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PureMathematics
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,2022,12(1),132-147
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.121018
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ExistenceofMildSolutionsfor
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ofSobolevType
YujieBai
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Dec.15
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DOI:10.12677/pm.2022.121018
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Abstract
Inthispaper,byutilizingtheresolventoperatortheoryandthefixedpointtheorem,
theexistenceofmildsolutionsfornonlocalproblemsofRiemann-Liouvillefractional
stochasticevolutionequationsofSobolev-typewithorder
α
∈
(1
,
2)
isdiscussedin
Hilbertspaces.
Keywords
Riemann-LiouvilleFractionalDerivative, StochasticFractionalEvolutionSystemsof
SobolevType, FixedPointTheorem,TheMeasureofNoncompactness
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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.
é
∀
α,β>
0,
e
•
3
ω
>
0
Ú
r
ë
Y
¼
ê
S
E
α,β
: [0
,
∞
)
→B
(
H
),
¦
S
E
α,β
(
t
)
´
(
M,ω
)
.
,
{
λ
α
:
Reλ>ω
}⊂
ρ
E
(
A
),
…
é
∀
x
∈
H
,
k
λ
α
−
β
R
(
λ
α
E,A
)
x
=
Z
∞
0
e
−
λt
S
E
α,β
(
t
)
xdt,Reλ>ω,
(2
.
2)
K
¡
{
S
E
α,β
(
t
)
}
t
>
0
´
d
Ž
f
é
(
A,E
)
)
¤
(
α,β
)-
ý
)
x
.
5
2.1
é
∀
1
<α<
2,
β
=
α
−
1
>
0,
Š
â
(2
.
2)
ª
Œ
λR
(
λ
α
E,A
)
x
=
Z
∞
0
e
−
λt
S
E
α,α
−
1
(
t
)
xdt,Reλ>ω,x
∈
H
,
K
¡
{
S
E
α,α
−
1
(
t
)
}
t
>
0
´
d
Ž
f
é
(
A,E
)
)
¤
(
M,ω
)
.
(
α,α
−
1)-
ý
)
x
.
A
O
/
,
β
= 1,
d
(2
.
2)
ª
Œ
•
λ
α
−
1
R
(
λ
α
E,A
)
x
=
Z
∞
0
e
−
λt
S
E
α,
1
(
t
)
xdt,Reλ>ω,x
∈
H
,
K
¡
{
S
E
α,
1
(
t
)
}
t
>
0
´
d
Ž
f
é
(
A,E
)
)
¤
(
α,
1)-
ý
)
x
.
DOI:10.12677/pm.2022.121018136
n
Ø
ê
Æ
x
Œ
'
é
∀
α,β,γ>
0,
d
c
g
α
(
λ
) =
λ
−
α
Ú
c
D
α
t
u
(
λ
) =
λ
α
b
u
(
λ
)
•
\
S
E
α,β
+
γ
(
λ
) =
λ
α
−
(
β
+
γ
)
E
(
λ
α
E
−
A
)
−
1
=
1
λ
γ
λ
α
−
β
E
(
λ
α
E
−
A
)
−
1
=
1
λ
γ
d
S
E
α,β
(
λ
)
=
\
(
g
γ
∗
S
E
α,β
)(
λ
)
.
Š
â
Laplace
C
†
•
˜
5
,
Œ
S
E
α,β
+
γ
(
t
) = (
g
γ
∗
S
E
α,β
)(
t
)
,t
>
0
.
1
<α<
2
,β
=
α
−
1
,γ
= 1
ž
,
k
S
E
α,α
(
t
) = (
g
1
∗
S
E
α,α
−
1
)(
t
) =
Z
t
0
S
E
α,α
−
1
(
s
)
ds,t
>
0
.
Ï
d
,
é
∀
x
∈
H
,
k
R
(
λ
α
E,A
)
x
=
Z
∞
0
e
−
λt
S
E
α,α
(
t
)
dt,Reλ>ω.
a
q
u
©
z
[6],
|
^
Laplace
C
†
5
Ÿ
(2.1)
ª
Ú
(2.2)
ª
,
·
‚
‰
Ñ
X
e
½
Â
.
