一类半正二阶 Neumann 问题正解的存在性 Existence of Positive Solutions for aClass of Semi-Positone Second OrderNeumann Problems

doi:10.12677/PM.2023.134104

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PureMathematicsnØêÆ,2023,13(4),987-995

PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.134104

˜aŒNeumann¯K)

•35

œœœkkk

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2023c319F¶¹^FÏµ2023c420F¶uÙFÏµ2023c427F

Á‡

©ïÄŒ¯K







−u

(t)+u(t) = λ(f(u(t))+w(t)),t∈[0,1],

(0) = u

(1) = 0

(P)

)•35§Ù¥λ•ëê§w∈C([0,1],R)÷v|w(t)|≤c,t∈[0,1],c•?¿~ê§

f∈C([0,∞),[0,∞))§…÷v‡‚ 5^‡§=f

:=lim

x→0

f(x)

=0f

∞

:=lim

x→∞

f(x)

=∞"ÏL$^

IþØÄ:½ny²•3~êλ

>0§0 <λ<λ

ž§¯K(P)•3˜‡)"

'…c

)§Œ§Neumann¯K§IþØÄ:½n

ExistenceofPositiveSolutionsfora

ClassofSemi-PositoneSecondOrder

NeumannProblems

LinyingShi

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Mar.19

,2023;accepted:Apr.20

,2023;published:Apr.27

,2023

©ÙÚ^:œk.˜aŒNeumann¯K)•35[J].nØêÆ,2023,13(4):987-995.

DOI:10.12677/pm.2023.134104

œk

Abstract

Weareconcernedwithexistenceofpositivesolutionsofsemi-positonesecondorder

problems







−u

(t)+u(t) = λ(f(u(t))+w(t)),t∈[0,1],

(0) = u

(1) = 0,

(P)

where λisapositiveparameter,w∈C([0,1],R),and|w(t)|≤c,t∈[0,1],cisapositive

constant, f∈C([0,∞),[0,∞)), andfissuperlinear,i.e,f

:=lim

x→0

f(x)

= 0,f

∞

:=lim

x→∞

f(x)

∞.Byusingﬁxedpointtheoremincones, weshowthatthereexistsaconstantλ

suchthat (P)hasapositivesolutionfor 0 <λ<λ

Keywords

Positive Solution,Semi-Positone, NeumannProblems,FixedPoint Theoremin Cones

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).

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

1.Úó

‡©•§Neumann >Š¯K)•35´˜a-‡¯K, ÚåNõÆö2•'

5, ¿®²¼˜)•35(J[1–7]. AO/, 2000 c,Jiang ÚLiu[8]$^IþØÄ:

½nïÄNeumann>Š¯K







(t)+Mu(t) = f(t,u(t)),t∈(0,1),

(0) = u

(1) = 0

(1.1)

)•35,Xe(J

½nA([[8],½n1]) bf∈C([0,1]×[0,∞),[0,∞)),0 <M<

.ee¡^‡ƒ˜¤á,

(A1)lim

u→0

max

t∈[0,1]

f(t,u)

= 0,lim

u→+∞

max

t∈[0,1]

f(t,u)

= +∞;

(A2)lim

u→0

max

t∈[0,1]

f(t,u)

= +∞,lim

u→+∞

max

t∈[0,1]

f(t,u)

= 0.

K¯K(1.1)–k˜‡).

DOI:10.12677/pm.2023.134104988nØêÆ

œk

2006cƒŸ8[9]$^ÿÀÝnØïÄNeumann>Š¯K







−u

(t)+η

u(t) = h(t)f(t,u(t)),t∈(0,1),

(0) = u

(1) = 0

(1.2)

)•35,Ù¥η>0,0 ≤δ<

,…b½e^‡:

(H1)f∈C([0,1]×[0,+∞),[0,+∞)) ´ëY¼ê,

(H2)h(t) ≥0,t∈[0,1]¿…0 <

1−δ

h(t)dt≤

h(t)dt<+∞.

¼Xe(J:

½nB([[9], ½n2]) b½(H1)–(H2) ¤á.e•3êaÚb¦0 ≤ϕ(a) ≤aA,ψ(b) ≥B,

K¯K(1.2)–k˜‡)u

∗

÷vmin{a,b}≤||u

∗

||≤max{a,b},Ù¥,

A= [max

0≤t≤1

G(t,s)h(s)ds]

−1

,B= [min

δ≤t≤1−δ

1−δ

G(t,s)h(s)ds]

−1

ϕ(r) = max{f(t,u) : (t,u) ∈[0,1]×[0,r]},ψ(r) = min{

f(t,u)

: (t,u) ∈[δ,1−δ]×[σr,r]},

G(t,s) •(1.2) éA‚5¯KGreen ¼ê,r•~ê.

