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PureMathematicsnØêÆ,2022,12(12),2045-2060
PublishedOnlineDecember2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1212221
äklÑž¢Caputo©ê‡©•§
-½5
ÁÁÁ•••
®e>ŒÆ§nÆ§®
ÂvFϵ2022c1026F¶¹^Fϵ2022c1128F¶uÙFϵ2022c125F
Á‡
©|^ØÄ:½n§ïáÕžmlÑ©Ùž¢©ê‘Ň©•§§Ì‡ïÄÕžm«
mSäkÙK$ÄÚlÑ©Ùž¢Caputo©ê‡ ©•§)•35!•˜5ÚìC-½5"
Ù¥§$^Ø NnÚMittag-Lefler¼ê°OO"•ª§$^‡y{y² ÑìC-½
5"
'…c
©ê‘Ň©•§§ìC-½5§ØÄ:½n
StabilityofCaputoFractionalDifferential
EquationswithDiscreteDelay
WeiZhu
SchoolofScience,BeijingUniversityofPostsandTelecommunications,Beijing
Received:Oct.26
th
,2022;accepted:Nov.28
th
,2022;published:Dec.5
th
,2022
Abstract
In this paper,we establish infinite time discretefractional stochastic differential equa-
©ÙÚ^:Á•.äklÑž¢Caputo©ê‡©•§-½5[J].nØêÆ,2022,12(12):2045-2060.
DOI:10.12677/pm.2022.1212221
Á•
tionswithdistributeddelaysbyusingthefixedpointtheorem.Wemainlystudythe
existence,uniquenessandasymptoticstability ofsolutionsofCaputofractionaldiffer-
ential equations withBrownian motion anddiscretedistributeddelays in infinitetime
intervals.Amongthem,thecompressionmappingprincipleandaccurateestimation
ofMittagLefflerfunctionareused.Finally,theasymptoticstabilityisprovedbythe
methodofcontradiction.
Keywords
FractionalStochasticDifferentialEquations,AsymptoticStability,FixedPoint
Theorems
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
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•Ä©ê‡©•§5Ÿ§ X-½5"[3]ÏL|©/A ^Mittag-Lefler¼êìCÐm§
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¦Š"éuŒ‡¼êf: [0,∞) →R§pfCaputo[4]êdeª‰Ñ
∂
p
t
f(t) =
1
Γ(1−p)
d
dt
Z
t
0
(t−s)
−p
(f(s)−f(0))ds,
Ù¥0<p61ÚΓ(s) :=
R
∞
0
t
s−1
e
−t
dt"
DOI:10.12677/pm.2022.12122212046nØêÆ
Á•
·‚òïıe©ê‡©•§



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
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
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
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


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



∂
β
t

x
i
(t)−
n
P
j=1
d
ij
x
j
(t−τ
j
(t))

=
n
P
j=1
c
ij
x
j
(t)+
n
P
j=1
a
ij
f
j
(x
j
(t))
+
n
P
j=1
b
ij
g
j
(x
j
(t−τ
j
(t)))+
n
P
j=1
l
ij
R
t
t−r(t)
h
j
(x
j
(s))ds
+
n
P
j=1
∂
γ
t
R
t
0
σ
ij
(s,x
j
(s),x
j
(s−τ(s)))dW
j
(s),t>0,
x
i
(t) = φ
i
(t),t∈[ϑ,0],
(1)
Ù¥§∂
β
t
Ú∂
γ
t
L«©êê§β ∈(
1
2
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2
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2
)§±9x(t)=φ(t),t∈[ϑ,0]´
•§(1)Щ^‡§Ù¥t7−→φ=(φ
1
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n
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T
Úx(t)=(x
1
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T
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n
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´† ²ƒ'•þ¶C =diag(c
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)
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)
n×n
!
C =(c
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)
n×n
!D=(d
ij
)
n×n
ÚL=(l
ij
)
n×n
L«ØÓÝ¶·‚˜f
j
!g
j
Úh
j
-¹
¼ê§Ù¥f(x(t))=(f
1
(x(t)),···,f
n
(x(t)))
T
∈R
n
,g(x(t))=(g
1
(x(t)),···,g
n
(x(t)))
T
∈
R
n
,h(x(t))=(h
1
(x(t)),···,h
n
(x(t)))
T
∈R
n
¶lÑžCò´Ú©ÙªžCò´.L«
•τ(t)Úr(t)"4·‚IPϑ=inf
t>0
{t−τ(t),t−r(t)}"dτ(t)Úr(t)´šKëY¼ê"d§
w(t) = (w
1
(t),w
2
(t),···,w
n
(t))
T
∈R
n
´Vǘm(Ω,F,P)¥n‘ÙK$Ä"
Ø©Ù{Ü©|„Xe"312!¥§·‚‰Ñ'uMittag-Lefler¼êb¿O˜
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(A1) ò´τ(t)§r(t)´ëY¼ê§÷vt−τ(t) →∞Út−r(t) →∞ast→∞"
(A2)Nf
j
(·)!g
j
(·)Úh
j
(·)§÷vf(0)≡0,g(0)≡0,h(0)≡0§σ
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(t,0,0)≡0Úä
kLipschitz~êÛLipschiz¼êα
j
§β
j
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Ù¥j= 1,2···n,
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andυ
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½Â2.1.[5]Mittag-Lefler¼êa.Vëê¼êd?êÐm½Â
E
α,β
(z) =
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X
k=0
z
k
Γ(kα+β)
,α,β>0,z∈C.
DOI:10.12677/pm.2022.12122212047nØêÆ
Á•
Ún2.1.[6]bE
α,β
´Mittag-Leffler¼ê§·‚Œ±
Z
∞
0
e
−st
t
β−1
E
α,β
(±at
α
)dt=
s
α−β
(s
α
∓a)
,<(α)>0,<(β)>0,<(s)>0,
,
L

