一类 Riemann-Liouville 分数阶中立型发展 方程的近似可控性 Approximate Controllability for a Class of Riemann-Liouville Fractional Neutral Evolution Equations

doi:10.12677/PM.2021.118175

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PureMathematicsnØêÆ,2021,11(8),1570-1584

PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.118175

˜aRiemann-Liouville©ê¥á.uÐ

•§CqŒ›5

888÷÷÷

∗

§§§ÚÚÚ

†

Ü“‰ŒÆêÆ†ÚOÆ,[‹=²

ÂvFÏµ2021c718F¶¹^FÏµ2021c819F¶uÙFÏµ2021c826F

Á‡

©$^Krasnoselskii ØÄ:½ny²Hilbert ˜m¥˜aRiemann-Liouville ©ê¥

á.uÐ•§CqŒ›5,¿‰ÑÄ–(JA^~f"

'…c

Riemann-Liouville©êê§¥á.uÐ•§§KrasnoselskiiØÄ:½n§CqŒ›5

ApproximateControllabilityforaClassof

Riemann-LiouvilleFractionalNeutral

EvolutionEquations

JihongWang

∗

,HeYang

†

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Jul.18

,2021;accepted:Aug.19

,2021;published:Aug.26

,2021

∗1˜Šö"

†ÏÕŠö"

©ÙÚ^:8÷,Ú.˜aRiemann-Liouville©ê¥á.uÐ•§CqŒ›5[J].nØêÆ,2021,11(8):

1570-1584.DOI:10.12677/pm.2021.118175

8÷§Ú

Abstract

Inthispaper, byusingtheKrasnoselskii ﬁxedpointtheorem, theapproximatecontrol-

labilityforaclassofnonlinearRiemann-Liouvillefractionalneutralevolutionequations

isinvestigatedinHilbertspaces.Anexampleisgiventoillustratetheapplicationof

theabstractconclusions.

Keywords

Riemann-LiouvilleFractionalDerivative,NeutralEvolutionEquations,Krasnoselskii

FixedPointTheorem,ApproximatelyControllability

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

1963 c, Kalman[1]ÄgJÑŒ›5Vg, duÙ3ÔnÆ+•2•A^¤•˜‡¹

ïÄ+•,Ù¥°(Œ›5®NõÆö¤ïÄ[2–4].´°(Œ›5´˜‡rVg,§‡¦

XÚUˆ¯k?¿‰ ½˜‡ªŽG, CqŒ›K‡¦XÚÃ•ªCuTªŽG, Ïd

3¢SA^¥CqŒ›5äk•\2•A^µ.©z[5]ïÄCaputo©ê¥á.•¼‡©

•§››XÚCqŒ› 5.2005 c,Heymans ÚPodlubny [6]•Ñ, Œ±òÔn¿Â8ÏuÊ

5|¥Riemann-Liouville ©êê½È©L«Ð©^‡, ùÐ©^‡•Ü·)º,

Ôny–.2014 c,©z[7]|^Darbo-SadovskiiØÄ:½ny²Banach˜mE¥ŽfŒ+

T(t);½š;œ¹eRiemann-Liouville ©ê¥á.uÐ•§











[x(t)−h(t,x(t))] = −Ax(t)+f(t,x(t)), t∈J

:= (0,b],0 <q<1

1−q

[x(t)−h(t,x(t))]|

t=0

= x

∈E

mild)•35,Ù¥

•qRiemann-Liouville.©êêŽf, I

1−q

•1−qRiemann-

Liouville.©êÈ©,−A: D(A) ⊂E→E´)ÛŒ+{T(t)}

t>0

Ã¡)¤.

DOI:10.12677/pm.2021.1181751571nØêÆ

8÷§Ú

2015c,©z[8]y²Banach˜mE¥Riemann-Liouville ©êuÐ•§











x(t) = Ax(t)+Bu(t)+f(t,x(t)),t∈J

,0 <q61

1−q

x(t)|

t=0

= x

∈E

CqŒ›5,Ù¥A: D(A) ⊂E→E´C

-Œ+{T(t)}

t>0

Ã¡)¤,››¼ê

u∈V:= L

([0,b],U),p>

,U•,˜‡Banach˜m.