½
Â
2.7
e
‘
Å
L
§
x
∈
C
(
I,
H
)
÷
v
È
©•
§
x
(
t
) =
S
E
α,α
−
1
(
t
)(
x
0
−
g
(
x
))+
S
E
α,α
(
t
)(
x
1
−
h
(
x
))
+
Z
t
0
S
E
α,α
(
t
−
s
)
f
(
s,x
(
s
))
ds
+
Z
t
0
S
E
α,α
(
t
−
s
)
σ
(
s,x
(
s
))
dW
(
s
)
,t
∈
I,
(2.3)
K
¡
x
•
š
Û
Ü
¯
K
(1.1)
mild
)
.
5
2.2
d
Laplace
C
†
•
˜
5
,
š
Û
Ü
¯
K
(1.1)
mild
)
•
Œ
±
¤
x
(
t
) =
S
E
α,α
−
1
(
t
)(
x
0
−
g
(
x
))+(
g
1
∗
S
E
α,α
−
1
)(
t
)(
x
1
−
h
(
x
))+
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
f
(
s,x
(
s
))
ds
+
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
σ
(
s,x
(
s
))
dW
(
s
)
,t
∈
I.
(2.4)
Ú
n
2.1
[8]
e
Ž
f
é
(
A,E
)
)
¤
˜
‡
(
M,ω
)
.
(
α,β
)-
ý
)
x
{
S
E
α,β
(
t
)
}
t
>
0
,
K
é
?
¿
γ>
0, (
A,E
)
•
)
¤
˜
‡
(
M
ω
γ
,ω
)
.
(
α,β
+
γ
)-
ý
)
x
{
S
E
α,β
+
γ
(
t
)
}
t
>
0
.
Ú
n
2.2
[8]
α>
0
,
1
<β
6
2.
e
{
S
E
α,β
(
t
)
}
t
>
0
´
d
Ž
f
é
(
A,E
)
)
¤
(
M,ω
)
.
(
α,β
)-
ý
)
x
,
K
é
∀
t>
0,
¼
ê
t
7→
S
E
α,β
(
t
)
3
B
(
H
)
¥
ë
Y
.
D
⊂
H
´
˜
‡
š
˜
k
.
4
à
8
,
P
DOI:10.12677/pm.2022.121018137
n
Ø
ê
Æ
x
Œ
'
γ
(
D
) := inf
{
ε>
0 :
D
3
H
¥
kk
•
ε
−
}
.
•
D
Hausdorff
š
;
5
ÿ
Ý
,
©
O
^
γ
(
·
)
Ú
γ
C
(
·
)
L
«
˜
m
H
Ú
C
(
I,
H
)
¥
Hausdorff
š
;
5
ÿ
Ý
.
e
B
⊂
C
(
I,
H
)
k
.
,
K
é
∀
t
∈
I,B
(
t
):=
{
u
(
t
):
u
∈
B
}
•
H
¥
k
.
f
8
…
γ
(
B
(
t
))
6
γ
C
(
B
).
Ú
n
2.3
[11]
S,T
´
Banach
˜
m
X
¥
š
˜
k
.
8
,
ρ
∈
R
,
K
š
;
5
ÿ
Ý
γ
(
·
)
÷
v
±
e
5
Ÿ
:
(1)
γ
(
S
) = 0
⇔
S
•
ƒ
é
;
8
;
(2)
S
⊂
T
=
⇒
γ
(
S
)
6
γ
(
T
);
(3)
γ
(
S
+
T
)
6
γ
(
S
)+
γ
(
T
),
Ù
¥
S
+
T
=
{
x
+
y
:
x
∈
S,y
∈
T
}
;
(4)
γ
(
S
∪
T
)
6
max
{
γ
(
S
)
,γ
(
T
)
}
;
(5)
γ
(
ρS
) =
|
ρ
|
γ
(
S
).
Ú
n
2.4
[12]
X
•
Banach
˜
m
,
Ž
f
P
:
D
(
P
)
⊂
X
→
X
ë
Y
k
.
.
e
é
?
¿
k
.
