Etude quantitative et qualitative des solutions de certaines équations intégrales

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N° d’ordre

:

REPUBLIQUE ALGERIENNE DEMOCRATIQUE & POPULAIRE

MINISTERE DE L’ENSEIGNEMENT SUPERIEUR & DE LA RECHERCHE

SCIENTIFIQUE

UNIVERSITE DJILLALI LIABES

FACULTE DES SCIENCES EXACTES

SIDI BEL ABBES

THESE

DE DOCTORAT

Présentée par

_B

_AGHDAD

_S

_AID

Spécialité : Mathématiques

Option : Analyse mathématiques et applications

Intitulée

« ……… »

Soutenue le 25/07/2019.

Devant le jury composé de :

Président :

Berhoun Farida Pr Univ. de SBA

Directeur de thèse : Benchohra Mouak Pr Univ. de SBA

Examinateurs :

Abbas Said Pr Univ. de Saida

Slimani Boualem Attou Pr Univ. de Tlemcen

Lazreg Jamal-Eddine MCA Univ. de SBA

Hedia Benaouda MCA Univ. de Tiaret

Année universitaire 2018/2019.

Etude quantitative et qualitative des

solutions de certaines équations intégrales

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République Algérienne Démocratique et Populaire

Minist`ere de l’Enseignement Sup´erieur et de la Recherche Scientifique

UNIVERSIT´E DJILLALI LIABES

FACULT´E DES SCIENCES EXACTES

SIDI BEL-ABB`ES

TH`

ESE DE DOCTORAT

Intitul´ee

´

Etude quantitative et qualitative des

solutions de certaines ´equations int´egrales

Pr´esent´ee par

BAGHDAD SAID

Thèse présentée et soutenue le:...

Devant le jury compos´e de:

Berhoun Farida Pr Univ. de SBA Pr´esidente

Benchohra Mouffak Pr Univ. de SBA Directeur de th`ese

Abbas Said Pr Univ. de Saida Examinateur

Slimani Boualem Attou Pr Univ. de Tlemcen Examinateur

Lazreg Jamal-Eddine MCA Univ. de SBA Examinateur

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Je d´

edie ce travail `

a

Mes chers fr`

eres et soeurs.

Mes enfants Wail et Sara.

Mes amis.

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Abstract

The objective of this thesis is to present both the results of existence and the stability of solutions of fractional integral equations and one fractional integral inclu-sion in different spaces. The main tool used is the technique associate to measures of noncompactness combining with fixed point theorems.

Keywords : Hadamard–Volterra–Stieltjes integral equations, Hadamard–Volterra– Stieltjes integral inclusions, Fr´echet spaces, Fixed-point theorems , Measure of non-compactness.

AMS Subject Classification : 26A33, 34G20, 35D40,45D05, 47H08, 54C60, 45G05,47H10.

R´

esum´

e

L’objectif de cette thèse est de présenter à la fois des résultats d’existence et la stabilité des solutions d’équations et inclusions intégrales d’ordre fractionnaire dans différents espaces. Le principal outil utilisé est la technique associée aux mesures de non-compacité et des théorèmes de points fixes.

Mots clé: Équations intégrales de type Hadamard–Volterra–Stieltjes, Inclusions intégrales de type Hadamard–Volterra–Stieltjes , espaces de Fréchet, théorèmes de points fixes , Mesures de non-compacité.

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Remerciements

Je remercie vivement Monsieur le Professeur BENCHOHRA MOUFFAK qui a dirigé cette thèse. Ses conseils multiformes et la richesse de ses connaissances ainsi que ses initiatives m’ont permis de mener à bien ce travail. Cette direction s’est caractérisé par une grande patience, une disponibilité permanante, des conseils abondants, un support et un suivi continu dans le but de mettre ce projet sous sa forme finale.

Je remercie madame Prof. Berhoun Farida qui m’a fait l’honneur de pr´esider le

jury.

J’adresse mes remerciements `a monsieur Prof. Abbas Said , monsieur Prof.

Sli-mani Boualem Attou, monsieur Dr. Lazreg Jamal-Eddine, monsieur Dr. Hedia Be-naouda d’avoir accept´e d’ˆetre membres du jury.

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List of Publications

1. S. Baghdad and M. Benchohra, Global Existence and Stability Results for Hadamard-Volterra-Stieltjes Integral Equations, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Vol. 68 No. 2 (2019), 1-14.

2. S. Baghdad and M. Benchohra, On Existence and Asymptotic Behaviour of Solutions of Hadamard-Volterra Integral Equations, Mathematical Sciences and Applications E-Notes, (accepted).

3. S. Baghdad and M. Benchohra, Global Existence Results for Hadamard-Volterra-Stieltjes Integral Inclusions, (submited).

4. S. Baghdad and M. Benchohra, Fractional Integral Equations in Fr´echet Spaces, (submited).

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Introduction

Linear and nonlinear integral equations and integral inclusions form an impor-tant class of problems in mathematics. There are different motivations for their study. Some equations describe mathematical models in physics, engineering or biology. There are also such equations whose interest lies in other branch of pure mathematics. It is known that integral inclusions are generalization of integral equa-tions and inequalities, and became a basic tool in optimal control theory, stochastic processes and dynamical systems. This theory is located within the mainstream of nonlinear analysis - or to put it more precisely - multi-valued analysis. Bearing in mind both mentioned aspects we are interested on a special class of integral equa-tions and integral inclusions, namely on fractional ones. That class comprises a lot of particular cases of this subject which can be encountered in research papers and monographs concerning the theory of integral equations or integral inclusions involv-ing different types of fractional derivatives as well as integral and their applications to real world problems see [6–8, 15–17, 39, 50, 51].

One of the most widely used techniques of proving that certain operator equa-tion has a soluequa-tion is to reformulate the problem as a fixed point problem and see if the latter can be solved via a fixed point argument. Measures of noncompact-ness play an important role in fixed point theory and have many applications in various branches of nonlinear analysis, including differential equations, integral and integro-differential equations, optimization, etc. Roughly speaking, a measure of noncompactness is a function defined on the family of all nonempty and bounded subsets of a certain metric space such that it is equal to zero on the whole family of relatively compact sets. This significant concept in mathematical science was defined by many authors in different manners [20, 28, 30–32]. In the last years there

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Contents 2 appeared many papers devoted to the applications of the measure noncompactness for establish some existence and stability results for various types of nonlinear inte-gral equations. A lot of these equations were considered in Banach spaces [33–37]. In some recent works on this direction, authors utilize a new method of a family of measures of noncompactness and fixed point theorems for condensing operators in Fr´echet spaces see [41, 57, 58]. The additional advantage of this works is the possibility of extension of the study for several problems to an unbounded domains. This thesis is devoted to study some fractional integral equations and one frac-tional integral inclusion in different spaces. We concentrate on the aspect of the stability of solutions in certain sense and the best possible assumptions ensuring the existence of solutions.

In the first Chapter, we collect all necessary definitions, lemmas and fixed point theorems. We present some functions spaces, fractional integral operators, multi-valued maps and the concept of measures of noncompactness.

The second chapter is devoted to study two types of fractional integral equa-tions. The first equation has the form

u(x, y) = µ(x, y)+ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1 f (t, s, u(t, s)) st dtds; (x, y) ∈ J,

where J = [1, +∞)×[1, b], r1, r2 > 0, µ : J → R, f : J ×R → R are given continuous

and bounded functions. The second one has the form

u(x, y) = µ(x, y)+ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1 f (t, s, u(t, s)) st dth1(y, t)dsh2(x, s); (x, y) ∈ J

where J = [1, +∞) × [1, b], r1, r2 > 0, µ : J → R is continuous and bounded

function, f : J × R → R is continuous function, h1 : [1, b] × [1, b] → R , h2 :

[1, +∞) × [1, +∞) → R are given functions. In the last, we give an example for each section.

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Contents 3 of solutions for Hadamard-Volterra-Stieltjes integral inclusion of the form

u(x, y) ∈µ(x, y) +HSIrF (s, t, u(s, t)); (x, y) ∈ J ;

where J = [1, +∞) × [1, b], µ : J → R is continuous and bounded function,

F : J × R → P(R) is a multivalued map, HS_Ir _{is Hadamard-Volterra-Stieltjes}

fractional integral of order r = (r1, r2), h1 : [1, b] × [1, b] → R , h2 : [1, +∞) ×

[1, +∞) → R are given functions, and P(R) is the family of all nonempty subsets of R and we present an illustrative example.

The fourth chapter contains investigations of the following fractional integral equation in Fr´echet space

u(x) = ϕ(x) + 1 Γ(r) Z x 1 lnx t r−1 f (t, u(t)) t dg(t); x ∈ J,

where J = [1, +∞), r > 0, ϕ : J → X is continuous function, f : J × X → X, g : J → R are given functions, (X, k · k) is a Banach space. An illustrative example completes this Chapter.

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Chapter 1 Preliminaries

This chapter is devoted to collect some definitions and auxiliary results which will be needed in further considerations.

1.1 Generalities and Notations

Let J = [1, +∞) × [1, b], BC be the Banach space of all real functions defined, continuous and bounded on J with the norm

kuk∞= sup

(x,y)∈J

|u(x, y)|.

Denote by L1_{(J, R) the Banach space of functions u : J → R that are Lebesgue}

integrable with norm

kukL1 =

Z Z

J

|u(x, y)|dydx.

