# Weak Solutions for Partial Random Hadamard Fractional Integral Equations with Multiple Delays

Authors: Saïd Abbas; Mouffak Benchohra; Yong Zhou; Ahmed Alsaedi
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

We present some results concerning the existence of weak solutions for some functional integral equations of Hadamard fractional order with random effects and multiple delays by applying Mönch’s and Engl’s fixed point theorems associated with the technique of measure of weak noncompactness.


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## Body

## 1. Introduction

Random differential equations arise in many applications and have been studied in the literature on bounded as well as unbounded internals of the real line for different aspects of the solution. See, for example, [1]. We refer the reader to the monograph [2] and the papers [1, 3] and the references therein. There are real-world phenomena with anomalous dynamics such as signals transmissions through strong magnetic fields, atmospheric diffusion of pollution, network traffic, and the effect of speculations on the profitability of stocks in financial markets, where the classical models are not sufficiently good to describe these features. In this case, the theory of fractional differential equations is a good tool for modeling such phenomena. Therefore, the study of the fractional differential equations with random parameters seems to be a natural one. For some fundamental results in the theory of fractional calculus and fractional differential equations, we refer the reader to the monographs of Abbas et al. [4, 5], Baleanu et al. [6], and Kilbas et al. [7] and a series of recent research articles [8–12] and the references therein.The measure of weak noncompactness is introduced by De Blasi [13]. The strong measure of noncompactness was developed first by Banaś and Goebel [14] and subsequently developed and used in many papers; see, for example, Akhmerov et al. [15], Alvàrez [16], and Guo et al. [17] and the references therein. In [18], the authors considered some existence results by applying the techniques of the measure of noncompactness. Recently, several researchers obtained other results by application of the technique of measure of weak noncompactness; see [5]. Existence of random solutions for functional differential and integral equations has extensively been studied in various papers; see [19, 20] and the references therein.In this paper, we discuss the existence of random solutions for the partial Hadamard fractional integral equation of the following form:(1)ux,y,w=∑i=1mbix,y,wux-ξi,y-μi,w+fx,y,IHσrux,y,w,ux,y,w,w;x,y∈J,w∈Ω,ux,y,w=Φx,y,w;ifx,y∈J~,w∈Ω,where J≔[1,a]×1,b, J~≔-ξ,a×-μ,b∖1,a×1,b, a,b>1, ξi,μi≥1 (i=1,…,m), ξ=maxi=1,…,m{ξi}, μ=maxi=1,…,m{μi}, σ=(1,1), r=(r1,r2)∈(0,∞)×(0,∞), bi:J×Ω→R (i=1,…,m), and f:J×E×E×Ω→E are given continuous functions, (Ω,A,ν) is a measurable space, and E is a real (or complex) Banach space with norm ·E and dual E∗, such that E is the dual of a weakly compactly generated Banach space X, IHσr is the left-sided mixed Hadamard integral of order r, and Φ:J~×Ω→E is a given continuous and measurable function such that (2)Φx,1,w=∑i=1mbix,1,wΦx-ξi,1-μi,w;x∈1,a,w∈Ω,Φ1,y,w=∑i=1mbi1,y,wΦ1-ξi,y-μi,w;y∈1,b,w∈Ω.