š
ƒ
é
;
8
S
⊂
D
(
P
),
k
γ
(
P
(
S
))
<γ
(
S
)
,
K
¡
P
´
v
à
N
.
Ú
n
2.5
[13]
σ
:
I
×
Ω
→
L
0
2
´
˜
‡
r
Œ
ÿ
N
,
e
Z
a
0
E
k
σ
(
θ
)
k
p
L
0
2
dθ<
+
∞
.
K
E
Z
t
0
σ
(
θ
)
dW
(
θ
)
p
6
L
σ
Z
t
0
E
k
σ
(
θ
)
k
p
L
0
2
dθ,
∀
t
∈
I,p
>
2
,
Ù
¥
L
σ
>
0
´
†
p
Ú
a
ƒ
'
~
ê
.
Ú
n
2.6
[1]( Marzur
½
n
)
D
´
Banach
˜
m
X
¥
˜
‡
;
f
8
.
K
§
à
4
•
conv
(
D
)
•
´
;
.
Ú
n
2.7
[1]( Krasnoselskii
Ø
Ä:½
n
)
B
•
Banach
˜
m
X
¥
˜
‡
š
˜
4
à
f
8
.
e
Ž
f
P,Q
:
B
→
X
÷
v
(i)
é
∀
x,y
∈
B
,
k
Px
+
Qy
∈
B
;
(ii)
P
´
Ø
Ž
f
;
(iii)
Q
´
ë
Y
Ž
f
,
K
P
+
Q
3
B
S
–
k
˜
‡
Ø
Ä:
.
Ú
n
2.8
[14](Sadovskii
Ø
Ä:½
n
)
X
•
Banach
˜
m
,
S
⊂
X
•
k
.
4
à
8
.
e
F
:
S
→
S
•
v
à
N
,
K
F
3
S
þ
–
•
3
˜
‡
Ø
Ä:
.
3.
Ì
‡
(
J
9
y
²
3ù
˜
Ü
©
,
·
‚
©
O
|
^
Krasnoselskii
Ø
Ä:½
n
Ú
Sadovskii
Ø
Ä:½
n
y
²
X
Ú
(1.1)
DOI:10.12677/pm.2022.121018138
n
Ø
ê
Æ
x
Œ
'
mild
)
•
3
5
.
é
?
¿
~
ê
r>
0,
½
Â
8
Ü
B
r
•
B
r
:=
{
x
∈
C
(
I,
H
) :
E
k
x
(
t
)
k
2
6
r,t
∈
I
}
.
•
y
²
©
Ì
‡
(
Ø
,
·
‚
Ú
\X
e
b
^
‡
:
(
F
1)
¼
ê
f
:
I
×
H
→
H
÷
v
:
(
i
)
é
∀
t
∈
I
,
f
(
t,
·
) :
H
→
H
ë
Y
,
é
∀
x
∈
H
,
f
(
·
,x
) :
I
→
H
Œ
ÿ
.
(
ii
)
•
3
¼
ê
m
∈
L
1
(
I,
R
+
),
¦
E
k
f
(
t,x
)
k
2
6
m
(
t
)
E
k
x
k
2
,
∀
t
∈
I,x
∈
H
.
(
F
2)
¼
ê
g,h
:
C
(
I,
H
)
→
H
ë
Y
,
…
•
3
~
ê
N
g
,N
h
>
0,
¦
é
∀
x,y
∈
C
(
I,
H
),
k
E
k
g
(
x
)
k
2
6
N
g
(
E
k
x
k
2
+1)
,E
k
g
(
x
)
−
g
(
y
)
k
2
6
N
g
E
k
x
−
y
k
2
,
E
k
h
(
x
)
k
2
6
N
h
(
E
k
x
k
2
+1)
,E
k
h
(
x
)
−
h
(
y
)
k
2
6
N
h
E
k
x
−
y
k
2
.
(
F
3)
¼
ê
σ
:
I
×
H
→
L
0
2
ë
Y
,
…
•
3
~
ê
N
σ
>
0,
¦
é
∀
t
∈
I,x,y
∈
H
,
k
E
k
σ
(
t,x
)
k
2
6
N
σ
(
E
k
x
k
2
+1)
,E
k
σ
(
t,x
)
−
σ
(
t,y
)
k
2
6
N
σ
E
k
x
−
y
k
2
.