Éþã©zéu,©ïÄŒ©Hilbert ˜mX¥š‚5©ê¥á.uÐ•§











[x(t)−h(t,x(t))] = Ax(t)+f(t,x(t))+Bu(t),t∈J

1−q

[x(t)−h(t,x(t))]|

t=0

= x

∈X

(1.1)

CqŒ›5,Ù¥A:D(A)⊂X→X´È½4‚5Žf,)¤X¥)ÛŒ+{T(t)}

t>0

, ››

¼êu∈L

(J,U),p>

,U•,˜‡Œ©g‡Hilbert ˜m, B: U→X•k.‚5Žf,¼ê

f,hò3©¥äN‰Ñ.

©3ŽfŒ+T(t) ´;Œ+œ¹e, $^Krasnoselskii ØÄ:½ny²¼êf,h÷v

,f^‡ž››XÚ(1.1) mild )•35. 3¼êf,h˜—k.…››XÚ(1.1) éA‚

5››XÚCqŒ›^‡e,y²››XÚ(1.1) CqŒ›5.

2.ý•£

(X,k·k) •Œ©Hilbert ˜m,J=[0,b]. PC(J,X) ´lJXþë Y¼êNU‰

êkxk

C(J,X)

= sup

t∈J

kx(t)k¤Banach˜m,L

(J,X)(1 6p<∞)´JþXŠp-•Bochner

ŒÈ¼êN¤Banach˜m, Ù‰ê•kxk

= (

kf(t)k

dt)

1−q

(J,X) = {x∈C(J

,X) : t

1−q

x(t) ∈C(J,X),0 <q<1,t∈J},

1−q

(J,X) U‰êkxk

1−q

= sup

t∈J

1−q

kx(t)k}¤Banach ˜m.X

:= (D(A

),k·k

) ´X¥

S˜m,Ù¥k·k

= kA

·k.Ø”˜„5,b∃N>1,¦

N:= sup

t∈J

kT(t)k<∞.

½Â2.1[8,9]¼êf: [0,+∞) →RqRiemann-Liouville.©êÈ©½Â•

f(t) =

Γ(q)

(t−s)

q−1

f(s)ds,t>0,q>0,

Ù¥ΓL«gamma ¼ê.

DOI:10.12677/pm.2021.1181751572nØêÆ

8÷§Ú

½Â2.2[8,9]¼êf: [0,+∞) →RqRiemann-Liouville.©êê½Â•

f(t) =

Γ(n−q)

(t−s)

n−q−1

f(s)ds,t>0,n−1 <q6n.

Ù¥n= [q]+1,[q] L«qêÜ©.

aqu©z[7,8] |^Laplace C†ÑXe½Â.

½Â2.3¡x∈C

1−q

(J,X) ´XÚ(1.1) mild ),…=x÷vÈ©•§











x(t) = t

q−1

(t)x

+h(t,x(t))+

(t−s)

q−1

(t−s)h(s,x(s))ds

(t−s)

q−1

(t−s)f(s,x(s))ds+

(t−s)

q−1

(t−s)Bu(s)ds,t∈J

1−q

[x(t)−h(t,x(t))]|

t=0

= x

(2.1)

Ù¥

(t) = q

∞

θξ

(θ)T(t

θ)dθ,

(θ) =

−1−

(θ

−

(θ) =

∞

n=1

(−1)

n−1

−qn−1

Γ(1+nq)

sin(nπq) ,θ∈(0,∞).

(θ)L«½Â3(0,∞) þVÇ—Ý¼ê,÷v

∞

(θ)dθ= 1,

∞

(θ)dθ=

Γ(1+ζ)

Γ(1+qζ)

, ζ∈[0,1].

u´‰½››¼ê, x(b,u) •XÚ(1.1) ƒAu››¼êumild )3ªŽž•t=b

Š.¡8Ü

(f) = {x(b,u) ∈X: u∈L

(J,U),x´XÚ(1.1) ƒAuumild)}

•XÚ(1.1)Œˆ8.

