扇形算子情形下时滞发展方程周期解的存在性 Existence of Periodic Solutions for Delayed Evolution Equation with Sector Operator

doi:10.12677/PM.2022.124075

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PureMathematicsnØêÆ,2022,12(4),653-664

PublishedOnlineApril2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.124075

÷/Žfœ/ež¢uÐ•§±Ï)•35

‰‰‰ééé

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c316F¶¹^FÏµ2022c418F¶uÙFÏµ2022c425F

Á‡

©?ØBanach˜mX¥ž¢uÐ•§

(t)+Au(t) = f(t,u(t),u(t−τ

),···,u(t−τ

)),t∈R,

±Ï)•35§Ù¥A:D(A)⊂X→X•÷/Žf§−A)¤X¥•ê-½)ÛŒ

+T(t)(t≥0),f:R×X

n+1

→XëY§'ut±ω•±Ï§τ

,···,τ

>0"·‚A^ØÄ:½n§

¼•§ω-±Ïmild)•35(J"

'…c

ž¢uÐ•§§ØÄ:½n§ω-±Ïmild)§•35

ExistenceofPeriodicSolutionsforDelayed

EvolutionEquationwithSectorOperator

QilinWei

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Mar.16

,2022;accepted:Apr.18

,2022;published:Apr.25

,2022

©ÙÚ^:‰é.÷/Žfœ/ež¢uÐ•§±Ï)•35[J].nØêÆ,2022,12(4):653-664.

DOI:10.12677/pm.2022.124075

‰é

Abstract

Thispaperdeals withtheexistence ofperiodicsolutionsfordelayedevolutionequation

inaBanachspaceX,

(t)+Au(t) = f(t,u(t),u(t−τ

),···,u(t−τ

)),t∈R,

whereA: D(A) ⊂X→Xisasectoroperatorand−Ageneratesaexponentialstability

analytic semigroup T(t)(t≥0)on X, f: R×X

n+1

→Xis a continuous functionmapping

anditisω-periodicint,τ

,···,τ

>0.Existenceresultsofω-periodicmildsolutions

areobtainedbyusingtheﬁxedpointtheorem.

Keywords

DelayedEvolutionEquation,FixedPointTheorem,ω-PeriodicMildSolutions,

Existence

2022byauthor(s)andHansPublishersInc.

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

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

1.Úó

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uÐ•§5?n,??ØÙ)•35!•˜5.U÷/ŽfeÄ–ž¢uÐ•§˜„/

(t)+Au(t) = f(t,u(t),u(t−τ

),···,u(t−τ

)),t∈R,(1)

DOI:10.12677/pm.2022.124075654nØêÆ

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Ù¥A:D(A)⊂X→X•X¥÷/Žf,−A)¤X¥•ê-½)ÛŒ+T(t)(t≥0),

f:R×X

n+1

→X•š‚5N,τ

>0(i=1,···,n)•~ê.3ù ‡©•§¥,žm±Ï

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)'ut

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(t)+Au(t) = f(t,u(t)),t≥0,(2)

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e)•3^‡e,•§(2)ω-±Ï;)•3•˜5ÚìC5.

¹ž¢Ä–uÐuÐ•§

(t)+Au(t) = f(t,u(t),u(t−τ)),t∈R,(3)

Ù±Ï)•35•˜ŠöïÄ,„©z[5–9].3©z[5]¥,J.Liu3˜„Banach˜mX¥

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A•Hilbert˜m½Žfœ/õž¢uÐ•§(1)±Ï¯K,ÙA ^)ÛŒ+nØ†˜‡¹ž

¢È©Øª¼±Ï)•35†ì?-½5.©z[7,8]?ØX•kSBanach˜m,

T(t)(t≥0) •C

-Œ+œ/, ž¢uÐ•§(3) ω-±Ïmild )•35.©z[7]ŠöA^üN

S“•{¼•†•Œ±Ïmild)•35.©z[8]$^ŽfŒ+nØ9Leray-Schauder

ØÄ:½n•§(3)±Ïmild)•3•˜5†ìC-½5.©z[9]ÏLé)Žf ±

Ïòÿ,|^ƒ'ØÄ:½n,¼ž¢uÐ•§(3)ω-±Ïmild)•35(J.

