RÉPUBLIQUE ALGÉRIENNE DÉMOCRATIQUE & POPULAIREMINISTERE DE L’ENSEIGNEMENT SUPÉRIEUR & DE LA RECHERCHE SCIENTIFIQUE

UNIVERSITE DJILLALI LIABBSFACULTE DES SCIENCES EXACTES

SIDI BEL-ABBÈS

THÈSEDE DOCTORAT

Présentée par:

HERIS AMEL

Spécialité : MathématiqueOption : Systèmes Dynamiques et Applications

IntituléeProblème de Darboux pour des équations

différentielles hyperboliques d’ordre fractionnaire

Soutenue le: . . . . . . . . . . . . . . . . .Devant le jury composé de:

Président

Ouahab Abdelghani, Pr. Univ. Djillali Liabes SBA.

Directeur de thèse

Benchohra Mouffak, Pr. Univ. Djilali Liabes SBA.

Examinateurs

Belmekki Mohamed Pr. Ecole Sup en SA. TlemcenAbbas Said M.C. (A). Univ. Tahar Moulay SaidaLazreg Jamal Eddine M.C. (A), Univ. Djillali Liabes SBASlimani Boualem Attou M.C. (A).Univ. de Tlemcen

ii

Acknowledgements

In The Name of Allah, Most Gracious, Most Merciful.

Praise be to Allah who gave me strength, inspiration and pru-dence to bring this thesis to a close. Peace be Upon His MessengerMuhammad and His Honorable Family.

First and foremost, I would like to express my sincere grati-tude to my supervisor Prof. Mouffak Benchohra for the continuoussupport of my study and research, for his patience, motivation, en-thusiasm, and immense knowledge. His guidance helped me in allthe time of research and writing of this thesis.

My special thank belongs to. Prof. Abdelghani Ouahab, Prof.Mohammed Belmekki, Dr. Said Abbas, Dr. Jamal-Eddine Lazregand Dr. Boualem Attou Slimani who in the midst of all their activ-ity, accepted to be members of this thesis committee.

I am also indebted to my beloved parents, my husband, sistersand brothers. A big thank may not enough for their encouragement,understanding, tolerance and prayers that acts as a giant supportfor my success. My thanks also go to all my friends and colleagues,wherever they may be, thank you.

iii

Publications1. M. Benchohra, J. Henderson and A. Heris, Fractional partial random differential

equations with delay, PanAmer. Math. J. 27 (1) (2017), 29-40.

2. M. Benchohra and A. Heris, Fractional partial random differential equationswith state-dependent delay, Annals West Univ. Timisoara-Math. (to appear).

3. M. Benchohra and A. Heris, Random impulsive partial hyperbolic fractional dif-ferential equations, Nonlinear Dyn. Syst. Theory (to appear).

Abstract

The objective of this thesis is to present the existence of random solutionsfor the fractional partial random differential equations in Banach spaces . Someequations present delay which may be fnite, infinite, or state-dependent. Ourresults will be obtained by means the measure of noncompactness and a ran-dom fixed point theorem with stochastic domain.

Key words:Random differential equation; left-sided mixed Riemann-Liouvilleintegral; Caputo fractional order derivative; Banach space; Darboux problem;measure of noncompactness .

AMS Subject Classification : 26A33.

iv

Résumé

L’objectif de cette thèse est de présenter, des résultats d’existence dessolutions du problème de Darboux pour des équations différentielles hyper-boliques d’ordre fractionnaire avec un effet dans un espace de Banach. On aconsidéré ce problème avec retard fini, infini et dépendant de l’etat. Nos résul-tats sont basées sur théorème du point fixe et la mesure de non compacité.

Mots clé: Equations différentielles; l’intégrale d’ordre fractionnaire au sens deRiemann-Liouville; dérivée de Caputo; espace de Banach; problème de Dar-boux; mesure de non compacité.

Classification AMS: 26A33.

v

Contents

Introduction 1

1 Preliminary 41.1 Some Notations and Definitions . . . . . . . . . . . . . . . . . . . . 41.2 Some Properties of Partial Fractional Calculus . . . . . . . . . . . 61.3 The phase space B . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Measure of noncompactness . . . . . . . . . . . . . . . . . . . . . . 81.5 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Fractional Partial Random Differential Equations 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Fractional Partial Random Differential Equations with Delay 173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Fractional Partial Random Differential Equations with finite Delay 17

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Fractional Partial Random Differential Equations with infiniteDelay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . 283.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Fractional Partial Random Differential Equations with State-DependentDelay 364.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vi

CONTENTS vii

4.2 Fractional Partial Random Differential Equations with finite Delay 364.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . 374.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Fractional Partial Random Differential Equations with infiniteDelay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . 444.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Random Impulsive Partial Hyperbolic Fractional Differential Equa-tions 535.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Bibliography 64

Introduction

Fractional calculus is generalization of ordinary differentiation and inte-gration to arbitrary non-integer order. The subject is as old as the differen-tial calculus, starting from some speculations of G.W. Lebeniz(1967) and L. Eu-ler(1730) and since then, it has continued to be developed up to nowadays. In-tegral equations are one of the most useful mathematical tools in both pure andapplied analysis. This is particulary true of problems in mechanical vibrationsand the related fields of engineering and mathematical physics. we can find nu-merous applications of differential and integral equation of fractional order infinance, hydrology, biophysics, thermodynamics, control theory, statistical me-chanics, astrophysics, cosmology and bioengineering ( [16,38,55,56,60]). Therehas been a significant development in ordinary and partial fractional differen-tial equations in recent years; see the monographs of Abbas et al. [10], Baleanuet al. [16], Kilbas et al. [44], Lakshmikantham et al, the papers of Abbas et al.

The theory of impulsive integer order differential equations and inclusionshave become important in some mathematical models of real processes andphenomena studied in physics, chemical technology, population dynamics, biotech-nology and economics. The study of impulsive fractional differential equationsand inclusions was initiated in the 1960’s by Milman and Myshkis [52, 53]. Atpresent the foundations of the general theory are already laid, and many ofthem are investigated in detail in the books of Aubin [13], Berhoun [20] and thereferences therein. There was an intensive development of the impulse theory,especially in the area of impulsive differential equations and inclusions withfixed moments; see for example [46, 57]. Recently in [2, 9], we have consideredsome classes of hyperbolic partial differential equations involving the Caputofractional derivative and impulses at fixed time. The theory of impulsive differ-ential equations and inclusions with variable time is relatively less developeddue to the difficulties created by the state-dependent impulses.Some interestingextensions to impulsive differential equations with variable times have beendone by Bajo and Liz [15], Abbas et al. [1], Belarbi and Benchohra [17], Ben-

1

CONTENTS 2

chohra et al. [18, 19], Frigon and O’Regan [28, 29, 30], Kaul et al. [41], Kaul andLiu [42, 43], Lakshmikantham et al. [47] and the references cited therein.

Functional differential equations with state-dependent delay appear fre-quently in applications as model of equations and for this reason the studyof this type of equations has received great attention in the last year; see forinstance [33, 34] and the references therein. The literature related to partialfunctional differential equations with state-dependent delay is limited; see forinstance [7, 37]. The literature related to ordinary and partial functional differ-ential equations with delay for which ρ(t, ·) = t or (ρ1(x, y, ·), ρ2(x, y, ·)) = (x, y)is very extensive; see for instance [5, 6, 32] and the references therein.

Random differential equations and random integral equations have beenstudied systematically in Ladde and Lakshmikantham [45] and Bharucha-Reid[21], respectively.They are good models in various branches of science and en-gineering since random factors and uncertainties have been taken into con-sideration. Hence, the study of the fractional differential equations with ran-dom parameters seem to be a natural one. We refer the reader to the mono-graphs [21, 45, 58], the papers [25, 26, 40] and the references therein.

Initial value problems for fractional differential equations with random pa-rameters have been studied by Lupulescu and Ntouyas [50]. The basic toolin the study of the problems for random fractional differential equations is totreat it as a fractional differential equation in some appropriate Banach space.In [51], authors proved the existence results for a random fractional equationunder a Carathéodory condition.

This thesis is devoted to the the existence of random solutions for the frac-tional partial random differential equations. Some equations present delaywhich may be finite, infinite, or state-dependent. The tools used include mea-sure of noncompactness and a random fixed point theorem with stochastic do-main. This thesis is arranged and organized as follows:

In Chapter 1, we introduce some notations, definitions,lemmas and fixedpoint theorems which are used throughout this thesis.

In chapter 2, we prove the existence of random solutions for the fractionalpartial random differential equations.

CONTENTS 3

In Chapter 3, in section two of chapter 3, we study the existence results forthe Darboux problem of partial fractional random differential equations withdelay. Later, we give similar results to nonlocal initial value problem.In Section 3.3, we prove the existence of random solutions for the fractionalpartial random differential equations with infinite delay.

In chapter 4, we shall be concerned to the existence for the fractional partialrandom differential equations with state-dependent delay.In Section 4.2, we study the existence results for the Darboux problem of partialfractional random differential equations with finite delay.In Section 4.3, we study the existence results for the Darboux problem of partialfractional random differential equations with infinite delay.

Finally, in Chapter 5, we discuss the existence of random solutions for theimpulsive partial fractional random differential equations.

Chapter 1

Preliminary

In this chapter, we introduce notations, definitions, and preliminary facts thatwill be used in the remainder of this thesis.

1.1 Some Notations and Definitions

Let J := [0, a] × [0, b], a, b > 0. Denote L1(J) the space of Bochner-integrablefunctions u : J → E with the norm

‖u‖L1 =

∫ a

0

∫ b

0

‖u(x, y)‖Edydx.

L∞(J) the Banach space of functions u : J → R which are essentially boundedwith the norm

‖u‖L∞ = infc > 0 : ‖u(x, y)‖ ≤ c, a.e. (x, y) ∈ J.

As usual, by AC(J) we denote the space of absolutely continuous functionsfrom J into E.

Let βE be the σ-algebra of Borel subsets of E. A mapping v : Ω → E is saidto be measurable if for any B ∈ βE, one has

v−1(B) = w ∈ Ω : v(w) ∈ B ⊂ A.

To define integrals of sample paths of random process, it is necessary todefine a jointly measurable map.

4

1.1 Some Notations and Definitions 5

Definition 1.1.1. A mapping T : Ω × E → E is called jointly measurable if forany B ∈ βE, one has

T−1(B) = (w, v) ∈ Ω× E : T (w, v) ∈ B ⊂ A× βE,

where A× βE is the direct product of the σ-algebras A and βE those defined inΩ and E respectively.

Lemma 1.1.1. Let T : Ω × E → E be a mapping such that T (·, v) is measurable forall v ∈ E, and T (w, ·) is continuous for all w ∈ Ω. Then the map (w, v) 7→ T (w, v) isjointly measurable.

Definition 1.1.2. A function f : J×E×Ω→ E is called random Carathéeodoryif the following conditions are satisfied:

(i) The map (x, y, w)→ f(x, y, u, w) is jointly measurable for all u ∈ E, and

(ii) The map u → f(x, y, u, w) is continuous for almost all (x, y) ∈ J andw ∈ Ω.

Let T : Ω × E → E be a mapping. Then T is called a random operator ifT (w, u) is measurable in w for all u ∈ E and it is expressed as T (w)u = T (w, u).In this case we also say that T (w) is a random operator on E. A random oper-ator T (w) on E is called continuous (resp. compact, totally bounded and com-pletely continuous) if T (w, u) is continuous (resp. compact, totally boundedand completely continuous) in u for all w ∈ Ω. The details of completely con-tinuous random operators in Banach spaces and their properties appear inItoh [40].

Definition 1.1.3. [27] Let P(Y ) be the family of all nonempty subsets of Y andC be a mapping from Ω intoP(Y ).A mapping T : (w, y) : w ∈ Ω, y ∈ C(w) →Y is called random operator with stochastic domain C if C is measurable (i.e.,for all closed A ⊂ Y, w ∈ Ω, C(w) ∩ A 6= ∅ is measurable) and for all openD ⊂ Y and all y ∈ Y, w ∈ Ω : y ∈ C(w), T (w, y) ∈ D is measurable. Twill be called continuous if every T (w) is continuous. For a random operatorT, a mapping y : Ω → Y is called random (stochastic) fixed point of T if forP−almost all w ∈ Ω, y(w) ∈ C(w) and T (w)y(w) = y(w) and for all openD ⊂ Y, w ∈ Ω : y(w) ∈ D is measurable.

1.2 Some Properties of Partial Fractional Calculus 6

1.2 Some Properties of Partial Fractional Calculus

In this section, we introduce notations, definitions and preliminary Lemmasconcerning to partial fractional calculus theory.

Let θ = (0, 0), r1, r2 > 0 and r = (r1, r2). For f ∈ L1(J), the expression

(Irθf)(x, y) =1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1f(s, t)dtds,

is called the left-sided mixed Riemann-Liouville integral of order r, where Γ(·)is the (Euler’s) Gamma function defined by Γ(ξ) =

∫∞0tξ−1e−tdt; ξ > 0.

In particular,

(Iθθu)(x, y) = u(x, y), (Iσθ u)(x, y) =

∫ x

0

∫ y

0

u(s, t)dtds; for almost all (x, y) ∈ J,

where σ = (1, 1).For instance, Irθu exists for all r1, r2 ∈ (0,∞), when u ∈ L1(J). Note also thatwhen u ∈ C, then (Irθu) ∈ C, moreover

(Irθu)(x, 0) = (Irθu)(0, y) = 0; x ∈ [0, a], y ∈ [0, b].

By 1− r we mean (1− r1, 1− r2) ∈ [0, 1)× [0, 1). Denote by D2xy := ∂2

∂x∂y, the

mixed second order partial derivative.

Definition 1.2.1. [61] Let r ∈ (0, 1]×(0, 1] and u ∈ L1(J). The Caputo fractional-order derivative of order r of u is defined by the expression

cDrθu(x, y) = (I1−r

θ D2xyu)(x, y).

The case σ = (1, 1) is included and we have

(cDσθu)(x, y) = (D2

xyu)(x, y); for almost all (x, y) ∈ J.