½Â2.4[10](CqŒ›5)XJK

(f) =X, Ù¥K

(f) L«K

(f) 4•, K¡XÚ(1.1)

3«mJþ´CqŒ›.

Ún2.1[7,11]é∀η∈(0,1], •3~êC

,¦

T(t)k6

,t∈J

Ún2.2[7]‚5Žfx{T

(t)}

t>0

äk±e5Ÿ:

DOI:10.12677/pm.2021.1181751573nØêÆ

8÷§Ú

1)é?¿‰½t>0 Ú∀x∈X,k

(t)xk6

Γ(q)

kxk.

2)é∀t>0,ŽfT

(t)´rëY,=é∀x∈XÚ0 <t

6b,k

)x−T

)xk→0(t

→t

Ún2.3[8]é∀x∈X,β∈(0,1),η∈(0,1],k

(t)x= A

1−β

(t)A

x,t∈J

(t)xk6

Γ(2−η)

qη

Γ(1+q(1−η))

kxk,t∈J

½n2.1 [12]( Krasnoselskii ØÄ:½n)D•Banach ˜mX¥˜‡š˜4àf8. e

ŽfQ

: D→X÷v

1)é∀x,y∈D,kQ

x+Q

y∈D;

2)Q

´Ø Žf;

3)Q

´ëYŽf,

3DS–k˜‡ØÄ:.

3.Ì‡(J

•y²©Ì‡(Ø,‰ÑXeb:

(H1)A)¤X¥;)ÛŒ+{T(t)}

t>0

(H2)¼êf: J×X→X÷vXe^‡:

1)é∀t∈J, ¼êf(t,·) : X→XëY,éz‡x∈X, ¼êf(·,x) : J→XrŒÿ.

2)•3¼êφ(t) ∈L

(J,R

),p>

,¦éa.e.t∈J,∀x∈C

1−q

(J,X),k

kf(t,x(t))k6φ(t),

Ù¥φ(t)÷vliminf

k→+∞

kφk

:= ρ<∞.

(H3)∃β∈(0,1),¼êh: J×X→X

ëY,…é∀x,y∈C

1−q

(J,X),∃H>0•~ê, ¦

h(t,x(t))−A

h(t,y(t))k6Ht

1−q

kx(t)−y(t)k,∀t∈J.

(H4)B: U→X•‚5k.Žf,…•3~êM

¦kBk6M

DOI:10.12677/pm.2021.1181751574nØêÆ

8÷§Ú

•Ä››XÚ(1.1)ƒA‚5©ê››XÚ







[x(t)−h(t,x(t))] = Ax(t)+f(t)+Bu(t),t∈J

1−q

[x(t)−h(t,x(t))]|

t=0

= x

∈X

(3.1)

½ÂŽfXe:

(b−s)

2q−2

(b−s)BB

∗

(b−s)ds,

R(λ,Λ

) = (λI+Λ

)

−1

,λ>0,

Ù¥B

∗

(b−s) ©OL«B,T

(b−s) ÝŽf.w,Λ

´k.‚5Žf.

Ún3.1©ê‚5››XÚ(3.1)CqŒ›…=3rŽfÿÀ¥,λ→0

,λR(λ,Λ

) →

dÚn3.1Œ, é∀λ>0, k

kR(λ,Λ

)k6

½Â¼ê

u(t) = (b−t)

q−1

∗

(b−t)R(λ,Λ

)p(x),t∈J

p(x) = x

−b

q−1

(b)x

−h(b,x(b))−

(b−s)

q−1

(b−s)f(s,x(s))ds

−

(b−s)

q−1

(b−s)h(s,x(s))ds.

