Sobolev型分数阶随机发展方程非局部问题 Mild解的存在性 Existence of Mild Solutions for Nonlocal Problems of Fractional Stochastic Evolution Equations of Sobolev Type

doi:10.12677/PM.2022.121018

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PureMathematicsnØêÆ,2022,12(1),132-147

PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.121018

Sobolev.©ê‘ÅuÐ•§šÛÜ¯K

Mild)•35

xxxŒŒŒ'''

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2021c1215F¶¹^FÏµ2022c117F¶uÙFÏµ2022c124F

Á‡

©|^ØÄ:½nÚý)ŽfnØ?ØHilbert ˜m¥Sobolev .α∈(1,2) Riemann-

Liouville©ê‘ÅuÐ•§šÛÜ¯Kmild)•35"

'…c

Riemann-Liouville©êê§Sobolev.©ê‘ÅuÐXÚ§ØÄ:½n§

š;5ÿÝ

ExistenceofMildSolutionsfor

NonlocalProblemsofFractional

StochasticEvolutionEquations

ofSobolevType

YujieBai

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Dec.15

,2021;accepted:Jan.17

,2022;published:Jan.24

,2022

©ÙÚ^:xŒ'.Sobolev.©ê‘ÅuÐ•§šÛÜ¯KMild)•35[J].nØêÆ,2022,12(1):132-147.

DOI:10.12677/pm.2022.121018

xŒ'

Abstract

Inthispaper,byutilizingtheresolventoperatortheoryandtheﬁxedpointtheorem,

theexistenceofmildsolutionsfornonlocalproblemsofRiemann-Liouvillefractional

stochasticevolutionequationsofSobolev-typewithorderα∈(1,2)isdiscussedin

Hilbertspaces.

Keywords

Riemann-LiouvilleFractionalDerivative, StochasticFractionalEvolutionSystemsof

SobolevType, FixedPointTheorem,TheMeasureofNoncompactness

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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Riemann-Liouville.©êuÐ•§šÛÜ¯Kmild)•35.

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Œ›5. 3©z[5]¥,Benchaabane<|^ŽfŒ+nØ!©ê‡È©Ú‘Å©ÛEâïá

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.©ê‘ÅuÐ•§mild)•3•˜5.

DOI:10.12677/pm.2022.121018133nØêÆ

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â·‚¤•,Sobolev.α∈(0,1)uÐ•§)•35®²2•ïÄ,'u

α∈(1,2) Riemann-Liouville .©ê‘ÅuÐ•§mild )•35ïÄ(Jƒé. É

þã©zéu, ©ò|^ý)ŽfnØÚØÄ:½nïÄHilbert˜m¥äkšÛÜ^‡

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





(Ex(t)) = Ax(t)+f(t,x(t))+σ(t,x(t))

dW(t)

d(t)

, t∈I

:= (0,a],

E(g

2−α

∗x)(0) = x

−g(x),E(g

2−α

∗x)

(0) = x

−h(x)

(1.1)

mild )•35, Ù¥1<α<2, D

´αRiemann-Liouville .©êê, G¼êx(t) 

ŠuHilbert ˜mH, A:D(A)⊆H→H´È½4‚5Žf,E:D(E)⊆H→H´4‚5Žf,

∈H,¼êf,g,h,σ´e©‰½·¼ê.

ïÄSobolev.‡©•§ž,·‚Ï~b:

(1)E,A´4‚5Žf;

(2)D(E) ⊂D(A),E´V;

(3)E

−1

´;Žf.

3ù«œ¹e,−AE

−1

´˜‡k.‚5Žf,§Œ±)¤˜‡˜—ëYŒ+,ë„©z[5][7].

©3ØbŽfE

−1

•35Ú;5œ¹e,$^Žfé(A,E))¤(α,α−1)-ý)x

α,α−1

(t)}

t>0

˜5ŸÚLaplaceC†5½ÂXÚ(1.1)mild),¿3vký)ŽfS

α,α−1

(t)

;5^‡e,|^ØÄ:½nÚý)ŽfnØy²XÚ(1.1)mild)•35.

2.ý•£

(Ω,F,{F

}

t>0

,P)L«‘kÈf{F

}

t>0

…÷v~5^‡VÇ˜m.3(Ω,F,{F

}

t>0

,P)

¥{W

}

t>0

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>0´˜

‡k.S,{e

}

k>0

´H˜‡IOX,÷v

= `

,k= 1,2,3···.

