# Existence and Attractivity Results for Coupled Systems of Nonlinear Volterra–Stieltjes Multidelay Fractional Partial Integral Equations

Authors: Saïd Abbas; Mouffak Benchohra; Naima Hamidi; Gaston N’Guérékata
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

We are concerned with some existence and attractivity results of a coupled fractional Riemann–Liouville–Volterra–Stieltjes multidelay partial integral system. We prove the existence of solutions using Schauder’s fixed point theorem; then we show that the solutions are uniformly globally attractive.


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## Body

## 1. Introduction

Fractional integral and fractional differential equations are among the most fast growing field in mathematics. They are used to describe many phenomena, especially the ones with long memory. Examples include but are not limited to viscoelasticity, viscoplasticity, biochemistry, control theory, mathematical psychology, mechanics, modeling in complex media (porous, etc.), and electromagnetism [1–4]. In recent years, there has been a significant development in ordinary and partial fractional integral equations; see, for instance, the monographs of Abbaset al. [5–7], Agarwalet al. [8], Kilbaset al. [9], Miller and Ross [10], Podlubny [11], Samkoet al. [12], and the papers [13–18] and the references therein.In this paper we study the existence and attractivity of solutions to the following coupled system of nonlinear fractional Riemann–Liouville–Volterra–Stieltjes quadratic multidelay partial integral equations:(1)
u
1
t
,
x
=
μ
1
α
t
,
x
+
1
Γ
r
1
Γ
r
2
∫
0
β
t
∫
0
x
β
t
-
s
r
1
-
1
x
-
y
r
2
-
1
×
f
1
t
,
x
,
s
,
y
,
u
1
γ
s
-
τ
1
,
y
-
ξ
1
,
u
2
γ
s
-
τ
1
,
y
-
ξ
1
,
…
,
u
1
γ
s
-
τ
m
,
y
-
ξ
m
,
u
2
γ
s
-
τ
m
,
y
-
ξ
m
d
y
d
s
g
t
,
s
u
2
t
,
x
=
μ
2
α
t
,
x
+
1
Γ
r
1
Γ
r
2
∫
0
β
t
∫
0
x
β
t
-
s
r
1
-
1
x
-
y
r
2
-
1
×
f
2
t
,
x
,
s
,
y
,
u
1
γ
s
-
τ
1
,
y
-
ξ
1
,
u
2
γ
s
-
τ
1
,
y
-
ξ
1
,
…
,
u
1
γ
s
-
τ
m
,
y
-
ξ
m
,
u
2
γ
s
-
τ
m
,
y
-
ξ
m
d
y
d
s
g
t
,
s
;
t
,
x
∈
J
,
(2)
u
1
t
,
x
=
Φ
1
t
,
x
u
2
t
,
x
=
Φ
2
t
,
x
;
t
,
x
∈
J
~
≔
-
T
,
∞
×
-
ξ
,
b
∖
0
,
∞
×
0
,
b
,where J≔R+×[0,b],b>0,r1,r2∈(0,∞),R+=[0,∞),τi,ξi≥0, i=1,…,m,T=maxi=1,…,m{τi}, ξ=maxi=1,…,m{ξi},α,β,γ:R+→R+,g:R+×R+→R,μj:J→R,fj:J′×R2m→R,j=1,2, are given continuous functions, limt→∞α(t)=∞,μj, j=1,2, are bounded, J′={(t,x,s,y)∈J2:s≤t,y≤x}, Φj:J~→R,j=1,2, are continuous and bounded functions with limt→∞Φj(t,x)=0,x∈[-ξ,b],μj(α(t),0)=Φj(t,0) for each t∈R+ and μj(α(0),x)=Φj(0,x), for each x∈[0,b], and Γ(·) is the (Euler’s) Gamma function defined by (3)Γζ=∫0∞tζ-1e-tdt;ζ>0.


