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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(7),4129-4141
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.117440
˜a‘š‚5>.^‡©•§)
õ)5
µµµyyyÚÚÚ
∗
§§§´´´ýýý
†
Ü“‰ŒÆ§êƆÚOÆ§[‹=²
ÂvFϵ2022c61F¶¹^Fϵ2022c627F¶uÙFϵ2022c74F
Á‡
‘š‚5>.^‡©•§>Š¯K3æ^àR|YYgj-\xù±9¦)‚/•þý
.•§|»•)•¡kX-‡A^"©$^ØÄ:•ê½nÚþe)•{§š‚5‘
•¼ê…3á?‡‚5O•ž§é¿©ëê§ïáþã¯K)•359õ)5
(J§ù•‡©•§>Š¯KêŠ)JønØ•{"•ÏL˜‡~f`²½n(Øk
5"
'…c
õ)5§)§þe)•{§ÿÀÝnØ
MultiplicityofPositiveSolutionsfor
aClassofSecond-OrderDifference
EquationwithNonlinearBoundary
Conditions
ZhengqiJing
∗
,YanqiongLu
†
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,Lanzhou Gansu
∗1˜Šö"
†ÏÕŠö"
©ÙÚ^:µyÚ,´ý .˜a‘š‚5>.^‡©•§)õ)5[J].A^êÆ?Ð,2022,11(7):
4129-4141.DOI:10.12677/aam.2022.117440
µyÚ§´ý
Received:Jun.1
st
,2021;accepted:Jun.27
th
,2022;published:Jul.4
th
,2022
Abstract
Boundaryvalueproblemsofdifferenceequationswithnonlinearboundaryconditions
have many important applicationssuch asin strengthening Bridges withpolyurethane
cement wireropes andsolving positiveradial solutionsof ellipticequationsin annular
domain.Inthispaper,byusingthefixedpointindextheoremandthemethodofupper
andlowersolutions, weobtaintheexistenceandmultiplicityofpositivesolutionsto
theaboveproblemsforsufficientlysmallparameterswhenthenonlineartermisa
positivefunctionandsuperlineargrowthatinfinity.Theresultsprovideatheoretical
methodfornumericalsolutionofboundaryvalueproblemsofdifferentialequations.
Finally,wegiveanexampletoillustratethevalidityofthemainresults.
Keywords
Multiplicity,PositiveSolution,UpperandLowerSolutions,Topological
DegreeTheory
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
‘š‚5>.^‡ ‡©•§/©•§>Š¯K´˜a-‡š‚5¯K,ÉNõÆö
ïÄ[1–11].1998c,DUNNINGERDRÚ°ÿ[1]$^þe)•{ÚÿÀÝnØïÄ‘š‚5
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αu(0)−βu
0
(0) = λg
1
(u(0)),
γu(1)+δu
0
(1) = λg
2
(u(1))
)õ)5,Ù¥λ>0,f:[0,∞)→(0,∞)ëY…f
∞
=lim
u→∞
f(u)/u=∞,g
1
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:[0,∞)→
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[2],AbdouK.Drame,DavidG.Costa[5],Shivaji[6,7],êX[9]óŠ9Ùë•©z.AO
DOI:10.12677/aam.2022.1174404130A^êÆ?Ð
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= λf(u),
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©•§>Š¯K
(
∆
2
u(k+1)+λa(k)f(u(k)) = 0,k∈[1,T]
Z
,
α∆u(0) = 0,u(T+1) = βu(k)
)•35.2014c,´ý ,êX[11]$^©ÜnØïÄ‘š‚5>.^‡©•§
(
−∆[p(k−1)∆y(k−1)]+q(k)y(k) = λa(k)f(y(k)),k∈I,
−∆y(0)+αg(y(0)) = 0,∆y(N)+βg(y(N+1)) = 0
)Û(,Ù¥α,β≥0•~ê,I:={1,······,N},p:{0,1,······,N}→(0,∞),q,a:I→
[0,∞),k∈Iž,a(k) >0.
Éþã©z[1,10,11]éu,©?Ø‘š‚5>.^‡©•§>Š¯K