{β

}

k>1

´˜‡ÕáÙK$Ä…÷v

hW(t),xi=

∞

k=1

,xiβ

(t),∀x∈H,t>0.

F

´d{W(s) : 0 6s6t})¤σ-“ê. -L

:= L

,Y). KL

´˜‡¢Œ©Hilbert

˜m, äk‰êkπk

= tr[πQπ

∗

], L

(Ω,H) L«Hilbert ˜mH¥¤krF

-Œÿ‘ÅCþ¤

˜m.C(I,L

(Ω,H)) ´lI:= [0,a] L

(Ω,H) ëYN˜m,…÷vÏ"^‡

sup

t∈I

Ekx(t)k

<∞.

DOI:10.12677/pm.2022.121018134nØêÆ

xŒ'

C(I,H) •C(I,L

(Ω,H)) 4f˜m, dŒÿÚF

-·AH-ŠL§x∈C(I,L

(Ω,H))

|¤,Ù‰ê½Â•

kxk

= (sup

t∈I

Ekx(t)k

)

,∀x∈C(I,H),

K(C(I,H),k·k

) •Banach˜m.B(H) :=B(H,H) L«lHgk.‚5ŽfNU

‰êk·k¤˜m.

½Â2.1 [8][1]é∀α>0, 

(t) =











α−1

Γ(α)

,t>0,

0,t60,

Ù¥Γ(·)L«Gamma¼ê.˜„/,¼êfÚgòÈ½Â•

(f∗g)(t) =

f(t−s)g(s)ds.

½Â2.2 [8][1]¼êu∈L

(I) α>0Riemann-Liouville.©êÈ©½Â•

u(t) := (g

∗u)(t) =

(t−s)u(s)ds,t>0.

AO/,-J

u(t) = u(t),dòÈ5Ÿ,È©Žf{J

}

α>0

÷vŒ+Æ

= J

α+β

,α,β>0.

½Â2.3[8][1]¼êu∈L

(I) ÷vg

n−α

∗u∈W

n,1

(I), u∈L

(I) α>0 Riemann-

Liouville.©êê½Â•

u(t) := D

n−α

∗u)(t) =

n−α

(t−s)u(s)ds,t>0,

Ù¥D

,n= [α]L«Œu½uα•ê.

½Â2.4 [6]¼êf½Â3R

þ.eÈ©

∞

−λt

f(t)dt

Âñ,KfLaplace C†•

f(λ) =

∞

−λt

f(t)dt.

DOI:10.12677/pm.2022.121018135nØêÆ

xŒ'

Riemann-Liouville.©êêLaplaceC†•

u(λ) = λ

bu(λ)−

n−1

k=0

n−α

∗u)

(k)

(0)λ

n−1−k

,(2.1)

Ù¥α>0,n= [α].

e¡,·‚0˜'u©êý)x{S

α,β

(t)}

t>0

ÄVg,•õ[!ë„©z[8][1].

½Â

(A) = {λ∈C: (λE−A) : D(E)∩D(A) →HŒ_,

(λE−A)

−1

∈B(H,D(E)∩D(A))}

•Žfé(A,E)ý)8,Ù¥D(A)ÚD(E) ©OL«ŽfAÚE½Â•. é∀λ>0,·

‚¡R(λE,A) := (λE−A)

−1

•ŽfAE-ý)Žf.

½Â2.5[9]{T(t)}

t>0

⊆B(H) ´˜‡C

-Œ+.e•3~êM>1,ω>0 ÷v

kT(t)k6Me

ωt

,t>0,

K¡{T(t)}

t>0

´(M,ω) .½•êk..

½Â2.6[10]A: D(A) ⊆H→H,E: D(E) ⊆H→H´Hilbert ˜mH¥4‚5Žf,

÷vD(A)∩D(E) 6= {0}.é∀α,β>0, e•3ω>0ÚrëY¼êS

α,β

: [0,∞) →B(H), ¦

α,β

(t)´(M,ω) .,{λ

: Reλ>ω}⊂ρ

(A),…é∀x∈H, k

α−β

R(λ

E,A)x=

∞

−λt

α,β

(t)xdt,Reλ>ω,(2.2)

K¡{S

α,β

(t)}

t>0

´dŽfé(A,E))¤(α,β)-ý)x.