ØÓuþã©z?ØAÏœ¹,©3˜„Banach˜mXþïÄ÷/Žfœ/eõž¢uÐ

•§±Ï¯K(1).$^SchauderØÄ:½n,¼š‚5‘f÷v˜g O•^‡žõž¢uÐ

•§(1)•35(J.$^BanachØ Nn÷/Žf œ/eõž¢uÐ•§(1)±Ï)

•˜5,ž± 3kSBanach˜mþ•›,¦ÙA^•\2•.

©1!‰ÑØ©¤IÎÒ9Ún,1n!‰Ñ•3•˜5½n9Ùy²,1o!‰Ñ

A^~f,1Ê!o(©9Ð".

2.ý•£

X•Banach˜m,A:D(A)⊂X→X•÷/Žf,−A)¤X¥•ê-½)ÛŒ

DOI:10.12677/pm.2022.124075655nØêÆ

‰é

+T(t)(t≥0).A

(α≥0)•ŽfA©ê˜,X

= (D(A

),k·k

),=X

•A

½Â•D(A

ã”‰êk·k

= kA

k¤Banach˜m.•3~êM≥1,M

≥1,δ>0¦

kT(t)k≤Me

−δt

,t≥0,(4)

T(t)k≤M

−α

−δt

,t>0.

E•,˜‡Banach˜m,Ù‰ê½Â•k·k

,…•30 <α<1¦

→E→X,

•3~êC>0,k

kxk

≤Ckxk

,x∈X

(R,X),C

(R,E),C

(R,X

)©O´N±ω•±ÏX,E,X

ŠëY¼êU‰êkuk

max

t∈[0,ω]

ku(t)k,kuk

=max

t∈[0,ω]

ku(t)k

,kuk

=max

t∈[0,ω]

ku(t)k

¤Banach˜m,´„C

(R,X

) →

(R,E).

h∈C

(R,X),•ÄX¥‚5uÐ•§

(t)+Au(t) = h(t),t∈R,(5)

ω-±Ï)•35.bŒ+T(t)(t≥0)•ê-½,=÷vª(4),dGelfandÌŒ»úª9(4)ª

r(T(ω)) =lim

n→∞

(ω)k≤lim

n→∞

√

−δ

= e

−δ

<1.

Ïd,1•ŽfT(ω)KŠ,lI−T(ω)kk._Žf(I−T(ω))

−1

,…

(I−T(ω))

−1

∞

n=0

(ω) =

∞

n=0

T(nω),(6)

••B,±eP

ρ= kI−T(ω))

−1

h∈C

(R,X)ž,dÙ•(J[10],´y•§(5)•3•˜ω-±Ï)

u(t) = (I−T(ω))

−1

t−ω

T(t−s)h(s)ds:= (Ph)(t),t∈R,(7)

…T)u∈C

(R,X) ∩C

(R,X

).é˜„h∈C

(R,X),d(7)ª(½u=Ph∈C

(R,X),

uØ˜½Œ‡,´•§(5)˜«ω-±Ï2Â),¡•ω-±Ïmild)[11].U(7)ª,‚5•

§(5)ω-±Ï)ŽfP: C

(R,X) →C

(R,X)•‚5k.Žf.

DOI:10.12677/pm.2022.124075656nØêÆ

‰é

Ó/,é÷/ŽfuÐ•§(1),aqu‚5•§(5),eu∈C

(R,X)÷vÈ©•§

u(t) = (I−T(ω))

−1

t−ω

T(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds,t∈R.(8)

K¡u••§(1)ω-±Ïmild).f:R×X

n+1

→X•C

-Nž,Umild)K5[[10],

Chapter3,Theorem1.5],•§(1)ω-±Ïmild)u∈C

(R,X)∩C

(R,X

)´;).

ÚÚÚnnn1([12]Arzela-Ascoli)8ÜD⊂C(J,X)ƒé;¿©7‡^‡´:D´ÝëY,¿…

é∀t∈J,8ÜD(t)´X¥ƒé;8.

ÚÚÚnnn2([13]BanachØ Nn)X•Ýþ˜m,Q•XþØ N .KQ3X¥k

…•k˜‡ØÄ:.

ÚÚÚnnn3([14]SchauderØÄ:½n)X•Banach˜m,D•X¥k.à48.eQ: D→D

ëY,KQ3D¥–•3˜‡ØÄ:.