∆
2
u(t−1)+λh(t)f(u(t)) = 0,t∈[1,T]
Z
,
αu(0)−β∆u(0) = λg
1
(u(0)),
γu(T+1)+δ∆u(T) = λg
2
(u(T))
(1)
)•359õ)5,Ù¥α,β,γ,δ≥0,%:=αδ+αγ(T+1)+ γβ>0•ëê,λ>0,[1,T]
Z
=
{1,2,···,T},h: [1,T]
Z
→[0,∞)…h6≡0;¿¼¯K(1))•35Úõ)5(J.
½n1.1be^‡
(A1)f: [0,∞) →(0,∞)ëY;
(A2)g
1
,g
2
: [0,∞) →(0,∞)ëY;
(A3)f
∞
=lim
u→∞
f(u)/u= ∞
¤á,K•3˜‡λ
∗
>0,¦0<λ<λ
∗
ž,¯K(1)–kü‡);λ= λ
∗
ž,¯K(1)–k˜
‡);λ>λ
∗
ž,¯K(1)Ã).
½n1.2bf≡0…(A2)Ú
(A4)(g
1
)
∞
=lim
u→∞
g
1
(u)/u= ∞, (g
2
)
∞
=lim
u→∞
g
2
(u)/u= ∞
¤á,K•3˜‡λ
∗
>0,¦0<λ<λ
∗
ž,¯K(1)–•3ü‡);λ=λ
∗
ž,¯K(1)–•
3˜‡);λ>λ
∗
ž,¯K(1)Ã).
DOI:10.12677/aam.2022.1174404131A^êÆ?Ð
µyÚ§´ý
2.ý•£
d©•§nØ•£[12],´•¯K(1))du¯K





∆
2
u(t−1)+λh(t)f(u(t)) = 0,t∈[1,T]
Z
,
αu(0)−β∆u(0) = 0,
γu(T+1)+δ∆u(T) = 0
(2)
)Ú¯K





∆
2
u(t−1) = 0,t∈[1,T]
Z
,
αu(0)−β∆u(0) = λg
1
(u(0)),
γu(T+1)+δ∆u(T) = λg
2
(u(T))
(3)
)ƒÚ.ÏLOŽŒ¯K(2))•
u
1
(t) = λ
T
X
s=1
G(t,s)h(s)f(u(s)),
¯K(3))•
u
2
(t) =
αg
2
(u(T))−γg
1
(u(0))
%
t+
(δ+γ(T+1))g
1
(u(0))+βg
2
(u(T))
%
.
ÏdØJy¯K(1)duÚ©•§
u(t) = λ(P(u(0),u(T))t+Q(u(0),u(T)))+λ
T
X
s=1
G(t,s)h(s)f(u(s)),(4)
Ù¥
P(s,t) =
αg
2
(t)−γg
1
(s)
%
,Q(s,t) =
(δ+γ(T+1))g
1
(s)+βg
2
(t)
%
(5)
…‚¼ê
G(t,s) =
1
%
(
(δ+γ(T+1)−γs)(β+αt),0 ≤t≤s≤T+1,
(δ+γ(T+1)−γt)(β+αs),0 ≤s≤t≤T+1.
(6)
Ún2.1G(t,s)´(6)¥½Â‚¼ê,KG(t,s)÷vXe5Ÿ:
(i)G(t,s) ≥0,t,s∈[0,T+1]
Z
;
(ii)G(t,s) ≥
1
T+1
G(τ,s),t∈[1,T]
Z
, s,τ∈[0,T+1]
Z
.
DOI:10.12677/aam.2022.1174404132A^êÆ?Ð
µyÚ§´ý
y²dG(t,s)½Â•
G(t,s) ≥
1
%
δβ≥0t,s∈[0,T+1]
Z
.
AO/,t∈[1,T]
Z
ž,k
G(t,s) ≥
1
%