52.1é∀1 <α<2,β= α−1 >0,Šâ(2.2)ªŒ

λR(λ

E,A)x=

∞

−λt

α,α−1

(t)xdt,Reλ>ω,x∈H,

K¡{S

α,α−1

(t)}

t>0

´dŽfé(A,E))¤(M,ω) .(α,α−1)-ý)x. AO/,β= 1,d

(2.2)ªŒ•

α−1

R(λ

E,A)x=

∞

−λt

α,1

(t)xdt,Reλ>ω,x∈H,

K¡{S

α,1

(t)}

t>0

´dŽfé(A,E))¤(α,1)-ý)x.

DOI:10.12677/pm.2022.121018136nØêÆ

xŒ'

é∀α,β,γ>0,dcg

(λ) = λ

−α

u(λ) = λ

bu(λ)•

α,β+γ

(λ) = λ

α−(β+γ)

E(λ

E−A)

−1

α−β

E(λ

E−A)

−1

α,β

(λ)

∗S

α,β

)(λ).

ŠâLaplaceC†•˜5,Œ

α,β+γ

(t) = (g

∗S

α,β

)(t),t>0.

1 <α<2,β= α−1,γ= 1 ž,k

α,α

(t) = (g

∗S

α,α−1

)(t) =

α,α−1

(s)ds,t>0.

Ïd,é∀x∈H, k

R(λ

E,A)x=

∞

−λt

α,α

(t)dt,Reλ>ω.

aqu©z[6],|^LaplaceC†5Ÿ(2.1)ªÚ(2.2)ª,·‚‰ÑXe½Â.

½Â2.7e‘ÅL§x∈C(I,H) ÷vÈ©•§

x(t) = S

α,α−1

(t)(x

−g(x))+S

α,α

(t)(x

−h(x))

α,α

(t−s)f(s,x(s))ds+

α,α

(t−s)σ(s,x(s))dW(s),t∈I,(2.3)

K¡x•šÛÜ¯K(1.1)mild).

52.2dLaplaceC†•˜5,šÛÜ¯K(1.1)mild)•Œ±¤

x(t) = S

α,α−1

(t)(x

−g(x))+(g

∗S

α,α−1

)(t)(x

−h(x))+

∗S

α,α−1

)(t−s)f(s,x(s))ds

∗S

α,α−1

)(t−s)σ(s,x(s))dW(s),t∈I.(2.4)

Ú n2.1[8]eŽfé(A,E) )¤˜‡(M,ω) .(α,β)-ý)x{S

α,β

(t)}

t>0

, Ké?¿

γ>0, (A,E)•)¤˜‡(

,ω) .(α,β+γ)-ý)x{S

α,β+γ

(t)}

t>0

Ún2.2[8]α>0,1<β62.e{S

α,β

(t)}

t>0

´dŽfé(A,E))¤(M,ω).

(α,β)-ý)x,Ké∀t>0, ¼êt7→S

α,β

(t)3B(H)¥ëY.

D⊂H´˜‡š˜k.4à8,P

DOI:10.12677/pm.2022.121018137nØêÆ

xŒ'

γ(D) := inf{ε>0 : D3H¥kk•ε−}.

•DHausdorﬀš;5ÿÝ,©O^γ(·)Úγ

(·)L«˜mHÚC(I,H)¥Hausdorﬀš

;5ÿÝ.eB⊂C(I,H)k.,Ké∀t∈I,B(t):={u(t):u∈B}•H¥k.f8…

γ(B(t)) 6γ

(B).

Ún2.3 [11]S,T´Banach˜mX¥š˜k.8,ρ∈R, Kš;5ÿÝγ(·)÷v±e

5Ÿ:

(1)γ(S) = 0 ⇔S•ƒé;8;

(2)S⊂T=⇒γ(S) 6γ(T);

(3)γ(S+T) 6γ(S)+γ(T),Ù¥S+T= {x+y:x∈S,y∈T};

(4)γ(S∪T) 6max{γ(S),γ(T)};

(5)γ(ρS) = |ρ|γ(S).

Ún2.4[12]X•Banach ˜m, ŽfP:D(P) ⊂X→XëYk.. eé?¿k.šƒé

;8S⊂D(P),k

γ(P(S)) <γ(S),

K¡P´vàN.

