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PureMathematicsnØêÆ,2021,11(4),685-693
PublishedOnlineApril2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.114083
Œn:>Š¯K)•35
ÚÚÚppp¸¸¸
Ü“‰ŒÆêƆÚOÆ,[‹=²
Email:dgf96520@163.com
ÂvFϵ2021c321F¶¹^Fϵ2021c423F¶uÙFϵ2021c430F
Á‡
$^IþØÄ:½n¼Œn:>Š¯K



ω
00
(t)+a(t)ω
0
(t)+b(t)ω(t)+λf(t,ω(t)) = 0,t∈(0,1),
ω(0) = 0,αω(η) = ω(1)
)•35(J§ Ù¥λ>0,0<η<1…÷v0<αη<1,f∈C([0,1]×[0,∞),(−∞,∞))
•3~êM¦f(t,ω) ≥−M¤á"
'…c
n:>Š¯K§Œ§)§ØÄ:½n
ExistenceofPositiveSolutions
Semi-PositiveSecond-Order
Three-PointBoundaryValueProblems
GaofengDu
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Email:dgf96520@163.com
©ÙÚ^:Úp¸.Œn:>Š¯K)•35[J].nØêÆ,2021,11(4):685-693.
DOI:10.12677/pm.2021.114083
Úp¸
Received:Mar.21
st
,2021;accepted:Apr.23
rd
,2021;published:Apr.30
th
,2021
Abstract
Byusingthefixed-pointtheoremincones,weobtaintheexistenceof positive solutions
forthesemi-positivesecond-orderthree-pointboundaryvalueproblems





ω
00
(t)+a(t)ω
0
(t)+b(t)ω(t)+λf(t,ω(t)) = 0,t∈(0,1),
ω(0) = 0,αω(η) = ω(1)
whereλ>0,0<η<1satisfies0<αη<1andf∈C([0,1] ×[0,∞),(−∞,∞)),with
f(t,ω) ≥−MforsomepositiveconstantsM.
Keywords
Three-PointBoundaryValueProblem,Semi-Positive,PositiveSolution,Fixed-Point
Theorem
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
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–L§¥,˜„‡‰•§ω
00
(t)=f(t,ω(t),ω
0
(t))N\˜½>.^‡,•Ò´`•§½)^
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„kDirichlet>.^‡,Robin>.^‡ÚNeumann>.^‡[1].Ù¥,õ:>Š¯K•@
dIl’inÚMoiseevm©ïÄ,3dÄ:þ,éuš‚5‘f≥0œ ¹®¹õ`D¤J[2–7].
DOI:10.12677/pm.2021.114083686nØêÆ
Úp¸
2003c,êX[8]A^ØÄ:½nÚƒ'‚5¯KÈ©/ª),?Øn:>Š¯K



u
00
(t)+a(t)u
0
(t)+b(t)u(t)+h(t)f(u) = 0,t∈(0,1),
u(0) = 0,αu(η) = u(1)
(1.1)
)•35,Ù¥0 <η<1,h,f,aÚb÷vXe^‡:
(H1)f∈C([0,∞),[0,∞));
(H2)h∈C([0,1],[0,+∞))¿…•3x
0
∈[0,1]¦h(x
0
) >0;
(H3)a∈C[0,1],b∈C([0,1],(−∞,0));
(H4)0 <αφ
1
(η) <1,
Ù¥,φ
1
(t)´>Š¯K



φ
00
1
(t)+a(t)φ
0
1
(t)+b(t)φ
1
(t) = 0,t∈(0,1),
φ
1
(0) = 0,φ
1
(1) = 1
(1.2)
).2002c,ƒŸ8[9]$^IþØÄ:½n,‡‚5Œš‚5n:>Š¯K



u
00
(t)+λf(t,u(t)) = 0,t∈(0,1),
u(0) = 0,u(1) = αu(η)
(1.3)
).Óž,?Øf(t,u)≥−M…÷v‡‚5^‡ž,é¿©λ>0,>Š¯K(1.3)–
•3˜‡),Ù¥λ>0´ëê,0<η<1,0 <α<1.ÄuþãóŠ,2007c,ƒŸ83©
z[10]¥qïÄŒ¯K(1.3)¥λ≡1ž)•35†õ)5.3dÄ:þ,2019c,Ÿ
A[11]E$^ØÄ:½nïÄDirichlet>.^ ‡eŒš‚5~‡©•§>Š¯K



y
00
(t)+b(t)y(t)+λf(t,y(t)) = 0,t∈(0,1),
y(0) = y(1) = 0
)•35,Ù¥λ>0•ëê,b∈C[0,1]…÷v−∞<b(t) <π
2
.
Éþã©zéu,©•ÄŒš‚5n:>Š¯K