Assume that (X, k · k) is an arbitrary Banach space with zero element θ. Denote

by Br(x) the closed ball centered at x and with radius r. The symbol Br stands

for the ball B(θ, r). When necessary we will also indicate the space by using the

notation Br(X). If A is a subset of X, then A and convA denote the closure and

convex closure of A , respectively. We denote the standard algebraic operations on sets by the symbols λ · A and A + B.

Let I = [a, b]; (−∞ ≤ a < b ≤ ∞) be a finite or infinite interval. We denote by AC[a, b] the space of functions f which are absolutely continuous on [a, b]. It is

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1.1. Generalities and Notations 5 known that AC[a, b] coincides with the space of primitives of Lebesgue summable functions

f ∈ AC[a, b] ⇔ f (x) = c +

Z x

a

g(t)dt, g ∈ L1(a, b), (1.1.1)

and therefore an absolutely continuous function f has a summable derivative f0

almost everywhere on [a, b]. Thus (1.1.1) yields

g(t) = f0(t) and c = f (a).

Definition 1.1.1 A function u : I → X is called simple if there is a finite sequence Dm ⊂ I; m = 1, · · · , p of measurable sets with I =

p S m=1 Dm, such that Di∩ Dj = ∅ for i 6= j, and u(t) = ym ∈ X for t ∈ Dm; m = 1, · · · , p,

i.e. u is constant on the measurable set Dm.

Definition 1.1.2 u : I → X is Bochner integrable if there is a sequence of simple functions un: I → X, n ∈ N such that limn→∞un(t) = u(t) a.e. in I and

lim

n→∞

Z

I

kun− ukX = 0.

Lemma 1.1.3 A measurable function u : I → X is Bochner integrable if and only if kukX : I → R+ is Lebesgue integrable. In this case we have

Z I u X ≤ Z I kukX.

For I = [1, +∞), we denote by L1_{(I, X) the Banach space of all measurable functions}

u : I → X which are Bochner integrable with the norm kuk =

Z

I

ku(t)kXdt.

More details for Bochner integral, we refer [79, 82].

The following compactness criterion in BC is particularly useful.

Lemma 1.1.4 (Corduneanu) [10, 11] Let M ⊂ BC. Then M is relatively compact in BC if the following conditions hold:

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1.2. Fractional integral operators 6 (i) M is uniformly bounded in BC,

(ii) the functions belonging to M are equicontinuous on any compact interval of J , (iii) the functions from M are equiconvergent, i.e., given > 0, there corresponds

T () > 0 such that |u(t, x) − lim

t→∞u(t, x)| < for any t ≥ T () and u ∈ M .

Lemma 1.1.5 (Gronwall) [67] Assume that the functions Φ, φ1, φ2 : R+ → R+ are

continuous functions such that Φ satisfies the following inequality: Φ(t) ≤ φ1(t) + Z t 0 φ2(τ )Φ(τ )dτ ; t ≥ 0, then Φ(t) ≤ φ1(t) + Z t 0 φ1(τ )φ2(τ ) exp Z t τ φ2(s)ds dτ ; t ≥ 0.

1.2 Fractional integral operators

The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past decades or so, due mainly to its demonstrated applications in nu-merous seemingly diverse and widespread fields of science and engineering. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables.

1.2.1 Hadamard fractional integrals

We present the definitions and some properties of the Hadamard type fractional integrals.

Definition 1.2.1 The Euler gamma function Γ is defined by the so-called Euler integral of the second kind

Γ(z) =

Z ∞

0

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1.2. Fractional integral operators 7 This integral is convergent for all complex z ∈ C with <(z) > 0 and the reduction formula

Γ(z + 1) = zΓ(z); (<(z) > 0) holds; it is obtained from (1.2.2) by integration by parts.

Let (a, b)(0 ≤ a < b ≤ ∞) be a finite or infinite interval of the half-axis R+ and let

<(α) > 0 and µ ∈ C, the left-sided and right-sided Hadamard fractional integrals of order α defined by (I_a+α f )(x) = 1 Γ(α) Z x a lnx t α−1 f (t)dt t (a < x < b) (1.2.3) and (I_b−α f )(x) = 1 Γ(α) Z b x ln t x α−1 f (t)dt t (a < x < b), (1.2.4)

respectively. When a = 0 and b = ∞, The fractional integrals, more general than those in (1.2.3) and (1.2.4) with µ ∈ C, are defined by

(I_0+,µα f )(x) = 1 Γ(α) Z x 0 t x µ lnx t α−1 f (t)dt t (x > 0) (1.2.5) and (I_−,µα f )(x) = 1 Γ(α) Z ∞ x x t µ ln t x α−1 f (t)dt t (x > 0), (1.2.6)

It can be directly verified that the Hadamard fractional integrals of the logarithmic functions [ln(x/a)]β−l and [ln(b/x)]β−l yield logarithmic functions of the same form. Property 1.2.1 If <(α) > 0, <(β) > 0 and 0 < a < b < ∞ ,then

Iα a+ ln t a β−1! (x) = Γ(β) Γ(β + α) lnx a β+α−1 , and Iα b− lnb t β−1! (x) = Γ(β) Γ(β + α) lnb x β+α−1 .

It can also be directly verified that the Hadamard type fractional integrals of the

power function xβ _{yield the same function, apart from a constant multiplication}

factor.

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1.2. Fractional integral operators 8 • If <(β + µ) > 0, then Iα 0+,µtβ (x) = (µ + β)−αxβ. In particular, if <(β) > 0, then Iα 0+tβ (x) = β −α_xβ_. • If <(β − µ) > 0, then Iα −,µtβ (x) = (µ − β)−αxβ. In particular, if <(β) < 0, then Iα −tβ (x) = (−β)−αxβ.

The Hadamard fractional integrals (1.2.3)- (1.2.4) with 0 < a < b < ∞ are defined on Lp_{(a, b), and the Hadamard type fractional integrals (1.2.5)- (1.2.6) on L}p

c(R+).

The next assertion is proved by using the Minkowski inequality.

Lemma 1.2.2 Let <(α) > 0, c ∈ R, µ ∈ C, 1 ≤ p ≤ ∞ and 0 < a < b < ∞. Then 1. The operators I_a+α and I_b−α are bounded in Lp(a, b) as follows

Iα a+f p ≤ k1kf kp and Iα b−f p ≤ k2kf kp, where k1 = 1 |Γ(α)| Z ln(b/a) 0 t<(α)−1et/pdt and k1 = 1 |Γ(α)| Z ln(b/a) 0 t<(α)−1e−t/pdt.

2. If <(µ) > c, the operator I_0+,µα is bounded in Lp_c_(R+) as follows

Iα 0+,µf Lpc ≤ kkf kL p c (k = [<(µ) − c] −<(α) ). In particular, if c < 0, the operator I₀₊α is bounded in Lp_c_(R+) by

Iα 0+f Lpc ≤ kkf kL p c (k = [−c] −<(α) ). 3. If <(µ) > −c, the operator Iα −,µ is bounded in Lpc(R+) by I−,µα f Lpc ≤ kkf kL p c (k = [<(µ) + c] −<(α) ). In particular, if c > 0, the operator Iα

− is bounded in Lpc(R+) by Iα −f Lpc ≤ kkf kL p c (k = c −<(α) ).

The Hadamard fractional integrals (1.2.3)- (1.2.4)- (1.2.5)-(1.2.6) satisfy the follow-ing semigroup property.

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1.2. Fractional integral operators 9 Property 1.2.3 Let <(α) > 0, <(β) > 0 and 1 ≤ p ≤ ∞.

• If 0 < a < b < ∞, then, for f ∈ Lp_{(a, b),}

Iα a+I β a+f = I α+β a+ (c ≤ 0) and Ib−α I β b−f = I α+β b− (c ≥ 0).

• If µ ∈ C, c ∈ R, a = 0 and b = ∞ then, for f ∈ Lp

c(R+), Iα 0+,µI β 0+,µf = I α+β 0+,µ (<(µ) ≤ c) and I α −,µI β −,µf = I−,µα+β (<(µ) ≤ −c). In particular, when µ = 0, Iα 0+I β 0+f = I α+β 0+ (c < 0) and I α −I β −f = I−α+β (c > 0).

More details information may be found in [47, 48, 63, 64].

1.2.2 Hadamard-Stieltjes fractional integrals

We begin by defining the variation of a function and what it means for a function to be of bounded variation. Let f : [a, b] → R be a function and Π = {x0, x1, ..., xn}

a partition of [a, b], we denote _ Π (f ) = n−1 X k=0 |f (xk+1) − f (xk)| and set b _ a (f ) = sup Π _ Π (f ), where the supremum is taken over all partitions of [a, b]. We clearly have 0 ≤ b W a (f ) ≤ ∞. The quantity b W a

(f ) is called the total variation of f over [a, b]. Definition 1.2.3 [21, 71] Let f : [a, b] → R is said to be of bounded variation on [a, b] if

b

W

a

(f ) is finite.

Theorem 1.2.4 If f : [a, b] → R is monotone on [a, b] then f is of bounded varia-tion on [a, b] and

b

W

a

(f ) = |f (b) − f (a)|.

The Stieltjes integral of the function f with respect to the function g is designated by

Z b

a

f (x)dg(x),

it is clear that the Riemann integral is a special case of the Stieltjes integral, obtained by setting g(x) = x. This integral exists under several conditions, we will restrict ourselves to only one theorem in this direction.