(δ+γ)β,s= 0,
(δ+γ)(β+α),s∈[1,T]
Z
,
(β+α)δ,s= T+1.
,˜•¡,éu?¿s,τ∈[0,T+1]
Z
,k
G(τ,s) ≤
1
%
(δ+γ(T+1)−γs)(β+αs)
ÏdG(t,s) ≥
1
T+1
G(τ,s)¤á.
Ún2.2¼êP(x,y)t+Q(x,y)÷vXe5Ÿ:
(i)P(x,y)t+Q(x,y) ≥min{Q(x,y),P(x,y)(T+1)+Q(x,y)}≥0,t∈[0,T+1]
Z
,x,y>0,
(ii)P(x,y)t+Q(x,y) ≤max{Q(x,y),P(x,y)(T+1)+Q(x,y)},t∈[0,T+1]
Z
,x,y>0,
(iii)P(x,y)t+Q(x,y) ≥
1
T+1
(P(x,y)τ+Q(x,y)),t∈[1,T]
Z
,τ∈[0,T+1]
Z
,x,y>0.
y²w,,éz‡x,y≥0…g
1
,g
2
´šK¼ê^‡e,éP(x,y)t+ Q(x,y)?1©Û,éu?
¿t∈[0,T+1]
Z
,τ∈[0,T+1]
Z
k
(1)P(x,y) ≥0ž,P(x,y)t+Q(x,y) ≥Q(x,y) ≥0;
(2)P(x,y) <0ž,P(x,y)t+Q(x,y) ≥P(x,y)(T+1)+Q(x,y) ≥0,
Ïd5Ÿ(i)¤á.ÓnŒy5Ÿ(ii)¤á.
ey5Ÿ(iii)¤á.éu?¿t∈[1,T]
Z
,τ∈[0,T+1]
Z
k
(1)P(x,y) ≥0ž,
P(x,y)t+Q(x,y) ≥P(x,y)+Q(x,y) =
1
T+1
(P(x,y)(T+1)+Q(x,y)(T+1))
≥
1
T+1
(P(x,y)(T+1)+Q(x,y)) ≥
1
T+1
(P(x,y)τ+Q(x,y));
(2)P(x,y) <0ž,
P(x,y)t+Q(x,y) ≥P(x,y)T+Q(x,y) ≥P(x,y)(T+1)+Q(x,y)
=
1
T+1
(P(x,y)+Q(x,y)(T+1)) =
1
T+1
(P(x,y)+Q(x,y)T+Q(x,y))
≥
1
T+1
Q(x,y) ≥
1
T+1
(P(x,y)τ+Q(x,y)),
Ïd5Ÿ(iii)¤á.
Ún2.3([14])E´Banach˜m,K⊂E´E¥˜‡I.é?¿r>0,PK
r
={u∈K|k
DOI:10.12677/aam.2022.1174404133A^êÆ?Ð
µyÚ§´ý
uk≤r}.bŽfT:
¯
K
r
→KëY,u∈∂K
r
ž,¦Tu6= u.
(i)eu∈∂K
r
ž,÷vkTuk≥kuk,Ki(T,K
r
,K) = 0.
(ii)eu∈∂K
r
ž,÷vkTuk≤kuk,Ki(T,K
r
,K) = 1.
-E= {u|u: [0,T+1]
Z
→R},KEU‰êkuk=max
t∈[0,T+1]
Z
|u(t) |¤Banach˜m.
½Â8ÜK
0
= {u∈E|u≥0},KK
0
•E¥šKI.½ÂEþIKXe:
K= {u∈E|u≥0,min
t∈[1,T]
Z
u(t) ≥
1
T+1
kuk}.(7)
´„K⊂K
0
,…(4)ŒŠØÄ:•§Tu= u,Ù¥ŽfT: E→E½ÂXe:
Tu(t) = λ(P(u(0),u(T))t+Q(u(0),u(T)))+λ
T
X
s=1
G(t,s)h(s)f(u(s)).(8)
Ún2.4T(K
0
) ⊂K,…ŽfT: K
0
→KëY.
y²éu?¿u∈K
0