(
F
4)
é
∀
t
∈
I
,
8
Ü
V
ε
:=
{
f
(
s,x
(
s
)) :
x
∈
B
r
,s
∈
[0
,t
−
ε
]
,ε
∈
(0
,t
)
}
´
;
.
©
O
½
Â
Ž
f
Q
1
,
Q
2
:
C
(
I,
H
)
→
C
(
I,
H
)
X
e
:
(
Q
1
x
)(
t
) :=
S
E
α,α
−
1
(
t
)(
x
0
−
g
(
x
))+(
g
1
∗
S
E
α,α
−
1
)(
t
)(
x
1
−
h
(
x
))
+
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
σ
(
s,x
(
s
))
dW
(
s
)
,
(
Q
2
x
)(
t
) :=
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
f
(
s,x
(
s
))
ds.
d
½
Â
2.7,
X
Ú
(1.1)
mild
)
d
u
Ž
f
Q
:=
Q
1
+
Q
2
Ø
Ä:
,
Ï
d
,
é
Ž
f
Q
A^
Krasnoselskii
Ø
Ä:½
n
y
²
Ù
3
B
r
þ
–
•
3
˜
‡
Ø
Ä:
.
Ú
n
3.1
e
^
‡
(
F
1)
−
(
F
3)
÷
v
,
…
Ø
ª
4
M
2
e
2
ωa
ω
2
(2
N
g
ω
2
+2
N
h
+
a
k
m
k
∞
+
aL
σ
N
σ
)
<
1(3.1)
¤
á
,
K
•
3
˜
‡
~
ê
r>
0,
¦
Q
:
B
r
→
B
r
.
DOI:10.12677/pm.2022.121018139
n
Ø
ê
Æ
x
Œ
'
y
²
:
w
,
B
r
•
C
(
I,
H
)
¥
š
˜
k
.
4
à
8
,
‡
é
∀
r>
0,
∃
x
∈
B
r
,
¦
E
k
(
Q
x
)(
t
)
k
2
>
r
.
d
Q
½
Â
,
k
r<E
k
(
Q
x
)(
t
)
k
2
6
4
E
(
k
S
E
α,α
−
1
(
t
)
kk
x
0
−
g
(
x
)
k
)
2
+4
E
(
k
(
g
1
∗
S
E
α,α
−
1
)(
t
)
kk
x
1
−
h
(
x
)
k
)
2
+4
E
k
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
f
(
s,x
(
s
))
ds
k
2
+4
E
k
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
σ
(
s,x
(
s
))
dW
(
s
)
k
2
6
4
M
2
e
2
ωa
E
k
x
0
−
g
(
x
)
k
2
+4
M
2
e
2
ωa
ω
2
E
k
x
1
−
h
(
x
)
k
2
+4
M
2
e
2
ωa
ω
2
Z
t
0
E
k
f
(
s,x
(
s
))
k
2
ds
+4
M
2
e
2
ωa
ω
2
L
σ
Z
t
0
E
k
σ
(
s,x
(
s
))
k
2
ds
6
4
M
2
e
2
ωa
ω
2
[(2
E
k
x
0
k
2
+2
N
g
(
r
+1))
ω
2
+(2
E
k
x
1
k
2
+2
N
h
(
r
+1))+
ra
k
m
k
∞
+
aL
σ
N
σ
(
r
+1)]
,
þ
ª
ü
>
Ó
Ø
±
r
,
r
→∞
ž
,
Œ
1
6
4
M
2
e
2
ωa
ω
2
(2
N
g
ω
2
+2
N
h
+
a
k
m
k
∞
+
aL
σ
N
σ
)
,
ù
†
(3.1)
ª
g
ñ
.
Ï
d
,
•
3
r>
0,
¦
Q
:
B
r
→
B
r
.
