotes the collision operator involved in the classical neutron transport theory (see, for example, [START_REF] Mokhtar-Kharroubi | Optimal spectral theory of the linear Boltzmann equation[END_REF]). It is given by

The operator K d is called the inelastic collision operator, it is given by

where each operator K j d ( j = 1, ..., l) describes an event in which a discrete energy E j is lost by a neutron at x with initial speed ρ j and final speed ρ. The speed ρ j is defined by

while M stands for the mass of neutron. The operator K e is called the elastic collision operator. It is given by

It describes the collisions which do not vary the kinetic energy of neutrons. It acts only through the angular part of the velocity variable .

The goal of this section is complete the analysis started in the papers [START_REF] Larsen | On the spectrum of the linear transport operatior[END_REF][START_REF] Sbihi | Analyse Spectrale De Modèles Neutroniques[END_REF][START_REF] Al-Izeri | On the asymptotic spectrum of a transport operator with elastic and inelastic collision operators[END_REF][START_REF] Al-Izeri | On the solution of a class of linear transport equation with partly elastic collision operators[END_REF] where several analysis of the spectral properties and the properties of solution to Cauchy problem governed by these operators for the problem. Our aim in this section to solve the stationary Problem (6.3.1)-(6.3.2) in L p -spaces (1 ≤ p < ∞). We introduce the functionnal setting of the problem (6.3.1)-(6.3.2) and we establish some existence results of the compactness to be able to solve the problem (6.3.1)-(6.3.2).

Notations

The aim of this section is to introduce the different notations and some preliminary results which we will need in the sequel.

We define the streaming operator T with absorbing boundary conditions (that is H = 0) by

Let ψ ∈ L p (D × V) and consider the resolvent equation for the operator T , (λ -T )ψ = ϕ (6.3.3)

where λ is a complex number and the unknown ψ must be sought in D(T ). Proceeding as in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF], one sees that, for Reλ > 0, the solution of Eq. (6.3.3) is formally given by

where s(x, v) = inf{s > 0; xsv D}. In fact, s(x, v) is the time required by a particle having the position x and the velocity v to go out of D.

Remark 6.3.1 As already observed, the operator T is nothing else but the operator T 0 (the boundary operator is H = 0), so the proof of Lemma 6.2.1 shows that T is dissipative and, for any λ ∈ C 0 (see (6.2.8)), the following estimate holds true

2

We note that, following (6.3.4), the resolvent set of the operator T contains the complex half plane C 0 .

We recall that In [START_REF] Appel | Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili[END_REF], Appell and De Pascale proved that the measure of weak non-compactness of De Blasi ω(•) can be expressed in L 1 -spaces as

for every subset M ∈ B L 1 (D × V) . Here |D| denotes the Lebesgue measure of D.

Compactness properties

In the paper [START_REF] Latrach | Existence results for a nonlinear transport equation in bounded geometry on L 1 -spaces[END_REF], the spectral properties of the operator T + K e + K d + K c was discussed in the space L 1 . In [START_REF] Sbihi | Analyse Spectrale De Modèles Neutroniques[END_REF], this analysis was extended to L p -spaces with p ∈ [1, +∞). Our objective in this section is to complete this analysis by investigating compactness properties of the operators (λ -T ) -1 K e and K e (λ -T ) -1 in the space L p for p ≥ 1 and λ ∈ C 0 .

Here we will split our discussion into two cases : the case 1 < p < ∞ and the case p = 1.

Case p ∈ (1, ∞)

In this subsection, we shall be interested in the compactness properties of the operators (λ-T ) -1 K e and K e (λ -T ) -1 on the spaces

Let us first note that the collision operator K c is a bounded linear operator on L p D × V defined by

where the kernel

It is clear that the collision operator K c is local in x. So, we may regard K c as an operator on L p (V; dv) depending on a parameter x ∈ D. One can define the mapping

where The elastic collision operator K e is defined by

where the kernel k e (•,

We introduce the following hypothesis:

(A7) for all ψ ∈ L p (S N-1 , dω), the set

is relatively compact set in L p (S N-1 , dω) and, for every, ψ ∈ L q (S N-1 , dω),

The hypothesis (A7) was introduced by M. Sbihi in [START_REF] Sbihi | Spectral theory of neutron transport semigroups with partly elastic collision operators[END_REF] (see also [START_REF] Sbihi | Analyse Spectrale De Modèles Neutroniques[END_REF]).

