一类二阶差分方程组 Robin 边值问题的正解 Positive Solutions of Robin Boundary Value Problems for a Class of Second-Order Difference System

doi:10.12677/PM.2022.126102

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PureMathematicsnØêÆ,2022,12(6),928-937

PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.126102

˜a©•§|Robin>Š¯K)

ÇÇÇ°°°²²²

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c53F¶¹^FÏµ2022c67F¶uÙFÏµ2022c614F

Á‡

©Ñué©•§|>Š¯Kš‚5‘ÍÜO•g•§)û˜aš‚5©•§|>Š¯

K)"$^šKþà¼êJensen ØªÚØÄ:•ênØ?Ø˜a©•§|

Robin>Š¯K











−∆

u(t−1) = f(t,u,v),t∈[1,T]

−∆

v(t−1) = g(t,u,v),t∈[1,T]

u(0) = ∆u(T) = 0,

v(0) = ∆v(T) = 0

)•35§Ù¥T≥2´˜‡ê§∆u(t)=u(t+1)−u(t)´c•©Žf§f,g:

[1,T]

×[0,∞)×[0,∞) →[0,∞)ëY"

'…c

JensenØª§)§©•§|§ØÄ:•ênØ

PositiveSolutionsofRobinBoundary

ValueProblemsforaClassofSecond-Order

DiﬀerenceSystem

HaiyiWu

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

©ÙÚ^:Ç°².˜a©•§|Robin>Š¯K)[J].nØêÆ,2022,12(6):928-937.

DOI:10.12677/pm.2022.126102

Ç°²

Received:May3

,2022;accepted:Jun.7

,2022;published:Jun.14

,2022

Abstract

Inthispaper,weconsiderthecoupledgrowthofnonlineartermsforboundaryvalue

problems of systemsof diﬀerenceequations,resolve the positive solutions ofboundary

valueproblemsforaclassofnonlineardiﬀerenceequations.AlsobyusingJensen’s

inequalityfornonnegativeconcavefunctionsandtheﬁxedpointindextheory,we

discusstheexistenceofpositivesolutionsofRobinboundaryvalueproblemsfora

classofsecond-orderdiﬀerencesystem











−∆

u(t−1) = f(t,u,v),t∈[1,T]

−∆

v(t−1) = g(t,u,v),t∈[1,T]

u(0) = ∆u(T) = 0,

v(0) = ∆v(T) = 0

whereT≥2istheinteger,∆u(t)=u(t+1) −u(t)istheforwarddiﬀerenceoperator,

f,g: [1,T]

×[0,∞)×[0,∞) →[0,∞)arecontinuous.

Keywords

Jensen’sInequality,PositiveSolutions,Second-OrderDiﬀerenceEquations,

FixedPointIndexTheory

2022byauthor(s)andHansPublishersInc.

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

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

1.Úó

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:= {a,a+1,···,b}.

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Æ¥Á•.!Fiobonacci ¯K!E|¯K. ,'u ©•§(|)>Š¯KïÄ%ƒé

DOI:10.12677/pm.2022.126102929nØêÆ

Ç°²

…ú, ¿…Œõê´æ^†‡©•§aq•{5ïÄ.Ù¥˜a´ïáGreen ¼êò¤ïÄ>

Š¯K=z•¦A½¼ê˜m¥ŽfØÄ:,,|^ØÄ:½n!þe)•{½ÿÀÝnØ¦

),,˜a´$^©ÜnØ!C©{!.:nØ(ÜØÄ:nØïÄ, ë„©z[1–10].

2009c, Henderson[9]3f,g,a,bäkfÍÜ^‡e, ÄuIþØÄ:½n?Ø©X











∆

u(n−1)+λa(n)f(u(n),v(n)) = 0,n∈[1,N−1]

∆

v(n−1)+µb(n)g(u(n),v(n)) = 0,n∈[1,N−1]

u(0) = βu(η),u(N) = αu(η) = 0,

v(0) = βv(η),v(N) = αv(η) = 0

)•35,Ù¥η∈[1,N−1]

,α,β,λ,µ>0,N≥4,…f,g,a,bšKëY.

3dÄ:þ,2011c,©[10]ïÄäk•˜„>.^‡©ê©XÚ)•35.