Š5¿´, ©z[8,9]ïÄ‘Neumann >.^‡…š‚5‘fšK¯K)•3

5.˜‡g,¯K´, eš‚5‘•Œœ/,UÄ)•35(J? (ƒ/`,·‚•ÄŒ

Neumann¯K







−u

(t)+u(t) = λ(f(u(t))+w(t)),t∈[0,1],

(0) = u

(1) = 0

(1.3)

)•35,λ•ëê.

©¡ob½:

(H1)f∈C([0,∞),[0,∞));

(H2)f

:=lim

x→0

f(x)

= 0,f

∞

:=lim

x→∞

f(x)

= ∞;

(H3)w∈C([0,1],R), …|w(t)|≤c, c•?¿~ê.

©Ì‡(JXe:

½n1.1b(H1)–(H3) ¤á,K•3~êλ

>0,¦0 <λ<λ

ž,¯K(1.3) –•3

˜‡).

5w(t)=0ž,¯K(1.3)Œòz•¯K(1.1) M=1žœ/.©b½|w(t)|≤c,-

h=f+ w, Ï•f≥0, ¤±h(t)≥−cáuŒœ/. ØJwÑ, w=0 ž, òz•Jiang Ú

Liu[8]óŠ.Ïd, ©í2JiangÚLiu[8]óŠ.

DOI:10.12677/pm.2023.134104989nØêÆ

œk

2.ý•£

-˜mX:= C[0,1], Ù3‰êkuk

∞

=max

t∈[0,1]

|u(t)|e¤Banach˜m, L

[0,1] 3‰êkyk

|y(t)|dte¤Banach˜m.

Ún2.1[10,11]X´˜‡Banach ˜m, …K´X¥˜‡I, =K⊂X. bΩ

, Ω

Xk.mf8,…k0 ∈Ω

,Ω

⊂Ω

F: K∩(Ω

\Ω

) →K

´ëYŽf…÷v

(i)kAuk≤kuk,u∈K∩∂Ω

,…kAuk≥kuk,u∈K∩∂Ω

;

(ii)kAuk≥kuk,u∈K∩∂Ω

,…kAuk≤kuk,u∈K∩∂Ω

KA3K∩(Ω

\Ω

)¥k˜‡ØÄ:.

Ún2.2-z÷v







−z

(t)+z(t) = λg(t),t∈[0,1],

(0) = z

(1) = 0,

Ù¥g∈L

[0,1],…

g(t)dt>0,K

z(t) ≥λσ||z||

∞

Ù¥σ=

cosh1

,M=max

0≤t≤1

G(t,s) =

cosh1

sinh1

,m=min

0≤t≤1

G(t,s) =

sinh1

y²Šâ©z[12]Œ•,

z(t) = λ

G(t,s)g(s)ds.

Ù¥

G(t,s) =







cosh(1−t)coshs

sinh1

,0 ≤s≤t≤1,

cosh(1−s)cosht

sinh1

,0 ≤t≤s≤1,

cosht=

−t

,sinht=

−e

−t

,w,G(t,s) ≥0,t∈[0,1],Œ±

z(t) = λ

G(t,s)g(s)ds≥λm

g(s)ds

= λσ

Mg(s)ds= λσmax

G(t,s)g(s)ds

= λσ||z||

∞

DOI:10.12677/pm.2023.134104990nØêÆ

œk

Ún2.3-u∈C[0,1], ÷v







−u

(t)+u(t) ≥−λc,t∈[0,1],

(0) ≥0, u

(1) ≥0,

Ku≥0,b½||u||

∞

≥

Mc+λσMc

u(t) ≥λσ||u||

∞

−λMc−λ

σMc.

y²v

(t)´‡©•§







−u

(t)+u(t) = −λc,t∈[0,1],

(0) = u

(1) = 0

•˜),K

−v

(t) = λ

G(t,s)cds≤λM

cds≤λMc.

Ïd,

−v

(t) ≤λMc.

-y(t) = u(t)−v

(t),Ky(t) ÷v







−y

(t)+y(t) ≥0,t∈[0,1],

(0) ≥0,y

(1) ≥0.

dÚn2.2Œ•§

y(t) ≥λσ||y||

∞

,t∈[0,1].

é?¿t∈[0,1],k

u(t) = y(t)+v

(t)

≥λσ||y||

∞

−λMc

= λσ||u−v

∞

−λMc

≥λσ(||u||

∞

−||v

∞

)−λMc

≥λσ(||u||

∞

−λMc)−λMc

≥λσ||u||

∞

−λMc−λ

σMc,t∈[0,1].

DOI:10.12677/pm.2023.134104991nØêÆ

œk

3.Ì‡(Jy²

½n1.1y²bf

= 0,f

∞

= ∞,…¯K







−u

(t)+u(t) = λ(f(u(t))+w(t)),t∈[0,1],

(0) = u

(1) = 0

du

u(t) = λ

G(t,s)(f(u(s))+w(s))ds,t∈[0,1].