Ún2.5 [13]σ: I×Ω →L

´˜‡rŒÿN,e

Ekσ(θ)k

dθ<+∞.K



σ(θ)dW(θ)



Ekσ(θ)k

dθ,∀t∈I,p>2,

Ù¥L

>0´†pÚaƒ'~ê.

Ún2.6[1]( Marzur½n)D´Banach˜mX¥˜‡;f8. K§à4•conv(D)

•´;.

Ún2.7[1]( Krasnoselskii ØÄ:½n)B•Banach ˜mX¥˜‡š˜4àf8. e

ŽfP,Q: B→X÷v

(i)é∀x,y∈B,kPx+Qy∈B;

(ii)P´Ø Žf;

(iii)Q´ëYŽf,

KP+Q3BS–k˜‡ØÄ:.

Ún2.8[14](SadovskiiØÄ:½n)X•Banach˜m,S⊂X•k.4à8.e

F:S→S•vàN, KF3Sþ–•3˜‡ØÄ:.

3.Ì‡(J9y²

3ù˜Ü©,·‚©O|^KrasnoselskiiØÄ:½nÚSadovskiiØÄ:½ny²XÚ(1.1)

DOI:10.12677/pm.2022.121018138nØêÆ

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mild)•35.

é?¿~êr>0,½Â8ÜB

•

:= {x∈C(I,H) : Ekx(t)k

6r,t∈I}.

•y²©Ì‡(Ø,·‚Ú\Xeb^‡:

(F1) ¼êf: I×H→H÷v:

(i)é∀t∈I,f(t,·) : H→HëY,é∀x∈H, f(·,x) : I→HŒÿ.

(ii)•3¼êm∈L

(I,R

),¦

Ekf(t,x)k

6m(t)Ekxk

,∀t∈I,x∈H.

(F2)¼êg,h: C(I,H) →HëY, …•3~êN

>0,¦é∀x,y∈C(I,H), k

Ekg(x)k

(Ekxk

+1),Ekg(x)−g(y)k

Ekx−yk

Ekh(x)k

(Ekxk

+1),Ekh(x)−h(y)k

Ekx−yk

(F3)¼êσ: I×H→L

ëY,…•3~êN

>0,¦é∀t∈I,x,y∈H, k

Ekσ(t,x)k

(Ekxk

+1),Ekσ(t,x)−σ(t,y)k

Ekx−yk

(F4) é∀t∈I, 8ÜV

:= {f(s,x(s)) : x∈B

,s∈[0,t−ε],ε∈(0,t)}´;.

©O½ÂŽfQ

: C(I,H) →C(I,H) Xe:

x)(t) := S

α,α−1

(t)(x

−g(x))+(g

∗S

α,α−1

)(t)(x

−h(x))

∗S

α,α−1

)(t−s)σ(s,x(s))dW(s),

x)(t) :=

∗S

α,α−1

)(t−s)f(s,x(s))ds.

d½Â2.7,XÚ(1.1)mild)duŽfQ:=Q

ØÄ:,Ïd,éŽfQA^

KrasnoselskiiØÄ:½ny²Ù3B

þ–•3˜‡ØÄ:.

Ún3.1e^‡(F1)−(F3) ÷v,…Øª

2ωa

(2N

+2N

+akmk

∞

+aL

) <1(3.1)

¤á,K•3˜‡~êr>0,¦Q: B

→B

DOI:10.12677/pm.2022.121018139nØêÆ

xŒ'

y²:w,B

•C(I,H)¥š˜k.4à8,‡é∀r>0,∃x∈B

,¦Ek(Qx)(t)k

r.dQ½Â,k

r<Ek(Qx)(t)k

64E(kS

α,α−1

(t)kkx

−g(x)k)

+4E(k(g

∗S

α,α−1

)(t)kkx

−h(x)k)

+4Ek

∗S

α,α−1

)(t−s)f(s,x(s))dsk

+4Ek

∗S

α,α−1

)(t−s)σ(s,x(s))dW(s)k

64M

2ωa

Ekx

−g(x)k

2ωa

Ekx

−h(x)k

2ωa

Ekf(s,x(s))k

ds+4

2ωa

Ekσ(s,x(s))k

2ωa

[(2Ekx

+2N

(r+1))ω

+(2Ekx

+2N

(r+1))+rakmk

∞

+aL

(r+1)],

þªü>ÓØ±r,r→∞ž,Œ

1 64

2ωa

(2N

+2N

+akmk

∞

+aL

ù†(3.1)ªgñ.Ïd,•3r>0,¦Q: B

→B

.