## 2. Preliminaries

LetC be the Banach space of all continuous functions from [-ξ,a]×[-μ,b] into E with the supremum (uniform) norm ·∞. By L∞(Ω,ν), we denote the Banach space of measurable functions u:Ω→C which are essentially bounded equipped with the norm (3)uL∞≔sup⁡essw∈Ω⁡uwC=inf⁡c>0:uwC≤cν—a.e.  inΩ.Denote by (E,w)=(E,σ(E,E∗)) the Banach space E with its weak topology.Definition 1.
A Banach spaceX is called weakly compactly generated (WCG, in short) if it contains a weakly compact set whose linear span is dense in X.Definition 2.
A functionh:E→E is said to be weakly sequentially continuous if h takes each weakly convergent sequence in E to a weakly convergent sequence in E (i.e., for any (un) in E with un→u in (E,w), one has h(un)→h(u) in (E,w)).Definition 3 (see [21]).
The functionu:J→E is said to be Pettis integrable on J if and only if there is an element uj∈E corresponding to each j⊂J such that ϕ(uj)=∫∫jϕ(u(s,t))dtds for all ϕ∈E∗, where the integral on the right-hand side is assumed to exist in the sense of Lebesgue (by definition, uj=∫∫ju(s,t)dtds).LetP(J,E) be the space of all E-valued Pettis integrable functions on J, and let L1J,R be the Banach space of Lebesgue measurable functions u:J→R. Define the class P1(J,E) by (4)P1J,E=u∈PJ,E:φu∈L1J,R;for  everyφ∈E∗.The space P1(J,E) is normed by (5)uP1=supφ∈E∗,φ≤1⁡∫1a∫1bφux,ydλx,y,where λ stands for a Lebesgue measure on J.The following result is due to Pettis (see [21], Theorem  3.4 and Corollary  3.41).Proposition 4 (see [21]).
Ifu∈P1(J,E) and h is a measurable and essentially bounded E-valued function, then uh∈P1(J,E).For all what follows, the sign “∫” denotes the Pettis integral.Let us recall the definitions of Pettis integral and Hadamard integral of fractional order.Definition 5 (see [7]).
The left-sided mixed Pettis Hadamard integral of orderq>0, for a function g∈P1([1,a],E), is defined as (6)IH1rgx=1Γq∫1xlnxsq-1gssds.Remark 6.
Letg∈P11,a,E. For every φ∈E∗, one has (7)φIH1rgx=IH1rφgx;for  a.e.x∈1,a.Definition 7.
Letr1,r2≥0, σ=(1,1), and r=(r1,r2). For w∈P1(J,E), define the left-sided mixed Pettis Hadamard partial fractional integral of order r by the expression (8)IHσrwx,y=1Γr1Γr2∫1x∫1yln⁡xsr1-1ln⁡ytr2-1ws,tstdtds.LetβE be the σ-algebra of Borel subsets of E. A mapping v:Ω→E is said to be measurable if, for any B∈βE, one has (9)v-1B=w∈Ω:vw∈B∈A.To define integrals of sample paths of random process, it is necessary to define a jointly measurable map.Definition 8.
A functionT:Ω×E→E is called jointly measurable if, for any B∈βE, one has (10)T-1B≔w,v∈Ω×E:Tw,v∈B∈A×βE,where A×βE is the direct product of the σ-algebras A and βE, those defined in Ω and E, respectively.Lemma 9 (see [22]).
A functionT:Ω×E→E is jointly measurable if T(·,u) is measurable for all u∈E and T(w,·) is continuous for all w∈Ω.Definition 10.
A functionf:J×E×E×Ω→E is called random Carathéodory if the following conditions are satisfied: (i)
The map(x,y,w)→f(x,y,u,v,w) is jointly measurable for all u,v∈E.(ii)
The map(u,v)→f(x,y,u,v,w) is continuous for all (x,y)∈J and w∈Ω.LetT:Ω×E→E be a mapping. Then T is called a random operator if T(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 operator T(w) on E is called continuous (resp., compact, totally bounded, and completely continuous) if T(w,u) is continuous (resp., compact, totally bounded, and completely continuous) in u for all w∈Ω. The details of completely continuous random operators in Banach spaces and their properties appear in Itoh [23].Definition 11 (see [24]).
LetP(Y) be the family of all nonempty subsets of Y and let C be a mapping from Ω into P(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≠∅} is measurable) and for all open D⊂Y and all y∈Y, {w∈Ω:y∈C(w),T(w,y)∈D} is measurable. T will be called continuous if every T(w) is continuous. For a random operator T, a mapping y:Ω→Y is called random (stochastic) fixed point of T if for P- almost all w∈Ω, y(w)∈C(w), and T(w)y(w)=y(w) and for all open D⊂Y, {w∈Ω:y(w)∈D} is measurable.Definition 12 (see [13]).
LetE be a Banach space, let ΩE be the bounded subsets of E, and let B1 be the unit ball of E. De Blasi measure of weak noncompactness is the map β:ΩE→[0,∞) defined by (11)βX=inf⁡ϵ>0:there  exists  a  weakly  compact  subsetΩofE:X⊂ϵB1+Ω.De Blasi measure of weak noncompactness satisfies the following properties:(a)
A⊂B⇒β(A)≤β(B).(b)
β(A)=0⇔A is weakly relatively compact.(c)
β(A∪B)=max⁡βA,βB.(d)
βA¯ω=βA (A¯ω denotes the weak closure of A).(e)
β(A+B)≤β(A)+β(B).(f)
β(λA)=λβ(A).(g)
βconvA=β(A).(h)
β(⋃λ≤hλA)=hβ(A).The next result follows directly from the Hahn-Banach theorem.Proposition 13.
LetE be a normed space, and x0∈E with x0≠0. Then, there exists φ∈E∗ with φ=1 and φ(x0)=x0.For a given setV of functions v:J→E, let us denote that (12)Vx,y=vx,y:v∈V;x,y∈J,VJ=vx,y:v∈V,x,y∈J.Lemma 14 (see [17]).
LetH⊂C be bounded and equicontinuous. Then the function (x,y)→β(H(x,y)) is continuous on J, and (13)βCH=maxx,y∈JβHx,y,β∫∫Jus,tdtds≤∫∫JβHs,tdtds,where H(x,y)={u(x,y):u∈H}; (x,y)∈J, and βC is De Blasi measure of weak noncompactness defined on the bounded sets of C.We will need the following fixed point theorems.Theorem 15 (see [25]).
LetQ be a nonempty, closed, convex, and equicontinuous subset of a metrizable locally convex vector space C(J,E) such that 0∈Q. Suppose that T:Q→Q is weakly, sequentially continuous. If the implication(14)V¯=conv¯0∪TV⟹Vis  relatively  weakly  compactholds for every subset V⊂Q, then the operator T has a fixed point.Theorem 16 (see [24]).
LetC:Ω→2Y be measurable with closed, convex, and solid C(w) (i.e., int⁡C(w)≠∅) for all w∈Ω. We assume that there exists measurable y0:Ω→Y with y0∈int⁡C(w) for all w∈Ω. Let T be a continuous random operator with stochastic domain C such that, for every w∈Ω,{y∈C(ω):T(w)y=y}≠∅. Then T has a stochastic fixed point.