## 2. Preliminaries

In this section, we recall some notations, definitions, and preliminary facts which will be used in this paper.L1([0,p]×[0,q]),p,q>0, will denote the space of all Lebesgue-integrable functions u:[0,p]×[0,q]→R equipped with the norm (4)u1=∫0p∫0qut,xdxdt.BC≔BC([-T,∞)×[-ξ,b]) will denote the usual Banach space of all bounded and continuous functions from [-T,∞)×[-ξ,b] into R equipped with the standard norm (5)uBC=supt,x∈-T,∞×-ξ,but,x.It is clear that the product space BC≔BC×BC turns out to be a Banach space if equipped with the norm (6)u1,u2BC=u1BC+u2BC.Definition 1 (see [19]).
Letr=(r1,r2)∈(0,∞)×(0,∞),θ=(0,0) and u∈L1([0,p]×[0,q]). The left-sided mixed Riemann–Liouville integral of order r of u is defined by (7)Iθrut,x=1Γr1Γr2∫0t∫0xt-τr1-1x-sr2-1uτ,sdsdτ,provided the integral exists.Example 2.
Letλ,ω∈(0,∞) and r=(r1,r2)∈(0,∞)×(0,∞), then (8)Iθrtλxω=Γ1+λΓ1+ωΓ1+λ+r1Γ1+ω+r2tλ+r1xω+r2,foralmostallt,x∈0,p×0,q.Ifu is a real-valued function defined on the interval [a,b], then we will use the symbol ⋁abu to denote the variation of u on [a,b]. We say that u is of bounded variation on the interval [a,b] whenever ⋁abu is finite. If w:[a,b]×[c,d]→R, then the symbol ⋁t=pqw(t,s) indicates the variation of the function t→w(t,s) on the interval [p,q]⊂[a,b], where s is arbitrarily fixed in the interval [c,d]. Analogously we define ⋁s=pqw(t,s). For more details on the properties of functions of bounded variation we refer the reader to [20].Ifu and φ are two real-valued functions defined on the interval [a,b], then under some appropriate conditions (see [20]) we can define the Stieltjes integral (in the Riemann–Stieltjes sense) (9)∫abutdφtof the function u with respect to φ. In this case we say that u is Stieltjes integrable on [a,b] with respect to φ. Several conditions are known to ensure Stieltjes integrability [20]. One of the most frequently used requires that u is continuous and φ is of bounded variation on [a,b].Now we recall a few properties of the Stieltjes integral included in the lemmas below.Lemma 3 (see [20, 21]).
Ifu is Stieltjes integrable on the interval [a,b] with respect to a function φ of bounded variation, then (10)∫abutdφt≤∫abutd⋁atφ.Lemma 4 (see [20, 21]).
Letu and v be Stieltjes integrable functions on the interval [a,b] with respect to a nondecreasing function φ such that u(t)≤v(t) for t∈[a,b]. Then (11)∫abutdφt≤∫abvtdφt.From now on, we will also consider Stieltjes integrals of the form(12)∫abutdsgt,sand Riemann–Liouville–Stieltjes integrals of fractional order of the form (13)1Γr∫0tt-sr-1usdsgt,s,where g:R+×R+→R,r∈(0,∞) and the symbol ds indicates the integration with respect to s.Let∅≠Ω⊂BC, and let G:Ω→Ω, and consider the solutions of equation(14)Gut,x=ut,x.In light of the definition of the attractivity of solutions of integral equations (for instance, [15]), we will introduce the following concept of attractivity of solutions for (14).Definition 5.
A solutions of (14) is said to be locally attractive if there exists a ball B(u0,η) in the space BC such that, for arbitrary solutions v=v(t,x) and w=w(t,x) of (14) belonging to B(u0,η)∩Ω, we have that, for each x∈[0,b],(15)limt→∞⁡vt,x-wt,x=0.When the limit (15) is uniform with respect to B(u0,η)∩Ω, solutions of (14) are said to be uniformly locally attractive (or equivalently that solutions of (14) are locally asymptotically stable).Definition 6 (see [15]).
The solutionv=v(t,x) of (14) is said to be globally attractive if (15) holds for each solution w=w(t,x) of (14). If condition (15) is satisfied uniformly with respect to the set Ω, solutions of (14) are said to be globally asymptotically stable (or uniformly globally attractive).Lemma 7 (see [22], p. 62).