∀k>0,B

= {x∈C

1−q

(J,X):kxk

1−q

6k},éAuþã››¼êu, ½ÂŽfF= F

→C

1−q

(J,X),Ù¥

x)(t) =

(t−s)

q−1

(t−s)f(s,x(s))ds+

(t−s)

q−1

(t−s)Bu(s)ds,t∈J

.(3.2)

x)(t) = t

q−1

(t)x

+h(t,x(t))+

(t−s)

q−1

(t−s)h(s,x(s))ds,t∈J

.(3.3)

½n3.1^‡(H1) −(H4) ¤á. XJ`<1,…λ→0

,λR(λ,Λ

)→0, KXÚ(1.1) –

k˜‡ØÄ:,Ù¥

`:= b

1−q

−β

kH+

1−q+qβ

1−β

Γ(1+β)

βΓ(1+qβ)

H+

Nρ

Γ(q)

(

p−1

pq−1

1−

λΓ

(q)(2q−1)



−β

kH+

Nρ

Γ(q)

(

p−1

pq−1

)

1−

pq−1

qβ

1−β

Γ(1+β)

Γ(1+qβ)β



<1(3.4)

y²µ´yB

´C

1−q

(J,X)¥k.4à8. e¡©oÚy²F3B

¥–k˜‡

ØÄ::

DOI:10.12677/pm.2021.1181751575nØêÆ

8÷§Ú

1˜Ú:y²•3k>0 ¦F(B

) ⊆B

bé∀k>0,Ñ•3x∈B

,¦é∀t∈J,kk6kFxk

1−q

.dF½ÂÚ^‡

(H2)−(H4),|^H¨olderØª,k

k6t

1−q

k(Fx)(t)k6

Γ(q)

k+b

1−q

[kA

−β

kHk+sup

t∈J

−β

kkh(t,0)k

]

1−q+qβ

1−β

Γ(1+β)

βΓ(1+qβ)



Hk+sup

s∈J

kh(s,0)k



Γ(q)

(

p−1

pq−1

1−

kφk

λΓ

(q)(2q−1)



q−1

Γ(q)

k+kA

−β

kHk+kA

−β

kkh(b,0)k

Γ(q)

(

p−1

pq−1

)

1−

pq−1

kφk

qβ

1−β

Γ(1+β)

βΓ(1+qβ)

(Hk+sup

s∈J

kh(s,0)k

)



3þãØªü>þ(.¿ÓžØ±k,-k→+∞,k`>1.ù†(3.4)ªgñ,¤±∃k>0,¦

F(B

) ⊆B

1Ú:y²F: B

→B

ëY.

{x

}

∞

m=1

⊂B

÷vlim

m→∞

−xk

1−q

= 0.ŠâfëY5,m→∞žk

kf(s,x

(s))−f(s,x(s))k→0,s∈J.

…

kf(s,x

(s))−f(s,x(s))k62φ(s) ∈L

(J,R

)

Ïdd^‡(H3) ÚLebesgue ››Âñ½n,k

1−q

k(Fx

)(t)−(Fx)(t)k

1−q

Γ(q)

(t−s)

q−1

kf(s,x

(s))−f(s,x(s))kds

1−q

−β

kHkx

−xk

1−q

1−q+qβ

1−β

Γ(1+β)

βΓ(1+qβ)

Hkx

−xk

1−q

λΓ

(q)

(t−s)

q−1

(b−s)

q−1

kp(x

)−p(x)kds

1−q

Γ(q)

(t−s)

q−1

kf(s,x

(s))−f(s,x(s))kds

1−q

−β

kHkx

−xk

1−q

1−q+qβ

1−β

Γ(1+β)

βΓ(1+qβ)

Hkx

−xk

1−q

λΓ

(q)(2q−1)



Γ(q)

(t−s)

q−1

kf(s,x

(s))−f(s,x(s))kds

+kA

−β

kHkx

−xk

1−q

qβ

1−β

Γ(1+β)

βΓ(1+qβ)

Hkx

−xk

1−q



DOI:10.12677/pm.2021.1181751576nØêÆ

8÷§Ú

kFx

−Fxk

1−q

→0(m→∞).

F3B

¥ëY.

1nÚ:ŽfF

´Ø .