,t∈[1,T]
Z
ž,dÚn2.1(ii)ÚÚn2.2(iii)Œ
Tu(t) ≥λ
1
T+1
{P(u(0),u(T))τ+Q(u(0),u(T))+
T
X
s=1
G(τ,s)h(s)f(u(s))}
≥
1
T+1
Tu(τ), τ∈[0,T+1]
Z
.
Ïd
min
t∈[1,T]
Z
Tu(t) ≥
1
T+1
kTuk
=Tu∈K,T(K
0
) ⊂K.qÏ•E•k•‘˜m,¤±dfëY5,´yT: K
0
→KëY.
3.)•35†Ø•35
½n3.1b(A1) −(A3).Kλ>0¿©ž,¯K(1)–•3˜‡);λ>0¿©Œ ž,¯
K(1)Ã).
y²-
M=max
(t,s)∈[0,T+1]
Z
×[0,T+1]
Z
G(t,s), m=min
(t,s)∈[1,T]
Z
×[1,T]
Z
G(t,s),
KdÚn2.1•,M,m>0,é?¿q>0,P
I(q) = Mmax
u∈K,kuk=q
T
X
s=1
h(s)f(u(s)) >0.
é?¿r
1
>0,PK
r
1
= {u∈K|kuk<r
1
}.é?¿u∈∂K
r
1
,•3¿©σ>0÷v
σ≤
r
1
2I(r
1
)
,σmax(Q(u(0),u(T)),P(u(0),u(T))+Q(u(0),u(T))) ≤r
1
/2.
DOI:10.12677/aam.2022.1174404134A^êÆ?Ð
µyÚ§´ý
λ≤σž,é?¿t∈[0,T+1]
Z
,dÚn2.2(ii)•
Tu(t) ≤
r
1
2
+σM
T
X
s=1
h(s)f(u(s)) ≤
r
1
2
+σI(r
1
) ≤r
1
,
=kTuk≤r
1
=kuk,u∈∂K
r
1
.dÚn2.3•
deg(T,K
r
1
,K) = 1.
鉽λ≤σ,Ïf
∞
=∞,•3~êp>0,¦é?¿u≥p,kf(u)≥ηu,Ù¥η>0¿©
Œ…÷v
λmη
T+1
T
X
s=1
h(s) ≥1.
r
2
≥max{(T+1)p,r
1
+1},PK
r
2
= {u∈K|kuk<r
2
}.u∈∂K
r
2
,ž,min
t∈[1,T]
Z
u(t) ≥
1
T+1
kuk≥
p.Ïd,é?¿t∈[1,T]
Z
,k
Tu(t) ≥λm
T
X
s=1
h(s)f(u(s)) ≥λmη
T
X
s=1
h(s)f(u(s))
≥
λmη
T+1
kuk
T
X
s=1
h(s) ≥kuk,
=kTuk≥kuk,u∈∂K
r
2
,dÚn2.3•
deg(T,K
r
2
,K) = 0.
dØÄ:•êŒ\5,deg(T,K
r
2
\
¯
K
r
1
,K)=−1,Ïd,ŽfT3K
r
2
\
¯
K
r
1
–•3˜‡ØÄ
:,=¯K(1)–•3˜‡).
d(A1),(A3)Œ•,•3˜‡~êc>0,¦é?¿u≥0,kf(u)≥cu.b¯K(4)•3
)u∈E,dÚn2.4•u∈K.¿©Œλ>0,¦
λmc
T+1
T
X
s=1
h(s) >1.
Ké?¿t∈[1,T]
Z
,k
u(t) ≥λmc
T
X
s=1
h(s)u(s) ≥
λmc
T+1
kuk
T
X
s=1
h(s) >kuk,
ù†bgñ.Ïd,λ¿©Œž,¯K(4)Ã).
4.õ‡)•35
•¼¯K(1)õ‡)•35,·‚Ú\¯K(1)þe)•{.
DOI:10.12677/aam.2022.1174404135A^êÆ?Ð
µyÚ§´ý
½Â4.1e¯u∈E÷v