ω
00
(t)+a(t)ω
0
(t)+b(t)ω(t)+λf(t,ω(t)) = 0,t∈(0,1),
ω(0) = 0,αω(η) = ω(1)
(1.4)
)•35.·‚5¿,©z[8]¥f´šK,·‚ïÄf>−Mœ/,¿…òí2©
z[9–12]óŠ.Ïd,Ø^‡(H3)-(H4)ƒ,©ob½±e^‡¤á:
(H5)−∞<inf{f(t,µ) : (t,µ) ∈[0,1]×[0,+∞)}= −M<0;
DOI:10.12677/pm.2021.114083687nØêÆ
Úp¸
(H6)
lim
µ→∞
min
δ≤t≤1−δ
f(t,µ)
µ
= +∞
.
51Šâ©z[8]Ã{†¯K(1.1)Green¼êäNLˆª,´Œ±?ØéA‚
5¯K)5ŸÚGreen¼ê5Ÿ,|^~êC´{,T¯K È©/ª).Ïd,©
ïÄÄuŒ¯K(1.4)éA‚5¯K)/ªÚGreen¼ê5Ÿ,A^ØÄ:½ny²¯K
(1.4))•35.
2.‚5>Š¯KGreen¼ê
•Äàg>Š¯K
(
ω
00
(t)+a(t)ω
0
(t)+b(t)ω(t) = 0,t∈(0,1),
ω(0) = 0,ω(1) = αω(η)
(2.1)
Green¼ê.Šâ©z[1]1ÊÙSN,Œ±>Š¯K(2.1)Green¼ê•
G(t,s) =
1
ρ
(
φ
1
(t)φ
2
(s),0 ≤t≤s≤1,
φ
1
(s)φ
2
(t),0 ≤s≤t≤1,
(2.2)
Ù¥,φ
1
(t)´>Š¯K(1.2)),φ
2
(t)´>Š¯K
(
φ
00
2
(t)+a(t)φ
0
2
(t)+b(t)φ
2
(t) = 0,t∈(0,1),
φ
2
(0) = 1,φ
2
(1) = 0
(2.3)
);ρ:=φ
0
1
(0)6=0•~ê.Šâ©z[8],φ
1
Úφ
2
•¯K(1.2)Ú(2.3)•˜)…kXe5
Ÿ:
5Ÿ2.1 [8]b(H3)¤á.φ
1
Úφ
2
÷v:
(i)φ
1
3t∈[0,1]î‚4O;
(ii)φ
2
3t∈[0,1]î‚4~.
e5ò|^¯K(2.1)Green¼ê5E¯K(2.1)éAšàg>Š¯K),·‚‰
ÑXeÚn.
Ún2.2 [8]b(H3)Ú(H4)¤á.y∈C[0,1],@o>Š¯K



ω
00
(t)+a(t)ω
0
(t)+b(t)ω(t)+y(t) = 0,t∈(0,1),
ω(0) = 0,αω(η) = ω(1)
(2.4)
duÈ©•§
ω(t) =
Z
1
0
G(t,s)p(s)y(s)ds+Aφ
1
(t),
Ù¥
A=
α
1−αφ
1
(η)
Z
1
0
G(η,s)p(s)y(s)ds, p(t) = exp

Z
t
0
a(s)ds

,
DOI:10.12677/pm.2021.114083688nØêÆ
Úp¸
G(t,s)•àg>Š¯K(2.1)Green¼ê.
52 AO/,y= 1ž,
ω
1
(t) =
Z
1
0
G(t,s)p(s)ds+A
1
φ
1
(t),
(2.5)
A
1
=
α
1−αφ
1
(η)
Z
1
0
G(η,s)p(s)ds.
e¡|^ù‡Ún5¼>Š¯K(2.1)Green¼êü‡-‡5Ÿ.
5Ÿ2.3 [8]G(t,s)´¯K(2.1)Green¼ê.é?¿t
0
∈(0,1],K•3q(t) ∈C[0,1],¦
G(t,s) ≥G(t
0
,s)q(t),t∈(0,1),s∈(0,1),
Ù¥q(t) = min