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1.2. Fractional integral operators 10 Theorem 1.2.5 The integral

Z b

a

f (x)dg(x),

exists if the function f is continuous on [a, b] and g is of finite variation on [a, b], and we have Z b a f (x)dg(x) ≤ sup x∈[a,b] |f (x)| b _ a (g).

Lemma 1.2.6 If the function f is continuous on [a, b] and if the function g has a Riemann integrable derivative g0 at every point of [a, b], then

Z b a f (x)dg(x) = Z b a f (x)g0(x)dx. For more details see [21, 69, 71].

In what follows, we consider the Hadamard-Stieltjes integral of order q > 0 for a function u of the form

HS_Iq au (x) = 1 Γ(q) Z x a lnx t q−1 u(t) t dg(t).

For functions of two or more variables there is various definitions of the variation are known. In our considerations we use one of them, the so called Vitali variation [80]. We give the definition of this sort of variation for functions of two variables and to the fundamental properties of functions with finite variation in this sense.

Let a domain ∆ = [a, b] × [c, d] ⊂ R2 be given. We consider a real function

p : ∆ −→ R defined on ∆. For a given subdomain ∆0 = [a1, b1] × [c1, d1] ⊂ ∆ ,

a ≤ a1 ≤ b1 ≤ b, c ≤ c1 ≤ d1 ≤ d we set mp(∆0) = p(b1, d1) − p(b1, c1) − p(a1, d1) + p(a1, c1); let us de define _ ∆ (p) = supX i |mp(∆0i)|.

Where the supremum is taken over all finite systems of nonoverlapping intervals ∆0i ⊂ ∆ (i.e. for the interiors

◦

∆0i of the intervals ∆0i we assume that

◦

∆0i∩

◦

∆0j = ∅

whenever i 6= j).

Definition 1.2.7 [80]The real function p : ∆ −→ R is of bounded variation on ∆ if W

∆

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1.3. Some properties of multi-valued maps 11 If p : ∆ = [a, b] × [c, d] −→ R and γ ∈ [c, d] is fixed, then we denote the usual variation of the function p(s, γ) in the interval [a, b] by

b W a p(·, γ). Similarly for d W c

p(α, ·) where α ∈ [a, b] is fixed.

Lemma 1.2.8 [80] Let p : ∆ = [a, b] × [c, d] −→ R be given such that W

∆

(p) < ∞,

b

W

a

p(·, γ0) < ∞ for some γ0 ∈ [c, d]. Then b

W

a

p(·, γ) < ∞ for all γ ∈ [c, d] and

b _ a p(·, γ) ≤_ ∆ (p) + b _ a p(·, γ0).

Definition 1.2.9 [1, 2, 47] Let q1, q2 > 0, r = (1, 1) and q = (q1, q2). For u ∈

L1_{(J, R), define the Hadamard-Stieltjes partial fractional integral of order q by the}

expression HS_Iq ru (x, y) = 1 Γ(q1)Γ(q2) Z x 1 Z y 1 lnx t q1−1 lny s q2−1 u(t, s) st dsg2(s, y)dtg1(t, x), where g1, g2 : [1, ∞) × [1, ∞) → R.

1.3 Some properties of multi-valued maps

Let (X, k · k) be a Banach space and A be a subset of X. We define

P(X) = {A ⊂ X : A 6= ∅} , Pcl(X) = {A ∈ P(X); A is closed},

Pcv(X) = {A ∈ P(X); A is convex}, Pbd(X) = {A ∈ P(X); A is bounded},

Pcp(X) = {A ∈ P(X); A is compact},

Pcl,cv(X) = {A ∈ P(X); A closed and convex},

Pcp,cv(X) = {A ∈ P(X); A compact and convex},

Prcp(X) = {A ∈ P(X); A relatively compact}.

Definition 1.3.1 [3, 6, 9] A multifunction ( also called multi-valued function, set-valued function) between two metric spaces X and Y is a map F : X → P(Y ). The graph of F is the subset of X × Y defined by

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1.3. Some properties of multi-valued maps 12 Given M ⊆ X, the image of M under F is the set

F (M ) = [

x∈M

F (x).

Likewise, given N ⊆ Y , the small pre-image of N under F is F₊−1 = {x ∈ X; F (x) ⊆ N } , while the large (or complete) pre-image of N under F is

F₋−1 = {x ∈ X; F (x) ∩ N 6= ∅} .

The usual notion of continuity of a single-valued function may be generalized to multifunction in several ways.

Definition 1.3.2 [22, 42, 44] A multifunction F : X → P(Y ) is called upper semi-continous at x ∈ X if, for any open set V ⊆ Y with F (x) ⊆ V , one may find an open neighbourhood U ⊆ X of x such that F (z) ⊆ V for all z ∈ U . Similarly, F is called lower semicontinous at x ∈ X if, for any open set V ⊆ Y with F (x) ∩ V 6= ∅, one may find an open neighbourhood U ⊆ X of x such that F (z) ∩ V 6= ∅ for all z ∈ U . A multifunction which is both upper semicontinuous and lower semicontinuous at x is simply called continuous at x.

In terms of sequences, semicontinuity of F : X → P(Y ) may be characterized as follows: F is upper semiconttinuous (respectively lower semicontinuous) at x if, for

any open set V ⊆ Y and any sequence (xn)n converging to x, from F (x) ⊆ V

(respectively F (x) ∩ V 6= ∅) it follows that F (xn) ⊆ V (respectively F (xn) ∩ V 6= ∅)

for sufficiently large n.

The mapping F has a fixed point if there exists x ∈ X such that x ∈ F (x).

Definition 1.3.3 A multifunction F : X → P(Y ) is called closed if its graph is a closed subset of X × Y .

Other equivalent characterizations are contained in the following

Lemma 1.3.4 The closedness of F : X → P(Y ) is equivalent to each of the follow-ing two conditions:

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1.3. Some properties of multi-valued maps 13 a) For any (x, y) ∈ X × Y with y /∈ F (x) there exist neighbourhoods U of x and V

of y, such that F (U ) ∩ V = ∅;

b) For any sequence (xn, yn) ∈ X × Y with xn → x and yn → y the relation

yn∈ F (xn) implies that y ∈ F (x).

Lemma 1.3.5 If F : X → Pcl(Y ) is upper semicontinuous, then F is closed.

Con-versely, if F : X → Pcp(Y ) is closed and locally compact, then F is upper

semicon-tinuous.

Definition 1.3.6 If F : X → Pcl(Y ) is a multivalued operator, a selection (or

selector) of F is a singlevalued operator f : X → Y such that f (x) ∈ F (x), for each x ∈ X.

In many fields of both the theory and applications of multifunctions it is extremely important to ensure the existence of selections with special additional properties. Given a multifunction F : X → P(Y ), we write

SF = {f : f (x) ∈ F (x) for all x ∈ X} ;

for the set of all selections of F . If X and Y are metric spaces and X is locally compact, we denote by C(X, Y ) the space of all continuous functions from X into Y , we write

SelCF = SF ∩ C(X, Y );

for the set of all continuous selections of F . This follows from the following important selection principle which is due to E. Michael and has found numerous applications.

Theorem 1.3.7 [22, 61] (Michael’s selections) Let X be a metric space, Y a

Ba-nach space, and F : X → Pcl,cv(Y ) a lower semicontinuous multifunction. Then F

admits continuous selection.

Let (X, d) be a metric space. For x ∈ X, A ⊆ X, and > 0, U(x) = {z ∈ X; d(z, x) < } ,

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1.3. Some properties of multi-valued maps 14 denotes the -neighbourhood of x, and

U(A) = {z ∈ X; %(z, A) < } ,

the -neighbourhood of A, where

%(z, A) = inf {d(z, x); x ∈ A} .

Definition 1.3.8 The (right and left) Hausdorff deviation and the Hausdorff dis-tance, respectively, of two sets A, B ⊆ X are defined by

H+(A, B) = sup {%(z, B); z ∈ A} = inf { > 0; A ⊆ U(B)} ,

H−(A, B) = H+(B, A),

H(A, B) = maxH+(A, B), H−(A, B) .

Recall that the system Pcl,bd(X), equipped with the Hausdorff metric, is a metric

space which is complete if X is complete.

Definition 1.3.9 [22, 61] Let us call a multifunction F : X → P(Y ) − δ−upper semicontinuous ( − δ−lower semicontinuous, − δ−continuous, respectively) at x ∈ X if, for any > 0, one may find a δ > 0 such that H+_{(F (z), F (x)) <}

(H−(F (z), F (x)) < , H(F (z), F (x)) < , respectively) for all z ∈ Uδ(x).

Lemma 1.3.10 The following holds :

• if F : X → P(Y ) is upper semicontinuous, then F is − δ−upper semicontin-uous;

• if F is − δ−lower semicontinuous, then F is lower semicontinuous;

• for F : X → Pcp(Y ), upper semicontinuity is equivalent to − δ−upper

semi-continuity;

• for F : X → Pcp(Y ), lower semicontinuity is equivalent to − δ−lower

semi-continuity.

From Lemma 1.3.10 it follows, in particular, that a compact-valued multifunction F is continuous if and only if F is continuous with respect to the Hausdorff metric.