LetD⊂BC. Then D is relatively compact in BC if the following conditions hold:(a)
D is uniformly bounded in BC.(b)
The functions belonging toD are almost equicontinuous on [-T,∞)×[-ξ,b],i.e., equicontinuous on every compact subset of [-T,∞)×[-ξ,b].(c)
The functions fromD are equiconvergent; that is, given ϵ>0,x∈[-ξ,b], there corresponds λ(ϵ,x)>0 such that |u(t,x)-limt→∞u(t,x)|<ϵ for any t≥λ(ϵ,x) and u∈D.


## 3. Existence and Attractivity Results

Definition 8.
By a solution to problem (1)-(2), we mean every coupled functions (u,v)∈BC such that (u,v) satisfies (1) on J and (2) on J~.We will use the following assumptions in the sequel:(H1)
There exist positive functionspj∈BC;j=1,2 such that (16)1+αtμjαt,x≤pjt,x;t,x∈J.(H2)
For allt1,t2∈R+ such that t1<t2, the function s↦g(t2,s)-g(t1,s) is nondecreasing on R+.(H3)
The functions↦g(0,s) is nondecreasing on R+.(H4)
The functionss↦g(t,s) and t↦g(t,s) are continuous on R+ for each fixed t∈R+ or s∈R+, respectively.(H5)
There exist continuous functionsqji:J′→R+; i=1,…,m, j=1,2 such that (17)1+∑i=1mu1i+u2ifjt,x,s,y,u11,u21,…,u1m,u2m≤∑i=1mq1it,x,s,yu1i+q2it,x,s,yu2i;for(t,x,s,y)∈J′, u1i,u2i∈R; i=1,…,m. Moreover, assume that (18)limt→∞⁡∫0βtβt-sr1-1qjit,x,s,ydsgt,s=0;i=1,…,m,j=1,2.Remark 9.
SetΦj∗≔sup(t,x)∈J~Φj(t,x), pj∗≔sup(t,x)∈Jpj(t,x),(19)qji∗≔supt,x∈J1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1qjit,x,s,ydyds⋁k=0sgt,k;for i=1,…,m and j=1,2. From the above assumptions, we infer that Φj∗,pj∗,qji∗ are finite.Theorem 10.
Assume that hypotheses(H1)-(H5) hold. Then problem (1)-(2) has at least one solution in the space BC. Moreover, solutions to problem (1)-(2) are uniformly globally attractive.Proof.
Define the operatorsNj:BC→BC; j=1,2 by(20)Njujt,x=Φjt,x;t,x∈J~,Njujt,x=μjαt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×fjt,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1,…,u1γs-τm,y-ξm,u2γs-τm,y-ξmdydsgt,s;t,x∈J,and consider the operator N:BC→BC such that, for any (u1,u2)∈BC,(21)Nu1,u2t,x=N1u1t,x,N2u2t,x.From the hypotheses above, we deduce that N(u) is continuous on [-T,∞)×[-ξ,b]. Now let us prove that N(u1,u2)∈BC for any uj∈BC;j=1,2. For arbitrarily fixed (t,x)∈J, we have (22)Njujt,x=μαt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×fjt,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1,…,u1γs-τm,y-ξm,u2γs-τm,y-ξmdydsgt,s≤pjt,x1+αt+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×∑i=1mq1it,x,s,yu1γs-τi,y-ξi+q2it,x,s,yu2γs-τi,y-ξidydsgt,s≤pjt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k≤pj∗+∑i=1mq1i∗+q2i∗,and for all (t,x)∈J~ and each uj∈BC,j=1,2, we have (23)Njujt,x=Φjt,x≤Φj∗.Thus,(24)NjujBC≤max⁡Φj∗,pj∗+∑i=1mqji∗≔ηj;j=1,2.Hence(25)Nu1,u2BC≤η1+η2≔η.
ThereforeN(u)∈BC. The problem of finding the solutions of the coupled system (1)-(2) is reduced to finding the solutions of the operator equation N(u1,u2)=(u1,u2). From (25), we infer that N transforms the ball Bη≔{(u1,u2)∈BC:(u1,u2)BC≤η} into itself. Now we will show that N:Bη→Bη satisfies the Schauder’s fixed point theorem [23]. The proof will be presented in several steps and cases.