1.3 The phase space BThe notation of the phase space B plays an important role in the study ofboth qualitative and quantitative theory for functional differential equations.A usual choice is a semi-normed space satisfying suitable axioms, which wasintroduced by Hale and Kato (see [31]). For further applications see for in-stance the books [35, 39, 48] and their references. For any (x, y) ∈ J denote

1.3 The phase space B 7

E(x,y) := [0, x]×0∪0× [0, y], furthermore in case x = a, y = b we write sim-ply E .Consider the space (B, ‖(·, ·)‖B) is a seminormed linear space of functionsmapping (−∞, 0]× (−∞, 0] into Rn, and satisfying the following fundamentalaxioms which were adapted from those introduced by Hale and Kato for ordi-nary differential functional equations:

(A1) If z : (−∞, a] × (−∞, b] → Rn continuous on J and z(x,y) ∈ B, for all(x, y) ∈ E , then there are constants H,K,M > 0 such that for any (x, y) ∈J the following conditions hold:

(i) z(x,y) is in B;

(ii) ‖z(x, y)‖ ≤ H‖z(x,y)‖B,

(iii) ‖z(x,y)‖B ≤ K sup(s,t)∈[0,x]×[0,y] ‖z(s, t)‖+M sup(s,t)∈E(x,y) ‖z(s,t)‖B,

(A2) For the function z(., .) in (A1), z(x,y) is a B-valued continuous function onJ.

(A3) The space B is complete.

Now, we present some examples of phase spaces [23, 24].

Example 1.3.1. Let B be the set of all functions φ : (−∞, 0] × (−∞, 0] → Rn

which are continuous on [−α, 0]× [−β, 0], α, β ≥ 0, with the seminorm

‖φ‖B = sup(s,t)∈[−α,0]×[−β,0]

‖φ(s, t)‖.

Then we have H = K = M = 1. The quotient space B = B/‖.‖B is isometricto the space C([−α, 0] × [−β, 0],Rn) of all continuous functions from [−α, 0] ×[−β, 0] into Rn with the supremum norm, this means that partial differentialfunctional equations with finite delay are included in our axiomatic model.

Example 1.3.2. Let Cγ be the set of all continuous functions φ : (−∞, 0] ×(−∞, 0]→ Rn for which a limit lim‖(s,t)‖→∞ e

γ(s+t)φ(s, t) exists, with the norm

‖φ‖Cγ = sup(s,t)∈(−∞,0]×(−∞,0]

eγ(s+t)‖φ(s, t)‖.

Then we have H = 1 and K = M = maxe−(a+b), 1.

1.4 Measure of noncompactness 8

Example 1.3.3. Let α, β, γ ≥ 0 and let

‖φ‖CLγ = sup(s,t)∈[−α,0]×[−β,0]

‖φ(s, t)‖+

∫ 0

−∞

∫ 0

−∞eγ(s+t)‖φ(s, t)‖dtds.

be the seminorm for the space CLγ of all functions φ : (−∞, 0]× (−∞, 0]→ Rn

which are continuous on [−α, 0]× [−β, 0] measurable on (−∞,−α]× (−∞, 0]∪(−∞, 0]× (−∞,−β], and such that ‖φ‖CLγ <∞. Then

H = 1, K =

∫ 0

−α

∫ 0

−βeγ(s+t)dtds, M = 2.

1.4 Measure of noncompactness

LetMX denote the class of all bounded subsets of a metric space X.

Definition 1.4.1. Let X be a complete metric space. A map α :MX → [0,∞) iscalled a measure of noncompactness onX if it satisfies the following propertiesfor all B,B1, B2 ∈MX .

(MNC.1) α(B) = 0 if and only if B is precompact (Regularity),

(MNC.2) α(B) = α(B) (Invariance under closure),

(MNC.3) α(B1 ∪B2) = α(B1) + α(B2) (Semi-additivity).

For more details on measure of noncompactness and its properties see [12].

Example 1.4.1. In every metric space X, the map φ : MX → [0,∞) withφ(B) = 0 if B is relatively compact and φ(B) = 1 otherwise is a measure ofnoncompactness, the so-called discrete measure of noncompactness [ [14], Ex-ample1, p. 19].

Lemma 1.4.1. [22] If Y is a bounded subset of Banach space X, then for each ε > 0,there is a sequence yk∞k=1 ⊂ Y such that

α(Y ) ≤ 2α(yk∞k=1) + ε.

Lemma 1.4.2. [54] uk∞k=1 ⊂ L1(J) is uniformly integrable, then α(uk∞k=1) ismeasurable and for each (x, y) ∈ J,

α

(∫ x

0

∫ y

0

uk(s, t)dtds

∞k=1

)≤ 2

∫ x

0

∫ y

0

α(uk(s, t)∞k=1)dtds.

1.5 Fixed Point Theorems 9

1.5 Fixed Point Theorems

Fixed point theory plays an important role in our existence results, thereforewe state the following lemma.

Lemma 1.5.1. [49] Let F be a closed and convex subset of a real Banach space, letG : F → F be a continuous operator and G(F ) be bounded. If there exist a constantk ∈ [0, 1) such that for each bounded subset B ⊂ F,

α(G(B)) ≤ kα(B),

then G has a fixed point in F.

Chapter 2

Fractional Partial RandomDifferential Equations

2.1 Introduction

We study in this chapter the existence of random solutions for the follow-ing fractional partial random differential equations

cDrθu(x, y, w) = f(x, y, u(x, y, w), w); for a.a. (x, y) ∈ J := [0, a]× [0, b], w ∈ Ω,

(2.1)with the initial conditions

u(x, 0, w) = ϕ(x,w); x ∈ [0, a],

u(0, y, w) = ψ(y, w); y ∈ [0, b],

ϕ(0, w) = ψ(0, w),

w ∈ Ω, (2.2)

where a, b > 0, θ = (0, 0), cDrθ is the fractional Caputo derivative of order

r = (r1, r2) ∈ (0, 1]× (0, 1], (Ω,A) is a measurable space, f : J × E × Ω → E isa given continuous function, (E, ‖ · ‖E) is a real Banach space, ϕ : [0, a] × Ω →E, ψ : [0, b]× Ω→ E are given are given functions such that ϕ(·, w) and ψ(·, w)are absolutely continuous functions for all w ∈ Ω, and ϕ(x, ·) and ψ(y, ·) aremeasurable for all x ∈ [0, a] and y ∈ [0, b] respectively, and C is the Banachspace of all continuous functions from J into E with the supremum (uniform)norm ‖ · ‖∞.

10

2.2 Existence Results 11

2.2 Existence Results

Definition 2.2.1. By a random solution of the random problem (2.1)-(2.2) wemean a measurable function u : Ω→ AC(J) that satisfies the equation (2.1) a.a.on J × Ω and the initial conditions (2.2) are satisfied.

Let h ∈ L1(J,Rn). We need the following lemma:

Lemma 2.2.1. [3, 10] A function u ∈ AC(J,Rn) is a solution of problemcDr

θu(x, y) = h(x, y); for a.a. (x, y) ∈ J := [0, a]× [0, b],

u(x, 0) = ϕ(x); x ∈ [0, a],

u(0, y) = ψ(y); y ∈ [0, b],

ϕ(0) = ψ(0).

if and only if u if and only if u(x, y) satisfies

u(x, y) = µ(x, y) + Irθh(x, y); for a.a. (x, y) ∈ J,

whereµ(x, y) = ϕ(x) + ψ(y)− ϕ(0).

Let us assume that the function f is random Carathéeodory on J × E × Ω.From the above Lemma, we have the following Lemma.

Lemma 2.2.2. Let 0 < r1, r2 ≤ 1. A function u ∈ Ω×AC is a solution of the randomfractional integral equation

u(x, y, w) = µ(x, y, w)+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x−s)r1−1(y−t)r2−1f(s, t, u(s, t, w), w)dtds,

(2.3)where

µ(x, y, w) = ϕ(x,w) + ψ(y, w)− ϕ(0, w),

if and only if u is a solution of the random problem (2.1)-(2.2).

The following hypotheses will be used in the sequel.

(H1) The functions w 7→ ϕ(x, 0, w) and w 7→ ψ(0, y, w) are measurable andbounded for a.e. x ∈ [0, a] and y ∈ [0, b] respectively,

(H2) The function f is random Carathéeodory on J × E × Ω,

2.2 Existence Results 12

(H3) There exist functions p1, p2 : J × Ω→ [0,∞) withpi(., w) ∈ AC(J, [0,∞))L∞(J, [0,∞)); i = 1, 2 such that

‖f(x, y, u, w)‖E ≤ p1(x, y, w) + p2(x, y, w)‖u‖E,

for all u ∈ E, w ∈ Ω and a.e. (x, y) ∈ J,

(H4) for any bounded B ⊂ E,

α(f(x, y, B,w)) ≤ p(x, y, w)α(B), for a.e. (x, y) ∈ J,

(H5) There exists a random function R : Ω→ (0,∞) such that

µ∗(w) +(p∗1(w) + p∗2(w)R(w))ar1br2

Γ(1 + r1)Γ(1 + r2)≤ R(w),

where

µ∗(w) = sup(x,y)∈J

‖µ(x, y, w)‖E, p∗i (w) = sup ess(x,y)∈Jpi(x, y, w); i = 1, 2.

Theorem 2.2.1. Assume that hypotheses (H1)− (H5) hold. If

` :=4q∗ar1br2

Γ(1 + r1)Γ(1 + r2)< 1,

then the problem (2.1)-(2.2) has a random solution defined on J.

Proof. From th hypothesis (H2), (H3), for each w ∈ Ω and almost all (x, y) ∈J, we have that f(x, y, u(x, y, w), w) is in L1. By using Lemma 2.2.2, the problem(2.1)-(2.2) is equivalent to the integral equation

u(x, y, w) = µ(x, y, w) +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(s, t, w), w)dtds;

for each w ∈ Ω and a.e. (x, y) ∈ J.

Define the operator N : Ω× C → C by

(N(w)u)(x, y) = µ(x, y, w)+

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(s, t, w), w)dtds.

Since the functions ϕ, ψ and f are absolutely continuous , then the function µand the indefinite integral are absolutely continuous for all w ∈ Ω and almost

2.2 Existence Results 13

all (x, y) ∈ J. Again, as the map µ is continuous for all w ∈ Ω and the indefiniteintegral is continuous on J, thenN(w) defines a mappingN : Ω×C → C.Henceu is a solution for the problem (2.1)-(2.2) if and only if u = (N(w))u. We shallshow that the operatorN satisfies all conditions of Lemma 1.5.1. The proof willbe given in several steps.

Step 1: N(w) is a random operator with stochastic domain on C.Since f(x, y, u, w) is random Carathéodory, the map w → f(x, y, u, w) is mea-surable in view of Definition 1.1.1.Similarly, the product (x − s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w) of a continuousand a measurable function is again measurable. Further, the integral is a limitof a finite sum of measurable functions, therefore, the map

w 7→ µ(x, y, w) +1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds,

is measurable. As a result, N is a random operator on Ω× C into C.

Let W : Ω→ P(C) be defined by

W (w) = u ∈ C : ‖u‖∞ ≤ R(w),

with W (w) bounded, closed, convex and solid for all w ∈ Ω. Then W is mea-surable by Lemma [ [27], Lemma 17]. Let w ∈ Ω be fixed, then from (H4), forany u ∈ w(w), we get

‖(N(w)u)(x, y)‖E

≤ ‖µ(x, y, w)‖E +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)‖f(s, t, u(s, t, w), w)‖Edtds

≤ ‖µ(x, y, w)‖E +1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p1(s, t, w)dtds

+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p2(s, t, w)‖u(s, t, w)‖Edtds

≤ µ∗(w) +p∗1(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

≤ µ∗(w) +(p∗1(w) + p∗2(w)R(w))ar1br2

Γ(1 + r1)Γ(1 + r2)

≤ R(w).

2.2 Existence Results 14

Therefore,N is a random operator with stochastic domainW andN(w) : W (w)→N(w). Furthermore, N(w) maps bounded sets into bounded sets in C.

Step 2: N(w) is continuous.Let un be a sequence such that un → u in C. Then, for each (x, y) ∈ J andw ∈ Ω, we have

‖(N(w)un)(x, y)− (N(w)u)(x, y)‖E ≤1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×‖f(s, t, un(s, t, w), w)− f(s, t, u(s, t, w), w)‖Edtds.

Using the Lebesgue Dominated Convergence Theorem, we get

‖N(w)un −N(w)u‖∞ → 0 as n→∞.

As a consequence of Steps 1 and 2, we can conclude that N(w) : W (w) →N(w) is a continuous random operator with stochastic domainW, andN(w)(W (w))is bounded.

Step 3: For each bounded subset B of W (w) we have

α(N(w)B) ≤ `α(B).

Let w ∈ Ω be fixed. From Lemmas 1.4.1 and 1.4.2, for any B ⊂ W and any

2.3 An Example 15

ε > 0, there exists a sequence un∞n=0 ⊂ B, such that for all (x, y) ∈ J, we have

α((N(w)B)(x, y))

= α

(µ(x, y) +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(s, t, w), w)dtds; u ∈ B

)≤ 2α

(∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s, t;w), w)dtds

∞n=1

)+ ε

≤ 4

∫ x

0

∫ y

0

α

((x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s, t, w), w)

∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)α (f(s, t, un(s, t, w), w)∞n=1) dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α (un(s, t, w)∞n=1) dtds+ ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (un∞n=1) + ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)α(B) + ε

≤ 4p∗2ar1br2

Γ(1 + r1)Γ(1 + r2)α(B) + ε

= `α(B) + ε.

Since ε > 0 is arbitrary, then

α(N(B)) ≤ `α(B).

It follows from Lemma 1.5.1 that for each w ∈ Ω, N has at least one fixedpoint in W. Since

⋂w∈Ω intW (w) 6= ∅ the hypothesis that a measurable selector

of intW exists holds. By Lemma 1.5.1, N has a stochastic fixed point, i.e., theproblem (2.1)-(2.2) has at least one random solution on C.

2.3 An Example

Let E = R, Ω = (−∞, 0) be equipped with the usual σ-algebra consisting ofLebesgue measurable subsets of (−∞, 0). Given a measurable function u : Ω→AC([0, 1]× [0, 1]), consider the following partial functional random differential

2.3 An Example 16

equation of the form

(cDrθu)(x, y, w) =

w2e−x−y−3

1 + w2 + 5|u(x, y, w)|; a.a. (x, y) ∈ J = [0, 1]× [0, 1], w ∈ Ω,

(2.4)with the initial conditions

u(x, 0, w) = x sinw; x ∈ [0, 1],

u(0, y, w) = y2 cosw; y ∈ [0, 1],w ∈ Ω, (2.5)

where (r1, r2) ∈ (0, 1]× (0, 1]. Set

f(x, y, u(x, y, w), w) =w2

(1 + w2 + 5|u(x, y, w)|)ex+y+3, (x, y) ∈ [0, 1]×[0, 1], w ∈ Ω.