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1.4. Some properties of Fr´echet spaces 15

1.4 Some properties of Fr´

echet spaces

A seminorm on vector space F is a map p : X → R such that 1. p(x) ≥ 0 ;

2. p(x + y) ≤ p(x) + p(y) ; 3. p(λx) = |λ|p(x),

for every x, y ∈ F and λ ∈ R. A family of seminorms P = {pα}α∈I on F defines a

unique topology TP compatible with the vector structure of F . The neighborhood

base BP of TP is determined by defining

S(∆, ) = {x ∈ X; p(x) < ; ∀p ∈ ∆},

BP = {S(∆, ); > 0 and ∆ a finite subset of P }.

The topology TP induced on F by P is the largest making all the seminorms

continu-ous but it is not necessarily Hausdorff. In fact (F , TP) is a locally convex topological

vector space and the local convexity of a topology on F is its subordination to a family of seminorms. Hausdorffness requires the further property

x = 0 ⇔ p(x) = 0; ∀p ∈ P.

Then it is metrizable if and only if the family of seminorms is countable. Con-vergence of a sequence (xn)n∈N in F is dependent on all the seminorms of P i.e

xn→ x ⇔ p(xn− x) → 0, ∀p ∈ P.

Completeness is if and only if we have convergence in F of every sequence (xn)n∈N

in F with

lim

n,m→∞p(xn− xm) = 0, ∀p ∈ P.

Definition 1.4.1 A set P of continuous seminorms on the locally convex space F is called fundamental system if for every continuous seminorm q there is p ∈ P and C > 0 so that q ≤ C · p.

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1.5. Measures of noncompactness 16 If a fundamental system of seminorms P is countable then we may assume that P = {k · kk; k ∈ N} where

k · k1 ≤ k · k2 ≤ k · k3 ≤ · · ·

This can be achieved by setting

kxkk = max

j=1,...,kpj(x),

where {pj; j ∈ N} is a given countable fundamental system of seminorms, we

put d(x, y) = ∞ X k=1 1 2k kx − ykk 1 + kx − ykk ,

so d(·, ·) is a translation invariant metric which generates an equivalent topology of F .

Definition 1.4.2 A Fr´echet space is a topological vector space F that is locally

convex, Hausdorff, metrizable and complete.

Remark 1.4.3 We notice that for a translation invariant metric d on a topological vector space F is complete if and only if F is complete with respect to d.

Definition 1.4.4 A function f : F → F is a contraction if for each n ≥ 1 there exists ln∈ [0, 1) such that

kf (u) − f (v)kn ≤ lnku − vkn for all u, v ∈ F .

More information about Fr´echet spaces see [45, 55, 56, 66].

1.5 Measures of noncompactness

Let X be a Banach space over the field K ∈ {R, C}. A nonnegative function φ defined on the bounded subsets of X will be called Sadovskij functional if it satisfies the following requirements (M, N ⊂ X bounded, λ ∈ K):

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1.5. Measures of noncompactness 17 (1.1) φ(M ∪ N ) = max{φ(M ), φ(N )}, (1.2) φ(M + N ) ≤ φ(M ) + φ(N ), (1.3) φ(λM ) = |λ|φ(M ), (1.4) φ(M ) ≤ φ(N ) for M ⊆ N , (1.5) φ([0, 1] · M ) = φ(M ), (1.6) φ(convM ) = φ(M ).

It is natural to call (1.1) the set additivity, (1.2) the algebraic subadditivity, (1.3) the homogeneity, (1.4) the monotonicity, (1.5) the absorption invariance, and (1.6) the convex closure invariance of φ. We remark that these axioms are not independent; for example, (1.4) follows from (1.1), and (1.5) follows from (1.6) if φ({0}) = 0. A particularly important additional property of a Sadovskij functional is

φ(M ) = 0 if and only if M is precompact.

which we call the regularity of φ. A regular Sadovskij functional is called measure of noncompactness. This name is motivated by the fact that, loosely speaking, the smaller φ(M ), the closer is M to being precompact (i.e., having compact closure). Apart from regularity, the most important property which plays a crucial role in both the theory and applications is the invariance property (1.6). There is another approach based on the Hausdorff distance H(M, N ) = max{D(M, N ), D(N, M )} of two bounded sets M, N ⊂ X, where, as usual,

D(M, N ) = inf{r > 0; M ⊆ N + Br(X)}.

A nonnegative function φ defined on the family of all bounded subsets of X is called set quantity if it satisfies (1.1), (1.2), (1.3) and (1.6). It is then shown that φ is a Sadovskij functional (in this terminology) which satisfies a Lipschitz condition

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1.5. Measures of noncompactness 18 with respect to the Hausdorff distance on X, and so φ is uniformly continuous. Now, suppose that N (X) is some family of bounded subsets of X with certain good additional properties (e.g., N (X) is stable under finite unions, algebraic sums, multiplication by scalars, and passing to the convex hull). Then

φN(M ) = dist (M, N (X)) = inf{H(M, N ); N ∈ N (X)},

is a set quantity, hence a Sadovskij functional. Clearly, φN(M ) = 0 if and only

if M ∈ N (X), and so φN is a measure of noncompactness if N (X) is the family of

all precompact subsets of X (see [20]).

1.5.1 Standard measures of noncompactness

We give now a list of the most known measures of noncompactness which arise over and over in applications and recall briefly some of their basic properties.

Definition 1.5.1 Let (E, d) be a complete metric space. The Kuratowski measure of noncompactness of a nonempty and bounded subsets Q of E, denoted by α(Q),is the infimum of all numbers δ > 0 such that Q can be covered by a finite number of sets with diameters < δ, i.e.,

α(Q) = inf ( δ > 0; Q ⊂ n [ i=1 Ai : Ai ⊂ E, diam(Ai) < δ, i = 1, 2, · · · , n; n ∈ N ) .

The following properties are consequences of Definition 1.5.1.

Proposition 1.5.1 Let A, B be nonempty and bounded subsets of complete metric space (E, d). Then

α(A) = 0 ⇔ A is compact. α(A) = α(A).

A ⊂ B ⇒ α(A) ≤ α(B).

α(A ∪ B) = max{α(A), α(B)}. α(A ∩ B) = min{α(A), α(B)}.

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1.5. Measures of noncompactness 19 The next result is a generalization of the well-known Cantor intersection theorem.

Theorem 1.5.2 Let (E, d) be a complete metric space. If (Fn) is a decreasing

sequence of nonempty, closed and bounded subsets of E such that lim

n→∞α(Fn) = 0,

then the intersection F∞ = ∞

T

n=1

Fn is nonempty and compact subset of E.

Other properties hold if E is a Banach space.

Proposition 1.5.2 Let A, B be nonempty and bounded subsets of a Banach space (X, k · k). Then

α(A + B) ≤ α(A) + α(B). α(A + x) = α(A); for all x ∈ X. α(λA) = |λ|α(A); for all λ ∈ K. α(A) = α(conv(A)).

For the proof of Propositions 1.5.1 and 1.5.2, we refer to [19].

Theorem 1.5.3 [19, 70] Let (X, k · k) be a Banach space. Let BX be the unit ball

in X. Then α(BX) = 0 if X is finite-dimensional, and α(BX) = 2 in the opposite

case.

In general, the computation of the exact value of α(A) is difficult. Another mea-sure of noncompactness, which seems to be more applicable, is so-called Hausdorff measure of noncompactness (or ball measure of noncompactness). It is defined as follows.

Definition 1.5.4 Let (E, d) be a complete metric space. The Hausdorff measure of noncompactness of a nonempty and bounded subsets Q of E, denoted by χ(Q),is the infimum of all numbers ε > 0 such that Q can be covered by a finite number of balls with radii < ε, i.e.,

χ(Q) = inf ( ε > 0 : Q ⊂ n [ i=1 B(xi, ri), xi ∈ E, ri < ε; i = 1, 2, · · · , n; n ∈ N ) .

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1.5. Measures of noncompactness 20 Definition 1.5.5 Let (X, k · k) be a Banach space. The Hausdorff measure of non-compactness of a nonempty and bounded subsets Q of X, denoted by χ(Q),is the infimum of all numbers ε > 0 such that Q has a finite ε-net in X, i.e.,

χ(Q) = inf {ε > 0 : Q ⊂ S + εBX, S ⊂ X, S is finite} .

The following properties follow from Definition 1.5.4.

Proposition 1.5.3 Let A, B be nonempty and bounded subsets of complete metric space (E, d). Then

χ(A) = 0 ⇔ A is compact. χ(A) = χ(A).

A ⊂ B ⇒ χ(A) ≤ χ(B).

χ(A ∪ B) = max{χ(A), χ(B)}. χ(A ∩ B) = min{χ(A), χ(B)}.

Proposition 1.5.4 Let A, B be nonempty and bounded subsets of a Banach space (X, k · k). Then

χ(A + B) ≤ χ(A) + χ(B). χ(A + x) = χ(A); for all x ∈ X. χ(λA) = |λ|χ(A); for all λ ∈ K. χ(A) = χ(conv(A)).

The next result shows the equivalence between the Kuratowski’s measure of noncompactness and the Hausdorff measure of noncompactness.

Theorem 1.5.6 [19] Let (E, d) be a complete metric space and Q be a nonempty and bounded subset of E. Then

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1.5. Measures of noncompactness 21 Remark 1.5.7 In the class of all infinite-dimensional spaces inequalities 1.5.7 are sharp.

We describe briefly another measure of noncompactness which is useful in applica-tions. At first, we need to recall the following concept.