é∀x,y∈B

,t∈J,d(3.3)ªÚ^‡(H3)Œ

1−q

k(F

x)(t)−(F

y)(t)k

1−q

−β

kkA

h(t,x(t))−A

h(t,y(t))k

1−q

(t−s)

q−1

1−β

(t−s)A

[h(s,x(s))−h(s,y(s))]kds

1−q

−β

kHkx−yk

1−q

1−q+qβ

1−β

Γ(1+β)

βΓ(1+qβ)

Hkx−yk

1−q

þªü>þ(.,k

x−F

1−q



1−q

HkA

−β

1−q+qβ

1−β

Γ(1+β)

βΓ(1+qβ)



kx−yk

1−q

(Ü(3.4)ªŒ, kF

x−F

1−q

<kx−yk

1−q

,F

´Ø .

Ω = {y∈C(J,X) : y(t) = t

1−q

x)(t),x∈B

1oÚ:y²Ω3C(J,X) ¥ƒé;.

Äky²Ω´ÝëY.

0 = t

6b, é∀y∈Ω,d^‡(H2) 2)Œ

ky(t

)−y(0)k

1−q

k(t

−s)

q−1

−s)f(s,x(s))kds

1−q

k(t

−s)

q−1

−s)Bu(s)kds

Γ(q)

(

p−1

pq−1

)

1−

kφk

λΓ

(q)(2q−1)



q−1

Γ(q)

+kA

−β

kHk+kA

−β

kkh(b,0)k

Γ(q)

(

p−1

pq−1

)

1−

pq−1

kφk

qβ

1−β

Γ(1+β)

βΓ(1+qβ)

(Hk+kh(s,0)k

)



→0(t

→0).

DOI:10.12677/pm.2021.1181751577nØêÆ

8÷§Ú

0 <t

6b,>0¿©ž,k

ky(t

)−y(t

1−q

Γ(q)

−s)

q−1

kf(s,x(s))+Bu(s)kds

Γ(q)

1−q

−s)

q−1

−t

1−q

−s)

q−1

kkf(s,x(s))+Bu(s)kds

−

1−q

−s)

q−1

−s)−T

−s)kkf(s,x(s))+Bu(s)kds

−

1−q

−s)

q−1

−s)−T

−s)kkf(s,x(s))+Bu(s)kds

i=1

éuI

ÚI

,UìÚn2.21), t

−t

→0,→0ž,k

1−q

Γ(q)

−s)

q−1

kf(s,x(s))+Bu(s)kds

→0.

−

1−q

−s)

q−1

−s)−T

−s)kkf(s,x(s))+Bu(s)kds.

2Nb

1−q

Γ(q)

−

−s)

q−1

kf(s,x(s))+Bu(s)kds.

→0.

éuI

,t

−t

→0 ž,dLebesgue ››Âñ½nŒ

Γ(q)

1−q

−s)

q−1

−t

1−q

−s)

q−1

kkf(s,x(s))+Bu(s)kds

→0.

éuI

,db^‡(H1) Œ•,t>0 žT

(t)´ÝëY,¤±t

−t

→0 ž,k

−

1−q

−s)

q−1

−s)−T

−s)kkf(s,x(s))+Bu(s)kds

1−q

sup

s∈[0,t

−]

−s)−T

−s)k

−

−s)

q−1

kf(s,x(s))+Bu(s)kds

→0.

DOI:10.12677/pm.2021.1181751578nØêÆ

8÷§Ú

Ïd,é∀x∈B

,t

−t

→0,→0ž,k

ky(t

)−y(t

)k6

i=1

→0.

¤±,8ÜΩ ´ÝëY.

Ùgy²é∀t∈J,8ÜΩ(t) = {y(t) : y∈Ω}3X¥ƒé;.

w,,Ω(0)3X ¥ƒé;.

t∈J

ž,é∀0 <<t,∀δ>0Š

,δ

(t)=qt

1−q

t−

∞

θ(t−s)

q−1

(θ)T((t−s)

θ)[f(s,x(s))+Bu(s)]dθds

=qt

1−q

T(

δ)

t−

∞

θ(t−s)

q−1

(θ)T((t−s)

θ−

δ)[f(s,x(s))+Bu(s)]dθds

dT(

δ) ;5,Ñ∀0 <<t,∀δ>0 8ÜΩ

(,δ)

(t) = {y

,δ

(t),y∈Ω}3X ¥ƒé;.