3.Ì‡(J

½½½nnn1A:D(A)⊂X→X•X¥÷/Žf,−A)¤X¥•ê-½)ÛŒ+

T(t)(t≥0).f: R×X

n+1

→X'ut±ω•±Ï,÷vLipschitz^‡.…÷ve^‡

(A1)•3~êC

>0 (i= 0,1,···,n),¦é?¿x

∈E(i= 1,···,n),k

kf(t,x

,···,x

)−f(t,y

,···,y

≤

i=0

−y

,t∈R,

(A2)CM

ρ·

i=0

1−α

<1,

K÷/Žfež¢uÐ•§(1)•3•˜ω-±Ïmild).

½½½nnn1yyy²²²·‚3C

(R,X)¥½ÂŽfQ

Qu= (P◦F)u,

Ù¥P•(7)ª¤½Â‚5uÐ•§±Ï¯K)Žf,F: C

(R,E) →C

(R,X)•Xe½ÂŽ

F(u)(t) = f(t,u(t),u(t−τ

),···,u(t−τ

)),t∈R,u∈C

(R,E).

l

Qu(t) = (I−T(ω))

−1

t−ω

T(t−s)f(t,u(t),u(t−τ

),···,u(t−τ

))ds,t∈R,u∈C

(R,E).

(9)

DOI:10.12677/pm.2022.124075657nØêÆ

‰é

´yQ:C

(R,E)→C

(R,X

)→C

(R,E).U(7)ª,Œ•÷/Žfež¢uÐ•§(1)ω±

Ïmild)duŽfQØÄ:.

db(A1)9(9)ª•,é∀u,v∈C

(R,X)9t∈R,k

kQu(t)−Qv(t)k

≤CkQu(t)−Qv(t)k

=Ck(I−T(ω))

−1

t−ω

T(t−s)(f(t,u(t),u(t−τ

),···,u(t−τ

))

−f(t,u(t),u(t−τ

),···,u(t−τ

)))dsk

≤Cρ

t−ω

T(t−s)k·k(f(t,u(t),u(t−τ

),···,u(t−τ

))

−f(t,u(t),u(t−τ

),···,u(t−τ

)))dsk

≤CM

t−ω

(t−s)

−α

ku(s)−v(s)k

ku(s−τ

)−v(s−τ

+···+C

ku(s−τ

)−v(s−τ

)ds

≤CM

ρ·

i=0

t−ω

(t−s)

−α

ds·ku−vk

=CM

ρ·

i=0

1−α

·ku−vk

d^‡(A2)Œ

kQu−Qvk

≤CM

ρ·

i=0

1−α

·ku−vk

<ku−vk

ù`²Q:C

(R,E)→C

(R,E)•Ø N.dBanachØ Nn•,ŽfQ3C

(R,E)¥•

3•˜ØÄ:u,TØÄ:•ž¢uÐ•§(1)ω-±Ïmild).

½½½nnn2A:D(A)⊂X→X•X¥÷/Žf,−A)¤X¥•ê-½)ÛŒ

+T(t)(t≥0)´;Œ+.e^‡(A2)9

(A3)f:R×E

n+1

→X,é∀t∈R,f(t,·,·,···,·,·):E

n+1

→XëY;é∀x

∈E,(i=

0,1,···,n),f(·,x

,···,x

) : R→XrŒÿ,

(A4)•3~êC

≥0(i= 0,1,···,n)9M≥0,¦é∀t∈R9x

∈E(i= 0,1,···,n),k

kf(t,x

,···,x

)k≤

i=0

+M,

é∀R>09x

∈E(i= 0,1,···,n),XJkx

≤R,k

kf(t,x

,···,x

)k≤R

i=0

+M:= h

(t),t∈R,

DOI:10.12677/pm.2022.124075658nØêÆ

‰é

…

liminf

r→∞

t−ω

(s)

(t−s)

ds=

i=0

1−α

¤á,K÷/Žfež¢uÐ•§(1)–•3˜‡ω-±Ïmild).

½½½nnn2yyy²²²ŽfQ´½n1y²¥¤½ÂŽf,e¡©oÚy².