Ún3.2e^‡(F2),(F3)÷v,KŽfQ

¥Ø .

y²:é∀x,y∈B

,t∈I,d(3.1) ª,k

Ek(Q

x)(t)−(Q

y)(t)k

63E(kS

α,α−1

(t)kkg(x)−g(y)k)

+3E(k(g

∗S

α,α−1

)(t)kkh(x)−h(y)k)

+3Ek

∗S

α,α−1

)(t−s)[σ(s,x(s))−σ(s,y(s))]dW(s)k

2ωa

+aL

)Ekx−yk

<Ekx−yk

ŽfQ

Ø .

Ún3.3e^‡(F1) ÷v,KQ

: B

→B

ëY.

DOI:10.12677/pm.2022.121018140nØêÆ

xŒ'

y²:S{x

}

n>1

⊂B

,é∀x∈B

,÷vlim

n→+∞

= x.KdQ

½Â,Œ•

Ek(Q

)(t)−(Q

x)(t)k

= Ek

∗S

α,α−1

)(t−s)[f(s,x

(s))−f(s,x(s))]dsk

2ωa

Ekf(s,x

(s))−f(s,x(s))k

2ωa

(2Ekf(s,x

(s))k

+2Ekf(s,x(s))k

)ds

64r

2ωa

m(s)ds.

5¿,¼ês7→m(s)3IþŒÈ,n→∞ž,

f(s,x

(s))−f(s,x(s))ds→0,Ïdd

Lebesgue››Âñ½n,ŽfQ

: B

→B

ëY.

½Â8ÜV:= {Q

x: x∈B

},V(t) := {(Q

x)(t) : x∈B

½n3.1e^‡(F1)−(F4) 9(3.1)ª¤á,KXÚ(1.1)3Iþ–•3˜‡mild ).

y²:dÚn3.1-3.3Œ•,·‚•I‡`²8ÜV3C(I,H) ¥ƒé;.

1˜Ú:y²8ÜV´ÝëY.

é∀x∈B

,0 6t

6a,k

Ek(Q

x)(t

)−(Q

x)(t

= Ek

∗S

α,α−1

)(t

−s)f(s,x(s))ds−

∗S

α,α−1

)(t

−s)f(s,x(s))dsk

62Ek

[(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)]f(s,x(s))dsk

+2Ek

∗S

α,α−1

)(t

−s)f(s,x(s))dsk

:= I

Ï•éuI

,d^‡(F1), k

= 2Ek

[(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)]f(s,x(s))dsk

k(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)k

Ekf(s,x(s))k

62r

k(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)k

m(s)ds,

…

k(g

∗S

α,α−1

)(t

−·)−(g

∗S

α,α−1

)(t

−·)k

m(s) 64

2ωa

m(s) ∈L

(I,R

DOI:10.12677/pm.2022.121018141nØêÆ

xŒ'

¤±,é∀t>0,dÚn2.1Œ(g

∗S

α,α−1

)(t) = S

α,α

(t).dÚn2.2•S

α,α

(t)‰êëY. Ïd,

→t

ž, (g

∗S

α,α−1

)(t

−s) −(g

∗S

α,α−1

)(t

−s)→0 uB(H), ÏddLebesgue ››Âñ½

nŒlim

→t

= 0.

éuI

,d^‡(F

),k

= 2Ek

∗S

α,α−1

)(t

−s)f(s,x(s))dsk

2ωa

Ekf(s,x(s))k

62r

2ωa

m(s)ds

→0(t

−t

→0).

Ïd,8ÜV3C(I,H) ¥ÝëY.

1Ú:y²8ÜV(t) 3H¥ƒé;.

t= 0 ž,V(0) ´ƒé;. •Iy²é∀t∈I

, 8ÜV(t) 3H¥ƒé;. é∀0 <ε<t,

½ÂŽfQ

Xeµ

x)(t) :=

t−ε

∗S

α,α−1

)(t−s)f(s,x(s))ds,

(t) := {(Q

x)(t) : x∈B

db^‡(F4)ÚÚn2.6Œ•conv(V

)´˜‡;8,l

ωa

(t−ε)conv(V

)•´˜‡;8,Ù

¥conv(V

)•V

4à•.dBochnerÈ©¥Š½n,Œ

x)(t) ∈

ωa

(t−ε)conv(V

),∀t∈I.