Definition 1.5.8 Let (E, d) be a complete metric space. Let Q be a nonempty and bounded subset of E.For ε > 0, the subset Q is said to be ε-discrete if the following property holds:

x, y ∈ Q, x 6= y ⇒ d(x, y) ≥ ε.

Remark 1.5.9 Let (E, d) be a complete metric space and Q be a nonempty and bounded subset of E. It is not hard to see that the set Q is relatively compact if and only if every ε-discrete subset of Q is finite for all ε > 0.

Definition 1.5.10 Let (E, d) be a complete metric space and Q be a nonempty and

bounded subset of E. Then the Istrˇatescu measure of noncompactness of Q, denoted

by β(Q),is defined by

β(Q) = inf {ε > 0 : has no infinite ε − discrete subsets} .

Remark 1.5.11 The above-mentioned properties of α and χ are also valid for β.

Note that there is no general formula for computing the value of β(BE) (Kottman

constant). However, some estimates exist in the literature for some particular spaces. Let us recall some results in this direction.

Theorem 1.5.12 [38] Let E be a Hilbert space. Then β(BE) =

√ 2.

Theorem 1.5.13 [65]Let E = lp_{, 1 ≤ p < ∞. Then}

β(BE) = 21/p.

Theorem 1.5.14 [49, 74] Let E = Lp_{, p ≥ 1. Then}

β(BE) =      21/p _{if 1 ≤ p ≤ 2,} 21−1/p _{if 2 ≤ p < ∞.}

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1.5. Measures of noncompactness 22 Now, we will mention another measure of noncompactness, namely inner Hausdorff

measure of noncompactness, denoted as χi which is very similar to the Hausdorff

measure of noncompactness χi, except that in this case the balls which cover the

set, have their center inside the set. It is defined as follows.

Definition 1.5.15 Let (E, d) be a complete metric space and Q be a nonempty and bounded subset of E. Then the inner Hausdorff measure of noncompactness of Q, denoted by χi(Q), is the infimum of all the numbers ε > 0 such that Q can be covered

by a finite number of balls with radii< ε and centers in Q, that is,

χi(Q) = inf ( ε > 0 : Q ⊂ n [ i=1 B(xi, ri), xi ∈ Q, ri < ε, i = 1, 2 · · · , n; n ∈ N ) .

Note that the measure χi(Q) does not have some properties of the measures α and

β. More precisely, if A and B are nonempty and bounded subsets of (E, d), then

A ⊂ B ; χi(A) ≤ χi(B),

χi(A ∪ B) 6= max {χi(A), χi(B)} .

Moreover, if E has the structure of a Banach space, and Q is a nonempty and bounded subset of E, then

χi(Q) 6= χi(conv(Q)).

More information can be found in [19, 20, 32, 62].

1.5.2 Some suitable measures of noncompactness

We will mention here the axiomatic approach for measure of noncompactness, de-veloped by Bana´s and Goebel [32] in 1980. Let (X, k · k) be a Banach space. We

denote by MX the collection of all nonempty and bounded subsets of X. We denote

by NX the collection of all relatively compact subsets of X.

Definition 1.5.16 A function µ : MX → R+ said to be measure of noncompactness

in the space X if it satisfies the following conditions:

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1.5. Measures of noncompactness 23 2. A ⊂ B ⇒ µ(A) ≤ µ(B).

3. µ(A) = µ(A) = µ(conv(A)).

4. µ(λA + (1 − λ)B) ≤ λµ(A) + (1 − λ)µ(B), for all λ ∈ [0, 1].

5. If (An) is a sequence of closed sets from MX such that An+1 ⊂ An for n =

1, 2, · · · and if lim

n→∞µ(An) = 0, then the set A∞= ∞

T

n=1

An6= ∅.

The family kerµ described in 1. is said to be the kernel of the measure of noncom-pactness µ.

Remark 1.5.17 Observe that from the axioms 1., 2., and 5. in the above definition, we have A∞ ∈ kerµ, which implies that A∞ is relatively compact.

In addition the measure of noncompactness µ is called

- Measure with maximum property if µ(A ∪ B) = max{µ(A), µ(B)}. - Homogeneous measure if µ(λA) = |λ|µ(A), λ ∈ R.

- Subadditive measure if µ(A + B) ≤ µ(A) + µ(B).

- Sublinear measure if it is homogeneous and subadditive. - Complete (or full) if kerµ = NX.

- Regular measure if it is full, sublinear and has a maximum property.

Notice that the Kuratowski’s measure α and the Hausdorff measure χ defined pre-viously are regulars measures of noncompactness.

Given Q ∈ M_C(I,R) and > 0, let

ω(Q, ) = sup{ω(u, ) : u ∈ Q}, where

ω(u, ) = sup{|u(t) − u(s)| : t, s ∈ I, |t − s| ≤ }, u ∈ Q. We have the following result, which is due to Bana´s and Goebel [17].

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1.5. Measures of noncompactness 24

Theorem 1.5.18 Let ω0 : MC(I,R) → R+ be the mapping defined by

ω0(Q) = lim

→0ω(Q, ), Q ∈ MC(I,R).

Then ω0 is a measure of noncompactness in MC(I,R) in the sense of Definition 1.5.16.

Moreover, we have

ω0(Q) = 2χ(Q), Q ∈ MC(I,R).

Let Q ∈ M_BC(R+,R), > 0, T > 0 and u ∈ Q be fixed. Let us define the following

quantities:

ωT(u, ) = sup{|u(t) − u(s)| : t, s ∈ [0, T ], |t − s| ≤ }, ωT(Q, ) = sup{ωT(u, ) : u ∈ Q}, ωT₀(Q) = lim →0ω T_{(Q, ),} ω0(Q) = lim T →∞ω T 0(Q).

Further, let us define the set functions a(Q), b(Q), c(Q) by putting

a(Q) = lim T →∞ sup u∈Q [sup{|u(t)| : t ≥ T }] , b(Q) = lim T →∞ sup u∈Q [sup{|u(t) − u(s)| : t, s ≥ T }] , c(Q) = lim sup t→∞ diam Q(t), where Q(t) = {u(t) : u ∈ Q}; t ≥ 0,

and diam Q(t) is the diameter of the set Q(t).

Finally, let us consider the functions µa, µb, µc defined on the family MBC(R+,R) by

µa(Q) = ω0(Q) + a(Q)

µa(Q) = ω0(Q) + b(Q)

µa(Q) = ω0(Q) + c(Q)

In [32], Bana´s proved the following result.

Theorem 1.5.19 The functions µa, µb, µc : MBC(R+,R) → R+ are measures of

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1.5. Measures of noncompactness 25

Lemma 1.5.20 1. Let A ⊆ C(I, X) be bounded, then ψX(A(t)) ≤ ψC(I,X)(A)

for all t ∈ I, where A(t) = {u(t); u ∈ A} ⊂ X. Furthermore, if A is equicon-tinuous on I, then ψX(A(t)) is continuous on I and

ψC(I,X)(A) = sup{ψX(A(t)); t ∈ I}.

2. If A ⊆ C(I, X) is bounded and equicontinuous, then ψX Z t 0 A(s)ds ≤ Z t 0 ψX(A(s))ds;

for all t ∈ I, where Z t 0 A(s)ds = Z t 0 u(s)ds; u ∈ A .

For the Fr´echet space F we accept the following definition of the family of mea-sures of noncompactness.

Definition 1.5.21 [41, 57, 58] A family of mappings µn : MF → R+ is said to be

a family of measures of noncompactness in the Fr´echet space F if it satisfies the following conditions

1. The family ker{µn} = {A ∈ MF; µn(A) = 0 for n ∈ N} is nonempty and

ker{µn} ⊂ NF.

2. µn(A) ≤ µn(B) for A ⊂ B, n ∈ N.

3. µn(ConvA) = µn(A) for n ∈ N.

4. If (Ai) is a sequence of closed sets from MF such that Ai+1 ⊂ Ai (i = 1, 2, · · · )

and if limi→∞µn(Ai) = 0 for each n ∈ N, then the intersection set A∞= ∞

T

i=1

Ai

is nonempty.

5. µn(λA) = |λ|µn(A) for λ ∈ R, n = 1, 2, · · ·

6. µn(A + B) ≤ µn(A) + µn(B) for n = 1, 2, · · ·

7. µn(A ∪ B) = max{µn(A), µn(B)} for n = 1, 2, · · ·

We call the family {µn}n∈N to be homogeneous, subadditive, sublinear, has the

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1.6. Fixed point theorems 26

Definition 1.5.22 The family of measures of noncompactness {µn}n∈N is said to

be regular if it is full (ker{µn} = NF), sublinear and has maximum property.

Remark 1.5.23 In Fr´echet space F we can also consider families of measures

{µT}T ≥0 indexed by nonnegative numbers instead of families {µn}n∈N indexed by

natural numbers see [57].

1.6 Fixed point theorems

Fixed point theorems have always a major role in various fields, specially, in fields of differential, integral and functional equations. Fixed point theorems constitute a topological tool for the qualitative investigations of solution of linear and nonlinear equations. The theory of fixed points is concerned with the conditions which guar-antee that a map T : E → E of a topological space E into it self admits one or more fixed points, that is, points x of E for which x = T x. In this section, we present some fixed point theorems involving an arbitrary measure of noncompactness in the

sense of Definition 1.5.16. So, if X is a Banach space, we denote by ψ : MX → R+

an arbitrary measure of noncompactness in X.