t
β−1
E
α,β
(±at
α
)

(s) =
s
α−β
(s
α
∓a)
,<(α)>0,<(β)>0,<(s)>0,
Ù¥<L«¢Ü§LL«.Ê.dC†"
Ún2.2.[7]XJ0<α<2,β´?¿˜‡ê,πα/2<µ<min{π,πα},,•3M>0÷v
E
α,β
(z) 6
M
1+|z|
,µ<|arg(z)|<π,|z|>0.
Ún2.3.[7]bE
α,β
´Mittag-Leffler¼ê,·‚Œ±
Z
t
0
E
α,β
(λs
α
)s
β−1
ds= t
β
E
α,β+1
(λt
α
),α>0,β>0.
Ún2.4.[7]bE
α,β
´Mittag-Leffler¼ê,·‚Œ±
1
Γ(γ)
Z
t
0
(t−s)
γ−1
E
α,β
(λs
α
)s
β−1
ds= t
β+γ−1
E
α,β+γ
(λt
α
),α>0,β>0,γ>0.
Ún2.5.[8]b(X,Σ,µ)´σk•ÿݘm,M´ëYÛÜ,[M] : [0,T]×Ω →R
+
´§g
C"eψ: X×[0,T]×Ω →R,T∈(0,∞)ŒÅÚÿþ§éuω∈Ω,
Z
X

Z
T
0
|ψ(x,t,ω)|
2
d[M](t,ω)

1
2
dµ(x)<∞,
@oéuA¤kω∈Ωéu¤kt∈[0,T],
Z
X
Z
t
0
ψ(x,r,ω)dM(r)dµ(x) =
Z
t
0
Z
X
ψ(x,t,ω)dµ(x)dM(r).
3©¥§·‚ò•Ä±e˜mþ•§(1))"½ÂS
φ
´F
0
·AL§˜mµϕ(t,ω):
[ϑ,∞)×Ω →R
n
÷vϕ∈C

[ϑ,∞),L
P
F
0
(Ω;R
n
)

‰ê½ÂXeµ
kϕk
p
:= sup
t>ϑ
E
n
X
i=1
|ϕ
i
(t)|
p
!
.
,,·‚-ϕ(t,·) = φ(t)3t∈[ϑ,0]þ…
P
n
i=1
E|ϕ
i
(t)|
p
→0t→∞,S
φ
´˜‡˜m"
3.̇(J
4˜c
i,i
(·)=c
i,i
(·)+λ
i
Ù¥λ
i
´ê…˜c
i,j
(·)=c
i,j
(·)Ù¥i6=j,i,j=1,2,···,n.XÚ(1)U
DOI:10.12677/pm.2022.12122212048nØêÆ
Á•
‰































∂
β
t

x
i
(t)−
n
P
j=1
d
ij
x
j
(t−τ
j
(t))

= −λ
i

x
i
(t)−
n
P
j=1
d
ij
x
j
(t−τ
j
(t))