Ú
n
3.2
e
^
‡
(
F
2)
,
(
F
3)
÷
v
,
K
Ž
f
Q
1
3
B
r
¥
Ø
.
y
²
:
é
∀
x,y
∈
B
r
,t
∈
I,
d
(3.1)
ª
,
k
E
k
(
Q
1
x
)(
t
)
−
(
Q
1
y
)(
t
)
k
2
6
3
E
(
k
S
E
α,α
−
1
(
t
)
kk
g
(
x
)
−
g
(
y
)
k
)
2
+3
E
(
k
(
g
1
∗
S
E
α,α
−
1
)(
t
)
kk
h
(
x
)
−
h
(
y
)
k
)
2
+3
E
k
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)[
σ
(
s,x
(
s
))
−
σ
(
s,y
(
s
))]
dW
(
s
)
k
2
6
3
M
2
e
2
ωa
ω
2
(
N
g
ω
2
+
N
h
+
aL
σ
N
σ
)
E
k
x
−
y
k
2
<E
k
x
−
y
k
2
.
Ž
f
Q
1
Ø
.
Ú
n
3.3
e
^
‡
(
F
1)
÷
v
,
K
Q
2
:
B
r
→
B
r
ë
Y
.
DOI:10.12677/pm.2022.121018140
n
Ø
ê
Æ
x
Œ
'
y
²
:
S
{
x
n
}
n
>
1
⊂
B
r
,
é
∀
x
∈
B
r
,
÷
v
lim
n
→
+
∞
x
n
=
x
.
K
d
Q
2
½
Â
,
Œ
•
E
k
(
Q
2
x
n
)(
t
)
−
(
Q
2
x
)(
t
)
k
2
=
E
k
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)[
f
(
s,x
n
(
s
))
−
f
(
s,x
(
s
))]
ds
k
2
6
M
2
e
2
ωa
ω
2
Z
t
0
E
k
f
(
s,x
n
(
s
))
−
f
(
s,x
(
s
))
k
2
ds
6
M
2
e
2
ωa
ω
2
Z
t
0
(2
E
k
f
(
s,x
n
(
s
))
k
2
+2
E
k
f
(
s,x
(
s
))
k
2
)
ds
6
4
r
M
2
e
2
ωa
ω
2
Z
t
0
m
(
s
)
ds.
5
¿
,
¼
ê
s
7→
m
(
s
)
3
I
þ
Œ
È
,
n
→∞
ž
,
Z
t
0
f
(
s,x
n
(
s
))
−
f
(
s,x
(
s
))
ds
→
0,
Ï
d
d
Lebesgue
›
›
Â
ñ
½
n
,
Ž
f
Q
2
:
B
r
→
B
r
ë
Y
.
½
Â
8
Ü
V
:=
{Q
2
x
:
x
∈
B
r
}
,V
(
t
) :=
{
(
Q
2
x
)(
t
) :
x
∈
B
r
}
.
½
n
3.1
e
^
‡
(
F
1)
−
(
F
4)
9
(3.1)
ª
¤
á
,
K
X
Ú
(1.1)
3
I
þ
–
•
3
˜
‡
mild
)
.
y
²
:
d
Ú
n
3.1-3.3
Œ
•
,
·
‚
•
I
‡
`
²
8
Ü
V
3
C
(
I,
H
)
¥
ƒ
é
;
.
1
˜
Ú
:
y
²
8
Ü
V
´
Ý
ë
Y
.
é
∀
x
∈
B
r
,
0
6
t
2
<t
1
6
a
,
k
E
k
(
Q
2
x
)(
t
1
)
−
(
Q
2
x
)(
t
2
)
k
2
=
E
k
Z
t
1
0
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
f
(
s,x
(
s
))
ds
−
Z
t
2
0
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)
f
(
s,x
(
s
))
ds
k
2
6
2
E
k
Z
t
2
0
[(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)]
f
(
s,x
(
s
))
ds
k
2
+2
E
k
Z
t
1
t
2
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
f
(
s,x
(
s
))
ds
k
2
:=
I
1
+
I
2
.