The functions w 7→ ϕ(x, 0, w) = x sinw and w 7→ ψ(0, y, w) = y2 cosw are mea-surable and bounded with

|ϕ(x, 0, w)| ≤ 1, |ψ(0, y, w)| ≤ 1,

hence, the conditions (H1) is satisfied.Clearly, the map (x, y, w) 7→ f(x, y, u, w) is jointly continuous for all u ∈ Rand hence jointly measurable for all u ∈ R. Also the map u 7→ f(x, y, u, w) iscontinuous for all (x, y) ∈ J and w ∈ Ω. So the function f is Carathéodory on[0, 1]× [0, 1]× R× Ω.For each u ∈ R, (x, y) ∈ [0, 1]× [0, 1] and w ∈ Ω, we have

|f(x, y, u, w)| ≤ 1 +5

e3|u|.

Hence the conditions (H3) is satisfied with p1(x, y, w) = p∗1 = 1 and p2(x, y, w) =p∗2 = 5

e3.

Also, the conditions (H4) is satisfied.We shall show that condition ` < 1 holds with a = b = 1. For each (r1, r2) ∈

(0, 1]× (0, 1] we get

` =4p∗2a

r1br2

Γ(1 + r1)Γ(1 + r2)

=20

e3Γ(1 + r1)Γ(1 + r2)

<20

e3

< 1.

Consequently, Theorem 2.2.1 implies that the problem (2.4)-(2.5) has a randomsolution defined on [0, 1]× [0, 1].

Chapter 3

Fractional Partial RandomDifferential Equations with Delay

3.1 Introduction

We study in this chapter existence results for the Darboux problem ofpartial fractional random differential equations with delay

3.2 Fractional Partial Random Differential Equationswith finite Delay

3.2.1 Introduction

In this section, we discuss the existence of random solutions for the followingfractional partial random differential equations with finite delay

(cDr0u)(x, y, w) = f(x, y, u(x,y), w), if (x, y) ∈ J := [0, a]× [0, b], w ∈ Ω, (3.1)

u(x, y, w) = φ(x, y, w), if (x, y) ∈ J := [−α, a]× [−β, b]\(0, a]× (0, b], w ∈ Ω,(3.2)

u(x, 0, w) = ϕ(x,w), x ∈ [0, a], u(0, y, w) = ψ(y, w), y ∈ [0, b], w ∈ Ω, (3.3)

where α, β, a, b > 0, cDr0 is the standard Caputo’s fractional derivative of order

−r = (r1, r2) ∈ (0, 1]× (0, 1], (Ω,A) is a measurable space, f : J ×C ×Ω→ E isa given function, φ : J ×Ω→ E is a given continuous function, ϕ : [0, a]×Ω→E,ψ : [0, b] × Ω → E are given absolutely continuous functions with ϕ(x,w) =φ(x, 0, w), ψ(y, w) = φ(0, y, w) for each x ∈ [0, a], y ∈ [0, b], w ∈ Ω and C :=

17

3.2 Fractional Partial Random Differential Equations with finite Delay 18

C([−α, 0]× [−β, 0], E) is the space of continuous functions on [−α, 0]× [−β, 0].

If u ∈ C(a,b) = C([−α, a] × [−β, b], E); a, b, α, β > 0 then for any (x, y) ∈ Jdefine u(x,y) by

u(x,y)(s, t, w) = u(x+ s, y + t, w),

for (s, t) ∈ [−α, 0]× [−β, 0]. Here u(x,y)(·, ·, w) represents the history of the statefrom time x− α up to the present time x and from time y − β up to the presenttime y.

Next we consider the following nonlocal initial value problem

(cDr0u)(x, y, w) = f(x, y, u(x,y), w), if (x, y) ∈ J := [0, a]× [0, b], w ∈ Ω, (3.4)

u(x, y, w) = φ(x, y, w), if (x, y) ∈ J := [−α, a]× [−β, b]\(0, a]× (0, b], w ∈ Ω,(3.5)

u(x, 0, w) +Q(u) = ϕ(x,w); x ∈ [0, a],

u(0, y, w) +K(u) = ψ(y, w); y ∈ [0, b],w ∈ Ω, (3.6)

where f, φ, ϕ, ψ are as in problem (3.1)-(3.3) and Q,K : C(J,E) → E aregiven continuous functions.

3.2.2 Existence Results

Lemma 3.2.1. Let 0 < r1, r2 ≤ 1, µ(x, y, w) = ϕ(x,w) + ψ(y, w) − ϕ(0, w). Afunction u ∈ Ω × C(a,b) is a solution of the random problem (3.1)-(3.3) if u satisfiescondition (3.2) for (x, y) ∈ J , w ∈ Ω and u is a solution of the equation

u(x, y, w) = µ(x, y, w) +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(x, y, u(s,t), w)dtds

for (x, y) ∈ J, w ∈ Ω

We will need to introduce the following hypotheses which are be assumedthere after:

(H1) The functions w 7→ ϕ(x, 0, w) and w 7→ ψ(0, y, w) are measurable andbounded for a.e. x ∈ [0, a] and y ∈ [0, b] respectively,

(H2) The function Φ is measurable for (x, y) ∈ J

(H3) The function f is random Carathéeodory on J × C × Ω,

3.2 Fractional Partial Random Differential Equations with finite Delay 19

(H4) There exist functions p1, p2 : J × Ω→ [0,∞) withpi(., w) ∈ AC(J, [0,∞))L∞(J, [0,∞)); i = 1, 2 such that

‖f(x, y, u, w)‖E ≤ p1(x, y, w) + p2(x, y, w)‖u‖C ,

for all u ∈ C,w ∈ Ω and a.e. (x, y) ∈ J,

(H5) For any bounded B ⊂ E,

α(f(x, y, B,w)) ≤ P2(x, y, w)α(B), for a.e. (x, y) ∈ J,

Set

µ∗(w) = sup(x,y)∈J

‖µ(x, y, w)‖E, p∗i (w) = sup ess(x,y)∈Jpi(x, y, w); i = 1, 2.

Theorem 3.2.1. Assume that hypotheses (H1)− (H5) hold. If

` :=4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)< 1,

then the problem (3.1)-(3.3) has a random solution defined on [−α, a]× [−β, b].

Proof. Define the operator N : Ω× C(a,b) → C(a,b) by

N(u)(x, y) =

φ(x, y, w), (x, y) ∈ J , w ∈ Ω,

µ(x, y) + 1Γ(r1)Γ(r2)

∫ x0

∫ y0

(x− s)r1−1(y − t)r2−1

×f(s, t, u(s,t), w)dtds, (x, y) ∈ J, w ∈ Ω.(3.7)

Since the functions ϕ, ψ and f are absolutely continuous , the function µand the indefinite integral are absolutely continuous for all w ∈ Ω and almostall (x, y) ∈ J. Again, as the map µ is continuous for all w ∈ Ω and the indefiniteintegral is continuous on J, thenN(w) defines a mappingN : Ω×C(a,b) → C(a,b).Hence u is a solution for the problem (3.1)-(3.3) if and only if u = (N(w))u. Weshall show that the operator N satisfies all conditions of Lemma 1.5.1. Theproof will be given in several steps.

Step 1: N(w) is a random operator with stochastic domain on C(a,b).Since f(x, y, u, w) is random Carathéodory, the mapw → f(x, y, u, w) is measur-able in view of Definition 1.1.1. Similarly, the product (x−s)r1−1(y−t)r2−1f(s, t, u(s,t), w)

3.2 Fractional Partial Random Differential Equations with finite Delay 20

of a continuous and a measurable function is again measurable. Further, the in-tegral is a limit of a finite sum of measurable functions, therefore, the map

w 7→ µ(x, y, w) +1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1f(s, t, u(s,t), w)dtds,

is measurable. As a result, N is a random operator on Ω× C(a,b) into C(a,b).

Let W : Ω→ P(C(a,b)) be defined by

W (w) = u ∈ C(a,b) : ‖u‖∞ ≤ R(w),

With R(·) is chosen appropriately. From instance, we assume that

R(w) ≥µ∗(w) + p∗1(w) ar1br2

Γ(1+r1)Γ(1+r2)

1− p∗2(w) ar1br2Γ(1+r1)Γ(1+r2)

.

with W (w) bounded, closed, convex and solid for all w ∈ Ω. Then W is mea-surable by Lemma [ [27], Lemma 17]. Let w ∈ Ω be fixed, then from (H4), forany u ∈ w(w), we get

‖(N(w)u)(x, y)‖E

≤ ‖µ(x, y, w)‖E +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)‖f(s, t, u(s,t), w)‖Edtds

≤ ‖µ(x, y, w)‖E +1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p1(s, t, w)dtds

+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p2(s, t, w)‖u(s,t)‖∞dtds

≤ µ∗(w) +p∗1(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

≤ µ∗(w) +(p∗1(w) + p∗2(w)R(w))ar1br2

Γ(1 + r1)Γ(1 + r2)

≤ R(w).

Therefore,N is a random operator with stochastic domainW andN(w) : W (w)→N(w). Furthermore, N(w) maps bounded sets into bounded sets in C(a,b).

3.2 Fractional Partial Random Differential Equations with finite Delay 21

Step 2: N(w) is continuous.Let un be a sequence such that un → u in C(a,b). Then, for each (x, y) ∈ J andw ∈ Ω, we have

‖(N(w)un)(x, y)− (N(w)u)(x, y)‖E ≤1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×‖f(s, t, un(s,t), w)− f(s, t, u(s,t), w)‖Edtds.

Using the Lebesgue Dominated Convergence Theorem, we get

‖N(w)un −N(w)u‖∞ → 0 as n→∞.

As a consequence of Steps 1 and 2, we can conclude that N(w) : W (w) →N(w) is a continuous random operator with stochastic domainW, andN(w)(W (w))is bounded.

Step 3: For each bounded subset B of W (w) we have

α(N(w)B) ≤ `α(B).

Let w ∈ Ω be fixed. From Lemmas 1.4.1 and 1.4.2, for any B ⊂ W and anyε > 0, there exists a sequence un∞n=0 ⊂ B, such that for all (x, y) ∈ J, we have

α((N(w)B)(x, y))

= α

(µ(x, y) +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(s,t), w)dtds; u ∈ B

)≤ 2α

(∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s,t), w)dtds

∞n=1

)+ ε

≤ 4

∫ x

0

∫ y

0

α

((x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s,t), w)

∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)α(f(s, t, un(s,t), w)∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α

(un(s,t)∞n=1

)dtds+ ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (un∞n=1) + ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)α(B) + ε

≤ 4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)α(B) + ε

= `α(B) + ε.

3.2 Fractional Partial Random Differential Equations with finite Delay 22

Since ε > 0 is arbitrary, then

α(N(B)) ≤ `α(B).

It follows from Lemma 1.5.1 that for each w ∈ Ω, N has at least one fixedpoint in W. Since

⋂w∈Ω intW (w) 6= ∅, and the hypothesis that a measurable

selector of intW exists holds. By Lemma 1.5.1, N has a stochastic fixed point,i.e., the problem (3.1)-(3.3) has at least one random solution on C(a,b).

Let us assume that the function f is random Carathéeodory on J × C × Ω.From the above Lemma, we have.

Lemma 3.2.2. Let 0 < r1, r2 ≤ 1, µ(x, y, w) = ϕ(x,w) + ψ(y, w) − ϕ(0, w). Afunction u ∈ Ω × C(a,b) is a solution of the random problem (3.4)-(3.6) if u satisfiescondition (3.5) for (x, y) ∈ J , w ∈ Ω, and the integral equation

u(x, y, w) = µ(x, y, w)−Q(u)−H(u)+

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(x, y, u(s,t), w)dtds

for (x, y) ∈ J, w ∈ Ω

The following hypotheses will be used in the sequel.

(H1) The functions w 7→ ϕ(x, 0, w) and w 7→ ψ(0, y, w) are measurable andbounded for a.e. x ∈ [0, a] and y ∈ [0, b] respectively,

(H2) The function Φ is measurable for (x, y) ∈ J

(H3) The function f is random Carathéeodory on J × C × Ω,

(H4) there exist constants d∗, d > 0 such that

‖Q(u)‖ ≤ d∗(1 + ‖u‖),

and‖Q(u)‖ ≤ d(1 + ‖u‖),

for u ∈ C(J,E),

(H5) There exist functions p1, p2 : J × Ω→ [0,∞) withpi(·, w) ∈ AC(J, [0,∞))L∞(J, [0,∞)); i = 1, 2 such that

‖f(x, y, u, w)‖E ≤ p1(x, y, w) + p2(x, y, w)‖u‖C ,

for all u ∈ C, w ∈ Ω, and a.e. (x, y) ∈ J,

3.2 Fractional Partial Random Differential Equations with finite Delay 23

(H6) There exist functions p3, p4 : J × Ω→ [0,∞) withpi(·, w) ∈ AC(J, [0,∞))L∞(J, [0,∞)); i = 3, 4 such that for each w ∈ Ω andfor any bounded B ⊂ C,

αc(f(x, y, B,w)) ≤ P2(x, y, w)αc(B), for a.e. (x, y) ∈ J,

αc(Q(B)) ≤ P3(x, y, w)αc(B), for a.e. (x, y) ∈ J,

and

αc(H(B)) ≤ P4(x, y, w)αc(B), for a.e. (x, y) ∈ J,

here α, αc designe respectively the measures of noncompactness on X andC,

(H7) There exists a random function R : Ω→ (0,∞) such that

µ∗(w) + (d+ d∗)(1 +R(w)) +(p∗1(w) + p∗2(w)R(w))ar1br2

Γ(1 + r1)Γ(1 + r2)≤ R(w),

where

µ∗(w) = sup(x,y)∈J

‖µ(x, y, w)‖E, p∗i (w) = sup ess(x,y)∈Jpi(x, y, w); i = 1, 2,

Theorem 3.2.2. Assume that hypotheses (H1)− (H7) hold. If

` := 2(p∗3(w) + p∗4(w)) +4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)< 1,

then the problem (3.4)-(3.6) has a random solution defined on [−α, a]× [−β, b].