∆
2
¯u(t−1)+λh(t)f(¯u(t)) ≤0,t∈[1,T]
Z
,
α¯u(0)−β∆¯u(0) ≥λg
1
(¯u(0)),
γ¯u(T+1)+δ∆¯u(T) ≥λg
2
(¯u(T)),
K¡¯u•¯K(1)þ);eu∈E÷v





∆
2
u(t−1)+λh(t)f(u(t)) ≥0,t∈[1,T]
Z
,
αu(0)−β∆u(0) ≤λg
1
(u(0)),
γu(T+1)+δ∆u(T) ≤λg
2
(u(T)),
K¡u•¯K(1)e).
e¡‰Ñ¯K(1))•35(J.
Ún4.1u,¯u∈E©O•¯K(1)e)Úþ),…÷vu≤¯u,K¯K(1)•3˜‡)u÷
vu(t) ≤u(t) ≤¯u(t),t∈[0,T+1]
Z
.
y²•9ϯK





∆
2
u(t−1)+λh(t)f
∗
(u(t)) = 0,t∈[1,T]
Z
,
αu(0)−β∆u(0) = λg
∗
1
(u(0)),
γu(T+1)+δ∆u(T) = λg
∗
2
(u(T)),
(9)
Ù¥
f
∗
(u(t)) =





f(¯u(t)),u(t) ≥¯u(t),
f(u(t)),u(t) ≤u(t) ≤¯u(t),
f(u(t)),u(t) ≤u(t),
g
∗
1
(u(t)) =





g
1
(¯u(t)),u(t) ≥¯u(t),
g
1
(u(t)),u(t) ≤u(t) ≤¯u(t),
g
1
(u(t)),u(t) ≤u(t),
g
∗
2
(u(t)) =