Ï
•
é
u
I
1
,
d
^
‡
(
F
1),
k
I
1
= 2
E
k
Z
t
2
0
[(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)]
f
(
s,x
(
s
))
ds
k
2
6
2
Z
t
2
0
k
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)
k
2
E
k
f
(
s,x
(
s
))
k
2
ds
6
2
r
Z
t
2
0
k
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)
k
2
m
(
s
)
ds,
…
k
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−·
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−·
)
k
2
m
(
s
)
6
4
M
2
e
2
ωa
ω
2
m
(
s
)
∈
L
1
(
I,
R
+
)
.
DOI:10.12677/pm.2022.121018141
n
Ø
ê
Æ
x
Œ
'
¤
±
,
é
∀
t
>
0,
d
Ú
n
2.1
Œ
(
g
1
∗
S
E
α,α
−
1
)(
t
) =
S
E
α,α
(
t
).
d
Ú
n
2.2
•
S
E
α,α
(
t
)
‰
ê
ë
Y
.
Ï
d
,
t
1
→
t
2
ž
, (
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)
→
0
u
B
(
H
),
Ï
d
d
Lebesgue
›
›
Â
ñ
½
n
Œ
lim
t
1
→
t
2
I
1
= 0.
é
u
I
2
,
d
^
‡
(
F
1
),
k
I
2
= 2
E
k
Z
t
1
t
2
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
f
(
s,x
(
s
))
ds
k
2
6
2
M
2
e
2
ωa
ω
2
Z
t
1
t
2
E
k
f
(
s,x
(
s
))
k
2
ds
6
2
r
M
2
e
2
ωa
ω
2
Z
t
1
t
2
m
(
s
)
ds
→
0(
t
2
−
t
1
→
0)
.
Ï
d
,
8
Ü
V
3
C
(
I,
H
)
¥
Ý
ë
Y
.
1
Ú
:
y
²
8
Ü
V
(
t
)
3
H
¥
ƒ
é
;
.
t
= 0
ž
,
V
(0)
´
ƒ
é
;
.
•
I
y
²
é
∀
t
∈
I
0
,
8
Ü
V
(
t
)
3
H
¥
ƒ
é
;
.
é
∀
0
<ε<t
,
½
Â
Ž
f
Q
ε
2
X
e
µ
(
Q
ε
2
x
)(
t
) :=
Z
t
−
ε
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
f
(
s,x
(
s
))
ds,
V
ε
2
(
t
) :=
{
(
Q
ε
2
x
)(
t
) :
x
∈
B
r
}
.
d
b
^
‡
(
F
4)
Ú
Ú
n
2.6
Œ
•
conv
(
V
ε
)
´
˜
‡
;
8
,
l
Me
ωa
ω
(
t
−
ε
)
conv
(
V
ε
)
•
´
˜
‡
;
8
,
Ù
¥
conv
(
V
ε
)
•
V
ε
4
à
•
.
d
Bochner
È
©
¥Š
½
n
,
Œ
(
Q
ε
2
x
)(
t
)
∈
Me
ωa
ω
(
t
−
ε
)
conv
(
V
ε
)
,
∀
t
∈
I.
Ï
d
,
é
∀
ε>
0,
8
Ü
V
ε
2
(
t
)
3
H
¥
ƒ
é
;
.
é
∀
x
∈
B
r
,
k
E
k
(
Q
2
x
)(
t
)
−
(
Q
ε
2
x
)(
t
)
k
2
=
E
k
Z
t
t
−
ε
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
f
(
s,x
(
s
))
ds
k
2
6
M
2
e
2
ωa
ω
2
Z
t
t
−
ε
E
k
f
(
s,x
(
s
))
k
2
ds
6
M
2
e
2
ωa
ω
2
r
Z
t
t
−
ε
m
(
s
)
ds.
¼
ê
s
7→
m
(
s
)
∈
L
1
([
t
−
ε,t
]
,
R
+
),
d
Lebesgue
›
›
Â
ñ
½
n
,
Œ
lim
ε
→
0
E
k
(
Q
2
x
)(
t
)
−
(
Q
ε
2
x
)(
t
)
k
2
= 0
,
DOI:10.12677/pm.2022.121018142
n
Ø
ê
Æ
x
Œ
'
=
t
∈
I
0
ž
,
•
3
˜
‡
ƒ
é
;
8
V
ε
2
(
t
)
?