Proof. Define the operator N : Ω× C(a,b) → C(a,b) by

(N(w)u)(x, y) =

φ(x, y, w), (x, y) ∈ J ,w ∈ Ω

µ(x, y, w)−Q(u)−H(u)

+ 1Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×f(s, t, u(s,t), w)dtds, (x, y) ∈ J, w ∈ Ω.(3.8)

3.2 Fractional Partial Random Differential Equations with finite Delay 24

Since the functions ϕ, ψ andQ,H and f are absolutely continuous , the functionµ and the indefinite integral are absolutely continuous for all w ∈ Ω and almostall (x, y) ∈ J. Again, as the map µ is continuous for all w ∈ Ω and the indefiniteintegral is continuous on J, thenN(w) defines a mappingN : Ω×C(a,b) → C(a,b).Hence u is a solution for the problem (3.4)-(3.6) if and only if u = (N(w))u. Weshall show that the operator N satisfies all conditions of Lemma 1.5.1. Theproof will be given in several steps.

Step 1: N(w) is a random operator with stochastic domain on C(a,b).Since f(x, y, u, w) is random Carathéodory, the map w → f(x, y, u, w) is mea-surable in view of Definition 1.1.1.Similarly, the product (x − s)r1−1(y − t)r2−1f(s, t, u(s,t), w) of a continuous anda measurable function is again measurable. Further, the integral is a limit of afinite sum of measurable functions, therefore, the map

w 7→ µ(x, y, w)−Q(u)−H(u)+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x−s)r1−1(y−t)r2−1f(s, t, u(s,t), w)dtds,

is measurable. As a result, N is a random operator on Ω× C(a,b) into C(a,b).

Let W : Ω→ P(C(a,b)) be defined by

W (w) = u ∈ C(a,b) : ‖u‖∞ ≤ R(w),

with W (w) bounded, closed, convex and solid for all w ∈ Ω. Then W ismeasurable by Lemma [ [27], Lemma 17]. Let w ∈ Ω be fixed, then from (H4),

3.2 Fractional Partial Random Differential Equations with finite Delay 25

for any u ∈ w(w), we get

‖(N(w)u)(x, y)‖E≤ ‖µ(x, y, w)‖E + ‖Q(u)‖+ ‖H(u)‖

+

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)‖f(s, t, u(s,t), w)‖Edtds

≤ ‖µ(x, y, w)‖E + d(1 + ‖u‖) + d∗(1 + ‖u‖)

+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p1(s, t, w)dtds

+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p2(s, t, w)‖u(s,t)‖∞dtds

≤ µ∗(w) + (d+ d∗)(1 +R(w)) +p∗1(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

≤ µ∗(w) + (d+ d∗)(1 +R(w)) +(p∗1(w) + p∗2(w)R(w))ar1br2

Γ(1 + r1)Γ(1 + r2)

≤ R(w).

Therefore,N is a random operator with stochastic domainW andN(w) : W (w)→N(w). Furthermore, N(w) maps bounded sets into bounded sets in C(a,b).

Step 2: N(w) is continuous.Let un be a sequence such that un → u in C(a,b). Then, for each (x, y) ∈ J andw ∈ Ω, we have

‖(N(w)un)(x, y)− (N(w)u)(x, y)‖E ≤ ‖Q(un)−Q(u)‖+ ‖H(un)−H(u)‖

+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1‖f(s, t, un(s,t), w)− f(s, t, u(s,t), w)‖Edtds.

Using the Lebesgue Dominated Convergence Theorem, we get

‖N(w)un −N(w)u‖∞ → 0 as n→∞.

As a consequence of Steps 1 and 2, we can conclude that N(w) : W (w) →N(w) is a continuous random operator with stochastic domainW, andN(w)(W (w))is bounded.

Step 3: For each bounded subset B of W (w) we have

αc(N(w)B) ≤ `αc(B).

3.2 Fractional Partial Random Differential Equations with finite Delay 26

Let w ∈ Ω be fixed. From Lemmas 1.4.1 and 1.4.2, for any B ⊂ W and anyε > 0, there exists a sequence un∞n=0 ⊂ B, such that for all (x, y) ∈ J, we have

αc((N(w)B)(x, y))

= α

(µ(x, y)−Q(u)−H(u)

+

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(s,t), w)dtds; u ∈ B

)≤ 2α

−Q(un)−H(un) +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s,t), w)dtds

∞n=1

+ ε

≤ 2α Q(un)+ 2α H(un)

+4

∫ x

0

∫ y

0

α

((x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s,t), w)

∞n=1

)dtds+ ε

≤ 2p3(s, t, w)α (un∞n=1) + 2p4(s, t, w)α (un∞n=1)

+4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)α(f(s, t, un(s,t), w)∞n=1

)dtds+ ε

≤ 2p3(s, t, w)αc(B) + 2p4(s, t, w)αc(B)

+4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α

(un(s,t)∞n=1

)dtds+ ε

≤ 2(p∗3(w) + p∗4(w))αc(B)

+

(4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (un∞n=1) + ε

≤ 2(p∗3(w) + p∗4(w))αc(B)

+

(4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)αc(B) + ε

≤(

2(p∗3(w) + p∗4(w)) +4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)

)αc(B) + ε

= `αc(B) + ε.

Since ε > 0 is arbitrary, then

αc(N(B)) ≤ `αc(B).

It follows from Lemma 1.5.1 that for each w ∈ Ω, N has at least one fixedpoint in W. Since

⋂w∈Ω intW (w) 6= ∅, and the hypothesis that a measurable

selector of intW exists holds. By Lemma 1.5.1, N has a stochastic fixed point,i.e., the problem (3.4)-(3.6) has at least one random solution on C(a,b).

3.2 Fractional Partial Random Differential Equations with finite Delay 27

3.2.3 An Example

Let E = R, Ω = (−∞, 0) be equipped with the usual σ-algebra consisting ofLebesgue measurable subsets of (−∞, 0). Consider the following partial func-tional random differential equation of the form

(cDrθu)(x, y, w) =

w2e−x−y−3

1 + w2 + 5|u(x, y, w)|; a.a. (x, y) ∈ J = [0, 1]× [0, 1], w ∈ Ω,

(3.9)u(x, y, w) = x sinw + y2 cosw, (x, y) ∈ [−1, 1]× [−2, 1]\(0, 1]× (0, 1], (3.10)

u(x, 0, w) = x sinw; x ∈ [0, 1], u(0, y, w) = y2 cosw; y ∈ [0, 1], w ∈ Ω. (3.11)

where (r1, r2) ∈ (0, 1]× (0, 1]. Set

f(x, y, u(x, y, w), w) =w2

(1 + w2 + 5|u(x− 1, y − 2, w)|)ex+y+3, (x, y) ∈ J, w ∈ Ω.

The functions w 7→ ϕ(x, 0, w) = x sinw , w 7→ ψ(0, y, w) = y2 coswand w 7→ Φ(x, y, w) = x sinw + y2 cosw are measurable and bounded with

|ϕ(x, 0, w)| ≤ 1, |ψ(0, y, w)| ≤ 1,

hence, the conditions (H1) is satisfied.Clearly, the map (x, y, w) 7→ f(x, y, u, w) is jointly continuous for all u ∈ Rand hence jointly measurable for all u ∈ R. Also the map u 7→ f(x, y, u, w) iscontinuous for all (x, y) ∈ J and w ∈ Ω. So the function f is Carathéodory on[0, 1]× [0, 1]× R× Ω.For each u ∈ R, (x, y) ∈ [0, 1]× [0, 1] and w ∈ Ω, we have

|f(x, y, u, w)| ≤ 1 +5

e3|u|.

Hence the conditions (H4) is satisfied with p1(x, y, w) = p∗1 = 1 and p2(x, y, w) =p∗2 = 5

e3.

Also, the conditions (H5) is satisfied.We shall show that condition ` < 1 holds with a = b = 1. For each (r1, r2) ∈

(0, 1]× (0, 1] we get

` =4p∗2a

r1br2

Γ(1 + r1)Γ(1 + r2)

=20

e3Γ(1 + r1)Γ(1 + r2)

<20

e3

< 1.

3.3 Fractional Partial Random Differential Equations with infinite Delay 28

Consequently, Theorem 3.2.1 implies that the problem (3.9) − (3.11) has a ran-dom solution defined on [−1, 1]× [−2, 1].

3.3 Fractional Partial Random Differential Equationswith infinite Delay

3.3.1 Introduction

We study in this section the existence of random solutions for the followingfractional partial random differential equations with infinite delay

(cDr0u)(x, y, w) = f(x, y, u(x,y), w), if (x, y) ∈ J := [0, a]× [0, b], w ∈ Ω, (3.12)

u(x, y, w) = φ(x, y, w), if (x, y) ∈ J := (−∞, a]× (−∞, b]\(0, a]× (0, b], w ∈ Ω,(3.13)

u(x, 0, w) = ϕ(x,w), x ∈ [0, a], u(0, y, w) = ψ(y, w), y ∈ [0, b], w ∈ Ω, (3.14)

where a, b > 0, cDr0 is the standard Caputo’s fractional derivative of order r =

(r1, r2) ∈ (0, 1] × (0, 1], (Ω,A) is a measurable space, f : J × B × Ω → E isa given function, φ : J × Ω → E is a given continuous function, ϕ : [0, a] ×Ω → E,ψ : [0, b] × Ω → E are given absolutely continuous functions withϕ(x,w) = φ(x, 0, w), ψ(y, w) = φ(0, y, w) for each x ∈ [0, a], y ∈ [0, b], w ∈ Ωand B is called a phase space that will be specified later. We denote by u(x,y) theelement of B defined by

u(x,y)(s, t, w) = u(x+ s, y + t, w); (s, t) ∈ (−∞, 0]× (−∞, 0],

here u(x,y)(·, ·, w) represents the history of the state from time −∞ up to thepresent time x and from time −∞ up to the present time y.

3.3.2 Existence Results

Let us assume that the function f is random Carathéeodory on J×B×Ω. Fromthe above Lemma, we have the following Lemma.

Let the space

∆ = u : (−∞, a]×(−∞, b]→ E : u(x,y) ∈ B for (x, y) ∈ E and u|J is continuous.

3.3 Fractional Partial Random Differential Equations with infinite Delay 29

Lemma 3.3.1. Let 0 < r1, r2 ≤ 1, µ(x, y, w) = ϕ(x,w) + ψ(y, w) − ϕ(0, w). Afunction u ∈ Ω × ∆ is a solution of the random problem (3.12)-(3.14) if u satisfiescondition (3.13) for (x, y) ∈ J , w ∈ Ω and u is a solution of the equation

u(x, y, w) = µ(x, y, w) +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(x, y, u(s,t), w)dtds

for (x, y) ∈ J, w ∈ Ω

The following hypotheses will be used in the sequel.

(H1) The functions w 7→ ϕ(x, 0, w) and w 7→ ψ(0, y, w) are measurable andbounded for a.e. x ∈ [0, a] and y ∈ [0, b] respectively,

(H2) The function Φ is measurable for (x, y) ∈ J

(H3) The function f is random Carathéeodory on J × B × Ω,

(H4) There exist functions p1, p2 : J × Ω→ [0,∞) withpi(., w) ∈ AC(J, [0,∞))L∞(J, [0,∞)); i = 1, 2 such that

‖f(x, y, u, w)‖E ≤ p1(x, y, w) + p2(x, y, w)‖u‖B,

for all u ∈ B, w ∈ Ω, and a.e. (x, y) ∈ J,

(H5) For any bounded B ⊂ E,

α(f(x, y, B,w)) ≤ P2(x, y, w)α(B), for a.e. (x, y) ∈ J,

wherep∗i (w) = sup ess(x,y)∈Jpi(x, y, w); i = 1, 2,

Theorem 3.3.1. Assume that hypotheses (H1)− (H6) hold. If

` :=4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)< 1,

then the problem (3.12)-(3.14) has a random solution defined on (−∞, a]× (−∞, b].

Proof.

Define the operator N : Ω×∆→ ∆ by

3.3 Fractional Partial Random Differential Equations with infinite Delay 30

(N(w)u)(x, y) =

φ(x, y, w), (x, y) ∈ J ,w ∈ Ω

µ(x, y, w) + 1Γ(r1)Γ(r2)

×∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×f(s, t, u(s,t), w)dtds, (x, y) ∈ J, w ∈ Ω.

(3.15)

Since the functions ϕ, ψ and f are absolutely continuous , then the functionµ and the indefinite integral are absolutely continuous for all w ∈ Ω and almostall (x, y) ∈ J. Again, as the map µ is continuous for all w ∈ Ω and the indefiniteintegral is continuous on J, then N(w) defines a mapping N : Ω × ∆ → ∆.Hence u is a solution for the problem (3.12)-(3.14) if and only if u = (N(w))u.

Let v(·, ·, ·) : (−∞, a]× (−∞, b]× Ω→ E be a function defined by,

v(x, y, w) =

φ(x, y, w), (x, y) ∈ J ′, w ∈ Ω,µ(x, y, w), (x, y) ∈ J, w ∈ Ω.

Then v(x,y) = φ for all (x, y) ∈ E . For each I continuous on J with I(x, y, w) = 0for each (x, y) ∈ E we denote by I the function defined by

I(x, y, w) =

0, (x, y) ∈ J ′, w ∈ Ω,I(x, y, w) (x, y) ∈ J, w ∈ Ω.

If u(·, ·, ·) satisfies the integral equation,

u(x, y, w) = µ(x, y, w)+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x−s)r1−1(y−t)r2−1f(s, t, u(s,t), w)dtds,

we can decompose u(·, ·, ·) as u(x, y, w) = I(x, y, w) + v(x, y, w); (x, y) ∈ J,which implies u(x,y) = I(x,y) +v(x,y), for every (x, y) ∈ J, and the function I(·, ·, ·)satisfies

I(x, y, w) =1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1f(s, t, I(s,t) + v(s,t), w)dtds.

SetC0 = I ∈ C(J,E) : I(x, y) = 0 for (x, y) ∈ E,

and let ‖.‖(a,b) be the seminorm in C0 defined by

‖I‖(a,b) = sup(x,y)∈E

‖I(x,y)‖B + sup(x,y)∈J

‖I(x, y)‖ = sup(x,y)∈J

‖I(x, y)‖, I ∈ C0.