Theorem 1.6.1 [19, 32, 52, 70] (Darbo) Let Ω be a nonempty, bounded, closed and convex subset of the Banach space X and let F : Ω −→ Ω be a continuous mapping. Assume that there exists a constant k ∈ [0, 1) such that ψ(F A) ≤ kψ(A) for any nonempty subset A of Ω. Then F has a fixed point in the set Ω.

Remark 1.6.2 Let us denote by Fix F the set of all fixed points of the operator F which belong to Ω. It can be shown that the set Fix F belongs to the family ker ψ.

Theorem 1.6.3 [43,52,59] (M¨onch) Let D be a bounded, closed and convex subset

of the Banach space X such that 0 ∈ D, and let F : D −→ D be a continuous mapping. If the implication

V = conv F (V ) or V = F (V ) ∪ {0} ⇒ ψ(V ) = 0

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1.6. Fixed point theorems 27 Theorem 1.6.4 [54, 62] Let Ω be a nonempty, bounded, closed and convex subset

of the Banach space X and let G : Ω −→ Pcl,cv(Ω) be a closed. Assume that there

exists a constant k ∈ [0, 1) such that ψ(GA) ≤ kψ(A) for any nonempty subset A of Ω. Then G has a fixed point in the set Ω.

Theorem 1.6.5 [44, 62, 76] Let Ω be a bounded, closed and convex subset of the

Banach space X such that 0 ∈ Ω, and let G : Ω −→ Pcv(Ω). Assume graph(G) is

closed, G maps compact sets into relatively compact sets, one has M ⊂ Ω, M = conv ({0} ∪ G(M )) ⇒ M is compact. Then there exists x ∈ Ω with x ∈ G(x).

Remark 1.6.6 Let us denote by Fix G the set of all fixed points of the operator G which belong to Ω. It can be shown that the set Fix G belongs to the family ker ψ.

Theorem 1.6.7 [41, 57, 68] Let Ω be a nonempty, bounded, closed, and convex

subset of a Fr´echet space F and let L : Ω → Ω be a continuous mapping. If L is a

contraction with respect to a family of measures of noncompactness {µn}n∈N i.e for

each n ∈ N and a nonempty A ⊂ Ω there exist a constants kn∈ [0, 1) such that

µn(L(A)) ≤ knµn(A),

then L has at least one fixed point in the set Ω.

The above Theorem is a generalization of the classical Darbo fixed point Theorem for the Fr´echet space.

Theorem 1.6.8 [52]Let Ω be a nonempty, bounded, closed, and convex subset of a Hausdorff locally convex space F such that 0 ∈ Ω, and let L be a continuous mapping of Ω into itself. If the implication

(V = conv L(V ) or V = L(V ) ∪ {0}) ⇒ V is relatively compact, holds for every subset V of Ω, then L has a fixed point.

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Chapter 2 Fractional Integral Equations in

Banach Spaces

2.1 Introduction

The classes of differential and integral equations of fractional order are one of the most useful mathematical tools in both pure and applied analysis, and various theoretical results have been obtained, see the works of Benchohra and al. [12, 13].

The aim of this chapter is to study the integal equations of Hadamard-Volterra type and another of Hadamard-Volterra-Stieltjes. These class comprises a lot of particular cases of fractional integral equations which can be encountred in research papers and monographs concerning the theory of integral equations and their appli-cations to real world problems see [1, 2, 12, 13, 18, 81].

Using the technique associated with measures of noncompactness and fixed point theorems we show that this equations has solutions being continuous and bounded functions on the interval [1, +∞) × [1, b], b > 1. Moreover, the choice of suitable measures of noncompactness allows us to assert that those solutions are asymptotic stable in certain sense which will be defined in the sequel.

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2.2. Auxiliary facts 29

2.2 Auxiliary facts

This section is devoted to collect some definitions and auxiliary results which will be needed in further considerations. At the beginning we present some basic facts concerning measures of noncompactness.

In what follows we will work in the Banach space BC consisting of all real functions defined, continuous and bounded on J . In order to define a measure of noncompactness in the space BC, let us fix a nonempty bounded subset Y of the space BC . For u ∈ Y , T ≥ 1, 1, 2 > 0, (x1, y1), (x2, y2) ∈ [1, T ] × [1, b] such that

|x2− x1| ≤ 1 and |y2− y1| ≤ 2.

We denote by ωT_(u,

1, 2) the modulus of continuity of the function u on the interval

[1, T ] × [1, b] i.e

ωT(u, 1, 2) = sup{|u(x2, y2) − u(x1, y1)|; (x1, y1), (x2, y2) ∈ [1, T ] × [1, b]}

ωT(Y, 1, 2) = sup{ωT(u, 1, 2); u ∈ Y }

ωT₀(Y ) = lim 1,2→0 ωT(Y, 1, 2) ω0(Y ) = lim T →∞ω T 0(Y )

If (t, s) is fixed from J , let us denote Y (t, s) = {u(t, s) : u ∈ Y } and diam Y (t, s) = sup {|u(t, s) − v(t, s)|; u, v ∈ Y }

Finally, consider the function ψ defined on the family MBC by the formula

ψ(Y ) = ω0(Y ) + lim

t7−→∞sup diam Y (t, s). (2.2.1)

It can be shown that the function ψ is a measure of noncompactness in the space BC. The kernel ker ψ consists of nonempty and bounded sets Y such that functions from Y are locally equicontinuous on J and the thickness of the bundle formed

by functions from Y tends to zero at infinity. This property will permit us to

characterize solutions of the integral equation considered in the next section. Now, let us assume that Ω is a nonempty subset of the space BC and F is an operator on Ω with values in BC. Consider the following equation

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2.3. On Existence and Asymptotic Behaviour of Solutions of

Hadamard-Volterra Integral Equations 30

Definition 2.2.1 The solution u = u(x, y) of Eq. (2.2.2) is said to be globally attractive if for each solution v = v(x, y) of Eq. (2.2.2) we have that

lim

x7−→∞(u(x, y) − v(x, y)) = 0.

In the case when this limit is uniform i.e when for each > 0 there exists T > 1 such that

|u(x, y) − v(x, y)| < .

For x ≥ T , we will say that solutions of Eq. (2.2.2) are uniformly globally attractive.

2.3 On Existence and Asymptotic Behaviour of

Solutions of Hadamard-Volterra Integral

Equa-tions

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Consider the following integral equation

u(x, y) = µ(x, y) + 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 f (t, s, u(t, s)) st dsdt; (x, y) ∈ J, (2.3.3)

where J = [1, +∞) × [1, b], r1, r2 > 0, µ : J → R is given continuous and bounded

function.

2.3.1 Main results

In this section we give two results for equation (2.3.3). The first one relies on the

Darbo fixed point theorem and the second one on the M¨onch fixed point theorem.

Equation (2.3.3) will be considered under the following assumptions : (H1) The function f is continuous and there exists b ∈ L1(J, R+) such that

|f (x, y, u1) − f (x, y, u2)| ≤ b(x, y)|u1− u2|; (x, y) ∈ J; u1, u2 ∈ R.

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[25] S. Baghdad and M. Benchohra, On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations, Mathematical Sciences and Applications E-Notes, (accepted).

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(H2) There exists continuous and bounded function g : J → R+ such that

Z x 1 Z y 1 lnx t r1−1 lny s r2−1 |f (t, s, 0)|dsdt ≤ g(x, y)

(H3) There exists continuous and bounded function h : J → R+ such that

Z x 1 Z y 1 lnx t r1−1 lny s r2−1 b(t, s)dsdt ≤ h(x, y) with khk < Γ(r1)Γ(r2)

Remark 2.3.1 In view of the assumption (H1) we infer that ∀u ∈ BC

|f (x, y, u)| ≤ b(x, y)kuk + |f (x, y, 0)|

Theorem 2.3.2 Under assumptions (H1) − (H3) Eq. (2.3.3) has at least one

so-lution u = u(x, y) in the space BC. Moreover, soso-lutions of Eq. (2.3.3) are globally attractive.

Proof : Consider the operator F on the space BC defined by :

(F u)(x, y) = µ(x, y) + 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 f (t, s, u(t, s)) st dsdt

Observe that in view of our assumptions, for any function u ∈ BC the function F u is continuous on J . Next, let us take an arbitrary function u ∈ BC. Using our assumptions, for a fixed (x, y) ∈ J we have

|F u(x, y)| ≤ |µ(x, y)| + 1

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Thus, we infer that the function F u is bounded on J . Then F u ∈ BC. We take

r = kµk + kgk

1 − khk

Γ(r1)Γ(r2)

We deduce that the operator F transforms the ball Br into itself.

Further, let (un) ⊂ Br such that un → u we get

|F un(x, y) − F u(x, y)| ≤ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 |f (t, s, u_n(t, s)) − f (t, s, u(t, s))| st dsdt ≤ kun− uk Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 b(t, s)dsdt, thus kF un− F uk ≤ khk · kun− uk Γ(r1)Γ(r2) .

Then when n → ∞ we obtain F un→ F u so F is continuous on Br.