¿
ª
C
u
V
(
t
),
V
(
t
)
3
H
¥
ƒ
é
;
.
d
Ascoli-Arzela
½
n
•
,
é
∀
t
∈
I
,
8
Ü
V
3
C
(
I,
H
)
¥
ƒ
é
;
.
d
Ú
n
2.7
•
,
Ž
f
Q
3
B
r
¥
•
3
˜
‡
Ø
Ä:
x
,
d
Ø
Ä:
=
•
X
Ú
(1.1)
mild
)
.
e
5
,
b
S
E
α,α
−
1
(
t
)(
t>
0)
3
˜
—
Ž
f
ÿ
À
¥
ë
Y
,
·
‚
Ú
\X
e
b
:
(
F
5)
é
∀
t
∈
I
,
8
Ü
{
Z
t
−
ε
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
ε
−
s
)
σ
(
s,x
(
s
))
dW
(
s
):
x
∈
B
r
,s
∈
[0
,t
−
ε
]
,ε
∈
(0
,t
)
}
´
;
.
½
n
3.2
e
^
‡
(
F
1)
−
(
F
5)
Ú
(3.1)
ª
¤
á
,
K
X
Ú
(1.1)
3
I
þ
–
•
3
˜
‡
mild
)
.
y
²
:
w
,
,
d
b
^
‡
(
F
1)
−
(
F
3),
Ž
f
Q
:
C
(
I,
H
)
→
C
(
I,
H
)
ë
Y
.
½
Â
Ž
f
Q
=
Q
1
+
Q
2
X
e
:
(
Q
1
x
)(
t
) :=
S
E
α,α
−
1
(
t
)(
x
0
−
g
(
x
))+(
g
1
∗
S
E
α,α
−
1
)(
t
)(
x
1
−
h
(
x
))
,
(
Q
2
x
)(
t
) :=
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
f
(
s,x
(
s
))
ds
+
Z
t
0
(
g
1
∗
S
E
α,α
−
1
)(
t
−
s
)
σ
(
s,x
(
s
))
dW
(
s
)
.
e
y
W
:=
{Q
2
x
:
x
∈
B
r
}
3
C
(
I,
H
)
¥
ƒ
é
;
.
d
Ascoli-Arzela
½
n
,
·
‚
I
‡
y
²
W
3
C
(
I,
H
)
¥
˜
—
k
.
…
Ý
ë
Y
,
W
(
t
)
3
H
¥
ƒ
é
;
.
a
q
u
Ú
n
3.1
y
²
•
W
´
˜
—
k
.
.
Ä
k
y
²
W
Ý
ë
Y
.
é
∀
x
∈
B
r
,
0
6
t
2
<t
1
6
a
ž
,
k
E
k
(
Q
2
x
)(
t
1
)
−
(
Q
2
x
)(
t
2
)
k
2
=
E
k
Z
t
2
0
[(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)]
f
(
s,x
(
s
))
ds
+
Z
t
2
0
[(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)]
σ
(
s,x
(
s
))
dW
(
s
)
+
Z
t
1
t
2
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
f
(
s,x
(
s
))
ds
+
Z
t
1
t
2
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
σ
(
s,x
(
s
))
dW
(
s
)
k
2
6
4
E
k
Z
t
2
0
[(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)]
f
(
s,x
(
s
))
ds
k
2
+4
E
k
Z
t
2
0
[(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
−
(
g
1
∗
S
E
α,α
−
1
)(
t
2
−
s
)]
σ
(
s,x
(
s
))
dW
(
s
)
k
2
+4
E
k
Z
t
1
t
2
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
f
(
s,x
(
s
))
ds
k
2
+4
E
k
Z
t
1
t
2
(
g
1
∗
S
E
α,α
−
1
)(
t
1
−
s
)
σ
(
s,x
(
s
))
dW
(
s
)
k
2
:= 4
4
X
i
=1
J
i
.
DOI:10.12677/pm.2022.121018143
n
Ø
ê
Æ
x
Œ
'
d
½
n
3.1
y
²
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