2020 c,3©[11]¥,Yang <ÄušKþà¼êJensen ØªÚØÄ:•ênØ, ïÄ

˜a[‚5~‡©•§|Robin>Š¯K











−((u

)

p−1

)

= f(t,u,v),t∈[0,1],

−((v

)

q−1

)

= g(t,u,v),t∈[0,1],

u(0) = u

(1) = 0,

v(0) = v

(1) = 0

)•35,Ù¥p,q>1,…f,g: [0,1]×[0,∞)×[0,∞) →[0,∞)ëY.

Éþã©zéu, ©|^šKþà¼ê#Nš‚5‘f,g·ÜO•,ÄušKþà¼ê

JensenØª(ÜØÄ:•ênØ5¼kO,ïÄXe©•§|Dirichlet>Š¯K











−∆

u(t−1) = f(t,u,v),t∈[1,T]

−∆

v(t−1)= g(t,u,v),t∈[1,T]

u(0) = u(T+1) = 0,

v(0) = v(T+1) = 0

(1)

)•35.

©|„Xe:1!•¹A‡ÐÚ(J, AO´dþà¼ê5Ÿ¤#ØªÚšKþ

à¼êJensenØª.1n!·‚k‰½©ob½,$^þà¼ê5•xš‚5‘f,g·ÜO

•.~X,f´g‚5O•,Kg´‡‚5O•.•, 3!¥,·‚‰Ñ©Ì‡(J9Ùy².

2.ý•£

©¤^˜m•:

DOI:10.12677/pm.2022.126102930nØêÆ

Ç°²

PE:= {u(t)|u: [0,T+1]

→R,u(0) = 0 = u(T+1)},P:= {u∈E|u(t) ≥0,t∈[0,T+1]

KE3‰êkuk= max

t∈[0,T+1]

|u(t)|e¤¢Banach ˜m,…P•Eþ˜‡I.

éu?¿(u,v) ∈E×E,

k(u,v)k:= max{kuk,kvk},

KE×E3‰êk(u,v)ke¤¢Banach˜m.

©XÚ(1)du







u(t) =

s=1

G(t,s)f(s,u(s),v(s)),t∈[1,T]

v(t) =

s=1

G(t,s)g(s,u(s),v(s)),t∈[1,T]

(2)

éuu(t),v(t) : [0,T+1]

→R,½ÂŽfA

: P×P→P,A: P×P→P×P,







(u,v)(t) :=

s=1

G(t,s)f(s,u

(s),v

(s)),t∈[1,T]

(u,v)(t) :=

s=1

G(t,s)g(s,u

(s),v

(s)),t∈[1,T]

(3)

…A(u,v)(t) := (A

(u,v)(t),A

(u,v)(t)),Ù¥G(t,s) := min{t,s},s,t∈[1,T]

•‚5¯K







−∆

u(t−1) = h(t),t∈[1,T]

u(0) = ∆u(T) = 0

‚¼ê.´y,ŽfA

,AëY, •§|(1) duŽf•§(u,v) = A(u,v).

©¦^Ì‡óäXe:

Ún1([8])E•¢Banach ˜m, P•E¥˜‡I, Ω ´Pþ˜‡k.m8, Žf

T: Ω∩P→PëY. e•3x

∈P\{0}, ¦

x−Tx6= λx

,x∈∂Ω∩P,λ≥0,

Ki(T,Ω∩P,P) = 0. ùpi´PþØÄ:•ê.

Ún2([8])E•¢Banach˜m,P•E¥˜‡I,Ω´Pþ˜‡k.m8,0 ∈Ω,

ŽfT: Ω∩P→PëY. e

x−λTx6= 0,x∈∂Ω∩P,λ∈[0,1],

Ki(T,Ω∩P,P) = 1.ùpi´PþØÄ:•ê.

Ún3([11])P×P´Banach˜m(E,k·k)˜‡I, ŽfT: P×P→P×PëY,

P×P

(T) <1,K•3P×PþŽfI−TëY_Žf(I−T)

−1

(I−T)

−1

P×P

∞

m=0

DOI:10.12677/pm.2022.126102931nØêÆ

Ç°²

Ù¥,r

P×P

(T)•ŽfT3P×PþÌŒ».

íØ1([11])P×P´Banach˜m(E,k·k)˜‡I, ŽfT: P×P→P×PëY,

P×P

(T) <1…x,x

∈P×P,÷vx≤Tx+x

,Kx≤(I−T)

−1

P×P

Ún4ω∈P´[0,T+1]

þþà¼ê,e•3[0,T+1]

þ˜:r,¦kωk= ω(r),

kωk<

t=1

ω(t)sin

πt

2T+1

.(4)

Ù¥,µ

t=1

tsin

πt

2T+1

>0.

y².dbŒ•,

t=1

ω(t)sin

πt

2T+1

t=1

ω(r·

+(1−

)·0)sin

πt

2T+1

t=r+1

ω((T+1)·

T+1

+(1−

T+1

)·0)sin

πt

2T+1

ω(r)

t=1

tsin

πt

2T+1

kωkµ

l(4)¤á.