−λ
i
n
P
j=1
d
ij
x
j
(t−τ
j
(t))+
n
P
j=1
˜c
ij
x
j
(t)+
n
P
j=1
a
ij
f
j
(x
j
(t))
+
n
P
j=1
b
ij
g
j
(x
j
(t−τ
j
(t)))+
n
P
j=1
l
ij
R
t
t−r(t)
h
j
(x
j
(s))ds
+
n
P
j=1
∂
γ
t
R
t
0
σ
ij
(s,x
j
(s),x
j
(s−τ(s)))dW
j
(s),t>0,
x
i
(t) = φ
i
(t),t∈[ϑ,0].
Ún3.1.^±e/ªL«Caputo©ê‘Ň©XÚ
(
∂
β
t
ϕ(t) = λϕ(s)+af(ϕ(t))+∂
α
t
R
t
0
B(s)dW
s
,
ϕ(0) = ϕ
0
,
Ù¥a´~ê…B(s) : [0,∞) →R,@o§ù‡XÚduÈ©•§
ϕ(s) =ϕ
0
E
β,1
(λt
β
)+a
Z
t
0
(t−s)
β−1
E
β,β
(λ(t−s)
β
)f(ϕ(s))ds
+
Z
t
0
(t−s)
β−α
E
β,β−α+1
(λ(t−s)
β
)B(s)dW
s
.
Proof.ÏL½ÂCaputo©ê꧷‚Œ±
1
Γ(1−β)
d
dt
Z
t
0
(t−s)
−β
(ϕ(s)−ϕ(0))ds=λϕ(s)+af(ϕ(t))
+
1
Γ(1−α)
d
dt
Z
t
0
(t−s)
−α
Z
s
0
B(u)dW
u
ds.
éªü>È©¿…¦^½n2.5,
1
Γ(1−β)
Z
t
0
(t−s)
−β
(ϕ(s)−ϕ(0))ds= ∗λ
Z
t
0
ϕ(s)ds+a
Z
t
0
f(ϕ(s))ds
+
1
Γ(2−α)
Z
t
0
(t−s)
1−α
B(s)dW
s
.
,
1
Γ(1−β)
Z
t
0
(t−s)
−β
ϕ(s)ds=
ϕ(0)
(1−β)Γ(1−β)
t
1−β
+λ
Z
t
0
ϕ(s)ds+a
Z
t
0
f(ϕ(s))ds
+
1
Γ(2−α)
Z
t
0
(t−s)
1−α
B(s)dW
s
.
DOI:10.12677/pm.2022.12122212049nØêÆ
Á•
3ü>¦^.Ê.dC†§·‚Œ±
1
Γ(1−β)
Γ(1−β)
s
1−β
ˆϕ(s) =
ϕ(0)
(1−β)Γ(1−β)
Γ(2−β)
s
2−β
+λ
1
s
ˆϕ(s)+a
1
s
ˆ
f(ϕ(s))
+
1
Γ(2−α)
Γ(2−α)
s
2−α
Z
∞
0
e
−st
B(t)dW
t
,
'u
R
t
0
(t−s)
1−α
B(s)dW
s
.Ê.dC†,·‚¦^Ún2.5§
Z
∞
0
e
−st
Z
t
0
(t−v)
1−α
B(v)dW
v
dt
=
Z
∞
0
B(v)dW
v
Z
∞
v
e
−st
(t−v)
1−α
dt
=
Z
∞
0
B(v)dW
v
Z
∞
0
e
−s(u+v)
u
1−α
du
=
Z
∞
0
e
−sv
B(v)dW
v
Z
∞
0
e
−su
u
1−α
du,
Ïd,
ˆϕ(s) =
s
β−1
s
β
−λ
ϕ(0)+
1
s
β
−λ
ˆ
f(ϕ(s))+
s
α−1
s
β
−λ
Z
∞
0
e
−st
B(t)dW
t
.
ü>_.Ê.dC†±9ŠâÚn2.1,·‚
ϕ(s) =ϕ
0
E
β,1
(λt
β
)+a
Z
t
0
(t−s)
β−1
E
β,β
(λ(t−s)
β
)f(ϕ(s))ds
+
Z
t
0
(t−s)
β−α
E
β,β−α+1
(λ(t−s)
β
)B(s)dW
s
.
•{B,P
˜
E
α,β
(t) =t
β−1
E
α,β
(−λ
i
t
α
)"Ïd§·‚Œ±ÏL†Ú n3.1y²aq•{-
•§(1)"
x
i
(t)=

φ
i
(0)−
n
X
j=1
d
ij
φ
j
(0−τ
j
(0))

E
β,1
(−λ
i
t
β
)+
n
X
j=1
d
ij
x
j
(t−τ
j
(t))
−λ
i
Z
t
0
˜
E
β,β
(t−s)
n
X
j=1
d
ij
x
j
(s−τ
j
(s))ds+
Z
t
0
˜
E
β,β
(t−s)
n
X
j=1
˜c
ij
x
j
(s)ds
+
Z
t
0
˜
E
β,β
(t−s)
n
X
j=1
a
ij
f
j
(x
j
(s))ds+
Z
t
0
˜
E
β,β
(t−s)
n
X
j=1
b
ij
g
j
(x
j
(s−τ
j
(s)))ds
+
Z
t
0
˜
E
β,β
(t−s)
n
X
j=1
l
ij
Z
s
s−r(s)
h
j
(x
j
(u))duds
+
Z
t
0
˜
E
β,β−γ+1
(t−s)
n
X
j=1
σ
ij
(s,x
j
(s),x
j
(s−τ(s)))dW
j
(s).
DOI:10.12677/pm.2022.12122212050nØêÆ
Á•
ÏL(Qϕ)(t) = φ(t)½ÂŽf§Ù¥t∈[−τ,0],t>0,i= 1,2,3,···,n,
(Qϕ)
i
(t)=

φ
i
(0)−
n
X
j=1
d
ij
φ
j
(0−τ
j
(0))

E
β,1
(−λ
i
t
β
)+
n
X
j=1
d
ij
ϕ
j
(t−τ
j
(t))
−λ
i
Z
t
0
˜
E
β,β
(t−s)
n
X
j=1
d
ij
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