3.3 Fractional Partial Random Differential Equations with infinite Delay 31

C0 is a Banach space with norm ‖.‖(a,b). Let the operator P : Ω × C0 → C0

be defined by

(P (w)I)(x, y) =1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x−s)r1−1(y− t)r2−1f(s, t, I(s,t) +v(s,t), w)dtds,

(3.16)for each (x, y) ∈ J. Then the operator N has a fixed point is equivalent to P hasa fixed point, and so we turn to proving that P has a fixed point.We shall showthat the operator P satisfies all conditions of Lemma 1.5.1. The proof will begiven in several steps.

Step 1: P (w) is a random operator with stochastic domain on C0.Since f(x, y, u, w) is random Carathéodory, the mapw → f(x, y, u, w) is measur-able in view of Definition 1.1.1. Similarly, the product (x−s)r1−1(y−t)r2−1f(s, t, u(s,t), w)of a continuous and a measurable function is again measurable. Further, the in-tegral is a limit of a finite sum of measurable functions, therefore, the map

w 7→ 1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1f(s, t, I(s,t) + v(s,t), w)dtds,

is measurable. As a result, P is a random operator on Ω× C0 into C0.

Let W : Ω→ P(C0) be defined by

W (w) = I ∈ C0 : ‖I‖(a, b) ≤ R(w),

With R(·) is chosen appropriately. From instance, we assume that

R(w) ≥((k‖φ(0, 0)‖+M‖φ‖)p∗2(w) + p∗1(w)) ar1br2

Γ(1+r1)Γ(1+r2)

1−Kp∗2(w) ar1br2Γ(1+r1)Γ(1+r2)

.

with W (w) bounded, closed, convex and solid for all w ∈ Ω. Then W is mea-surable by Lemma [ [27], Lemma 17]. Let w ∈ Ω be fixed, then from (H4), for

3.3 Fractional Partial Random Differential Equations with infinite Delay 32

any u ∈ w(w), we get

‖(P (w)I)(x, y)‖

≤∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)‖f(s, t, I(s,t) + v(s,t), w)‖dtds

≤ p∗1(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R∗(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

≤ (p∗1(w) + p∗2(w)R∗(w))ar1br2

Γ(1 + r1)Γ(1 + r2)

≤ R(w).

where

‖I(s,t) + v(s,t)‖B ≤ ‖I(s,t)‖B + ‖v(s,t)‖B≤ KR(w) +K‖φ(0, 0)‖+M‖φ‖B := R∗(w).

Therefore, P is a random operator with stochastic domain W and P (w) :W (w) → W (w). Furthermore, P (w) maps bounded sets into bounded sets inC0.

Step 2: P (w) is continuous.Let In be a sequence such that In → u in C0. Then, for each (x, y) ∈ J andw ∈ Ω, we have

‖(P (w)In)(x, y)− (P (w)I)(x, y)‖E ≤1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×‖f(s, t, In(s,t) + vn(s,t), w)− f(s, t, I(s,t) + v(s,t), w)‖Fdtds.

Using the Lebesgue Dominated Convergence Theorem, we get

‖P (w)In − P (w)I‖∞ → 0 as n→∞.

As a consequence of Steps 1 and 2, we can conclude that P (w) : W (w) →W (w) is a continuous random operator with stochastic domainW, and P (w)(W (w))is bounded.

Step 3: For each bounded subset B of W (w) we have

α(P (w)B) ≤ `α(B).

3.3 Fractional Partial Random Differential Equations with infinite Delay 33

Let w ∈ Ω be fixed. From Lemmas 1.4.1 and 1.4.2, for any B ⊂ W and anyε > 0, there exists a sequence In∞n=0 ⊂ B, such that for all (x, y) ∈ J, we have

α(P (w)B)(x, y))

= α

(∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, I(s,t) + v(s,t), w)dtds; I ∈ B

)≤ 2α

(∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, In(s,t) + vn(s,t), w)dtds

∞n=1

)+ ε

≤ 4

∫ x

0

∫ y

0

α

((x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, In(s,t) + vn(s,t), w)

∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)α(f(s, t, In(s,t) + vn(s,t), w)∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α

(In(s,t) + vn(s,t)∞n=1

)dtds+ ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α(un(s,t)∞n=1

)+ ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (un∞n=1) + ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (In∞n=1) + ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)α(B) + ε

≤ 4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)α(B) + ε

= `α(B) + ε.

Since ε > 0 is arbitrary, then

α(P (B)) ≤ `α(B).

It follows from Lemma 1.5.1 that for each w ∈ Ω, P has at least one fixedpoint in W. Since

⋂w∈Ω intW (w) 6= ∅ the hypothesis that a measurable selector

of intW exists holds. By Lemma 1.5.1, N has a stochastic fixed point, i.e., theproblem (3.12)-(3.14) has at least one random solution.

3.3 Fractional Partial Random Differential Equations with infinite Delay 34

3.3.3 An Example

Let E = R, Ω = (−∞, 0) be equipped with the usual σ-algebra consisting ofLebesgue measurable subsets of (−∞, 0). Consider the following partial func-tional random differential equation of the form

(cDrθu)(x, y, w) =

cex+y−γ(x+y)‖u(x,y)‖(ex+y + e−x−y)(1 + w2 + ‖u(x,y))‖

; a.a. (x, y) ∈ J, w ∈ Ω,

(3.17)u(x, y, w) = x sinw + y2 cosw, (x, y) ∈ (−∞, 1]× (−∞, 1]\(0, 1]× (0, 1], w ∈ Ω,

(3.18)u(x, 0, w) = x sinw; x ∈ [0, 1], u(0, y, w) = y2 cosw; y ∈ [0, 1], w ∈ Ω. (3.19)

where J = [0, 1]× [0, 1]c = 8Γ(r1+1)Γ(r2+1)

and γ a positive real constant.Let

Bγ = u ∈ C((−∞, 0]× (−∞, 0],R) : lim‖(θ,η)‖→∞

eγ(θ+η)u(θ, η) exists in R.

The norm of Bγ is given by

‖u‖γ = sup(θ,η)∈(−∞,0]×(−∞,0]

eγ(θ+η)|u(θ, η)|.

LetE := [0, 1]× 0 ∪ 0 × [0, 1],

and u : (−∞, 1]× (−∞, 1]→ R such that u(x,y) ∈ Bγ for (x, y) ∈ E, then

lim‖(θ,η)‖→∞

eγ(θ+η)u(x,y)(θ, η) = lim‖(θ,η)‖→∞

eγ(θ−x+η−y)u(θ, η)

= e−γ(x+y) lim‖(θ,η)‖→∞

eγ(θ+η)u(θ, η) <∞.

Hence u(x,y) ∈ Bγ. Finally we prove that

‖u(x,y)‖γ = K sup|u(s, t)| : (s, t) ∈ [0, x]×[0, y]+M sup‖u(s,t)‖γ : (s, t) ∈ E(x,y),

where K = M = 1 and H = 1.If x+ θ ≤ 0, y + η ≤ 0 we get

‖u(x,y)‖γ = sup|u(s, t)| : (s, t) ∈ (−∞, 0]× (−∞, 0],

and if x+ θ ≥ 0, y + η ≥ 0 then we have

‖u(x,y)‖γ = sup|u(s, t)| : (s, t) ∈ [0, x]× [0, y].

3.3 Fractional Partial Random Differential Equations with infinite Delay 35

Thus for all (x+ θ, y + η) ∈ [0, 1]× [0, 1], we get

‖u(x,y)‖γ = sup|u(s, t)| : (s, t) ∈ (−∞, 0]×(−∞, 0]+sup|u(s, t)| : (s, t) ∈ [0, x]×[0, y].

Then

‖u(x,y)‖γ = sup‖u(s,t)‖γ : (s, t) ∈ E+ sup|u(s, t)| : (s, t) ∈ [0, x]× [0, y].

(Bγ, ‖.‖γ) is a Banach space. We conclude that Bγ is a phase space. Set

f(x, y, u(x,y), w) =cex+y−γ(x+y)‖u(x,y)‖

(ex+y + e−x−y)(1 + w2 + ‖u(x,y))‖, (x, y) ∈ [0, 1]× [0, 1].

The functions w 7→ ϕ(x, 0, w) = x sinw , w 7→ ψ(0, y, w) = y2 coswand w 7→ Φ(x, y, w) = x sinw + y2 cosw are measurable and bounded with

|ϕ(x, 0, w)| ≤ 1, |ψ(0, y, w)| ≤ 1,

hence, the conditions (H1) is satisfied.Clearly, the map (x, y, w) 7→ f(x, y, u, w) is jointly continuous for all u ∈ Bγand hence jointly measurable for all u ∈ Bγ. Also the map u 7→ f(x, y, u, w) iscontinuous for all (x, y) ∈ J and w ∈ Ω. So the function f is Carathéodory on[0, 1]× [0, 1]× Bγ × Ω.For each u ∈ Bγ, (x, y) ∈ [0, 1]× [0, 1] and w ∈ Ω, we have

|f(x, y, u(x,y))| ≤ 1 +1

c‖u‖B.

Hence the conditions (H4) is satisfied with p1(x, y, w) = p∗1 = 1 and p2(x, y, w) =p∗2 = 1

c.

Also, the conditions (H5) is satisfied.We shall show that condition ` < 1 holds with a = b = 1. For each (r1, r2) ∈

(0, 1]× (0, 1] we get

` =4p∗2a

r1br2

Γ(1 + r1)Γ(1 + r2)

=4

cΓ(1 + r1)Γ(1 + r2)

<1

2< 1.

Consequently, Theorem 3.3.1 implies that the problem (3.17)− (3.19) has a ran-dom solution defined on (−∞, 1]× (−∞, 1].

Chapter 4

Fractional Partial RandomDifferential Equations withState-Dependent Delay

4.1 Introduction

In this chapter, we shall be concerned to the existence for the followingfractional partial random differential equations with state-dependent delay

4.2 Fractional Partial Random Differential Equationswith finite Delay

4.2.1 Introduction

In this section, we shall be concerned with the existence of solutions for thefollowing fractional partial random differential equations:

(cDr0u)(x, y, w) = f(x, y, u(ρ1(x,y,u(x,y),w),ρ2(x,y,u(x,y),w)), w), if J := [0, a]×[0, b], w ∈ Ω,

(4.1)u(x, y, w) = φ(x, y, w), if (x, y) ∈ J := [−α, a]× [−β, b]\(0, a]× (0, b], w ∈ Ω,

(4.2)u(x, 0, w) = ϕ(x,w), x ∈ [0, a], u(0, y, w) = ψ(y, w), y ∈ [0, b], w ∈ Ω, (4.3)

where α, β, a, b > 0, cDr0 is the standard Caputo’s fractional derivative of order

r = (r1, r2) ∈ (0, 1] × (0, 1], (Ω,A) is a measurable space, f : J × C × Ω →

36

4.2 Fractional Partial Random Differential Equations with finite Delay 37

E, ρ1, ρ2 : J × C × Ω → E are given functions, φ : J × Ω → E is a givencontinuous function, ϕ : [0, a]× Ω→ E,ψ : [0, b]× Ω→ E are given absolutelycontinuous functions with ϕ(x,w) = φ(x, 0, w), ψ(y, w) = φ(0, y, w) for eachx ∈ [0, a], y ∈ [0, b], w ∈ Ω, (E, ‖ · ‖E) a separable Banach space, and C :=C([−α, 0]× [−β, 0], E) is the space of continuous functions on [−α, 0]× [−β, 0].

If u ∈ C([−α, a] × [−β, b], E); a, b, α, β > 0 then for any (x, y) ∈ J defineu(x,y) by

u(x,y)(s, t, w) = u(x+ s, y + t, w), for (s, t) ∈ [−α, 0]× [−β, 0].

Here u(x,y)(·, ·, w) represents the history of the state u.

4.2.2 Existence Results

Let us assume that the function f is random Carathéeodory on J×C×Ω. Fromthe above Lemma, we have the following Lemma.

Lemma 4.2.1. Let 0 < r1, r2 ≤ 1, µ(x, y, w) = ϕ(x,w) + ψ(y, w) − ϕ(0, w). Afunction u ∈ Ω × C(a,b) is a solution of the random problem (4.1)-(4.3) if u satisfiescondition (4.2) for (x, y) ∈ J , w ∈ Ω and u is a solution of the equation

u(x, y, w) = µ(x, y, w)+

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)dtds

for (x, y) ∈ J, w ∈ Ω

SetR :=R(ρ−1 ,ρ−2 )

= (ρ1(s, t, u, w), ρ2(s, t, u, w)) : (s, t) ∈ J, u(s,t)(·, ·, w) ∈ C,w ∈ Ω, ρi(s, t, u, w) ≤ 0; i = 1, 2.

We always assume that ρi : J × C × Ω → E; i = 1, 2 are continuous and thefunction (s, t) 7−→ u(s,t) is continuous fromR into C.

Let us introduce the following hypotheses which are assumed after.

(H1) The functions w 7→ ϕ(x, 0, w) and w 7→ ψ(0, y, w) are measurable andbounded for a.e. x ∈ [0, a] and y ∈ [0, b] respectively,

(H2) The function Φ is measurable for (x, y) ∈ J

(H3) The function f is random Carathéeodory on J × C × Ω,

4.2 Fractional Partial Random Differential Equations with finite Delay 38

(H4) There exist functions p1, p2 : J × Ω→ [0,∞) withpi(·, w) ∈ AC(J, [0,∞))L∞(J, [0,∞)); i = 1, 2 such that

‖f(x, y, u, w)‖E ≤ p1(x, y, w) + p2(x, y, w)‖u‖C ,

for all u ∈ C, w ∈ Ω and a.e. (x, y) ∈ J,

(H5) For any bounded B ⊂ E,

α(f(x, y, B,w)) ≤ P2(x, y, w)α(B), for a.e. (x, y) ∈ J,

(H6) There exists a random function R : Ω→ (0,∞) such that

µ∗(w) +(p∗1(w) + p∗2(w)R(w))ar1br2

Γ(1 + r1)Γ(1 + r2)≤ R(w),

where

µ∗(w) = sup(x,y)∈J

‖µ(x, y, w)‖E, p∗i (w) = sup ess(x,y)∈Jpi(x, y, w); i = 1, 2,

Theorem 4.2.1. Assume that hypotheses (H1)− (H6) hold. If

` :=4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)< 1,

then the problem (4.1)-(4.3) has a random solution defined on [−α, a]× [−β, b].