Ún5([7])Ψ

: [0,∞)→[0,∞) ´ëYþà¼ê, Ψ

: [1,T]

→[0,∞) ´‰½¼ê,

>0,K



i=1

(i)

i=1



≥

i=1

(Ψ

(i))

i=1

Ún6ϕ

(t) = sin

πt

2T+1

,t∈[1,T]

t=1

G(t,s)ϕ

(t) = ϕ

(s),s∈[1,T]

Ù¥,λ

= 4sin

4T+2

y². Ï•λ

s=1

G(t,s)ϕ

(s)=ϕ

(t),t∈[1,T]

.5¿,‚¼êé¡5,G(t,s)=

G(s,t),t,s∈[1,T]

.¤±λ

s=1

G(s,t)ϕ

(s) = ϕ

(t),t∈[1,T]

.Ïd(Ø¤á.

Ún7ψ: [0,∞) →[0,∞) ëY…3[0,∞) þà,KψüN4O.

y².é?¿x>x

≥0,duψþà, K

ψ(x

) ≥ψ(x

−x

x−x

(ψ(x)−ψ(x

)),

DOI:10.12677/pm.2022.126102932nØêÆ

Ç°²

(ÜψšK5,lim

x→∞

ψ( x)

≥0.lψ(x

) ≥ψ(x

3.Ì‡(J9y²

©ob½µ

(H1)¼êf,g: [1,T]

×[0,∞)×[0,∞) →[0,∞) ëY;

(H2)¼êψ

,ψ

: [0,∞) →[0,∞)ëY,ψ

•þà¼ê,…•3~êα>2(T+1)

sin

4T+2

,c>

0,¦

(i)f(t,x,y) ≥ψ

(y)−c,g(t,x,y) ≥ψ

(x)−c,t∈[1,T]

,x,y∈[0,∞);

(ii)ψ

(ψ

(z)) ≥αz−c,z∈[0,∞);

(H3)•3~êa

≥0,r>0,…r

P×P

) <1,k

f(t,u,v) ≤a

u+b

v, g(t,u,v) ≤c

u+d

v,x,y∈[0,r],t∈[1,T]

Ù¥,ŽfT

: P×P→P×P,

(u,v)(t) := (

s=1

G(t,s)(a

u(t)+b

v(t)),

s=1

G(t,s)(c

u(t)+d

v(t))).

©Ì‡(JXe:

½n1b½(H1)-(H3) ¤á,•§|(1) –•3˜‡).

y².PM

:= {(u,v) ∈P×P: (u,v) = A(u,v)+λ(ω

,ω

),λ≥0},Ù¥ω

(t) := (2T+1)t−t

e(u

) ∈M

,K•3λ

≥0,d

(t) =

s=1

G(t,s)f(s,u

(s),v

(s))+λ

, v

(t) =

s=1

G(t,s)g(s,u

(s),v

(s))+λ

,(5)

(Ü(2),(3)Œ,(u

) = A(u

)+λ

(ω

,ω

d(H1) Œ•,∆

(t−1)≤0,lu

3[1,T]

þ´þà.ÓnŒ,v

3[1,T]

þ•´þ

à.(Ü(5) ,

(t) ≥

s=1

G(t,s)f(s,u

(s),v

(s)), v

(t) ≥

s=1

G(t,s)g(s,u

(s),v

(s)),

db½(H2)¥(i),

(t) ≥

s=1

G(t,s)ψ

(s))−c

, v

(t) ≥

s=1

G(t,s)ψ

(s))−c

,(6)

DOI:10.12677/pm.2022.126102933nØêÆ

Ç°²

Ïd,

(t) ≥

s=1

G(t,s)ψ

(

τ=1

G(s,τ)ψ

(τ)))−c

.(7)

τ=1

G(s,τ) =

τ=1

τ+

τ=s+1

(T+1−s)s

-h(s) =

(T+1−s)s

,Kh•Š

,•ŒŠ

(T+1)

.5¿T≥2,¤±

τ=1

G(s,τ) ≥1.