Step 1 (N is continuous). Let {(un,vn)}n∈N be a sequence such that un→u and vn→v in Bη. Then, for each (t,x)∈[-T,∞)×[-ξ,b], we have(26)N1unt,x-N1ut,x≤1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f1t,x,s,y,unγs-τ1,y-ξ1,vnγs-τ1,y-ξ1,…,unγs-τm,y-ξm,vnγs-τm,y-ξm-f1t,x,s,y,uγs-τ1,y-ξ1,vγs-τ1,y-ξ1,…,uγs-τm,y-ξm,vγs-τm,y-ξmdyds⋁k=0sgt,k.
Case 1. Assume that (t,x)∈J~∪([0,a]×[0,b]);a>0, then, since (un,vn)→(u,v) as n→∞ and f1,g,γ are continuous, (26) implies (27)Nun-NuBC→0asn→∞.
Case 2. Let (t,x)∈(a,∞)×[0,b];a>0, then from (H5) and (26) we obtain(28)N1unt,x-N1ut,x≤2Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k≤∑i=1m2Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×q1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k.Since t→∞, then (28) gives (29)N1un-N1uBC→0asn→∞.Let us show that N2 is continuous in the same way as continuity of N1.
Step 2 (N(Bη) is uniformly bounded). This fact is obvious because N(Bη)⊂Bη and Bη is a bounded set.
Step 3 (N(Bη) is equicontinuous on every compact subset [0,a]×[0,b] of J,a>0). Let (t1,x1),(t2,x2)∈[0,a]×[0,b], t1<t2,x1<x2, and let (u,v)∈Bη. Without loss of generality, let us assume that β(t1)≤β(t2). Then we obtain (30)
N
1
u
t
2
,
x
2
-
N
1
u
t
1
,
x
1
≤
μ
1
α
t
2
,
x
2
-
μ
1
α
t
1
,
x
1
+
1
Γ
r
1
Γ
r
2
∫
0
β
t
2
∫
0
x
2
β
t
2
-
s
r
1
-
1
x
2
-
y
r
2
-
1
×
f
1
t
2
,
x
2
,
s
,
y
,
u
γ
s
-
τ
1
,
y
-
ξ
1
,
v
γ
s
-
τ
1
,
y
-
ξ
1
,
…
,
u
γ
s
-
τ
m
,
y
-
ξ
m
,
v
γ
s
-
τ
m
,
y
-
ξ
m
-
f
1
t
1
,
x
1
,
s
,
y
,
u
γ
s
-
τ
1
,
y
-
ξ
1
,
v
γ
s
-
τ
1
,
y
-
ξ
1
,
…
,
u
γ
s
-
τ
m
,
y
-
ξ
m
,
v
γ
s
-
τ
m
,
y
-
ξ
m
d
y
d
s
⋁
k
=
0
s
g
t
2
,
k
+
1
Γ
r
1
Γ
r
2
∫
β
t
1
β
t
2
∫
0
x
2
β
t
2
-
s
r
1
-
1
x
2
-
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r
2
-
1
×
f
1
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1
,
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1
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s
,
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,
…
,
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γ
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-
ξ
m
,
v
γ
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-
τ
m
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-
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y
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⋁
k
=
0
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g
t
2
,
k
+
1
Γ
r
1
Γ
r
2
∫
0
β
t
1
∫
0
x
1
β
t
2
-
s
r
1
-
1
x
2
-
y
r
2
-
1
-
β
t
1
-
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r
1
-
1
x
1
-
y
r
2
-
1
×
f
1
t
1
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1
,
s
,
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,
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1
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-
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1
,
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γ
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1
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-
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1
,
…
,
u
γ
s
-
τ
m
,
y
-
ξ
m
,
v
γ
s
-
τ
m
,
y
-
ξ
m
d
y
d
s
⋁
k
=
0
s
g
t
1
,
k
+
1
Γ
r
1
Γ
r
2
∫
0
β
t
1
∫
x
1
x
2
β
t
2
-
s
r
1
-
1
x
2
-
y
r
2
-
1
×
f
1
t
1
,
x
1
,
s
,
y
,
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γ
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1
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1
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⋁
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=
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,
k
.Thus (31)N1ut2,x2-N1ut1,x1≤μ1αt2,x2-μ1αt1,x1+1Γr1Γr2∫0βt2∫0x2βt2-sr1-1x2-yr2-1×f1t2,x2,s,y,uγs-τ1,y-ξ1,vγs-τ1,y-ξ1,…,uγs-τm,y-ξm,vγs-τm,y-ξm-f1t1,x1,s,y,uγs-τ1,y-ξ1,vγs-τ1,y-ξ1,…,uγs-τm,y-ξm,vγs-τm,y-ξmdyds⋁k=0sgt2,k+1Γr1Γr2∫βt1βt2∫0x2βt2-sr1-1x2-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt2,k+1Γr1Γr2∫0βt1∫0x1βt2-sr1-1x2-yr2-1-βt1-sr1-1x1-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt1,k+1Γr1Γr2∫0βt1∫x1x2βt2-sr1-1x2-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt1,kUsing continuity of the functions μ1,α,β,γ,f1,g,h,qi,i=1,…m, and since t1→t2 and x1→x2, the right-hand side of the above inequality tends to zero. The equicontinuity of N1 for the cases t1<t2<0, x1<x2<0 and t1≤0≤t2, x1≤0≤x2 is immediate.