Proof. Define the operator N : Ω× C(a,b) → C(a,b) by

(N(w)u)(x, y) =

φ(x, y, w), (x, y) ∈ J ,w ∈ Ω

µ(x, y, w) + 1Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×f(s, t, u(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)),w)dtds, (x, y) ∈ J, w ∈ Ω.(4.4)

Since the functions ϕ, ψ and f are absolutely continuous, the function µ andthe indefinite integral are absolutely continuous for all w ∈ Ω and almost all(x, y) ∈ J. Again, as the map µ is continuous for all w ∈ Ω and the indefiniteintegral is continuous on J, thenN(w) defines a mappingN : Ω×C(a,b) → C(a,b).Hence u is a solution for the problem (4.1)-(4.3) if and only if u = (N(w))u. Weshall show that the operator N satisfies all conditions of Lemma 1.5.1. Theproof will be given in several steps.

4.2 Fractional Partial Random Differential Equations with finite Delay 39

Step 1: N(w) is a random operator with stochastic domain on C(a,b).Since f(x, y, u, w) is random Carathéodory, the map w → f(x, y, u, w) is mea-surable in view of Definition 1.1.1. Similarly, the function (s, t) 7→ (x−s)r1−1(y−t)r2−1f(s, t, u(s,t), w) is measure as the product of a continuous and a measurablefunction. Further, the integral is a limit of a finite sum of measurable functions,therefore, the map

w 7→ µ(x, y, w)+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x−s)r1−1(y−t)r2−1f(s, t, u(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)dtds,

is measurable. As a result, N is a random operator on Ω× C(a,b) into C(a,b).

Let W : Ω→ P(C(a,b)) be defined by

W (w) = u ∈ C(a,b) : ‖u‖∞ ≤ R(w),

with R(·) is chosen appropriately. From instance, we assume that

R(w) ≥µ∗ + p∗1(w) ar1br2

Γ(1+r1)Γ(1+r2)

1− p∗2(w) ar1br2Γ(1+r1)Γ(1+r2)

.

Clearly, W (w) is bounded, closed, convex and solid for all w ∈ Ω. Then W ismeasurable by Lemma 17 of [27]. Let w ∈ Ω be fixed, then from (H4), for anyu ∈ w(w), we get

‖(N(w)u)(x, y)‖E

≤ ‖µ(x, y, w)‖E +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)‖f(s, t, u(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)‖Edtds

≤ ‖µ(x, y, w)‖E +1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p1(s, t, w)dtds

+1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1p2(s, t, w)‖u(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t)))‖Edtds

≤ µ∗(w) +p∗1(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

≤ µ∗(w) +(p∗1(w) + p∗2(w)R(w))ar1br2

Γ(1 + r1)Γ(1 + r2)

≤ R(w).

4.2 Fractional Partial Random Differential Equations with finite Delay 40

Therefore,N is a random operator with stochastic domainW andN(w) : W (w)→N(w). Furthermore, N(w) maps bounded sets into bounded sets in C(a,b).

Step 2: N(w) is continuous.Let un be a sequence such that un → u in C(a,b). Then, for each (x, y) ∈ J andw ∈ Ω, we have

‖(N(w)un)(x, y)− (N(w)u)(x, y)‖E ≤1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×‖f(s, t, un(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)− f(s, t, u(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)‖Edtds.

Using the Lebesgue Dominated Convergence Theorem, we get

‖N(w)un −N(w)u‖∞ → 0 as n→∞.

As a consequence of Steps 1 and 2, we can conclude that N(w) : W (w) →N(w) is a continuous random operator with stochastic domainW, andN(w)(W (w))is bounded.

Step 3: For each bounded subset B of W (w) we have

α(N(w)B) ≤ `α(B).

Let w ∈ Ω be fixed. From Lemmas 1.4.1 and 1.4.2, for any B ⊂ W and any

4.2 Fractional Partial Random Differential Equations with finite Delay 41

ε > 0, there exists a sequence un∞n=0 ⊂ B, such that for all (x, y) ∈ J, we have

α((N(w)B)(x, y))

= α

(µ(x, y) +

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)

×f(s, t, u(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)dtds; u ∈ B)

≤ 2α

(∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)dtds

∞n=1

)+ ε

≤ 4

∫ x

0

∫ y

0

α

((x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)

∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)α(f(s, t, un(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t))), w)∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α

(un(ρ1(s,t,u(s,t)),ρ2(s,t,u(s,t)))

∞n=1

)dtds+ ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (un∞n=1) + ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)α(B) + ε

≤ 4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)α(B) + ε

= `α(B) + ε.

Since ε > 0 is arbitrary, then

α(N(B)) ≤ `α(B).

It follows from Lemma 1.5.1 that for each w ∈ Ω, N has at least one fixedpoint in W. Since

⋂w∈Ω intW (w) 6= ∅, and a measurable selector of intW ex-

ists, Lemma 1.5.1 implies that N has a stochastic fixed point, i.e., the problem(4.1)-(4.3) has at least one random solution on C(a,b).

4.2.3 An Example

Let E = R, Ω = (−∞, 0) be equipped with the usual σ-algebra consisting ofLebesgue measurable subsets of (−∞, 0). Consider the following partial func-

4.2 Fractional Partial Random Differential Equations with finite Delay 42

tional random differential equation of the form

(cDr0u)(x, y) =

|u(x− σ1(u(x, y, w)), y − σ2(u(x, y, w)), w)|+ 2

ex+y+4(1 + w2 + 5|u(x− σ1(u(x, y, w)), y − σ2(u(x, y, w)), w)|), if (x, y) ∈ J,

(4.5)u(x, y, w) = x sinw+y2 cosw, (x, y) ∈ [−1, 1]×[−2, 1]\(0, 1]×(0, 1], w ∈ Ω, (4.6)

u(x, 0, w) = x sinw; x ∈ [0, 1], u(0, y, w) = y2 cosw; y ∈ [0, 1], w ∈ Ω, (4.7)

where σ1 ∈ C(R, [0, 1]), σ2 ∈ C(R, [0, 2]). Set

ρ1(x, y, ϕ, w) = x− σ1(ϕ(0, 0, w)),

ρ2(x, y, ϕ, w) = y − σ2(ϕ(0, 0, w)),

where (x, y) ∈ J = [0, 1]× [0, 1], ϕ(·, ·, w) ∈ C([−1, 0]× [−2, 0],R), w ∈ Ω,

f(x, y, ϕ, w) =|ϕ|+ 2

(ex+y+4)(1 + w2 + 5|ϕ|), (x, y) ∈ J, ϕ ∈ C([−1, 0]× [−2, 0],R).

The functions w 7→ ϕ(x, 0, w) = x sinw , w 7→ ψ(0, y, w) = y2 coswand w 7→ Φ(x, y, w) = x sinw + y2 cosw are measurable and bounded with

|ϕ(x, 0, w)| ≤ 1, |ψ(0, y, w)| ≤ 1,

hence, the conditions (H1) is satisfied.Clearly, the map (x, y, w) 7→ f(x, y, u, w) is jointly continuous for all u ∈ Rand hence jointly measurable for all u ∈ R. Also the map u 7→ f(x, y, u, w) iscontinuous for all (x, y) ∈ J and w ∈ Ω. So the function f is Carathéodory on[0, 1]× [0, 1]× R× Ω.For each ϕ ∈ C([−1, 0]× [−2, 0],R), (x, y) ∈ [0, 1]× [0, 1] and w ∈ Ω, we have

|f(x, y, u, w)| ≤ 1 +5

e3|u|.

Hence the conditions (H4) is satisfied with

p1(x, y, w) = p∗1 = 1, p2(x, y, w) = p∗2 =5

e3.

Also, the conditions (H5) is satisfied.

4.3 Fractional Partial Random Differential Equations with infinite Delay 43

We shall show that condition ` < 1 holds with a = b = 1. For each (r1, r2) ∈(0, 1]× (0, 1] we get

` =4p∗2a

r1br2

Γ(1 + r1)Γ(1 + r2)

=20

e3Γ(1 + r1)Γ(1 + r2)

<20

e3

< 1.

Consequently, Theorem 4.2.1 implies that the problem (4.5) − (4.7) has a ran-dom solution defined on [−1, 1]× [−2, 1].

4.3 Fractional Partial Random Differential Equationswith infinite Delay

4.3.1 Introduction

In this section, we shall be concerned with the existence of solutions for thefollowing fractional partial random differential equations:

(cDr0u)(x, y, w) = f(x, y, u(ρ1(x,y,u(x,y),w),ρ2(x,y,u(x,y),w))), if (x, y) ∈ J, w ∈ Ω, (4.8)

u(x, y, w) = φ(x, y, w), if (x, y) ∈ J := (−∞, a]× (−∞, b]\(0, a]× (0, b], w ∈ Ω,(4.9)

u(x, 0, w) = ϕ(x,w), x ∈ [0, a], u(0, y, w) = ψ(y, w), y ∈ [0, b], w ∈ Ω, (4.10)

where α, β, a, b > 0, J := [0, a]× [0, b] cDr0 is the standard Caputo’s fractional

derivative of order r = (r1, r2) ∈ (0, 1] × (0, 1], (Ω,A) is a measurable space,f : J × B × Ω→ E, ρ1, ρ2 : J × B × Ω→ E are given functions, φ : J × Ω→ Eis a given continuous function, ϕ : [0, a] × Ω → E,ψ : [0, b] × Ω → E are givenabsolutely continuous functions with ϕ(x,w) = φ(x, 0, w), ψ(y, w) = φ(0, y, w)for each x ∈ [0, a], y ∈ [0, b], w ∈ Ωand B is called a phase space that will bespecified later. We denote by u(x,y) the element of B defined by

u(x,y)(s, t, w) = u(x+ s, y + t, w); (s, t) ∈ (−∞, 0]× (−∞, 0],

here u(x,y)(·, ·, w) represents the history of the state from time −∞ up to thepresent time x and from time −∞ up to the present time y.

4.3 Fractional Partial Random Differential Equations with infinite Delay 44

4.3.2 Existence Results

Let us assume that the function f is random Carathéeodory on J × B × Ω.Let the space

∆ = u : (−∞, a]×(−∞, b]→ E : u(x,y) ∈ B for (x, y) ∈ E and u|J is continuous.

From the above Lemma, we have.

Lemma 4.3.1. Let 0 < r1, r2 ≤ 1, µ(x, y, w) = ϕ(x,w) + ψ(y, w) − ϕ(0, w). Afunction u ∈ Ω × ∆ is a solution of the random problem (4.8)-(4.10) if u satisfiescondition (4.9) for (x, y) ∈ J , w ∈ Ω, and the integral equation

u(x, y, w) = µ(x, y, w)+

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w), w)dtds

for (x, y) ∈ J, w ∈ Ω

SetR′ :=R′(ρ−1 ,ρ−2 )

= (ρ1(s, t, u, w), ρ2(s, t, u, w)) : (s, t, u, w) ∈ J×B×Ω, ρi(s, t, u, w) ≤ 0; i = 1, 2.

We always assume that ρi : J × B × Ω → E; i = 1, 2 are continuous and thefunction (s, t) 7−→ u(s,t) is continuous fromR into B.

We will need to introduce the following hypothesis:

(Cφ) There exists a continuous bounded function L :R′(ρ−1 ,ρ−2 ) → (0,∞) suchthat

‖φ(s,t)‖B ≤ L(s, t)‖φ‖B, for any(s, t) ∈ R′.

In the sequel we will make use of the following generalization of a consequenceof the phase space axioms ( [36], Lemma 2.1).

Lemma 4.3.2. If u ∈ Ω, then

‖u(s,t)‖B = (M + L′)‖φ‖B +K sup(θ,η)∈[0,max0,s]×[0,max0,t]

‖u(θ, η)‖,

whereL′ = sup

(s,t)∈R′L(s, t).

The following hypotheses will be used in the sequel.

4.3 Fractional Partial Random Differential Equations with infinite Delay 45

(H1) The functions w 7→ ϕ(x, 0, w) and w 7→ ψ(0, y, w) are measurable andbounded for a.e. x ∈ [0, a] and y ∈ [0, b] respectively,

(H2) The function Φ is measurable for (x, y) ∈ J

(H3) The function f is random Carathéeodory on J × B × Ω,

(H4) There exist functions p1, p2 : J × Ω→ [0,∞) withpi(·, w) ∈ AC(J, [0,∞))L∞(J, [0,∞)); i = 1, 2 such that

‖f(x, y, u, w)‖E ≤ p1(x, y, w) + p2(x, y, w)‖u‖B,

for all u ∈ B, w ∈ Ω and a.e. (x, y) ∈ J,

(H5) For any bounded B ⊂ E,

α(f(x, y, B,w)) ≤ P2(x, y, w)α(B), for a.e. (x, y) ∈ J,

wherep∗i (w) = sup ess(x,y)∈Jpi(x, y, w); i = 1, 2,

Theorem 4.3.1. Assume that hypotheses (H1)− (H5) and (Cφ) hold. If

` :=4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)< 1,

then the problem (4.8)-(4.10) has a random solution defined on (−∞, a]× (−∞, b].

(N(w)u)(x, y) =

φ(x, y, w), (x, y) ∈ J ,w ∈ Ω

µ(x, y, w) + 1Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×f(s, t, u(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)dtds, (x, y) ∈ J, w ∈ Ω.(4.11)

Since the functions ϕ, ψ and f are absolutely continuous , then the functionµ and the indefinite integral are absolutely continuous for all w ∈ Ω and almostall (x, y) ∈ J. Again, as the map µ is continuous for all w ∈ Ω and the indefiniteintegral is continuous on J, then N(w) defines a mapping N : Ω × ∆ → ∆.Hence u is a solution for the problem (4.8)-(4.10) if and only if u = (N(w))u.

Let v(·, ·, ·) : (−∞, a]× (−∞, b]× Ω→ E be a function defined by,

v(x, y, w) =

φ(x, y, w), (x, y) ∈ J ′, w ∈ Ω,µ(x, y, w), (x, y) ∈ J, w ∈ Ω.

4.3 Fractional Partial Random Differential Equations with infinite Delay 46

Then v(x,y) = φ for all (x, y) ∈ E . For each I continuous on J with I(x, y, w) = 0for each (x, y) ∈ E we denote by I the function defined by

I(x, y, w) =

0, (x, y) ∈ J ′, w ∈ Ω,I(x, y, w) (x, y) ∈ J, w ∈ Ω.