Under condition (A7), we have

where K e (x, ρ) is the operator defined by

We recall also the following lemma owing to M. Sbihi [START_REF] Sbihi | Spectral theory of neutron transport semigroups with partly elastic collision operators[END_REF].

Lemma 6.3.1 An elastic collision operator K e satisfying (A7) can be approximated in the norm operator topology by collision operators with kernels of the form

where α i ∈ L ∞ (D × I), f i ∈ L p (S N-1 ) and g i ∈ L q (S N-1 ), i ∈ J with q denotes the exponent conjugate of p while J is a finite set. Proposition 6.3.1 Let λ ∈ C 0 . If K e is a regular collision operator in L p D × I × S N-1 satisfying (A7), then the operators (λ -T ) -1 K e is compact.

Proof. Since K e is regular operator, according to Lemma 6.3.2, it is enough to establish the theorem for a collision operator with a kernel of the form

where α i ∈ L ∞ (D × I), f i ∈ L p (S N-1 ) and g i ∈ L q (S N-1 ).

where the kernel k e (•, •, •, •) is a nonnegative measurable function on D × I × S N-1 × S N-1 .

We introduce the following hypotheses:

(A8)

Proceeding as in the proof of Lemma 3.2 in [START_REF] Latrach | Existence results for a nonlinear transport equation in bounded geometry on L 1 -spaces[END_REF] we prove the following proposition. The proof is omitted.

Proposition 6.3.2 Let K e be a regular collision operator in L 1 (D×V) and assume that K e satisfies assumptions (A8) and (A9). Then, for any λ ∈ C 0 , (λ -T )K e is a Dunford-Pettis operator.

Next, let K c be the bounded linear operator defined by

Existence results

In this section we discuss existence results for the boundary value problem (6. Proof. For the proof we refer, for example, to [START_REF] Latrach | Existence results for a nonlinear transport equation in bounded geometry on L 1 -spaces[END_REF]. We are now ready to state our existence result. where N(r), N (r) and N (r) denote the upper bounds of N f , N g and N -σ , respectively, on B p r .

Let θ 1 be the real number defined by So, for any λ ∈ C θ 1 , we have

Next, let λ ∈ C λ r and consider the sequence (M n ) ∈N be the sequence of nonempty closed bounded convex subsets of B 1 r defined by

We note that for λ ∈ C λ r , so using the convexity of B 1 r and the inclusion F λ (B 1 r ) + G λ (B 1 r ) ⊂ B 1 r we check easily by induction that (M n ) n∈N is a decreasing sequence of B(L 1 D × V ) (in the sens of the inclusion).

Next, using the properties of ω(•), we get

Let Θ 3 be a nonnegative real number such that

Hence, for all λ satisfying Reλ > Θ 3 and all subset M n of B 1 r , we have ω(F λ (M n ) + G λ (M n )) ≤ γω(M n ). (6.3.9)

It follows from the estimate above that, for any λ satisfying Re(λ) ≥ Θ 3 , ω(M n+1 ) ≤ γω(M n ) and therefore ω(M n ) ≤ γ n ω(M 0 ). The fact that γ ∈ (0, 1) implies lim n→∞ ω(M n ) = 0. Now invoking Definition 2.1.11 (e), we infer that M = ∞ n=1 M n is a nonempty weakly compact subset of B 1 r . Moreover, the inclusion F λ (M) + G λ (M) ⊆ M is obvious.

To conclude the proof, it suffices to consider the fixed point problem ψ = (F λ )(ψ) + G λ (ψ) on the set M.We first note that F λ (M) + G λ (M) ⊆ M (see above). Moreover, we established above that F λ is ws-compact on B 1 r , so it is ws-compact on M. Since F λ (M) + G λ (M) ⊆ M, we have F λ (M) ⊆ M and therefore F λ (M) is relatively weakly compact. Further, we proved above that G λ is ww-compact and α-contractive. Hence all the conditions of Theorem 2.1.4 are satisfied which ends the proof.