We can also prove that(32)N2u,vt2,x2-N2u,vt1,x1≤μ2αt2,x2-μ2αt1,x1+1Γr1Γr2∫0βt2∫0x2βt2-sr1-1x2-yr2-1×f2t2,x2,s,y,uγs-τ1,y-ξ1,vγs-τ1,y-ξ1,…,uγs-τm,y-ξm,vγs-τm,y-ξm-f2t1,x1,s,y,uγs-τ1,y-ξ1,vγs-τ1,y-ξ1,…,uγs-τm,y-ξm,vγs-τm,y-ξmdyds⋁k=0sgt2,k+1Γr1Γr2∫βt1βt2∫0x2βt2-sr1-1x2-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt2,k+1Γr1Γr2∫0βt1∫0x1βt2-sr1-1x2-yr2-1-βt1-sr1-1x1-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt1,k+1Γr1Γr2∫0βt1∫x1x2βt2-sr1-1x2-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt1,k→0,ast1→t2,x1→x2.Hence (33)Nu,vt2,x2-Nu,vt1,x1≤N1ut2,x2-N1ut1,x1+N2vt2,x2-N2vt1,x1→0,ast1→t2,x1→x2.
Step 4 (N(Bη) is equiconvergent). Let (t,x)∈J and u∈Bη, then we get (34)Nu,vt,x≤μ1αt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f1t,x,s,y,uγs-τ1,y-ξ1,vγs-τ1,y-ξ1,…,uγs-τm,y-ξm,vγs-τm,y-ξmdydsgt,s+μ2αt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f2t,x,s,y,uγs-τ1,y-ξ1,vγs-τ1,y-ξ1,…,uγs-τm,y-ξm,vγs-τm,y-ξmdydsgt,s≤p1t,x1+αt+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k+p2t,x1+αt+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k≤p1∗1+αt+p2∗1+αt+∑i=1m2Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×q1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k.Now, since α(t)→∞ as t→∞, we conclude that, for each x∈[0,b], we obtain (35)Nu,vt,x→0,ast→+∞.Also, for each x∈[-ξ,0], we get (36)Nu,vt,x≤Φ1t,x+Φ2t,x→0,ast→+∞.Then, for each x∈[-ξ,b], we get (37)Nu,vt,x→0,ast→+∞.Hence, (38)Nu,vt,x-Nu,v+∞,x→0,ast→+∞.In view of Steps 1 to 4, along with the Lemma 7, we deduce that N:Bη→Bη is continuous and compact. From an application of Schauder’s theorem [23], we conclude that N has a fixed point (u,v) which is a solution of the coupled system (1)-(2).