## 3. Existence Results

Let us start by defining what we mean by a random solution of problem (1).Definition 17.
By a random solution of problem (1), we mean a measurable function u:Ω→C([-ξ,a]×[-μ,b]) that satisfies the integral equation u(x,y,w)=∑i=1mbi(x,y,w)u(x-ξi,y-μi,w)+fx,y,IHσru(x,y,w),u(x,y,w),w on J×Ω, as well as u(x,y,w)=ϕ(x,y,w) on J~×Ω.The following hypotheses will be used in the sequel:H1
The functionsw↦bi(x,y,w), i=1,…,m, are bounded for a.e. (x,y)∈J, and bi(·,·,w)∈L∞(J,R).H2
The functionf is random Carathéodory on J×E×E×Ω for each w∈Ω.H3
For a.e.(x,y)∈J, as well as all w∈Ω, the function u→f(x,y,IHσu,u,w) is weakly, sequentially continuous.H4
There exist functionsp1,p2,p3:J×Ω→[0,∞) with pi(.,w)∈L∞(J,[0,∞)), i=1,2,3, for each w∈Ω, such that, for all φ∈E∗, one has (15)φfx,y,u,v,w≤p1x,y,wφ+p2x,y,wuE+p3x,y,wvE1+φ,for allu,v∈E and a.e. (x,y)∈J.H5
For allu∈E, there exists a continuous function ψ:[0,∞)→[0,∞) with ψ(0)=0, such that, for each φ∈E∗,(x1,y1),(x2,y2)∈J and any w∈Ω, one has (16)∑i=1mbix1,y1,wux1-ξi,y1-μi,w-bix2,y2,wux2-ξi,y2-μi,w+φfx-1,y1,ux1,y1,vx1,y1,w-fx2,y2,ux2,y2,vx2,y2,w≤ψx1-x2+y1-y2φ1+φ+uE+vE.H6
There exists a functionq:J×Ω→[0,∞) with q(·,·,w)∈L∞(J,[0,∞)) for each w∈Ω such that, for any bounded B⊂E, (17)αfx,y,IHσB,B,w≤qx,y,wαB,for  a.e.x,y∈J,whereIHσB≔{IHσu(x,y):u(x,y)∈B;(x,y)∈J}.H7
There exists a random functionR:Ω→(0,∞) such that (18)p1∗w+mb∗+p3∗wRw+p2∗wRwlog⁡ar1log⁡br2Γ1+r1Γ1+r2≤Rw,where(19)b∗=maxi=1,…,mess⁡supx,y,w∈J×Ω⁡bix,y,pi∗w=ess⁡supx,y∈J⁡pix,y,w;i=1,2,3.Set (20)q∗=ess⁡supx,y,w∈J×Ω⁡qx,y,w.Theorem 18.
Assume that hypotheses(H1)–(H7) hold. If(21)l≔mb∗+q∗<1,then problem (1) has a random solution defined on [-ξ,a]×[-μ,b].Proof.
Define the operatorN:Ω×C→C by (22)Nwux,y=∑i=1mbix,y,wux-ξi,y-μi,w+fx,y,IHσrux,y,w,ux,y,w,w;x,y∈J,Nwux,y=Φx,y,w;x,y∈J~. The functions Φ,bi, i=1,…,m, are continuous for all w∈Ω. Again, as the function f is continuous on J, N(w) defines a mapping N:Ω×C→C. Thus, u is a solution for problem (1) if and only if u=(N(w))u. We shall show that the operator N satisfies all conditions of Theorem 16. The proof will be given in several steps.
Step  1 (N(w) is a random operator with stochastic domain on C). Since f(x,y,u,v,w) is random Carathéodory, the map w→f(x,y,u,v,w) is measurable in view of Definition 10. Therefore, the map (23)w⟼∑i=1mbix,y,wux-ξi,y-μi,w+fx,y,IHσrux,y,w,ux,y,w,wis measurable. As a result, N is a random operator on Ω×C×C into C.
LetW:Ω→P(C) be defined by (24)Ww=u∈C:uC≤Rw,ux1,y1,w-ux2,y2,wE≤ψx1-x2+y1-y2.Clearly, the subset W(w) is closed, convex, end equicontinuous for all w∈Ω. Then W is measurable by Lemma  17 in [24]. Therefore, N is a random operator with stochastic domain W.
Step  2 (N(w) is continuous). Let {un} be a sequence such that un→u in C. Then, there exists ϕ∈E∗ such that (N(w)un)(x,y)E=ϕ((N(w)un)(x,y)), and (N(w)u)(x,y)E=ϕ((N(w)u)(x,y)).