If u(·, ·, ·) satisfies the integral equation,

u(x, y, w) = µ(x, y, w) +1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

× f(s, t, u(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)dtds,

we can decompose u(·, ·, ·) as u(x, y, w) = I(x, y, w) + v(x, y, w); (x, y) ∈ J,which implies u(x,y) = I(x,y) +v(x,y), for every (x, y) ∈ J, and the function I(·, ·, ·)satisfies

I(x, y, w) =1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×f(s, t, I(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + v(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)dtds.

SetC0 = I ∈ C(J,E) : I(x, y) = 0 for (x, y) ∈ E,

and let ‖.‖(a,b) be the seminorm in C0 defined by

‖I‖(a,b) = sup(x,y)∈E

‖I(x,y)‖B + sup(x,y)∈J

‖I(x, y)‖ = sup(x,y)∈J

‖I(x, y)‖, I ∈ C0.

C0 is a Banach space with norm ‖.‖(a,b). Let the operator P : Ω × C0 → C0

be defined by

(Pw)(x, y) =1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×f(s, t, I(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + v(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)dtds. (4.12)

for each (x, y) ∈ J. Then the operator N has a fixed point is equivalent to P hasa fixed point, and so we turn to proving that P has a fixed point.We shall showthat the operator P satisfies all conditions of Lemma 1.5.1. The proof will begiven in several steps.

Step 1: P (w) is a random operator with stochastic domain on C0.Since f(x, y, u, w) is random Carathéodory, the map w → f(x, y, u, w) is mea-surable in view of Definition 1.1.1.

4.3 Fractional Partial Random Differential Equations with infinite Delay 47

Similarly, the product (x− s)r1−1(y− t)r2−1f(s, t, u(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w) ofa continuous and a measurable function is again measurable. Further, the inte-gral is a limit of a finite sum of measurable functions, therefore, the map

w 7→ 1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×f(s, t, I(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + v(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)dtds,

is measurable. As a result, P is a random operator on Ω× C0 into C0.

Let W : Ω→ P(C0) be defined by

W (w) = I ∈ C0 : ‖I‖(a, b) ≤ R(w),

With R(·) is chosen appropriately. From instance, we assume that

R(w) ≥((k‖φ(0, 0)‖+ (M + L′)‖φ‖B)p∗2(w) + p∗1(w)) ar1br2

Γ(1+r1)Γ(1+r2)

1−Kp∗2(w) ar1br2Γ(1+r1)Γ(1+r2)

.

with W (w) bounded, closed, convex and solid for all w ∈ Ω. Then W is mea-surable by Lemma [ [27], Lemma 17]. Let w ∈ Ω be fixed, then from (H4), forany u ∈ w(w), we get

‖(P (w)I)(x, y)‖ ≤∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)

×‖f(s, t, I(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + v(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)‖dtds

≤ p∗1(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R∗(w)

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

≤ (p∗1(w) + p∗2(w)R∗(w))ar1br2

Γ(1 + r1)Γ(1 + r2)

≤ R(w).

where

‖I(s,t) + v(s,t)‖B ≤ ‖I(s,t)‖B + ‖v(s,t)‖B≤ KR(w) +K‖φ(0, 0)‖+ (M + L′)‖φ‖B := R∗(w).

4.3 Fractional Partial Random Differential Equations with infinite Delay 48

Therefore, P is a random operator with stochastic domain W and P (w) :W (w) → W (w). Furthermore, P (w) maps bounded sets into bounded sets inC0.

Step 2: P (w) is continuous.Let In be a sequence such that In → I in C0. Then, for each (x, y) ∈ J andw ∈ Ω, we have

‖(P (w)In)(x, y)− (P (w)I)(x, y)‖E

≤ 1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×‖f(s, t, In(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + vn(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)

−f(s, t, I(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + v(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)‖Edtds

≤ 1

Γ(r1)Γ(r2)

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

×‖f(s, t, In(s,t) + vn(s,t), w)− f(s, t, I(s,t) + v(s,t), w)‖Edtds.

Using the Lebesgue Dominated Convergence Theorem, we get

‖P (w)In − P (w)I‖∞ → 0 as n→∞.

As a consequence of Steps 1 and 2, we can conclude that P (w) : W (w) →W (w) is a continuous random operator with stochastic domainW, and P (w)(W (w))is bounded.

Step 3: For each bounded subset B of W (w) we have

α(P (w)B) ≤ `α(B).

Let w ∈ Ω be fixed. From Lemmas 1.4.1 and 1.4.2, for any B ⊂ W and any

4.3 Fractional Partial Random Differential Equations with infinite Delay 49

ε > 0, there exists a sequence In∞n=0 ⊂ B, such that for all (x, y) ∈ J, we have

α(P (w)B)(x, y))

= α

(∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)

×f(s, t, I(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + v(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)dtds; I ∈ B)

≤ 2α

(∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)

×f(s, t, In(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)) + vn(ρ1(s,t,u(s,t),w),ρ2(s,t,u(s,t),w)), w)dtds

∞n=1

)+ ε

≤ 4

∫ x

0

∫ y

0

α

((x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, In(s,t) + vn(s,t), w)

∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)α(f(s, t, In(s,t) + vn(s,t), w)∞n=1

)dtds+ ε

≤ 4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α

(In(s,t) + vn(s,t)∞n=1

)dtds+ ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α(un(s,t)∞n=1

)+ ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (un∞n=1) + ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (In∞n=1) + ε

≤(

4

∫ x

0

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)α(B) + ε

≤ 4p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)α(B) + ε

= `α(B) + ε.

Since ε > 0 is arbitrary, then

α(P (B)) ≤ `α(B).

It follows from Lemma 1.5.1 that for each w ∈ Ω, P has at least one fixedpoint in W. Since

⋂w∈Ω intW (w) 6= ∅ the hypothesis that a measurable selector

of intW exists holds. By Lemma 1.5.1, N has a stochastic fixed point, i.e., theproblem (4.8)-(4.10) has at least one random solution on.

4.3 Fractional Partial Random Differential Equations with infinite Delay 50

4.3.3 An Example

Let E = R, Ω = (−∞, 0) be equipped with the usual σ-algebra consisting ofLebesgue measurable subsets of (−∞, 0). Consider the following partial func-tional random differential equation of the form

(cDr0u)(x, y) =

cex+y−γ(x+y)

ex+y + e−x−y

× |u(x− σ1(u(x, y)), y − σ2(u(x, y)))|1 + w2 + |u(x− σ1(u(x, y)), y − σ2(u(x, y)))|

; a.a.(x, y) ∈ [0, 1]× [0, 1], w ∈ Ω,

(4.13)u(x, y, w) = x sinw + y2 cosw, (x, y) ∈ (−∞, 1]× (−∞, 1]\(0, 1]× (0, 1], w ∈ Ω,

(4.14)u(x, 0, w) = x sinw; x ∈ [0, 1], u(0, y, w) = y2 cosw; y ∈ [0, 1], w ∈ Ω. (4.15)

where c = 8Γ(r1+1)Γ(r2+1)

and γ a positive real constant.Let

Bγ = u ∈ C((−∞, 0]× (−∞, 0],R) : lim‖(θ,η)‖→∞

eγ(θ+η)u(θ, η) exists in R.

The norm of Bγ is given by

‖u‖γ = sup(θ,η)∈(−∞,0]×(−∞,0]

eγ(θ+η)|u(θ, η)|.

LetE := [0, 1]× 0 ∪ 0 × [0, 1],

and u : (−∞, 1]× (−∞, 1]→ R such that u(x,y) ∈ Bγ for (x, y) ∈ E , then

lim‖(θ,η)‖→∞

eγ(θ+η)u(x,y)(θ, η) = lim‖(θ,η)‖→∞

eγ(θ−x+η−y)u(θ, η)

= e−γ(x+y) lim‖(θ,η)‖→∞

eγ(θ+η)u(θ, η) <∞.

Hence u(x,y) ∈ Bγ. Finally we prove that

‖u(x,y)‖γ = K sup|u(s, t)| : (s, t) ∈ [0, x]×[0, y]+M sup‖u(s,t)‖γ : (s, t) ∈ E(x,y),

where K = M = 1 and H = 1.If x+ θ ≤ 0, y + η ≤ 0 we get

‖u(x,y)‖γ = sup|u(s, t)| : (s, t) ∈ (−∞, 0]× (−∞, 0],

4.3 Fractional Partial Random Differential Equations with infinite Delay 51

and if x+ θ ≥ 0, y + η ≥ 0 then we have

‖u(x,y)‖γ = sup|u(s, t)| : (s, t) ∈ [0, x]× [0, y].

Thus for all (x+ θ, y + η) ∈ [0, 1]× [0, 1], we get

‖u(x,y)‖γ = sup|u(s, t)| : (s, t) ∈ (−∞, 0]×(−∞, 0]+sup|u(s, t)| : (s, t) ∈ [0, x]×[0, y].

Then

‖u(x,y)‖γ = sup‖u(s,t)‖γ : (s, t) ∈ E+ sup|u(s, t)| : (s, t) ∈ [0, x]× [0, y].

(Bγ, ‖.‖γ) is a Banach space. We conclude that Bγ is a phase space.Set

ρ1(x, y, ϕ, w) = x− σ1(ϕ(0, 0, w)),

ρ2(x, y, ϕ, w) = y − σ2(ϕ(0, 0, w)),

where (x, y) ∈ J, ϕ(·, ·, w) ∈ Bγ, w ∈ Ω,

f(x, y, ϕ, w) =cex+y−γ(x+y)|ϕ|

(ex+y + e−x−y)(1 + w2 + |ϕ|), (x, y) ∈ [0, 1]×[0, 1], ϕ ∈ Bγ, w ∈ Ω.

The functions w 7→ ϕ(x, 0, w) = x sinw , w 7→ ψ(0, y, w) = y2 coswand w 7→ Φ(x, y, w) = x sinw + y2 cosw are measurable and bounded with

|ϕ(x, 0, w)| ≤ 1, |ψ(0, y, w)| ≤ 1,

hence, the conditions (H1) is satisfied.Clearly, the map (x, y, w) 7→ f(x, y, u, w) is jointly continuous for all u ∈ Bγand hence jointly measurable for all u ∈ Bγ. Also the map u 7→ f(x, y, u, w) iscontinuous for all (x, y) ∈ J and w ∈ Ω. So the function f is Carathéodory on[0, 1]× [0, 1]× Bγ × Ω.For each u ∈ Bγ, (x, y) ∈ [0, 1]× [0, 1] and w ∈ Ω, we have

|f(x, y, u(x,y))| ≤ 1 +1

c‖u‖B.

Hence the conditions (H4) is satisfied with p1(x, y, w) = p∗1 = 1 and p2(x, y, w) =p∗2 = 1

c.

Also, the conditions (H5) is satisfied.

4.3 Fractional Partial Random Differential Equations with infinite Delay 52

We shall show that condition ` < 1 holds with a = b = 1. For each (r1, r2) ∈(0, 1]× (0, 1] we get

` =4p∗2a

r1br2

Γ(1 + r1)Γ(1 + r2)

=4

cΓ(1 + r1)Γ(1 + r2)

<1

2< 1.

Consequently, Theorem 4.3.1 implies that the problem (4.13)− (4.15) has a ran-dom solution defined on (−∞, 1]× (−∞, 1].

Chapter 5

Random Impulsive PartialHyperbolic Fractional DifferentialEquations

5.1 Introduction

In this chapter, we discuss the existence of random solutions for the followingimpulsive partial fractional random differential equations:

cDrxku(x, y, w) = f(x, y, u(x, y, w), w); if (x, y) ∈ Jk, k = 0, . . . ,m,w ∈ Ω,

u(x+k , y, w) = u(x−k , y, w) + Ik(u(x−k , y, w)); if y ∈ [0, b], k = 1, . . . ,m,w ∈ Ω,

u(x, 0, w) = ϕ(x,w); x ∈ [0, a], w ∈ Ω,

u(0, y, w) = ψ(y, w); y ∈ [0, b], w ∈ Ω,

ϕ(0, w) = ψ(0, w),(5.1)

where J0 = [0, x1] × [0, b], Jk := (xk, xk+1] × [0, b]; k = 1, . . . ,m, a, b > 0, θk =(xk, 0); k = 0, . . . ,m, cDr

xkis the fractional Caputo derivative of order r =

(r1, r2) ∈ (0, 1] × (0, 1], 0 = x0 < x1 < · · · < xm < xm+1 = a, (Ω,A) is ameasurable space, f : J × E × Ω → E; Ik : E → E; k = 1, . . . ,m are givencontinuous functions, ϕ : [0, a] × Ω → E and ψ : [0, b] × Ω → E are givenabsolutely continuous functions. Here u(x+

k , y, w) and u(x−k , y, w) denote theright and left limits of u(x, y, w) at x = xk, respectively.

53

5.2 Existence Results 54

5.2 Existence Results

Let the space

PC =u : J → E : u ∈ C(Jk); k = 0, 1, . . . ,m, and there

exist u(x−k , y) and u(x+k , y); k = 1, . . . ,m,

with u(x−k , y) = u(xk, y) for each y ∈ [0, b].

We need the following auxiliary lemma.

Lemma 5.2.1. [8] Let 0 < r1, r2 ≤ 1, µ(x, y) = ϕ(x) + ψ(y) − ϕ(0) and let f :J × E → E be continuous. A function u ∈ PC(J) is a solution of the fractionalintegral equation

u(x, y) =

µ(x, y) + 1Γ(r1)Γ(r2)

∫ x0

∫ y0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t))dtds;

if (x, y) ∈ [0, x1]× [0, b],

µ(x, y) +∑k

i=1(Ii(u(x−i , y))− Ii(u(x−i , 0)))

+ 1Γ(r1)Γ(r2)

∑ki=1

∫ xixi−1

∫ y0

(xi − s)r1−1(y − t)r2−1f(s, t, u(s, t))dtds

+ 1Γ(r1)Γ(r2)

∫ xxk

∫ y0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t))dtds;

if (x, y) ∈ (xk, xk+1]× [0, b], k = 1, . . . ,m,

if and only if u is a solution of the problem

cDrxku(x, y) = f(x, y, u(x, y)); if (x, y) ∈ Jk, k = 0, . . . ,m,

u(x+k , y) = u(x−k , y) + Ik(u(x−k , y)); if y ∈ [0, b], k = 1, . . . ,m,

u(x, 0) = ϕ(x); x ∈ [0, a],

u(0, y) = ψ(y); y ∈ [0, b],

ϕ(0) = ψ(0).