Now, we take a nonempty Y ⊂ Br, for T ≥ 1, (x1, y1), (x2, y2) ∈ [1, T ] × [1, b]

with |x2− x1| ≤ 1 and |y2− y1| ≤ 2; for each 1, 2 > 0. Fix arbitrarily u ∈ Y we

have |F u(x2, y2) − F u(x1, y1)| = 1 Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx2 t r1−1 lny2 s r2−1 f (t, s, u(s, t)) st dsdt − 1 Γ(r1)Γ(r2) Z x1 1 Z y1 1 lnx1 s r1−1 lny1 t r2−1f (t, s, u(s, t)) st dtds ≤ 1 Γ(r1)Γ(r2) sup (x1,y1),(x2,y2)∈[1,T ]×[1,b] |u(x2, y2) − u(x1, y1)| Z x2 1 Z y2 1 lnx2 t r1−1 lny2 s r2−1 b(t, s)dsdt ≤ khk Γ(r1)Γ(r2) sup (x1,y1),(x2,y2)∈[1,T ]×[1,b] |u(x2, y2) − u(x1, y1)|, Thus ω0(F Y ) ≤ kω0(Y ). (2.3.1)

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Further, for u, v ∈ Y and an arbitrary fixed (x, y) ∈ [1, T ] × [1, b] we obtain |F u(x, y) − F v(x, y)| = 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 f (t, s, u(t, s)) st dsdt − 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 f (t, s, v(t, s)) st dsdt ≤ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 |f (t, s, u(s, t)) − f (t, s, v(s, t))|dsdt ≤ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 b(t, s)|u(t, s) − v(t, s)|dsdt ≤ khk Γ(r1)Γ(r2) sup u,v∈Y |u(s, t) − v(s, t)|, Then lim

x−→∞ sup diam (F Y )(x, y) ≤ k limx−→∞ sup diam Y (x, y). (2.3.2)

Observe, that linking (2.3.1), (2.3.2) and the definition of the measure of noncom-pactness ψ given by the formula (2.2.1), we obtain

ψ(F Y ) ≤ kψ(Y ).

Finally, in view of the Darbo fixed point theorem we deduce that F has at least one fixed point in Br which is a solution of equation (2.3.3). Moreover, taking into

account the fact that the set Fix F ∈ kerψ and the characterisation of sets belonging to ker ψ (Remark (1.6.2)) we conclude that solutions of equation (2.3.3) are globally attractive in the sense of Definition (2.2.1).

2

Now we will formulate an other result. by applying M¨onch’s Theorem.

We have F : Br → Br continuous, let Y ⊂ Br with Y = F (Y ) ∪ {0}. Then

for all u in Y , there exist v in Y such that u = F v. For T ≥ 1, (x1, y1), (x2, y2) ∈

[1, T ] × [1, b] such that |x2− x1| ≤ 1 , |y2− y1| ≤ 2 , 1, 2 > 0 and u, v ∈ Y we get

|u(x2, y2) − u(x1, y1)| = 1 Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx t r1−1 lny s r2−1 f (t, s, v(t, s)) st dsdt − 1 Γ(r1)Γ(r2) Z x1 1 Z y1 1 lnx t r1−1 lny s r2−1 f (t, s, v(t, s)) st dsdt ≤ supv∈A|v(x2, y2) − v(x1, y1)| Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx t r1−1 lny s r2−1 b(t, s)dsdt.

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T →∞sup_u∈A|u(x2, y2) − u(x1, y1)| ≤

khk Γ(r1)Γ(r2) lim T →∞sup_v∈A|v(x2, y2) − v(x1, y1)|, then ω0(Y ) ≤ kω0(Y ). (2.3.3)

Next, let u, v, w, z ∈ Y such that u = F v and w = F z, for x, y ∈ J we have |u(x, y) − w(x, y)| = 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1f (t, s, v(t, s)) st dsdt − 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 f (t, s, z(t, s)) st dsdt ≤ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx t r1−1 lny s r2−1 |f (t, s, v(t, s)) − f (t, s, z(t, s))|dsdt ≤ khk Γ(r1)Γ(r2) |v(x, y) − z(x, y)|. Then lim

x−→∞ sup diam (Y )(x, y) ≤ k limx−→∞ sup diam Y (x, y) (2.3.4)

From the estimates (2.3.3) and (2.3.4) we infer that ψ(Y ) ≤ kψ(Y ).

Since k < 1, we obtain ψ(Y ) = 0. Combining the above result and Theorem 1.6.3 we complete the proof.

2

2.3.2 An example

We consider the following Hadamard- Volterra integral equation u(x, y) = x + 1 y2_ex + 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1 e−t+ s t2_s3 _{+ 1} p u2_{(s, t) + e}−tdtds st , (2.3.5)

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where (x, y) ∈ J = [1, +∞) × [1, b], b > 1 and r1, r2 > 0. Set µ(x, y) = x + 1 y2_ex and f (t, s, u(t, s)) = e−t+ s t2_s3_{+ 1} p u2_{(t, s) + e}−t_{; (t, s) ∈ J.}

It is clear that equation (2.3.5) can be written as equation (2.3.3). Let us show that conditions (H1) − (H3) hold. |f (t, s, u1(t, s)) − f (t, s, u2(t, s))| ≤ e−t+ s t2_s3_{+ 1} q u2 2(t, s) + e−t− q u2 1(t, s) + e−t ≤ e −t_{+ s} t2_s3_{+ 1} q u2 2(t, s) − q u2 1(t, s) ≤ e −t_{+ s} t2_s3_{+ 1}|u2(t, s) − u1(t, s)| . We take b(t, s) = e −t_{+ s} t2_s3_{+ 1} , f (t, s, 0) = e−t + s t2_s3_{+ 1}e −1 2t and we have : Z x 1 Z y 1 lnx s r1−1 lny t r2−1 e−t + s t2_s3_{+ 1}dtds ≤ (ln x)r1_{(ln y)}r2 Z x 1 Z y 1 e−t+ s t2_s3 dtds ≤ (ln x)r1_{(ln y)}r2 Z x 1 Z y 1 e−t s3 + 1 t2_s2 dtds ≤ (ln x)r1_{(ln y)}r2 −e−y+ e−1 − 1 2x2 + 1 2 + −1 y + 1 −1 x + 1 = h(x, y). We have also Z x 1 Z y 1 lnx s r1−1 lny t r2−1 |f (t, s, 0)|dtds ≤ (ln x)r1_{(ln y)}r2 Z x 1 Z y 1 e−t+ s t2_s3_{+ 1}e −1 2tdtds ≤ (ln x)r1_{(ln y)}r2 Z x 1 Z y 1 e−t+ s t2_s3 e −1 2tdtds ≤ (ln x)r1_{(ln y)}r2 Z x 1 Z y 1 e−32t s3 + e−12t s2 ! dtds ≤ (ln x)r1_{(ln y)}r2 −2 3e −3 2y+ 2 3e −3 2 − 1 2x2 + 1 2 + −1 x + 1 −2e−12y + 2e− 1 2 = g(x, y).

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2.4. Global Existence Results for Hadamard–Volterra–Stieltjes Integral

Equations 36

It is clear that h(x, y), g(x, y) a bounded functions on J . And we have also

k = khk Γ(r1)Γ(r2) = 1 2e + 1 Γ(r1)Γ(r2) < 1, for arbitraries r1, r2.

Consequently from Theorem 2.3.2 the Eq. (2.3.5) has at least solution in BC and solutions of equation (2.3.5) are globally attractive.

2.4 Global Existence Results for Hadamard–Volterra–

Stieltjes Integral Equations

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Consider the following integral equation

u(x, y) = µ(x, y) + Z x 1 Z y 1 lnx s r1−1 lny t r2−1f (t, s, u(t, s)) stΓ(r1)Γ(r2) dth1(y, t)dsh2(x, s); (x, y) ∈ J (2.4.6)

where J = [1, +∞) × [1, b], r1, r2 > 0, µ : J → R is continuous and bounded

function, f : J × R → R is continuous function, h1 : [1, b] × [1, b] → R , h2 :

[1, +∞) × [1, +∞) → R are given functions.

In this section we give two results for (2.4.6). The first one relies on the Darbo

fixed point theorem and the second one on the M¨onch fixed point theorem.

2.4.1 Main results

Equation (2.4.6) will be considered under the following assumptions :

(H1) The function f is continuous and there exists a continuous and bounded

func-tion g : [1, +∞) × [1, b] −→ R+ such that |f (x, y, u1) − f (x, y, u2)| ≤ g(x, y)|u1− u2| |u1| + |u2| + 1 ; (x, y) ∈ J ; u1, u2 ∈ R.

(2) _{[24] S. Baghdad and M. Benchohra, Global Existence and Stability Results for Hadamard-Volterra-Stieltjes}

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Equations 37

(H2) The function t 7→ h1(y, t) is continuous and of bounded variation on [1, b] for

each fixed y ∈ [1, b], and as the function s 7→ h2(x, s) is continuous and of

bounded variation on [1, +∞) for each x ∈ [1, +∞). (H3) There exist a constant λ, η > 0 such that

sup x≥1;1≤y≤b Z x 1 Z y 1 lnx s r1−1 |f (s, t, 0)|dt α=t _ α=1 h1(y, α)ds β=s _ β=1 h1(x, β) ≤ λ. And sup x≥1 Z x 1 lnx s r1−1 ds β=s _ β=1 h1(x, β) ≤ η. With k = ηkgk ln b Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) < 1.

Remark 2.4.1 In view of the assumption (H1) we infer that for each u ∈ X

|f (x, y, u)| ≤ g(x, y)|u|

|u| + 1 + |f (x, y, 0)|; (x, y) ∈ J ; u ∈ R.