Ïd,é∀ε>0, 8ÜV

(t)3H¥ƒé;.é∀x∈B

Ek(Q

x)(t)−(Q

x)(t)k

= Ek

t−ε

∗S

α,α−1

)(t−s)f(s,x(s))dsk

2ωa

t−ε

Ekf(s,x(s))k

2ωa

t−ε

m(s)ds.

¼ês7→m(s) ∈L

([t−ε,t],R

),dLebesgue››Âñ½n,Œ

lim

ε→0

Ek(Q

x)(t)−(Q

x)(t)k

= 0,

DOI:10.12677/pm.2022.121018142nØêÆ

xŒ'

=t∈I

ž, •3˜‡ƒé;8V

(t) ?¿ªCuV(t), V(t) 3H¥ƒé;. dAscoli-Arzela

½n•,é∀t∈I,8ÜV3C(I,H) ¥ƒé;.

dÚn2.7•,ŽfQ3B

¥•3˜‡ØÄ:x,dØÄ:=•XÚ(1.1)mild).

e5,bS

α,α−1

(t)(t>0) 3˜—ŽfÿÀ¥ëY,·‚Ú\Xeb:

(F5) é∀t∈I, 8Ü{

t−ε

∗S

α,α−1

)(t−ε−s)σ(s,x(s))dW(s):x∈B

,s∈[0,t−ε],ε∈

(0,t)}´;.

½n3.2e^‡(F1)−(F5) Ú(3.1)ª¤á,KXÚ(1.1)3Iþ–•3˜‡mild ).

y²:w,,db^‡(F1)−(F3), ŽfQ: C(I,H) →C(I,H) ëY.

½ÂŽfQ= Q

Xe:

x)(t) := S

α,α−1

(t)(x

−g(x))+(g

∗S

α,α−1

)(t)(x

−h(x)),

x)(t) :=

∗S

α,α−1

)(t−s)f(s,x(s))ds+

∗S

α,α−1

)(t−s)σ(s,x(s))dW(s).

eyW:= {Q

x: x∈B

}3C(I,H)¥ƒé;.dAscoli-Arzela½n,·‚I‡y²W3C(I,H)

¥˜—k.…ÝëY,W(t) 3H¥ƒé;.aquÚn3.1y²•W´˜—k..

Äky²WÝëY.é∀x∈B

,0 6t

6až, k

Ek(Q

x)(t

)−(Q

x)(t

= Ek

[(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)]f(s,x(s))ds

[(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)]σ(s,x(s))dW(s)

∗S

α,α−1

)(t

−s)f(s,x(s))ds

∗S

α,α−1

)(t

−s)σ(s,x(s))dW(s)k

64Ek

[(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)]f(s,x(s))dsk

+4Ek

[(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)]σ(s,x(s))dW(s)k

+4Ek

∗S

α,α−1

)(t

−s)f(s,x(s))dsk

+4Ek

∗S

α,α−1

)(t

−s)σ(s,x(s))dW(s)k

:= 4

i=1

DOI:10.12677/pm.2022.121018143nØêÆ

xŒ'

d½n3.1y²Œt

→t

ž,J

→0.,˜•¡, d^‡(F3), k

k(g

∗S

α,α−1

)(t

−s)−(g

∗S

α,α−1

)(t

−s)k

Ekσ(s,x(s))k

64aL

(r+1)

2ωa

d(g

∗S

α,α−1

)(t)3B(H) ¥ëYŒ•,t

→t

ž,(g

∗S

α,α−1

)(t

)−(g

∗S

α,α−1

)(t

) →0,

ÏdŠâLebesgue››Âñ½nŒlim

→t

= 0.éuJ

,d^‡(F3), k

2ωa

Ekσ(s,x(s))k

2ωa

(r+1)(t

−t

)

→0(t

→t

Ïd,8ÜW3C(I,H) ¥ÝëY.