Consider the space

PC = PC(J × Ω) =u : J × Ω→ E : u(·, ·, w) is continuous on Jk;k = 0, 1, . . . ,m, and there exist u(x−k , y, w)

and u(x+k , y, w); k = 1, . . . ,m, with

u(x−k , y, w) = u(xk, y, w) for each y ∈ [0, b], w ∈ Ω.

This set is a Banach space with the norm

‖u‖PC = sup(x,y)∈J

‖u(x, y, w)‖E.

we have the following lemma.

5.2 Existence Results 55

Lemma 5.2.2. Let 0 < r1, r2 ≤ 1, µ(x, y, w) = ϕ(x,w) + ψ(y, w) − ϕ(0, w). Afunction u ∈ PC is a solution of the random fractional integral equation

u(x, y, w) =

µ(x, y, w) + 1Γ(r1)Γ(r2)

∫ x0

∫ y0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds;

if (x, y) ∈ [0, x1]× [0, b], w ∈ Ω,

µ(x, y, w) +∑k

i=1(Ii(u(x−i , y, w))− Ii(u(x−i , 0, w)))

+ 1Γ(r1)Γ(r2)

∑ki=1

∫ xixi−1

∫ y0

(xi − s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds

+ 1Γ(r1)Γ(r2)

∫ xxk

∫ y0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds;

if (x, y) ∈ (xk, xk+1]× [0, b], k = 1, . . . ,m, w ∈ Ω,(5.2)

if and only if u is a solution of the random problem (5.1).

The following hypotheses will be used in the sequel.

(H1) The functions w 7→ ϕ(x, 0, w) and w 7→ ψ(0, y, w) are measurable andbounded for almost each x ∈ [0, a] and y ∈ [0, b] respectively,

(H2) The function f is random Carathéeodory on J × E × Ω,

(H3) There exist functions p1, p2, p3 : J × Ω→ [0,∞) withpi(·, w) ∈ L∞(J, [0,∞)); i = 1, 2, 3 such that

‖f(x, y, u, w)‖E ≤ p1(x, y, w) + p2(x, y, w)‖u‖E,

and‖Ik(u)‖E ≤ p3(x, y, w)‖u‖E,

for all u ∈ E, w ∈ Ω and almost each (x, y) ∈ J,

(H4) For any bounded B ⊂ E,

α(f(x, y, B,w)) ≤ p2(x, y, w)α(B), for almost each (x, y) ∈ J,

andα(Ik(B)) ≤ p3(x, y, w)α(B), for almost each (x, y) ∈ J.

Setµ∗(w) = sup

(x,y)∈J‖µ(x, y, w)‖E,

p∗i (w) = sup ess(x,y)∈Jpi(x, y, w); i = 1, 2, 3.

5.2 Existence Results 56

Remark 5.2.1. Conditions (H3) and (H4) are equivalent [12].

Theorem 5.2.1. Assume that hypotheses (H1)− (H3) hold. If

` := 2mp∗3(w) +4(m+ 1)p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)< 1,

then the problem (5.1) has a random solution defined on J.

Proof. By Lemma 5.2.2, the problem (5.1) is equivalent to the integral equa-tion

u(x, y, w) =

µ(x, y, w) + 1Γ(r1)Γ(r2)

∫ x0

∫ y0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds;

if (x, y) ∈ [0, x1]× [0, b], w ∈ Ω,

µ(x, y, w) +∑k

i=1(Ii(u(x−i , y, w))− Ii(u(x−i , 0, w)))

+ 1Γ(r1)Γ(r2)

∑ki=1

∫ xixi−1

∫ y0

(xi − s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds

+ 1Γ(r1)Γ(r2)

∫ xxk

∫ y0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds;

if (x, y) ∈ (xk, xk+1]× [0, b], k = 1, . . . ,m, w ∈ Ω,

for each w ∈ Ω and almost each (x, y) ∈ J.

Define the operator N : PC → PC by

(Nu)(x, y) = µ(x, y, w) +k∑i=1

(Ii(u(x−i , y, w))− Ii(u(x−i , 0, w)))

+1

Γ(r1)Γ(r2)

k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds

+1

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds.

Since the functions ϕ, ψ and Ik and f are absolutely continuous, the function µand the indefinite integral are absolutely continuous for all w ∈ Ω and almostall (x, y) ∈ J. Again, as the maps µ and Ik are continuous for all w ∈ Ω and theindefinite integral is continuous on J, then N(w) defines a mapping N : PC →PC. Hence u is a solution for the problem (5.1) if and only if u = Nu. We shallshow that the operatorN satisfies all conditions of Lemma 1.5.1. The proof willbe given in several steps.

Step 1: N is a random operator with stochastic domain on PC.Since f(x, y, u, w) is random Carathéodory, the map w → f(x, y, u, w) is mea-surable in view of Definition 1.1.1.

5.2 Existence Results 57

Similarly, the product (x − s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w) of a continuousand a measurable function is again measurable. Further, the integral is a limitof a finite sum of measurable functions and Ik is measurable. Therefore, themap

w 7→ µ(x, y, w) +k∑i=1

(Ii(u(x−i , y, w))− Ii(u(x−i , 0, w)))

+1

Γ(r1)Γ(r2)

k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds

+1

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1f(s, t, u(s, t, w), w)dtds

is measurable. As a result, N is a random operator from PC into PC.

Let W : Ω→ P(PC) be defined by

W (w) = u ∈ PC : ‖u‖PC ≤ R(w),

with R(·) being chosen appropriately. For instance, we assume that

R(w) ≥µ∗ +

(m+1)p∗1(w)ar1br2

Γ(1+r1)Γ(1+r2)

1− 2mp∗3(w)− (m+ 1)p∗2(w) ar1br2Γ(1+r1)Γ(1+r2)

.

The set W (w) is bounded, closed, convex and solid for all w ∈ Ω. Then W ismeasurable (Lemma 17 ( [27]). Let w ∈ Ω be fixed, then from (H4), for any

5.2 Existence Results 58

u ∈ w(w), we get

‖(Nu)(x, y)‖E

≤ ‖µ(x, y, w)‖E +k∑i=1

‖Ii(u(x−i , y, w))‖+ ‖Ii(u(x−i , 0, w))‖

+ +1

Γ(r1)Γ(r2)

k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1‖f(s, t, u(s, t, w), w)‖Edtds

+1

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1‖f(s, t, u(s, t, w), w)‖Edtds,

≤ ‖µ(x, y, w)‖E +k∑i=1

(p3(x, y, w)‖u‖+ (p3(xi, 0, w))‖u‖)

+1

Γ(r1)Γ(r2)

k∑i=1

(∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1p1(s, t, w)dtds

+

∫ xi

xi−1

∫ y

0

(x− s)r1−1(y − t)r2−1p2(s, t, w)‖u(s, t, w)‖Edtds)

+1

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1p1(s, t, w)dtds

+1

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1p2(s, t, w)‖u(s, t, w)‖Edtds

≤ µ∗(w) + 2mp∗3(w)R(w)

+k∑i=1

(p∗1(w)

Γ(r1)Γ(r2)

∫ xi

xi−1

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R(w)

Γ(r1)Γ(r2)

∫ xi

xi−1

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

)+

p∗1(w)

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

+p∗2(w)R(w)

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1dtds

≤ µ∗(w) + 2mp∗3(w)R(w) +(p∗1(w) + p∗2(w)R(w))(m+ 1)ar1br2

Γ(1 + r1)Γ(1 + r2)

≤ R(w).

Therefore, N is a random operator with stochastic domain W and N : W (w)→

5.2 Existence Results 59

W (w). Furthermore, N maps bounded sets into bounded sets in PC.Step 2: N is continuous.Let un be a sequence such that un → u in PC. Then, for each (x, y) ∈ J andw ∈ Ω, we have

‖(Nun)(x, y)− (N(w)u)(x, y)‖E

≤k∑i=1

(‖Ii(un(x−i , y, w))− Ii(u(x−i , y, w))‖+ ‖Ii(un(x−i , 0, w))− Ii(u(x−i , 0, w))‖)

+1

Γ(r1)Γ(r2)

k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1

× ‖f(s, t, un(s, t, w), w)− f(s, t, u(s, t, w), w)‖Edtds

+1

Γ(r1)Γ(r2)

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1

× ‖f(s, t, un(s, t, w), w)− f(s, t, u(s, t, w), w)‖Edtds.

Using the Lebesgue dominated convergence theorem, we get

‖Nun −Nu‖∞ → 0 as n→∞.

As a consequence of Steps 1 and 2, we can conclude that N : W (w)→ W (w)is a continuous random operator with stochastic domain W, and N(W (w)) isbounded.Step 3: For each bounded subset B of W (w) we have

α(NB) ≤ `α(B).

Let w ∈ Ω be fixed. From Lemmas 1.4.1 and 1.4.2, for any B ⊂ W and anyε > 0, there exists a sequence un∞n=0 ⊂ B, such that for all (x, y) ∈ J, we have

5.2 Existence Results 60

α((NB)(x, y))

= α

µ(x, y, w) +

k∑i=1

(Ii(u(x−i , y, w))− Ii(u(x−i , 0, w)))

+k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(s, t, w), w)dtds

+

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, u(s, t, w), w)dtds; u ∈ B

≤ α

k∑i=1

(Ii(un(x−i , y, w))− Ii(un(x−i , 0, w)))

+k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s, t, w), w)dtds

+

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)f(s, t, un(s, t, w), w)dtds

∞n=1

+ ε

≤ α

k∑i=1

(Ii(un(x−i , y, w))− Ii(un(x−i , 0, w)))

∞n=1

+2k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)αf(s, t, un(s, t, w), w)∞n=1dtds

+2

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)αf(s, t, un(s, t, w), w)∞n=1dtds+ ε

≤ 2mp3(x, y, w)α (un(s, t, w)∞n=1)

+4k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α (un(s, t, w)∞n=1) dtds

+4

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)α (un(s, t, w)∞n=1) dtds+ ε

≤ 2mp3(x, y, w)α (un∞n=1)

+

(4

k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)

)α (un∞n=1) dtds

+

(4

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dsdt

)α (un∞n=1) + ε

5.3 An Example 61

≤ 2mp3(x, y, w)α(B)

+

(4

k∑i=1

∫ xi

xi−1

∫ y

0

(xi − s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)α(B)

+

(4

∫ x

xk

∫ y

0

(x− s)r1−1(y − t)r2−1

Γ(r1)Γ(r2)p2(s, t, w)dtds

)α(B) + ε

≤(

2mp∗3(w) +4(m+ 1)p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)

)α(B) + ε

= `α(B) + ε.

Since ε > 0 is arbitrary, we haven

α(N(B)) ≤ `α(B).

It follows from Lemma 1.5.1 that for each w ∈ Ω, N has at least one fixedpoint inW. Since

⋂w∈Ω intW (w) 6= ∅, there exists a measurable selector of intW ,

thus N has a stochastic fixed point, i.e., the problem (5.1) has at least one ran-dom solution.

5.3 An Example

Let E = R, Ω = (−∞, 0) be equipped with the usual σ-algebra consisting ofLebesgue measurable subsets of (−∞, 0). Given a measurable function u : Ω→AC([0, 1] × [0, 1]), consider the following impulsive partial fractional randomdifferential equations of the formcDr

xku(x, y, w) = w2e−x−y−3

1+w2+5|u(x,y,w)| ; if (x, y) ∈ Jk, k = 0, . . . ,m,

u(x+k , y, w) = u(x−k , y, w) + w2

(1+w2+10|u(x,y,w)|)ex+y+10 ; if y ∈ [0, 1], k = 1, . . . ,m,

(5.3)where J = [0, 1]× [0, 1], w ∈ Ω, (r1, r2) ∈ (0, 1]× (0, 1] with the initial conditions

u(x, 0, w) = x sinw; x ∈ [0, 1],

u(0, y, w) = y2 cosw; y ∈ [0, 1].w ∈ Ω, (5.4)

Set

f(x, y, u(x, y, w), w) =w2

(1 + w2 + 5|u(x, y, w)|)ex+y+10, (x, y) ∈ [0, 1]×[0, 1], w ∈ Ω,

5.3 An Example 62

and

Ik(u(x−k , y, w)) =w2

(1 + w2 + 10|u(x, y, w)|)ex+y+10, y ∈ [0, 1], k = 1, . . . ,m, w ∈ Ω.

The functions w 7→ ϕ(x, 0, w) = x sinw and w 7→ ψ(0, y, w) = y2 cosw are mea-surable and bounded with

|ϕ(x, 0, w)| ≤ 1, |ψ(0, y, w)| ≤ 1,

hence, the condition (H1) is satisfied.Clearly, the map (x, y, w) 7→ f(x, y, u, w) is jointly continuous for all u ∈ Rand hence jointly measurable for all u ∈ R. Also the map u 7→ f(x, y, u, w) iscontinuous for all (x, y) ∈ J and w ∈ Ω. So the function f is Carathéodory on[0, 1]× [0, 1]× R× Ω.For each u ∈ R, (x, y) ∈ [0, 1]× [0, 1] and w ∈ Ω, we have

|f(x, y, u, w)| ≤ 1 +5

e10|u|,

and|Ik(u)| ≤ 10

e10|u|.

Hence the condition (H4) is satisfied with

p1(x, y, w) = p∗1(w) = 1, p2(x, y, w) = p∗2(w) =5

e10, p3(x, y, w) = p∗3(w) =

10

e10.

We shall show that condition ` < 1 holds with a = b = 1. Indeed, if we assume,for instance, that the number of impulses m = 3, then we have

` = 2mp∗3(w) +4(m+ 1)p∗2(w)ar1br2

Γ(1 + r1)Γ(1 + r2)

=60

e10+

80

e10Γ(1 + r1)Γ(1 + r2)< 1,

which is satisfied for each (r1, r2) ∈ (0, 1] × (0, 1]. Consequently, Theorem 5.2.1implies that the problem (5.3)-(5.4) has a random solution defined on [0, 1] ×[0, 1].

Conclusion and Perspective

In this thesis, we have considered the problem of existence results of ex-istence of random solutions for the fractional partial random differential equa-tions in Banach spaces . Some equations present delay which may be finite, in-finite, or state-dependent. Our results will be obtained by means the measureof noncompactness and a random fixed point theorem with stochastic domain.

We project to look for similar problems in the case when the impulses arevariable, that is they depend on the state variable.

63

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