Step 5 (the uniform global attractivity of solutions). Now let us study the stability of solutions of the coupled system (1)-(2). Let (u1,u2) and (v1,v2) be two solutions of (1)-(2). Then, for each (t,x)∈[-T,∞)×[-ξ,b], we obtain (39)u1,u2t,x-v1,v2t,x=Nu1,u2t,x-Nv1,v2t,x≤1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f1t,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1,…,u1γs-τm,y-ξm,u2γs-τm,y-ξm-f1t,x,s,y,v1γs-τ1,y-ξ1,v2γs-τ1,y-ξ1,…,v1γs-τm,y-ξm,v2γs-τm,y-ξmdydsgt,s+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f2t,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1,…,u1γs-τm,y-ξm,u2γs-τm,y-ξm-f2t,x,s,y,v1γs-τ1,y-ξ1,v2γs-τ1,y-ξ1,…,v1γs-τm,y-ξm,v2γs-τm,y-ξmdydsgt,s.Thus(40)u1,u2t,x-v1,v2t,x=Nu1,u2t,x-Nv1,v2t,x≤1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f1t,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1),…,u1γs-τm,y-ξm,u2γs-τm,y-ξm)-f1t,x,s,y,v1γs-τ1,y-ξ1,v2γs-τ1,y-ξ1,…,v1γs-τm,y-ξm,v2γs-τm,y-ξmdyds⋁k=0sgt,k+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f2t,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1),…,u1γs-τm,y-ξm,u2γs-τm,y-ξm)-f2t,x,s,y,v1γs-τ1,y-ξ1,v2γs-τ1,y-ξ1,…,v1γs-τm,y-ξm,v2γs-τm,y-ξmdyds⋁k=0sgt,k.Hence(41)u1,u2t,x-v1,v2t,x=Nu1,u2t,x-Nv1,v2t,x≤2Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×∑i=1mq1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k≤2Γr1Γr2∑i=1m∫0βt∫0xβt-sr1-1x-yr2-1×q1it,x,s,y+q2it,x,s,ydyds⋁k=0sgt,k.By using (41) and (H5), we obtain (42)limt→∞⁡u1,u2t,x-v1,v2t,x=0.Therefore, all solutions of the coupled system (1)-(2) are uniformly globally attractive.
LetBC¯≔BCn (product space) be the Banach space equipped with the following norm: (43)u1,u2,…,unBC¯=∑k=1nukBC.From the above theorem, we deduce the following consequence.Corollary 11.
Consider the system of nonlinear fractional Riemann–Liouville–Volterra–Stieltjes quadratic multidelay partial integral equations of the form(44)u1t,x=μ1αt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f1t,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1,…,unγs-τ1,y-ξ1,…,u1γs-τm,y-ξm,u2γs-τm,y-ξm,…,unγs-τm,y-ξmdydsgt,su2t,x=μ2αt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f2t,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1,…,unγs-τ1,y-ξ1,…,u1γs-τm,y-ξm,u2γs-τm,y-ξm,…,unγs-τm,y-ξmdydsgt,s⋮unt,x=μnαt,x+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×fnt,x,s,y,u1γs-τ1,y-ξ1,u2γs-τ1,y-ξ1,…,unγs-τ1,y-ξ1,…,u1γs-τm,y-ξm,u2γs-τm,y-ξm,…,unγs-τm,y-ξmdydsgt,s;t,x∈J,(45)u1t,x=Φ1t,xu2t,x=Φ2t,x⋮unt,x=Φnt,x;t,x∈J~≔-T,∞×-ξ,b∖0,∞×0,b,where r1,r2∈(0,∞), τi,ξi≥0; i=1,…,m, T=maxi=1,…,m{τi}, ξ=maxi=1,…,m{ξi},α,β,γ:R+→R+, g:R+×R+→R, μj:J→R, fj:J′×R2m→R, j=1,…,n are given continuous functions, limt→∞α(t)=∞,μj; j=1,…,n are bounded, J′={(t,x,s,y)∈J2:s≤t,y≤x}, Φj:J~→R; j=1,2 are continuous and bounded functions with limt→∞Φj(t,x)=0;x∈[-ξ,b],μj(α(t),0)=Φj(t,0) for each t∈R+ and μj(α(0),x)=Φj(0,x); for each x∈[0,b].