Thus, for each(x,y)∈J and w∈Ω, one has (25)Nwunx,y-Nwux,yE=ϕNwunx,y-Nwux,y≤∑i=1mbix,y,wunx-ξi,y-μi,w-ux-ξi,y-μi,wE+ϕfx,y,IHσrunx,y,w,unx,y,w,w-fx,y,IHσrux,y,w,ux,y,w,w.Using the Lebesgue Dominated Convergence Theorem, we get (26)Nwun-Nwu∞⟶0asn⟶∞.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 domain W, and N(w)(W(w)) is bounded.
Step  3 (for every w∈Ω,{u∈W(w):N(w)u=u}≠∅). For this, we apply Theorem 15. The proof will be given in several claims.
Claim  1 (N(w) maps W(w) into itself). Let w∈Ω be fixed, and let u∈W(w),(x,y)∈J. Assume that (N(w)u)(x,y)≠0. Then there exists ϕ∈E∗ such that (N(w)u)(x,y)E=ϕ((N(w)u)(x,y)). Thus, we get (27)Nwux,yE≤∑i=1mbix,y,wux-ξi,y-μi,wE+ϕfx,y,IHσrux,y,w,ux,y,w,w≤∑i=1mbix,y,wux-ξi,y-μi,wE+p1x,y,w+p2x,y,wIHσrux,y,wE+p3x,y,wux,y,wE≤mb∗u∞+p1∗w+p2∗wΓr1Γr2∫1x∫1ylog⁡xsr1-1log⁡ytr2-1us,t,wEdtds+p3∗wRw≤p1∗w+mb∗+p3∗wRw+p2∗wRwlog⁡ar1log⁡br2Γ1+r1Γ1+r2≤Rw.Next, for any fixed w∈Ω, let (x1,y1),(x2,y2)∈J such that x1<x2 and y1<y2, and let u∈W(w), where (N(w)u)(x1,y1)-(N(w)u)(x2,y2)≠0. Then there exists ϕ∈E∗ such that (N(w)u)(x1,y1)-(N(w)u)(x2,y2)E=ϕ((N(w)u)(x1,y1)-(N(w)u)(x2,y2)) and φ=1. Thus, one has (28)Nwux2,y2-Nwux1,y1E=ϕNwux2,y2-Nwux1,y1≤ψx1-x2+y1-y2. Hence N(W(w))⊂W(w). Therefore, N(w):W(w)→N(w) maps W(w) into itself.
Claim  2 (N(w) is weakly, sequentially continuous). Let (un) be a sequence in W(w) and let (un(x,y,w))→u(x,y,w) in (E,ω) for any w∈Ω, and each (x,y)∈J. Fix (x,y)∈J, and since f satisfies assumption (H6), one has that f(x,y,IHσun(x,y,w),un(x,y,w),w) converges weakly uniformly to (29)fx,y,IHσux,y,w,ux,y,w,w.
Hence, the Lebesgue Dominated Convergence Theorem for Pettis integral implies that(Nun)(x,y,w) converges weakly uniformly to (N(w)u)(x,y) in (E,ω). We do it for any w∈Ω, and each (x,y)∈J, so N(w)(un)→N(w)(u). Then N:W(w)→W(w) is weakly, sequentially continuous.
Claim  3 (implication (14) holds). Let V be a subset of W(w) such that V¯=conv¯(N(w)(V)∪{0}). Obviously, V(x,y,w)⊂conv¯((N(w)V)(x,y)∪{0}). Further, as V is bounded and equicontinuous, by Lemma  3 in [26], the function (x,y,w)→u(x,y,w)=β(V(x,y,w)) is continuous on J×Ω. Since the function μ is continuous on J×Ω, the set {μ(x,y,w);(x,y)∈J,w∈Ω}⊂E is compact. From Lemma 14 and the properties of the measure β, for any w∈Ω, as well as each (x,y)∈J, one has (30)vx,y,w≤βNwVx,y∪0≤βNwVx,y=β∑i=1mbix,y,wux-ξi,y-μi,w+fx,y,IHσrux,y,w,ux,y,w,w≤∑i=1mbix,y,wβVx,y,w+qx,y,wβVx,y,w≤mb∗vx,y,w+q∗vx,y,w≤mb∗+q∗vC.Thus, (31)vC≤lvC.From (21), we get vC=0, that is; v(x,y,w)=β(V(x,y,w))=0, for any w∈Ω, and each (x,y)∈J. Hence, Theorem  2 in [27] shows that V is weakly relatively compact in C.
As consequence of Claims 1–3 and from Theorem15, it follows that, for every w∈Ω, {u∈W(w):N(w)u=u}≠∅. Apply now Theorem 16; Steps 1–3 show that, for each w∈Ω,N has at least one fixed point in W. Since ⋂w∈Ωint⁡W(w)≠∅, and a measurable selector of int⁡W exists, the operator N has a stochastic fixed point; that is, problem (1) has at least one random solution defined on [-ξ,a]×[-μ,b].