φ
1
(t)
|φ
1
|
0
,
φ
2
(t)
|φ
2
|
0

,t∈[0,1],|·|
0
´þ(.‰ê.
5Ÿ2.4 [8]G(t,s)´¯K(2.1)Green¼ê,é?¿s,t∈[0,1],G(t,s)÷vG(t,s) ≤G(s,s).
3.̇(J
•y²©Ì‡(J§I‡XeÚn:
Ún3.1[13]E´Banach˜m,K⊂E´I.bΩ
1
9Ω
2
´E¥k.m8,θ∈Ω
1
,…
Ω
1
⊂Ω
2
,-T: K∩(Ω
2
\Ω
1
) →K´ëYŽf.ee^‡ƒ˜¤á:
(i)kTuk≤kuk,u∈K∩∂Ω
1
,kTuk≥kuk,u∈K∩∂Ω
2
;
(ii)kTuk≥kuk,u∈K∩∂Ω
1
,kTuk≤kuk,u∈K∩∂Ω
2
,
@oT3K∩(Ω
2
\Ω
1
)¥–k˜‡ØÄ:.
Ún3.2[1]b(H3)Ú(H4)¤á.y∈C[0,1]…y≥0,é?¿t
0
∈(0,1),-|ω|
0
= ω(t
0
),
@o>Š¯K(2.4)•˜)ω(t)÷v
ω(t) ≥|ω|
0
q(t),t∈[0,1].
53 é?¿δ∈(0,
1
2
),•3γ
δ
>0,KÚn3.2Œí2•
ω(t) ≥γ
δ
|ω|
0
,t∈[δ,1−δ].
¯¢þ,•I-
γ
δ
= min

q(t)|t∈[δ,1−δ],δ∈(0,
1
2
)

.
DOI:10.12677/pm.2021.114083689nØêÆ
Úp¸
d53´•0 <γ
δ
<1.-
K=

ω∈C[0,1] : ω(t) ≥0,min
δ≤t≤1−δ
ω(t) ≥γ
δ
|ω|
0

.
KK´C[0,1]¥šKI,…T(K) ⊂K.-
Λ = max{f(t,µ)+M: (t,µ) ∈[0,1]×[0,1]}.
½n3.3 b^‡(H5)Ú(H6)¤á.K
0 <λ<min

1−αφ
1
(η)
1−αφ
1
(η)+α

Z
1
0
G(s,s)p(s)Λds

−1
,
γ
δ

R
1
0
G(t,s)p(s)ds+A
1
φ
1
(t)

−1
M

ž,¯K(1.4)–•3˜‡)ω
∗
∈C[0,1].
y²E9ϼê
g(t,µ) =f(t,µ)+M,(t,µ) ∈[0,1]×[0,+∞),
g(t,µ) =



g(t,µ),(t,µ) ∈[0,1]×[0,+∞),
g(t,0),(t,µ) ∈[0,1]×(−∞,0],
=g(t,max{µ,0}),(t,µ) ∈[0,1]×(−∞,+∞).
w,,g: [0,1]×(−∞,+∞) →[0,+∞)´ëY.
•Än:>Š¯K



ω
00
(t)+a(t)ω
0
(t)+b(t)ω(t)+λg(t,ω(t)−m(t)) = 0,t∈(0,1),
ω(0) = 0,αω(η) = ω(1)
(3.1)
),Ù¥m(t) =λMω
1
(t),ω
1
(t)X52¥(2.5)ª¤«.ØJy²,ω
∗
´>Š¯K(1.4))
…=ω:= ω
∗
+m´>Š¯K(3.1)),…ω(t) >m(t),0 <t<1.
½ÂŽfT: C[0,1] →C[0,1]Xe:
(Tω)(t) :=
Z
1
0
G(t,s)p(s)λg(s,ω(s)−m(s))ds+Aφ
1
(t)
=
Z
1
0
G(t,s)p(s)λg(s,ω(s)−m(s))ds
+