Suppose that(H2)-(H4) and the following assumptions are verified: (H1′)
There exist positive functionspj∈BC, j=1,…,n, such that (46)1+αtμjαt,x≤pjt,x;t,x∈J.(H2′)
There exist continuous functionsqji:J′→R+, i=1,…,m, j=1,…,n, such that (47)1+∑i=1m∑j=1nujifjt,x,s,y,u11,u21,…,un1,…,u1m,u2m,…,unm,unm≤∑i=1m∑j=1nqjit,x,s,yuji;for(t,x,s,y)∈J′, uji∈R, i=1,…,m, j=1,…,n. Moreover, assume that (48)limt→∞⁡∫0βtβt-sr1-1qjit,x,s,ydsgt,s=0;i=1,…,m,j=1,…,n.Then problem (44)-(45) has at least one solution in the space BC¯. In addition, the solutions are uniformly globally attractive.


## 4. An Example

To illustrate our results, we consider the following coupled system of nonlinear fractional order Riemann–Liouville–Volterra–Stieltjes quadratic multidelay partial integral equation(49)u1t,x=x2e-t1+t+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f1t,x,s,y,u1γs-12,y-1,u1γs-14,y-2,u2γs-12,y-1,u2γs-14,y-2dydsgt,su2t,x=x2e-t1+t+1Γr1Γr2∫0βt∫0xβt-sr1-1x-yr2-1×f2t,x,s,y,u1γs-12,y-1,u1γs-14,y-2,u2γs-12,y-1,u2γs-14,y-2dydsgt,s;t,x∈J,(50)u1t,x=x2e-tu2t,x=x21+t2;t,x∈J~≔-12,∞×-2,1∖0,∞×0,1,where J≔R+×[0,1], r1=1/4, r2=1/2, α(t)=β(t)=γ(t)=t; t∈R+,(51)μαt,x=x2e-t1+t;t,x∈R+×0,1,f1t,x,s,y,u1,u2,v1,v2=xs-3/4u1+u2+v1+v2sin⁡tsin⁡s1+y2+t21+u1+u2+v1+v2;ift,x,s,y∈J′,s≠0,y∈0,1,u1,u2,v1,v2∈R,f1t,x,0,y,u1,u2,v1,v2=0;ift,x∈J,y∈0,1,u1,u2,v1,v2∈R,f2t,x,s,y,u1,u2,v1,v2=xs-3/4u1+u2+v1+v2sin⁡te-s1+u1+u2+v1+v2;if(t,x,s,y)∈J′, y∈[0,1]andu1,u2,v1,v2∈R, (52)J′=t,x,s,y∈J2:s≤t,y≤x,gt,s=s,t,s∈R+2.First, we can see that limt→∞α(t)=∞ and limt→∞Φj(t,x)=0;j=1,2. Next, the assumption (H1) is satisfied with pj(t,x)=x2e-t and consequently pj∗=1. Also, it is clear that the function g satisfies assumptions (H2)-(H4).Finally, the functionsfj, j=1,2, satisfy the assumption (H5). Indeed, fj are continuous and satisfy the inequality (53)fjt,x,s,y,u1,u2,v1,v2≤q1t,x,s,yu1+u2+q2t,x,s,yv1+v21+u1+u2+v1+v2;(t,x,s,y)∈J′, u1,u2,v1,v2∈R. Also, we have (54)q1t,x,s,y=xs-3/4sin⁡tsin⁡s1+y2+t2;t,x,s,y∈J′,y∈0,1,s≠0,q1t,x,0,y=0;t,x∈J,y∈0,1.and (55)q2t,x,s,y=xs-3/4sin⁡te-s;t,x,s,y∈J′,y∈0,1.For i=1,2, we have also (56)∫0tt-sr1-1qit,x,s,ydsgt,s≤∫0tt-s-3/4xs-3/4sin⁡tsin⁡sds⋁k=0sgt,k≤xsin⁡t∫0tt-s-3/4s-3/4ds≤xΓ21/4πsin⁡tt≤xΓ21/4πt→0ast→∞,and(57)qi∗≔supt,x∈J1Γr1Γr2∫0t∫0xt-sr1-1x-yr2-1qit,x,s,ydyds⋁k=0sgt,k≤supt,x∈JxΓ1/4πsin⁡tt=Γ1/4π.Consequently, Theorem 10 implies that the coupled system (49)-(50) has a solution defined on [-1/2,∞)×[-2,1]; moreover solutions of this system are uniformly globally attractive.



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