1˜Ú,QD

⊂D

é∀R>0,

= {u∈C

(R,E)|ku(t)k

≤R,t∈R},

K∃R>0,¦QD

⊂D

.ÄK,é∀R>0,t∈R,∃u

∈D

,¦R<kQu

(t)k

R<kQu

(t)k

≤Ck(I−T(ω))

−1

k·

t−ω

T(t−s)k·kf(s,u

(s),u

(s−τ

),···,u

(s−τ

))kds

≤CM

ρ·

t−ω

i=0

(t−s)

=CM

ρ·

t−ω

(s)

(t−s)

ds,

þªüàÓ¦

1 ≤CM

ρ·

t−ω

(s)

(t−s)

≤CM

i=0

1−α

†b^‡(A2)gñ!∃R>0,¦QD

⊂D

1Ú,Q: D

→D

ëY.

{u

}∈D

,÷vlim

n→∞

= u∈D

.db(A3)¥f(t,x

,···,x

)'ux

(i=0,···,n)ë

Y5•

f(t,u

(t),u

(t−τ

),···,u

(t−τ

)) →f(t,u(t),u(t−τ

),···,u(t−τ

)),(n→∞),



kf(t,u

(t),u

(t−τ

),···,u

(t−τ

))−f(t,u(t),u(t−τ

),···,u(t−τ

))k≤2h

(t),t∈R,

¼ê

(s)

(t−s)

LebesgueŒÈ,dLebesgue››Âñ½nŒ•

DOI:10.12677/pm.2022.124075659nØêÆ

‰é

kQu

(t)−Qu(t)k

≤CkQu

(t)−Qu(t)k

≤Ck(I−T(ω))

−1

k·

t−ω

T(t−s)k·kf(t,u

(t),u

(t−τ

),···,u

(t−τ

))

−f(t,u(t),u(t−τ

),···,u(t−τ

))kds

≤CM

ρ·

t−ω

(t−s)

·kf(t,u

(t),u

(t−τ

),···,u

(t−τ

))

−f(t,u(t),u(t−τ

),···,u(t−τ

))kds

≤CM

ρ·

t−ω

(s)

(t−s)

→0,(n→∞).

Ïd,Q: D

→D

ëY.

1nÚ,H(t) = {Qu(t)|u∈D

}ëY.

é∀0 ≤t

≤ω9∀u∈D

Qu(t

)−Qu(t

) = (I−T(ω))

−1

−ω

T(t

−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds

−(I−T(ω))

−1

−ω

T(t

−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds

= (I−T(ω))

−1

−ω

(T(t

−s)−T(t

−s))f(s,u(s),u(s−τ

),···,u(s−τ

))ds

−(I−T(ω))

−1

−ω

T(t

−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds

+(I−T(ω))

−1

T(t

−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds

:= I

l

kQu(t

)−Qu(t

≤kI

+kI

•Iyt

−t

→0ž,kI

k→0(i= 1,2,3)=Œ.drëYŒ+5Ÿ,k

≤CkI

≤Ck(I−T(ω))

−1

k·

−ω

k(T(t

−s)−T(t

−s))·f(s,u(s),u(s−τ

),···,u(s−τ

))k

≤CMρ·

−ω

k(T(t

−t

)−I)·f(s,u(s),u(s−τ

),···,u(s−τ

))k

→0,(t

−t

→0),

DOI:10.12677/pm.2022.124075660nØêÆ

‰é

≤CkI

≤Ck(I−T(ω))

−1

k·

−ω

T(t

−s)·f(s,u(s),u(s−τ

),···,u(s−τ

))k

≤CM

ρ·

−ω

(s)

−s)

→0,(t

−t

→0),

≤CkI

≤Ck(I−T(ω))

−1

k·

T(t

−s)·f(s,u(s),u(s−τ

),···,u(s−τ

))k

≤CM

ρ·

(s)

−s)

→0,(t

−t

→0).

=kQu(t

)−Qu(t

→0(t

−t

→0),†u∈D

ÀÃ',H(t)ÝëY.

1oÚ,2yH(t) = {Qu(t)|u∈D

}ƒé;.