½Â

K= {u∈X|u(t) ≥λσ||u||

∞

−λMc−λ

σMc,t∈[0,1]}

du||u||

∞

≥

Mc+λσMc

,¤±u(t) ≥0,KK•X¥I.eu∈K,(ÜÚn2.3Œ•,

Lu(t) = λ

G(t,s)((f(u(s))+w(s)))ds

= λ

G(t,s)(f(u(s)+w(s)+c)ds−λ

G(t,s)cds

≥λm

(f(u(s))+w(s)+c)ds−λMc

= λσ

M(f(u(s)+w(s)+c)ds−λMc

≥λσ||Lu||

∞

+λσMc−λMc

≥λσ||Lu||

∞

−λMc−λ

σMc.

L(K) ⊂K,´•L´ëY, eyL´;Žf.

S⊂C[0,1] •k.8,df+wëY5•,•3D>0, k

f(u)+w≤D,u∈S.

éuu∈Sk

||Lu||

∞

=max

t∈[01]



G(t,s)((f(u(s))+w(s)))ds



≤λ

MDds

≤λMD.

DOI:10.12677/pm.2023.134104992nØêÆ

œk

Ïd,L(S)3C[0,1]þ˜—k..é?¿t

∈[0,1](t



Lu(t

)−Lu(t

)



G(t

,s)(f(u(s))+w(s))ds−λ

G(t

,s)(f(u(s))+w(s))ds



≤λD



G(t

,s)−G(t

,s)



ds.

dG(t,s) ëY5Œ•,é?¿ε>0,•3δ(ε) >0,¦e|t

−t

|<δ,K|Lu(t

)−Lu(t

)|<ε.

,L(S)3C[0,1]þÝëY. dArz`ela-Ascoli½n•,L(S) 3C[0,1]þ´ƒé;, Ïd,L´

ëY.

(a)ef

= 0,Ké?¿η>0, •3H

>1,1 <u≤H

ž,kf(u) ≤ηu, ÷v

λM(η+c) ≤1.

Ïd,eu∈K…||u||

∞

= H



Lu(t)



G(t,s)((f(u(s))+w(s)))ds



≤λ

G(t,s)



((f(u(s))+w(s))



≤λM

(



ηu(s)



+c)ds

≤λM

(η||u||

∞

+c||u||

∞

)ds

≤||u||

∞

.(3.1)

=λ≤

M(η+c)

:= λ

Ω

:= {u∈X: ||u||

∞

d(3.1)•

||Lu||

∞

≤||u||

∞

,u∈K∩∂Ω

.(3.2)

(b)ef

∞

= ∞,Ké?¿µ>0, •3

>0,u≥

ž,kf(u) ≥µu,…÷v

mµσ−λ

µmMc−λ

µm

c−λmc≥1.

-H

= max{2H

+λMc+λ

σMc

λσ

}…Ω

:= {u∈X: ||u||

∞

},Ku∈K,||u||

∞

= H

u(t) ≥λσ||u||

∞

−λMc−λ

σMc≥

DOI:10.12677/pm.2023.134104993nØêÆ

œk

Ïd



Lu(t)



G(t,s)(f(u(s))+w(s))ds



≥λ



G(t,s)(µu(s)−c)ds



≥λm

[µ(λσ||u||

∞

−λMc−λ

σMc)−c]ds

= λ

µmσ||u||

∞

−λ

µmMc−λ

µm

c−λmc

≥λ

mµσ||u||

∞

−(µλ

mMc+λ

µm

c+λmc)||u||

∞

≥(λ

mµσ−λ

µmMc−λ

µm

c−λmc)||u||

∞

≥||u||

∞

Ïd

||Lu||

∞

≥||u||

∞

,u∈K∩∂Ω

.(3.3)

l,d(3.2),(3.3)ÚÚn2.1Œ•,L3K∩(Ω

\Ω

)¥k˜‡ØÄ:,¦H

≤||u||

∞

≤H

Ïd,0 ≤λ≤λ

žk˜‡).

4.A^

~•Ä¯K







−u

(t)+u(t) = λ(2u

+sin2t),t∈[0,1],

(0) = u

(1) = 0

(4.1)

)•35,Ù¥λ>0 •ëê.

)ùpf(u) = 2u

,w(t) = sin2t,w,f(u) ≥0,…f÷vf

= 0,f

∞

= ∞,^‡(H1)-(H2)

¤á,¿…|sin2t|≤1,t∈[0,1],^‡(H3) ¤á.

Šâ½n1.1•3~êλ

>0,¦0 <λ<λ

ž,¯K(4.1)•3˜‡).

Ä7‘8

I[g,‰ÆÄ7(1OÒ:12061064).

ë•©z

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