dψ

üN4OÚÚn5Œ,

(

τ=1

G(s,τ)ψ

(τ)))≥ψ



τ=1

G(s,τ)ψ

(τ))

τ=1

G(s,τ)



≥

τ=1

G(s,τ)ψ

(ψ

(τ)))

τ=1

G(s,τ)

≥

(T+1)

τ=1

G(s,τ)ψ

(ψ

(τ))),

(Ü(H2)¥(ii)9(7)Œ,

(t)≥

(T+1)

s=1

G(t,s)

τ=1

G(s,τ)ψ

(ψ

(τ)))−c

≥

8α

(T+1)

s=1

G(t,s)

τ=1

G(s,τ)u

(τ)−c

(t) ≥

8α

(T+1)

s=1

G(t,s)

τ=1

G(s,τ)u

(τ)−c

.(8)

Pµ

t=1

sin

πt

2T+1

>0. é(8) üàÓž¦±sin

πt

2T+1

, 2lt=1Tþ¦Ú,¿dÚn6

Œ,

t=1

(t)sin

πt

2T+1

≥

8α

(T+1)

t=1



s=1

G(t,s)

τ=1

G(s,τ)u

(τ)sin

πt

2T+1



−µ

8α

(T+1)

s=1



t=1

G(t,s)sin

πt

2T+1

τ=1

G(s,τ)u

(τ)



−µ

8α

(T+1)

s=1

τ=1

G(s,τ)sin

πs

2T+1

(τ)−µ

8α

(T+1)

τ=1

s=1

G(s,τ)sin

πs

2T+1

(τ)−µ

8α

(T+1)

t=1

sin

πt

2T+1

(t)−µ

DOI:10.12677/pm.2022.126102934nØêÆ

Ç°²

l

t=1

(t)sin

πt

2T+1

≤

(T+1)

8α−(T+1)

.(9)

Ún4,

t=1

(t)sin

πt

2T+1

≤

(T+1)

rλ

[8α−(T+1)

]

u

k.,eyv

k..dÚn4 Œ•,

t=1

(t)sin

πt

2T+1

d(6)Ú(9)(ÜÚn69Ún4Œí•,

(T+1)

8α−(T+1)

≥

t=1

(t)sin

πt

2T+1

≥

t=1

s=1

G(t,s)ψ

(s))sin

πt

2T+1

−µ

s=1

t=1

G(t,s)ψ

(s))sin

πt

2T+1

−µ

t=1

(t))sin

πt

2T+1

−µ

t=1



(t)

·kv



sin

πt

2T+1

−µ

≥

(kv

t=1

(t)sin

πt

2T+1

−µ

rλ

(kv

k)−µ

Ïd,

(kv

k) <

(T+1)

rλ

(8α−(T+1)

)

rλ

d(H2)Œ•,lim

z→∞

(z) = ∞,K•3c

>0,¦kv

k<c

,M

k..

-R>sup{(u,v):(u,v)∈M

},K•3ω

6=0,¦(u,v)6=A(u,v)+λ(ω

,ω

).dÚn1 ,

?¿(u,v) ∈∂B

∩P×P,λ6= 0,

i(A,B

∩P×P,P×P) = 0.(10)

={(u,v)∈

∩P×P:(u,v)=λA(u,v),λ∈[0,1]},eyM

={0}.e(u

)∈

DOI:10.12677/pm.2022.126102935nØêÆ

Ç°²

,(u

)∈

∩P×P,K(u

)=λ

A(u

).du¯K(1)duŽf•§(u

A(u

),(Ü(2),é,‡λ

∈[0,1],

(t) ≤

s=1

G(t,s)f(s,u

(s),v

(s)), v

(t) ≤

s=1

G(t,s)g(s,u

(s),v

(s)),

d(H3)•,

(t) ≤

s=1

G(t,s)(a

), v

(t) ≤

s=1

G(t,s)(c

l

) ≤T

).(11)

díØ1Ú(10)Œ•, u

= v

= 0,M

= {0}.dÚn2Œ•,éu?¿(u,v) ∈B

∩P×P,λ∈

[0,1],(u,v) 6= λA(u,v).K

i(A,B

∩P×P,P×P) = 1.(12)

(Ü(10),(12),

i(A,(B

)∩P×P,P×P) = −1.

KA3(B

)∩P×P–k˜‡ØÄ:.

Ä7‘8

I[g,‰ÆÄ7“cÄ7]Ï‘8(11801453,11901464),[‹Ž“c‰EÄ7Oy]

Ï(20JR10RA100).

ë•©z

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