g
2
(¯u(t)),u(t) ≥¯u(t),
g
2
(u(t)),u(t) ≤u(t) ≤¯u(t),
g
2
(u(t)),u(t) ≤u(t).
´„¯K(9)duÚ©•§
u(t) = λ(P
∗
(u(0),u(T))t+Q
∗
(u(0),u(T)))+λ
T
X
s=1
G(t,s)h(s)f
∗
(u(s)),
Ù¥
P
∗
(s,t) =
αg
∗
2
(t)−γg
∗
1
(s)
%
,
Q
∗
(s,t) =
(δ+γ(T+1))g
∗
1
(s)+βg
∗
2
(t)
%
,
DOI:10.12677/aam.2022.1174404136A^êÆ?Ð
µyÚ§´ý
G(t,s)´ƒéA‚¼ê.-
T
∗
u(t) = λ(P
∗
(u(0),u(T))t+Q
∗
(u(0),u(T)))+λ
T
X
s=1
G(t,s)h(s)f
∗
(u(s)),
dÚn2.4´yŽfT
∗
: E→EëY.duf
∗
,g
∗
1
,g
∗
2
´k.,ÏdT
∗
k..ŠâSchauderØÄ:½
n•,T
∗
k˜‡ØÄ:u,=u•¯K(9)˜‡).
e¡y²¯K(9))u÷vu(t) ≤u(t) ≤¯u(t).
Äky²u(t) ≤¯u(t).b•3t
0
∈[0,T+1]
Z
,¦u(t
0
) >¯u(t
0
).Kk±eo«œ¹:
(i)u(t) >¯u(t),t∈[0,T+1]
Z
.k
f
∗
(u(t)) = f(¯u(t)),g
∗
1
(u(0)) = g
1
(¯u(0)),g
∗
2
(u(T)) = g
2
(¯u(T)),
Ïd
∆
2
(¯u−u)(t−1) ≤0,α(¯u−u)(0)−β∆(¯u−u)(0) ≥0,γ(¯u−u)(T+1)+δ∆(¯u−u)(T) ≥0,
Šâ4ŒŠn[13]•,éu?¿t∈[0,T+1]
Z
k¯u(t) ≥u(t),gñ!
(ii)u(t) >¯u(t),t∈[a,b]
Z
,a,b∈[0,T+1],…u(a−1) ≤¯u(a−1),u(b+1) ≤¯u(b+1).
K
∆
2
(¯u−u)(t−1) ≤0,t∈[a,T+1]
Z
,
α(¯u−u)(a−1)−β∆(¯u−u)(a−1) ≥0,γ(¯u−u)(b+1)+δ∆(¯u−u)(b) ≥0.
2gd4ŒŠn[13]•,éu?¿t∈[a,b]
Z
,¯u(t) ≥u(t),gñ!
$^ƒq•{Œ?n±eü«œ¹:
(iii)u(t) >¯u(t),t∈[0,a−1]
Z
,a∈[0,T+1]
Z
,…u(a) ≤¯u(a).
(iv)u(t) >¯u(t),t∈[a+1,T+1]
Z
,a∈[0,T+1]
Z
,…u(a) ≤¯u(a).
ÓnŒyu(t)≤u(t),t∈[0,T+1]
Z
.lu(t)≤u(t)≤¯u(t).Ïdf
∗
= f,g
∗
1
= g
1
,g
∗
2
= g
2
,=u´
¯K(1)˜‡).
du·‚•Ä´¯K(1)),5½u<0ž,kf(u) = f(0),g
1
(u) = g
1
(0),g
2
(u) = g
2
(0).
Ún4.2(A1)-(A3)¤á,-I⊂(0,∞)•;f8,eλ∈I,K•3~êb
I
>0,¦¯K(1)¤
k)u÷vkuk≤b
I
.
y²b{u
n
}
∞
n=1
´¯K(1)),…lim
n→∞
|u
n
|6=∞,λ
n
∈I.dÚn2.4•u
n
∈K.df
∞
=
∞•,Àη>0¿©Œ÷v
λ
n
η
T
T
P
s=1
h(s) ≥2.K•3˜‡p>0,¦u≥pž,f(u) ≥ηu.
Ïlim
n→∞
|u
n
|=∞,é?¿p>0,•3N>0,¦n≥Nž,kmin
t∈[1,T]
u
n
(t)≥
1
T+1
ku
n
k≥p.Ï
d
u
n
(t) ≥λ
n
T
X
s=1
h(s)f(u
n
(s)) ≥
λ
n
η
T+1
ku
n
k
T
X
s=1
h(s) ≥2 ku
n
k,
gñ!Ïdb†Ø,·K(.
DOI:10.12677/aam.2022.1174404137A^êÆ?Ð
µyÚ§´ý
-Γ={λ>0|¯K(1)•3˜‡)},…-λ
∗
=supΓ.d½n3.1•Γ6=∅,…0<λ
∗
<∞.e
yλ
∗
∈Γ.λ
n
∈Γ…λ
n
→λ
∗
.Ï•λ
n
´k.,¤±dÚn4.2•λ
n
ƒéA)u
n
´k..dÚ©
ŽfΓëY5•λ
∗
∈Γ.
-λ
∗
´¯K(1)˜‡)u
∗
¤éAëê,…½Â
˜
f(u(t)) =