## 4. An Example

Let(32)E=l1=w=w1,w2,…,wn,…:∑n=1∞wn<∞be the Banach space with norm wE=∑n=1∞wn and let Ω=-∞,0 be equipped with the usual σ-algebra consisting of Lebesgue measurable subsets of -∞,0. Given a measurable function u:Ω→C([-7/2,e]×[-5,e]), consider the functional random integral problem of the following form:(33)ux,y,w=x3e-y17+w2ux-3,y-43,w+xy210+w2ux-2,y-6,w+111+w2+x2+y2ux-92,y-54,w+w2e-x-y-31+w2+ux,y,w+IHσrux,y,w,x,y∈J=1,e×1,e,w∈Ω,ux,y,w=Φx,y,w,x,y∈J~,w∈Ω,where J~≔[-7/2,e]×[-5,e]∖1,e×1,e, m=3, and r=(r1,r2); r1,r2∈(0,∞), u=u1,u2,…,un,…, f=(f1,f2,…,fn,…), and (34)fnx,y,IHσru,u,w=w2e-x-y-31+w2+unx,y,w+IHσrunx,y,w;(x,y)∈J=[0,1]×[0,1], w∈Ω, n∈N, and Φ:J~→E is a continuous and measurable function such that ϕ=(ϕ1,ϕ2,…,ϕn,…), where (35)ϕnx,1,w=x317+w2ϕnx-3,-13,w+111+w2+x2ϕnx-92,-14,w;x∈0,1,n∈N,ϕn1,y,w=111+w2+y2ϕn-72,y-54,w;y∈0,1,n∈N.Set (36)b1x,y,w=x3e-y17,b2x,y=xy210,b3x,y,w=111+x2+y2.Then, b∗=1/10. For each u,v∈E, (x,y)∈0,1×0,1, and w∈Ω, one has (37)fx,y,u,v,wE≤1+1e3uE+vE.Hence condition (H4) is satisfied with p1∗=1 and p2∗=p3∗=1/e3.A simple computation shows that all other conditions of Theorem18 are satisfied. Consequently, Theorem 18 implies that problem (33) has a random solution defined on [-7/2,e]×[-5,e].



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