é∀t∈R,u∈D

,0 <δ<ω.

u)(t)=(I−T(ω))

−1

t−δ

t−ω

T(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds

=T(δ)(I−T(ω))

−1

t−δ

t−ω

T(t−s−δ)f(s,u(s),u(s−τ

),···,u(s−τ

))ds,

lk

k(Q

u)(t)k

≤k(I−T(ω))

−1

t−δ

t−ω

kT(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

))k

≤ρ

t−δ

t−ω

T(t−s)k·kf(s,u(s),u(s−τ

),···,u(s−τ

))k

≤M

t−δ

t−ω

(s)

(t−s)

ds:= T<∞,

dŽfT

(δ)´X

¥;ŽfŒ•,é∀t∈R,(Q

)(t)3X

¥ƒé;.

DOI:10.12677/pm.2022.124075661nØêÆ

‰é

é∀u∈D

,t∈R,k

kQu(t)−Q

u(t)k

=k(I−T(ω))

−1

t−ω

T(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds

−(I−T(ω))

−1

t−δ

t−ω

T(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

))dsk

=k(I−T(ω))

−1

(

t−ω

T(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

))ds

−

t−δ

t−ω

T(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

)))dsk

≤k(I−T(ω))

−1

t−δ

T(t−s)f(s,u(s),u(s−τ

),···,u(s−τ

))kds

≤M

t−δ

(s)

(t−s)

ds.

lŒ•,H(t)3E¥ƒé;.

nþ¤ã,dArzela-Ascoli½nŒ•,8ÜH3C

(R,E)¥ƒé;,Q:D

→D

•ëYŽ

f.dëYŽfSchauderØÄ:½n•,Q3D

¥–•3˜‡ØÄ:,TØÄ:= •÷/Ž

fež¢uÐ•§(1)ω-±Ïmild).

4.A^

Ω⊂R

(Ω×R

n+1

),g(x,t,ξ

,ξ

,···,ξ

)'ut±ω•

±Ï,τ>0•~ê,•Äõž¢Ô.>Š¯K







∂u

∂t

−∆u= g(x,t,u(x,t),u(x,t−τ

),···,u(x,t−τ

)),(x,t) ∈Ω×R

∂Ω

= 0,

(10)

žmω-±Ï)•3,k

½½½nnn3ee^‡¤á

(A5)•3M

≥0,M≥0(i= 0,1,···,n),g∈C

(Ω×R

n+1

),g(x,t,ξ

,ξ

,···,ξ

) 'ut±ω

•±Ï.eg÷v^‡:

|g(x,t,ξ

,ξ

,···,ξ

)|≤

i=0

+M,(11)

(A6)CM

k(I−T(ω))

−1

1−α

i=0

DOI:10.12677/pm.2022.124075662nØêÆ

‰é

K÷/Žfœ/ež¢Ô.•§žm±Ï¯K(10)–•3˜‡žmω-±Ïmild).

yyy²²²X= L

(Ω)(p>N),ŠX¥ŽfA:

D(A) = W

2,p

(Ω)∩W

1,p

(Ω),Au= −∆u.(12)

U[10,Chapter 7,Theorem3.6],−A+λ

I)¤X¥Ø )ÛŒ+S

(t)(t≥0),Ù¥λ

•LaplaceŽ

f−∆3>.^‡u|

∂Ω

=0e1˜AŠ.dS

(t)(t≥0)Ø 5,−A)¤Œ+T

(t)=

−λ

(t)÷v•ê-½^‡.UT

(t))Û5†Aý)ª;5,T

(t)•´X¥;Œ+.½

Âš‚5Nf: R×X

n+1

→X:

f(t,v

,···,v

) = g(·,t,v

(·),v

(·),···,v

(·)),(13)

Kd(12)ª´„,f:R×X

n+1

→XëY,…÷v^‡(A2).ù,÷/Žfœ/ež¢Ô.

>Š¯K(10)z•X¥÷/Žfež¢uÐ•§(1).U½n2,éA•§(10)•3žmω-±

Ïmild)u∈C

(R,X)(3Œ+T

(t)¿Âe).

5.o(

-Œ+cJe,·‚$^)ÛŽfŒ+nØ!ëYŽfƒ'ØÄ:½nïÄš‚

5‘f÷v˜gO•^‡ž,ž¢uÐ•§ω-±Ïmild)•35†•˜5,¿‰ÑA^~

f.Cc5,é¹ž¢‘uÐ•§±Ï)ïÄ¤J´a,¹ž¢‘óÀuÐ•§±Ï)ïÄ

,¹ž¢‘óÀuÐ•§±Ï)ïÄkX2•A^µÚy¢¿Â.

ë•©z

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