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PureMathematics
n
Ø
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,2023,13(4),987-995
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.134104
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λ
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w
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n
ExistenceofPositiveSolutionsfora
ClassofSemi-PositoneSecondOrder
NeumannProblems
LinyingShi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Mar.19
th
,2023;accepted:Apr.20
th
,2023;published:Apr.27
th
,2023
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[J].
n
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,2023,13(4):987-995.
DOI:10.12677/pm.2023.134104
œ
k
Abstract
Weareconcernedwithexistenceofpositivesolutionsofsemi-positonesecondorder
problems
−
u
00
(
t
)+
u
(
t
) =
λ
(
f
(
u
(
t
))+
w
(
t
))
,t
∈
[0
,
1]
,
u
0
(0) =
u
0
(1) = 0
,
(
P
)
where
λ
isapositiveparameter,
w
∈
C
([0
,
1]
,
R
)
,and
|
w
(
t
)
|≤
c,t
∈
[0
,
1]
,
cisapositive
constant,
f
∈
C
([0
,
∞
)
,
[0
,
∞
))
, and
f
issuperlinear,i.e,
f
0
:=lim
x
→
0
f
(
x
)
x
= 0
,f
∞
:=lim
x
→∞
f
(
x
)
x
=
∞
.
Byusingfixedpointtheoremincones, weshowthatthereexistsaconstant
λ
0
>
0
suchthat (P)hasapositivesolutionfor
0
<λ<λ
0
.
Keywords
Positive Solution,Semi-Positone, NeumannProblems,FixedPoint Theoremin Cones
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
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h
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t
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≥
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,
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<
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−
δ
δ
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=max
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|
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|
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[0
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.
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[10,11]
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´
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‡
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m
,
…
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´
X
¥
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‡
I
,
=
K
⊂
X
.
b
Ω
1
, Ω
2
´
X
k
.
m
f
8
,
…
k
0
∈
Ω
1
,
Ω
1
⊂
Ω
2
.
-
F
:
K
∩
(Ω
2
\
Ω
1
)
→
K
´
ë
Y
Ž
f
…
÷
v
(i)
k
Au
k≤k
u
k
,u
∈
K
∩
∂
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,
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Au
k≥k
u
k
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∈
K
∩
∂
Ω
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;
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(ii)
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Au
k≥k
u
k
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∈
K
∩
∂
Ω
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,
…
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Au
k≤k
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k
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∈
K
∩
∂
Ω
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∩
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.
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-
z
÷
v
−
z
00
(
t
)+
z
(
t
) =
λg
(
t
)
,t
∈
[0
,
1]
,
z
0
(0) =
z
0
(1) = 0
,
Ù
¥
g
∈
L
1
[0
,
1]
,
…
R
1
0
g
(
t
)
dt>
0
,
K
z
(
t
)
≥
λσ
||
z
||
∞
,
Ù
¥
σ
=
m
M
=
1
cosh1
2
,M
=max
0
≤
t
≤
1
G
(
t,s
) =
cosh1
2
sinh1
,m
=min
0
≤
t
≤
1
G
(
t,s
) =
1
sinh1
.
y
²
Š
â
©
z
[12]
Œ
•
,
z
(
t
) =
λ
Z
1
0
G
(
t,s
)
g
(
s
)
ds.
Ù
¥
G
(
t,s
) =
cosh(1
−
t
)cosh
s
sinh1
,
0
≤
s
≤
t
≤
1
,
cosh(1
−
s
)cosh
t
sinh1
,
0
≤
t
≤
s
≤
1
,
cosh
t
=
e
−
t
+
e
t
2
,
sinh
t
=
e
t
−
e
−
t
2
,
w
,
G
(
t,s
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≥
0
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∈
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,
1]
,
Œ
±
z
(
t
) =
λ
Z
1
0
G
(
t,s
)
g
(
s
)
ds
≥
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Z
1
0
g
(
s
)
ds
=
λσ
Z
1
0
Mg
(
s
)
ds
=
λσ
max
Z
1
0
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(
t,s
)
g
(
s
)
ds
=
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||
z
||
∞
.
DOI:10.12677/pm.2023.134104990
n
Ø
ê
Æ
œ
k
Ú
n
2.3
-
u
∈
C
[0
,
1],
÷
v
−
u
00
(
t
)+
u
(
t
)
≥−
λc,t
∈
[0
,
1]
,
u
0
(0)
≥
0
, u
0
(1)
≥
0
,
K
u
≥
0
,
b
½
||
u
||
∞
≥
Mc
+
λσMc
σ
,
K
u
(
t
)
≥
λσ
||
u
||
∞
−
λMc
−
λ
2
σMc.
y
²
v
0
(
t
)
´
‡
©•
§
−
u
00
(
t
)+
u
(
t
) =
−
λc,t
∈
[0
,
1]
,
u
0
(0) =
u
0
(1) = 0
•
˜
)
,
K
−
v
0
(
t
) =
λ
Z
1
0
G
(
t,s
)
cds
≤
λM
Z
1
0
cds
≤
λMc.
Ï
d
,
−
v
0
(
t
)
≤
λMc.
-
y
(
t
) =
u
(
t
)
−
v
0
(
t
),
K
y
(
t
)
÷
v
−
y
00
(
t
)+
y
(
t
)
≥
0
,t
∈
[0
,
1]
,
y
0
(0)
≥
0
,y
0
(1)
≥
0
.
d
Ú
n
2.2
Œ
•
§
y
(
t
)
≥
λσ
||
y
||
∞
,t
∈
[0
,
1]
.
é
?
¿
t
∈
[0
,
1]
,
k
u
(
t
) =
y
(
t
)+
v
0
(
t
)
≥
λσ
||
y
||
∞
−
λMc
=
λσ
||
u
−
v
0
||
∞
−
λMc
≥
λσ
(
||
u
||
∞
−||
v
0
||
∞
)
−
λMc
≥
λσ
(
||
u
||
∞
−
λMc
)
−
λMc
≥
λσ
||
u
||
∞
−
λMc
−
λ
2
σMc,t
∈
[0
,
1]
.
DOI:10.12677/pm.2023.134104991
n
Ø
ê
Æ
œ
k
3.
Ì
‡
(
J
y
²
½
n
1.1
y
²
b
f
0
= 0,
f
∞
=
∞
,
…
¯
K
−
u
00
(
t
)+
u
(
t
) =
λ
(
f
(
u
(
t
))+
w
(
t
))
,t
∈
[0
,
1]
,
u
0
(0) =
u
0
(1) = 0
d
u
u
(
t
) =
λ
Z
1
0
G
(
t,s
)(
f
(
u
(
s
))+
w
(
s
))
ds,t
∈
[0
,
1]
.
½
Â
K
=
{
u
∈
X
|
u
(
t
)
≥
λσ
||
u
||
∞
−
λMc
−
λ
2
σMc,t
∈
[0
,
1]
}
du
||
u
||
∞
≥
Mc
+
λσMc
σ
,
¤
±
u
(
t
)
≥
0
,
K
K
•
X
¥
I
.
e
u
∈
K
,
(
Ü
Ú
n
2.3
Œ
•
,
Lu
(
t
) =
λ
Z
1
0
G
(
t,s
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DOI:10.12677/pm.2023.134104993
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(
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∞
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2
mMc
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λ
3
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+
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∞
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(
λ
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λ
3
µm
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c
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||
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.
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ì
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š
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>
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¯
K
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ó
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