f(u
∗
(t)+ε),u(t) ≥u
∗
(t)+ε,
f(u(t)),−ε≤u(t) ≤u
∗
(t)+ε,
f(−ε),u(t) ≤−ε,
˜g
1
(u(t)) =





g
1
(u
∗
(t)+ε),u(t) ≥u
∗
(t)+ε,
g
1
(u(t)),−ε≤u(t) ≤u
∗
(t)+ε,
g
1
(−ε),u(t) ≤−ε,
˜g
2
(u(t)) =





g
2
(u
∗
(t)+ε),u(t) ≥u
∗
(t)+ε,
g
2
(u(t)),−ε≤u(t) ≤u
∗
(t)+ε,
g
2
(−ε),u(t) ≤−ε.
-
˜
T
λ
(u(t)) = λ(
˜
P(u(0),u(T))t+
˜
Q(u(0),u(T)))+λ
T
X
s=1
G(t,s)h(s)
˜
f(u(s)),
Ù¥
˜
P(s,t) =
α˜g
2
(t)−γ˜g
1
(s)
%
,
˜
Q(s,t) =
(δ+γ(T+1))˜g
1
(s)+β˜g
2
(t)
%
.
•
Ω = {u∈E|−ε<u(t) <u
∗
(t)+ε}.
Ún4.3u∈E,•3˜‡¿©ε>0,0 <λ<λ
∗
ž÷v
˜
T
λ
u= u,Ku∈
¯
Ω.
y²w,u≥0.•y²u≤u
∗
+ ε,Äk`²u
∗
+ ε´¯K(1)˜‡þ).Ï•u
∗
≥0,¤
±•3˜‡~êc>0,¦t∈[1,T]
Z
ž,kf(u
∗
(t))>c.ÏL˜—ëY5,•3˜‡ ε
0
,¦
0 ≤ε≤ε
0
ž,k
|f(u
∗
(t)+ε)−f(u
∗
(t)) |<c(λ
∗
−λ)/λ.
K
∆
2
(u
∗
+ε)(t−1) = ∆
2
u
∗
(t−1) = −λ
∗
h(t)f(u
∗
(t))
= −λh(t)f(u
∗
(t)+ε)+λ[h(t)f(u
∗
(t)+ε)−h(t)f(u
∗
(t))]+(λ−λ
∗
)h(t)f(u
∗
(t))
<−λh(t)f(u
∗
(t)+ε)+ch(t)(λ
∗
−λ)+ch(t)(λ−λ
∗
) = −λh(t)f(u
∗
(t)+ε).
Ïd
∆
2
(u
∗
+ε)(t−1)+λh(t)f(u
∗
(t)+ε) ≤0.
Ón,ε>0¿©ž,dg
1
,g
2
˜—ëY5Œ
α(u
∗
(0)+ε)−β∆(u
∗
(0)+ε) ≥λg
1
(u
∗
(0)+ε),γ(u
∗
(T+1)+ε)+δ∆(u
∗
(T)+ε) ≥λg
2
(u
∗
(T)+ε).
DOI:10.12677/aam.2022.1174404138A^êÆ?Ð
µyÚ§´ý
u
∗
+ε´¯K(1)˜‡þ).ŠâÚn4.1•u≤(u
∗
+ε).
½n1.1y²-0<λ<λ
∗
,Ï•u
∗
Ú0©O´¯K(1)þ)Úe),¤±dÚn4.1•,¯
K(1)•3˜‡)u
λ
÷v0≤u
λ
≤u
∗
,Ïd0<λ≤λ
∗
ž,¯K(1)•3˜‡);λ>λ
∗
ž,¯
K(1)Ã).?˜Ú,dÚn4.3•,0 <λ<λ
∗
ž,u
λ
∈Ω.
PB(u
λ
,R)•E¥±u
λ
•%,R•Œ»¥.é¿©ŒR,d
˜
T
λ
3λ;«mþk.•,
deg(I−
˜
T
λ
,B(u
λ
,R),0) = 1.
e•3u∈∂Ω,¦u=
˜
T
λ
u,Kf=
˜
f,g=˜g,=u•¯K(1)1‡).bé¤k u∈∂Ω,ku6=
˜
T
λ
u,deg(I−
˜
T
λ
,Ω,0)k½Â.dÚn4.3•,
˜
T
λ
3B(u
λ
,R)\ΩþvkØÄ:,dÿÀÝƒØ5•
deg(I−T
λ
,Ω,0) = deg(I−
˜
T
λ
,Ω,0) = 1.
,˜•¡,dÚn4.2•,é?¿‰½λ∈I,¯K(1)ŒU)k.,Ïd
deg(I−T
λ
,B(0,M),0) = c,
Ù¥c´~ê,M>0•¿©Œ~ê.Ï•λ>λ
∗
ž,¯K(1)Ã),¤±c=0.dÿÀÝƒØ5
Υ,
deg(I−T
λ
,B(0,M)\Ω,0) = −1,
Ïd0 <λ<λ
∗
ž,¯K(1)•31‡).
½n1.2y²b¯K(1)¥f≡0.3ù«œ¹e,¯K(1)duŽf•§u= Tu,Ù¥
Tu(t) = λ(
αg
2
(u(T))−γg
1
(u(0))
%
t+
(δ+γ(T+1))g
1
(u(0))+βg
2
(u(T))
%
).
e^±eI“O3(7)¥¤½ÂI
K
1
= {u∈E|u= at+b,a,b∈R,t∈[0,T+1]
Z
},
KT(K
0
) ⊂K
1
ëY.?˜Ú,eu∈K
1
,Pkuk= max{u(0),u(T+1)}.
|^^‡(A4)¥g
1
Úg
2
ƒAOO†kc^‡(A3)fO,e¡y²†½n1.1aq,d
?Ñ.
~4.1•lÑ>Š¯K