Theorem 2.4.2 Under assumptions (H1)−(H3) the integral equation (2.4.6) has at

least one solution u = u(x, y). Moreover, solutions of (2.4.6) are globally attractive. Proof : Consider the operator F defined on the space BC in the following way :

(F u)(x, y) = µ(x, y) + 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1f (t, s, u(t, s)) st dth1(y, t)dsh2(x, s); Observe that in view of our assumptions, for any function u ∈ BC the function F u is continuous on J . Next, let us take an arbitrary function u ∈ BC. Using our

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Equations 38

assumptions, for a fixed (x, y) ∈ J we have |F u(x, y)| ≤ |µ(x, y)|

+ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1 |f (t, s, u(s, t))| st dth1(y, t)dsh2(x, s) ≤ |µ(x, y)| + 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1 × g(s, t)|u(s, t)| |u(s, t)| + 1 + |f (s, t, 0)| dth1(y, t)dsh2(x, s) ≤ kµk + ln b Γ(r1)Γ(r2) kgk.kuk kuk + 1 α=b _ α=1 h2(y, α) Z x 1 lnx s r1−1 ds β=s _ β=1 h1(x, β) + Z x 1 Z y 1 lnx s r1−1 |f (s, t, 0)|dt α=t _ α=1 h2(y, α)ds β=s _ β=1 h1(x, β) ≤ kµk + ln b Γ(r1)Γ(r2) kgk.kuk kuk + 1 α=b _ α=1 h2(y, α) Z x 1 lnx s r1−1 ds β=s _ β=1 h1(x, β) + Z x 1 Z y 1 lnx s r1−1 |f (s, t, 0)|dt α=t _ α=1 h2(y, α)ds β=s _ β=1 h1(x, β) ≤ kµk + ln b Γ(r1)Γ(r2) " kgk α=b _ α=1 h2(y, α)η + λ # . Hence we obtain kF uk ≤ kµk + ln b Γ(r1)Γ(r2) " kgk α=b _ α=1 h2(y, α)η + λ # .

Thus, we infer that the function F u is bounded on J . Then F u ∈ BC. We take r = kµk + ln b Γ(r1)Γ(r2) " kgk α=b _ α=1 h2(y, α)η + λ # . We deduce that the operator F transforms the ball Br into itself.

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2.4. Global Existence Results for Hadamard–Volterra–Stieltjes Integral Equations 39 |F un(x, y) − F u(x, y)| ≤ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1|f (t, s, u_n(s, t)) − f (t, s, u(s, t))| st × dth1(y, t)dsh2(x, s) ≤ 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1g(t, s)|u_n(s, t) − u(s, t)| |un(s, t)| + |u(s, t)| + 1 × dth1(y, t)dsh2(x, s) ≤ kgk ln bkun− uk Γ(r1)Γ(r2)(kunk + kuk + 1) α=b _ α=1 h2(y, α) Z x 1 lnx s r1−1 ds β=s _ β=1 h1(x, β). Consequently kF un− F uk ≤ ηkgk ln bkun− uk Γ(r1)Γ(r2) α=b _ α=1 h2(y, α).

Then when n → ∞ we obtain F un→ F u so F is continuous on Br.

Now, we take a nonempty Y ⊂ Br, for T ≥ 1, (x1, y1), (x2, y2) ∈ [1, T ] × [1, b] with

|x2− x1| ≤ 1 and |y2− y1| ≤ 2; for each 1, 2 > 0. Fix arbitrarily u in Y we have

|F u(x2, y2) − F u(x1, y1)| = 1 Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx2 s r1−1 lny2 t r2−1 f (t, s, u(s, t)) st dth1(y2, t)dsh2(x2, s) − 1 Γ(r1)Γ(r2) Z x1 1 Z y1 1 lnx1 s r1−1 lny1 t r2−1 f (t, s, u(s, t)) st dth1(y1, t)dsh2(x1, s) ≤ 1 Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx2 s r1−1 lny2 t r2−1 × sup (x1,y1),(x2,y2)∈[1,T ]×[1,b] |f (t, s, u(x2, y2)) − f (t, s, u(x1, y1))| dth1(y2, t)dsh2(x2, s) ≤ 1 Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx2 s r1−1 lny2 t r2−1 g(s, t) × sup (x1,y1),(x2,y2)∈[1,T ]×[1,b] |u(x2, y2) − u(x1, y1)| dth1(y2, t)dsh2(x2, s) ≤ kgk ln b Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) sup (x1,y1),(x2,y2)∈[1,T ]×[1,b] |u(x2, y2) − u(x1, y1)| × Z x2 1 lnx2 s r1−1 dsh2(x2, s) ≤ ηkgk ln b Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) sup (x1,y1),(x2,y2)∈[1,T ]×[1,b] |u(x2, y2) − u(x1, y1)| .

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Equations 40

Thus

ω0(F Y ) ≤ kω0(Y ). (2.4.7)

Further, for u, v ∈ Y and an arbitrary fixed (x, y) ∈ [1, T ] × [1, b] we obtain

|F u(x, y) − F v(x, y)| = 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1f (t, s, u(s, t)) st dth1(y, t)dsh2(x, s) − 1 Γ(r1)Γ(r2) Z x 1 Z y 1 lnx s r1−1 lny t r2−1 f (t, s, v(s, t)) st dth1(y, t)dsh2(x, s) ≤ ln b Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) Z x 1 Z y 1 lnx s r1−1 |f (t, s, u(s, t)) − f (t, s, v(s, t))| dsh2(x, s) ≤ ln b Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) Z x 1 Z y 1 lnx s r1−1 g(t, s) |u(s, t) − v(s, t)| |u(t, s)| + |v(t, s)| + 1dsh2(x, s) ≤ η ln bkgk Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) sup u,v∈Y |u(s, t) − v(s, t)|. Then lim

x−→∞ sup diam (F Y )(x, y) ≤ k limx−→∞ sup diam Y (x, y). (2.4.8)

Observe, that linking (2.4.7), (2.4.8) and the definition of the measure of noncom-pactness ψ given by the formula (2.2.1), we obtain

ψ(F Y ) ≤ kψ(Y ).

Finally, in view of the Darbo fixed point theorem we deduce that F has at least

one fixed point in Br which is a solution of equation (2.4.6). Moreover, taking

into account the fact that the set F ix F ∈ ker ψ and the characterization of sets belonging to ker ψ (Remark 1.6.2 ) we conclude that all solutions of equation (2.4.6) are globally attractive in the sense of Definition (2.2.1).

2 Now we will give an other result using M¨onch’s fixed point Theorem.

Equation (2.4.6) will be considered under the following assumptions :

(C1) The function f is continuous and there exists a continuous and bounded

func-tion g : [1, +∞) × [1, b] −→ R+ such that

|f (x, y, u1) − f (x, y, u2)| ≤

g(x, y)|u1− u2|

|u1| + |u2| + 1

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Equations 41

(C2) The function t 7→ h1(y, t) is continuous and of bounded variation on [1, b] for

each fixed y ∈ [1, b], and as the function s 7→ h2(x, s) is continuous and of

bounded variation on [1, +∞) for each x ∈ [1, +∞). (C3) There exist a constant λ, η > 0 such that

sup x≥1;1≤y≤b Z x 1 Z y 1 lnx s r1−1 |f (s, t, 0)|dt α=t _ α=1 h1(y, α)ds β=s _ β=1 h1(x, β) ≤ λ. And sup x≥1 Z x 1 lnx s r1−1 ds β=s _ β=1 h1(x, β) ≤ η. With k = ηkgk ln b Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) < 1.

Theorem 2.4.3 Under assumptions (C1) − (C3) equation (2.4.6) has at least one

solution u = u(x, y) in the space BC. Moreover, solutions of (2.4.6) are globally attractive.

Proof : We have F : Br → Br continuous, let Y ⊂ Br with Y = F (Y ) ∪ {0}. Then

for all u in Y , there exist v in Y such that u = F v.

For T ≥ 1, (x1, y1), (x2, y2) ∈ [1, T ] × [1, b] such that |x2− x1| ≤ 1 , |y2− y1| ≤ 2

, 1, 2 > 0 and u, v ∈ Y we get |u(x2, y2) − u(x1, y1)| = 1 Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx2 s r1−1 lny2 t r2−1 f (t, s, v(s, t)) st dth1(y2, t)dsh2(x2, s) − 1 Γ(r1)Γ(r2) Z x1 1 Z y1 1 lnx1 s r1−1 lny1 t r2−1 f (t, s, v(s, t)) st dth1(y1, t)dsh2(x1, s) ≤ ln b Γ(r1)Γ(r2) Z x2 1 Z y2 1 lnx2 s r1−1 g(s, t) |v(x2, y2) − v(x1, y1)| dth1(y2, t)dsh2(x2, s) ≤ ln b.kgk Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) sup v∈Y |v(x2, y2) − v(x1, y1)| Z x2 1 lnx2 s r1−1 dsh2(x2, s) ≤ η ln b.kgk Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) sup v∈Y |v(x2, y2) − v(x1, y1)| .

In view of our assumptions, we have sup u∈Y |u(x2, y2) − u(x1, y1)| ≤ η ln b.kgk Γ(r1)Γ(r2) α=b _ α=1 h2(y, α) sup v∈Y |v(x2, y2) − v(x1, y1)| .