é∀x∈B

ky(t)−y

,δ

(t)k

6qb

1−q

θ(t−s)

q−1

(θ)T((t−s)

θ)[f(s,x(s))+Bu(s)]dθdsk

+qb

1−q

∞

θ(t−s)

q−1

(θ)T((t−s)

θ)[f(s,x(s))+Bu(s)]dθds

−

t−

∞

θ(t−s)

q−1

(θ)T((t−s)

θ)[f(s,x(s))+Bu(s)]dθdsk

6qb

1−q

(t−s)

q−1

kf(s,x(s))+Bu(s)kds

θξ

(θ)dθ

+qb

1−q

t−

(t−s)

q−1

kf(s,x(s))+Bu(s)kds

∞

θξ

(θ)dθ

6qb

1−q

(t−s)

q−1

kf(s,x(s))+Bu(s)kds

θξ

(θ)dθ

1−q

Γ(q)

t−

(t−s)

q−1

kf(s,x(s))+Bu(s)kds

→0(→0,δ→0).

=t∈J

ž, •3˜‡ƒé;8Ω

(,δ)

(t) ?¿%CΩ(t),Ω(t) 3X ¥ƒé;. dAscoli-Arzela

½n•8ÜΩ ´C(J,X) ¥ƒé;8.¤±F

: B

→B

´ëYŽf,Ïdd½n2.1•,F3

þ–k˜‡ØÄ:x∈B

,dØÄ:=•XÚ(1.1) mild ).



½n3.2^‡(H1)−(H4) ¤á,fÚA

h˜—k.. XJƒA‚5XÚ(3.1)3JþCq

DOI:10.12677/pm.2021.1181751579nØêÆ

8÷§Ú

y²:d½n3.1 •,∀λ>0,x

∈X, •3XÚ(1.1) mild )x

∈C

1−q

(J,X) ¦

(t)=t

q−1

(t)x

+h(t,x

(t))+

(t−s)

q−1

(t−s)h(s,x

(s))ds

(t−s)

q−1

(t−s)[f(s,x

(s))+Bu(s)]ds

Ù¥

u(t) = (b−t)

q−1

∗

(b−t)R(λ,Λ

)p(x

),t∈J

p(x

)=x

−b

q−1

(b)x

−h(b,x

(b))−

(b−s)

q−1

(b−s)f(s,x

(s))ds

−

(b−s)

q−1

(b−s)h(s,x

(s))ds

I−Λ

R(λ,Λ

) = λR(λ,Λ

)



(b) = b

q−1

(b)x

+h(b,x

(b))+

(b−s)

q−1

(b−s)h(s,x

(s))ds

(b−s)

q−1

(b−s)f(s,x

(s))ds

(b−s)

q−1

(b−s)(b−s)

q−1

∗

(b−s)R(λ,Λ

)



−b

q−1

(b)x

−h(b,x

(b))−

(b−τ)

q−1

(b−τ)f(τ,x

(τ))dτ

−

(b−τ)

q−1

(b−τ)h(τ,x

(τ))dτ



= x

−p(x

(b))+Λ

R(λ,Λ

)p(x

(b))

= x

−λR(λ,Λ

)p(x

(b))(3.5)

dfÚA

h˜—k.5Œ•, {f(·,x

(·)):λ>0}Ú{h(·,x

(·)):λ>0}3L

(J,X) ¥k., •

3fS, Ø”•{f(·,x

(·)) : λ>0}Ú{h(·,x

(·)) : λ>0}, 3L

Ú{h(·)}.Ïd,dT

(t);5Œ•,λ→0

ž,k

(b−s)

q−1

(b−s)[f(s,x

(s))−ω(s)]ds→0

(b−s)

q−1

1−β

(b−s)A

[h(s,x

(s))−h(s)]ds→0

DOI:10.12677/pm.2021.1181751580nØêÆ

8÷§Ú

η= x

−b

q−1

(b)x

−h(b)−

(b−s)

q−1

(b−s)h(s)ds−

(b−s)

q−1

(b−s)ω(s)ds

kp(x

(b))−ηk

6kA

−β

kkA

h(b,x

(b))−A

h(b)k+

(b−s)

q−1

1−β

(b−s)A

kh(s,x

(s))−h(s)kds

(b−s)

q−1

(b−s)kf(s,x

(s))−ω(s)kds

→0(λ→0

)