∆
2
u(t−1)+λt(u
2
(t)+1) = 0,t∈[1,8]
Z
,
αu(0)−β∆u(0) = λe
u(0)
,
γu(9)+δ∆u(8) = λu
3
(8),
(10)
Ù¥h(t) = t∈[1,8]
Z
…h(t) 6≡0,f(u) = u
2
+1.N´y
lim
u→+∞
f(u)
u
=lim
u→+∞
u
2
+1
u
= +∞
DOI:10.12677/aam.2022.1174404139A^êÆ?Ð
µyÚ§´ý
¤á,=^‡(A1)-(A3)¤á.Kd½n1.1•é¿©λ>0,•3˜‡λ
∗
>0,¦0<λ<
λ
∗
ž,¯K(10)–kü‡);λ= λ
∗
ž,¯K(10)–k˜‡);λ>λ
∗
ž,¯K(10)Ã).
ef(u) ≡0…ØJy
lim
u→+∞
g
1
(u)
u
=lim
u→+∞
e
u
u
= +∞,lim
u→+∞
g
2
(u)
u
=lim
u→+∞
u
3
u
= +∞
¤á,=^‡(A2)Ú(A4)¤á.Ïdd½n1.2•é¿©λ>0,•3˜‡λ
∗
>0,¦0<λ<
λ
∗
ž,¯K(10)–kü‡);λ= λ
∗
ž,¯K(10)–k˜‡);λ>λ
∗
ž,¯K(10)Ã).
Ä7‘8
I[g,‰ÆÄ7“cÄ7(No11901464, No11801453),Ü“‰ŒÆ“c“‰ïUåJ,
Oy‘8(NWNU-LKQN-2020-20),[‹Ž“c‰EÄ7Oy‘8(21JR1RA230),[‹ŽpÆ
M#UåJ,‘8(2021A-006).
ë•©z
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DOI:10.12677/aam.2022.1174404141A^êÆ?Ð

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