α
1−αφ
1
(η)
Z
1
0
G(η,s)p(s)λg(s,ω(s)−m(s)ds

φ
1
(t).
N´yT:C[0,1]→C[0,1]•ëYŽf,…ω∈C[0,1]´>Š¯K(3.1))…=
ω∈C[0,1] ´TØÄ:. Ïd, •Iy²T3C[0,1] ¥•3ØÄ:ω, …ω(t) >m(t),0 <t<1.
DOI:10.12677/pm.2021.114083690nØêÆ
Úp¸
Ω
1
= {ω∈K:|ω|
0
≤1},eyé?¿ω∈∂Ω
1
,ω≥Tω.eØ,,K•3ω
0
∈∂Ω
1
,¦
ω
0
≤Tω
0
.Ï•ω
0
(t)−m(t) ≤1,0 ≤t≤1,d5Ÿ2.4Œ
ω
0
(t) ≤(Tω
0
)(t)
=
Z
1
0
G(t,s)p(s)λg(s,ω
0
(s)−m(s))ds
+

α
1−αφ
1
(η)
Z
1
0
G(η,s)p(s)λg(s,ω
0
(s)−m(s))ds

φ
1
(t)
≤
Z
1
0
G(t,s)p(s)λmax
0≤s≤1
g(s,ω
0
(s)−m(s))ds
+
α
1−αφ
1
(η)
Z
1
0
G(η,s)p(s)λmax
0≤s≤1
g(s,ω
0
(s)−m(s))ds
≤
Z
1
0
G(t,s)p(s)λ

max
0≤s≤1
f(s,max{ω
0
(s)−m(s),0})+M

ds
+
α
1−αφ
1
(η)
Z
1
0
G(η,s)p(s)λ

max
0≤s≤1
f(s,max{ω
0
(s)−w(s),0})+M

ds
≤
Z
1
0
G(s,s)p(s)λΛds+
α
1−αφ
1
(η)
Z
1
0
G(s,s)p(s)λΛds
≤λ
Z
1
0
G(s,s)p(s)Λds

1+
α
1−αφ
1
(η)

<1,
ù†|ω
0
|
0
= 1gñ.
,˜•¡,-S
λ
= {ω∈K: Tω≤ω},m
λ
= sup{|ω|
0
: ω∈S
λ
}.
eym
λ
<+∞.eØ,,K•3S{ω
n
}
∞
n=1
⊂K,¦Tω
n
≤ω
n
…|ω
n
|
0
→∞(n→∞).
Ï•ω
n
∈K,
min
δ≤t≤1−δ
ω
n
(t) ≥γ
δ
|ω
n
|
0
,
Ïd
min
δ≤t≤1−δ
(ω
n
(t)−m(t)) ≥γ
δ
|ω
n
|
0
−|m|
0
→+∞(n→∞).
Šâ^‡(H6),Kk
lim
n→∞
min
δ≤t≤1−δ
g(t,ω
n
(t)−m(t))
ω
n
(t)−m(t)
=lim
n→∞
min
δ≤t≤1−δ
g(t,ω
n
(t)−m(t))
ω
n
(t)−m(t)
=lim
n→∞
min
δ≤t≤1−δ
f(t,ω
n
(t)−m(t))+M
ω
n
(t)−m(t)
= +∞.
Ïd,•3N>0,¦n>Nž,
min
δ≤t≤1−δ
(ω
n
(t)−m(t)) ≥
γ
δ
2
|ω
n
|
0
,
DOI:10.12677/pm.2021.114083691nØêÆ
Úp¸
min
δ≤t≤1−δ
g(t,ω
n
(t)−m(t))
ω
n
(t)−m(t)
≥
4(
R
1
0
G(η,s)p(s)ds)
−1
λγ
δ
,
¤±
|ω
n
|
0
=max
0≤t≤1
|ω
n
(t)|≥max
0≤t≤1
(Tω
n
)(t) ≥(Tω
n
)(η)
=
Z
1
0
G(η,s)p(s)λg(s,ω
n
(s)−m(s))ds
+