(Ü(3.5)ªÚÚn3.1 Œ

(b)−x

k6kλR(λ,Λ

)p(x

(b))k

6kλR(λ,Λ

)kkp(x

(b))−ηk+kλR(λ,Λ

)ηk

→0(λ→0

)

4.A^

~X= U:= L











[x(t,z)−

ν(z,τ)x(t,τ)dτ] = ∂

x(t,z)+u(t,z)+e

−t

sinx(t,z),(t,z) ∈(0,b]×[0,π],

x(t,0) = x(t,π) = 0,t∈[0,b],

[x(t,z)−

ν(z,τ)x(t,τ)dτ]|

t=0

= x

(z),z∈[0,π],

(4.1)

Ù¥0 <q<1, q= β=

,J:=[0,b],b>0 ´~ê,u∈L

(J,U) •››¼ê.

Ax=

∂

∂z

Ù¥D(A) ={x∈X:x

∈X,x(0) =x(π)=0}, KA´;)ÛŒ+{T(t)}

t>0

Ã¡)

¤,…•3N>1¦kT(t)k6N. u´

T(t)x=

∞

n=1

−n

hx,e

DOI:10.12677/pm.2021.1181751581nØêÆ

8÷§Ú

Ù¥

(z) = (

)

sin(nz).

ŽfA

: D(A

) ⊂X→X½Â•:

∞

n=1

nhx,e

Ù¥D(A

) := {x(·) ∈X:

∞

n=1

nhx,e

∈X}. ^‡(H1) ÷v.

•y²XÚ(4.1)CqŒ›, Ú\Xeb^‡:

(P)¼êν÷vXe^‡:

1)¼êνŒÿ,ν(0,θ) = ν(π,θ) = 0,…

(z,τ)dτdz<∞.

2)¼ê∂

ν(θ,z) Œÿ,…

H:=



(∂

ν(z,τ))

dτdz



<∞.

é∀t∈J,-

x(t)(z) = x(t,z),

f(t,x(t))(z) = f(t,x(t,z)) = e

−t

sinx(t,z),

h(t,x(t))(z) =

ν(z,τ)x(t,τ)dτ,

(νx)(z) =

ν(z,τ)x(τ)dτ,

Bu(t)(z) = u(t,z),

dfLˆªŒ

kf(t,x(t,z))k6e

−t

^‡(H2) ¤á.qÏ•

hν(x),e

i= h

ν(z,τ)x(τ)dτ,e

(z)(

ν(z,τ)x(τ)dτ)dz,

KdŽfA

½ÂŒ

ν(x)k= k

∞

n=1

nhν(x),e

N´y^‡(H3) ¤á…A

ν˜—k.. dBu(t)(z)=u(t,z) ,kBu(t)(z)k6ku(t,z)kŒB

DOI:10.12677/pm.2021.1181751582nØêÆ

8÷§Ú

•ü ŽfI,^‡(H4) ÷v.XJλR(λ,Λ

) →0(λ→0

),KdÚn3.1•XÚ(4.1) ƒA‚

5XÚ3«mJþCqŒ›.Ïd,d½n3.2Œ•,››XÚ(4.1) CqŒ›.

Ä7‘8

I[Ä7”“c‰Æ]Ï‘8(No.12061062)"

ë•©z

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8÷§Ú

[12]Zhou,Y.andFeng,J.(2010)ExistenceofMildSolutionsforFractionalNeutralEvolution

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DOI:10.12677/pm.2021.1181751584nØêÆ