Ùg,y²é∀t∈I,8ÜW(t) := {(Q

x)(t) : x∈B

}3H¥ƒé;.t= 0ž, W(0) ´ƒ

é;,•Iy²é∀t∈I

,W(t) ´ƒé;.é∀0 <ε<t,½ÂŽfQ

Xe:

x)(t) := (Q

1,ε

x)(t)+(Q

2,ε

x)(t),

(t) := {(Q

x)(t) : x∈B

},∀0 <ε<t,

Ù¥

1,ε

x)(t) := S

α,α−1

(ε)

t−ε

∗S

α,α−1

)(t−ε−s)f(s,x(s))ds,

2,ε

x)(t) := S

α,α−1

(ε)

t−ε

∗S

α,α−1

)(t−ε−s)σ(s,x(s))dW(s).

d½n3.1y²Œ•

2ωa

(t−ε)conv(V

1,ε

x)(t) ∈

2ωa

(t−ε)conv(V

),∀t∈I.

Ïd,é∀ε>0,8Ü{(Q

1,ε

x)(t):x∈B

}3H¥ƒé;.qdb^‡(F5)Œ8Ü

DOI:10.12677/pm.2022.121018144nØêÆ

xŒ'

{(Q

2,ε

x)(t) : x∈B

}3H¥ƒé;.Ïd,8ÜW

(t)3H¥ƒé;.é∀x∈B

Ek(Q

x)(t)−(Q

x)(t)k

= Ek

∗S

α,α−1

)(t−s)f(s,x(s))ds

−S

α,α−1

(ε)

t−ε

∗S

α,α−1

)(t−ε−s)f(s,x(s))ds

∗S

α,α−1

)(t−s)σ(s,x(s))dW(s)

−S

α,α−1

(ε)

t−ε

∗S

α,α−1

)(t−ε−s)σ(s,x(s))dW(s)k

62Ek

∗S

α,α−1

)(t−s)f(s,x(s))ds

−S

α,α−1

(ε)

t−ε

∗S

α,α−1

)(t−ε−s)f(s,x(s))dsk

+2Ek

∗S

α,α−1

)(t−s)σ(s,x(s))dW(s)

−S

α,α−1

(ε)

t−ε

∗S

α,α−1

)(t−ε−s)σ(s,x(s))dW(s)k

:= 2(K

éuK

t−ε

α,α−1

(ε)−Ik

Ek(g

∗S

α,α−1

)(t−ε−s)f(s,x(s))k

+3sup

06s6t−ε

k(g

∗S

α,α−1

)(t−ε−s)−(g

∗S

α,α−1

)(t−s)k

t−ε

Ekf(s,x(s))k

2ωa

t−ε

Ekf(s,x(s))k

ds.

aqu½n3.1y²,ε→0

ž,K

→0.ÓnK

→0.Ïd

lim

ε→0

Ek(Q

x)(t)−(Q

x)(t)k

= 0,

=t∈I

ž,•3˜‡ƒé;8W

(t)?¿ªCuW(t), W(t)3H¥ƒé;. dAscoli-Arzela

½nŒ•,8ÜW3C(I,H) ¥ƒé;,Ïd,γ

(W) = γ

)) = 0.

qÏ•

Ek(Q

x)(t)−(Q

y)(t)k

62kS

α,α−1

(t)k

Ekg(x)−g(y)k

+2k(g

∗S

α,α−1

)(t)k

Ekh(x)−h(y)k

2ωa

)Ekx−yk

DOI:10.12677/pm.2022.121018145nØêÆ

xŒ'

dÚn2.3Ú(3.1)ª,k

(Q(B

)) 6γ

))+γ

))

2ωa

)γ

)

6γ

ù¿›XQ:B

→B

´˜‡vàN. dÚn2.8 •, ŽfQ3B

¥•3˜‡ØÄ:x,

dØÄ:=•XÚ(1.1)mild).

4.(Ø

ÛÜ¯Kmild)•35.·‚Äk½ÂdŽfé(A,E))¤(α,α−1)-ý)x{S

α,α−1

(t)}

t>0

Ùg, 3Øb{S

α,α−1

(t)}

t>0

;œ¹e, ·‚|^KrasnoselskiiØÄ:½nÚSadovskiiØÄ:

½ny²XÚ(I) mild )•35.

Ä7‘8

I[g,‰ÆÄ7“cÄ7‘8(No.11701457)"

ë•©z

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xŒ'

[6]Yang,H.(2020)ExistenceResultsofMildSolutionsfortheFractionalStochasticEvolution

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