α
1−αφ
1
(η)
Z
1
0
G(η,s)p(s)λg(s,ω
n
(s)−m(s))ds

φ
1
(t)
≥
Z
1
0
G(η,s)p(s)λg(s,ω
n
(s)−m(s))ds
≥

ω
n
(s)−m(s)

Z
1
0
G(η,s)p(s)λ
g(s,ω
n
(s)−m(s))
ω
n
(s)−m(s)
ds
≥min
δ≤s≤1−δ
(ω
n
(s)−m(s))
Z
1
0
G(η,s)p(s)λmin
δ≤s≤1−δ
g(s,ω
n
(s)−m(s))
ω
n
(s)−m(s)
ds
≥
γ
δ
2
|ω
n
|
0
Z
1
0
G(η,s)p(s)dsλ
4(
R
1
0
G(η,s)p(s)ds)
−1
λγ
δ
= 2|ω
n
|
0
,
ù´˜‡gñ.m
λ
<+∞.
-δ=2 +m
λ
,@oδ>1,¿…é?¿ω∈∂Ω
δ
,kTω≥ω,ŠâÚn3.1,TkØÄ:
ω∈Ω
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[1]êX.š‚5~‡©•§šÛܯK[M].®:‰Æч,2004.
DOI:10.12677/pm.2021.114083692nØêÆ
Úp¸
[2]Il’in,V.A.andMoiseev,E.I.(1987)NonlocalBoundaryValueProblemoftheFirstKind
foraSturm-LiouvilleOperatorinItsDifferentialandFiniteDifferenceAspects.Journalof
DifferentialEquations,23,803-810.
[3]Gupta,C.P. (1992)Solvability of aThree-Point NonlinearBoundaryValue Problemfora Sec-
ondOrder OrdinaryDifferentialEquation. Journal ofMathematicalAnalysis andApplications,
168,540-551.https://doi.org/10.1016/0022-247X(92)90179-H
[4]Ma,R.(1997) ExistenceTheorems fora SecondOrder Three-PointBoundary ValueProblem.
JournalofMathematicalAnalysisandApplications,212,430-442.
https://doi.org/10.1006/jmaa.1997.5515
[5]Zhang,Q.andJiang,D.(2010)MultipleSolutionstoSemipositoneDirichletBoundaryValue
ProblemswithSingularDependentNonlinearitiesforSecondOrderThree-PointDifferential
Equations.ComputersandMathematicswithApplications,59,2516-2527.
https://doi.org/10.1016/j.camwa.2009.12.036
[6]Gao,C.,Zhang,F.andMa,R.(2017)ExistenceofPositiveSolutionsofSecond-OrderPe-
riodicBoundaryValueProblemswithSign-ChangingGreen’sFunction.ActaMathematicae
ApplicataeSinica,EnglishSeries,33,263-268.https://doi.org/10.1007/s10255-017-0657-2
[7]Ma, R.,Gao, C.and Chen,R. (2010)Existenceof PositiveSolutionsofNonlinear Second-Order
PeriodicBoundaryValueProblems.BoundaryValueProblems,2010,ArticleNo.626054.
https://doi.org/10.1155/2010/626054
[8]Ma,R.andWang,H.(2003)PositiveSolutionsofNonlinearThree-PointBoundary-Value
Problems.JournalofMathematicalAnalysisandApplications,279,216-227.
https://doi.org/10.1016/S0022-247X(02)00661-3
[9]Yao,Q.(2002)AnExistenceTheoremofPositiveSolutionforaSuperlinearSemipositone
Second-OrderThree-PointBVP.JournalofMathematicalStudy,35,32-35.
[10]ƒŸ8.Œn:>Š¯K)Ú)[J].A^êÆÆ,2007,30(2):209-217.
[11]ŸA,ƒ,o[.˜aŒ~‡©•§>Š¯K)•35[J].ìÀŒÆÆ(n
Ƈ),2019,54(10):7-12.
[12]?á^,SŒ%.'uŒn:>Š¯K)•35[J].ñÜ“‰ŒÆÆ(g,‰Æ‡),
2004,32(3):31-33.
[13]Guo,D.andLakshmikantham,V.(1998)NonlinearProblemsinAbstractCones.Academic
Press,SanDiego.
DOI:10.12677/pm.2021.114083693nØêÆ

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