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PureMathematics
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,2022,12(7),1205-1216
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.127132
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ExistenceandMultiplicityofPositive
SolutionsforaClassofDifferential
EquationswithNonlinearBoundary
Conditions
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[J].
n
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,2022,12(7):1205-1216.
DOI:10.12677/pm.2022.127132
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CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Jun.14
th
,2022;accepted:Jul.15
th
,2022;published:Jul.22
nd
,2022
Abstract
Inthispaper, weareconcernedwiththeexistenceandmultiplicityofpositivesolutions
forsecondordernonlineardifferentialequationsboundaryvalueproblems
u
00
(
t
)
−
k
2
u
(
t
)+
h
(
t
)
f
(
u
(
t
)) = 0
,
0
<t<
1
,
u
(0) = 0
,u
0
(1)
−
g
(
u
(1)) =
b,
(
P
)
where
b
isapositiveparameter,
k>
0
,
a
∈
C
([0
,
1]
,
(0
,
∞
))
,f,g
∈
C
([0
,
∞
)
,
(0
,
∞
))
. When
f
and
g
satisfytheproperconditions,weprovethatthereexistsapositivenumber
b
∗
,suchthat(P)haszero,exactlyoneandatleasttwopositivesolutionsaccording
to
b>b
∗
,b
=
b
∗
and
0
<b<b
∗
,respectively. Theproofofthemainresultsisbasedon
topologicaltheoryandthemethodofupperandlowersolutions.
Keywords
NonlinearBoundaryConditions, PositiveSolutions,TopologicalDegree,Upperand
LowerSolutions
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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(
t
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1
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0,
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r
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k
u
k
<r
1
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d
(H3)
Œ
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1
<r
1
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3
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2
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g
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u
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≤
k
3
u,
ρ
=min
{
ρ
1
,ρ
2
}
,
0
≤
u
≤
ρ
…
b<
k
3
ρ
ž
,
é
?
¿
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∈
∂K
r
1
,
K
k
Tu
(
t
) =
ϕ
(
t
)(
b
+
g
(
u
(1)))+
Z
1
0
G
(
t,s
)
h
(
s
)
f
(
u
(
s
))
ds
≤
1
k
b
+
1
k
g
((
u
(1))+
ML
Z
1
0
f
(
u
(
s
))
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1
3
ρ
+
1
3
ρ
+
1
3
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r
1
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â
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n
2
Œ
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i
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r
1
,K
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.
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,
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3
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¦
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≥
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u
)
≥
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,
¿
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σmηl
≥
1
.
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r
2
≥
max
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σ
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1
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r
2
=
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∈
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:
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u
k
<r
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}
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J
u
∈
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r
2
,
K
min
t
∈
[
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4
,
3
4
]
u
(
t
)
≥
σ
k
u
k≥
p,
é
?
¿
u
∈
∂K
r
2
,
K
k
Tu
(
t
) =
ϕ
(
t
)(
b
+
g
(
u
(1)))+
Z
1
0
G
(
t,s
)
h
(
s
)
f
(
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(
s
))
ds
≥
Z
3
4
1
4
G
(
t,s
)
h
(
s
)
f
(
u
(
s
))
ds
≥
mησl
k
u
k
≥k
u
k
,
DOI:10.12677/pm.2022.1271321209
n
Ø
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X
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W
ù
¿
›
X
k
Tu
k≥k
u
k
,u
∈
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d
Ú
n
2
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(
T,K
r
2
,K
) = 0
.
Š
â
Ø
Ä:
•
ê
Œ
\
5
•
i
(
T,K
r
2
\
K
r
1
,K
) =
−
1
.
ù
L
²
T
3
K
r
2
\
K
r
1
¥
k
˜
‡
Ø
Ä:
,
=
¯
K
(1)
–
k
˜
‡
)
.
5
¿
,
ϕ
´
¯
K
u
00
(
t
)
−
k
2
u
(
t
) = 0
,
0
<t<
1
,
u
(0) = 0
,u
0
(1) = 1
)
.
u
´
¯
K
(1)
)
…
=
v
=
u
−
(
b
+
g
(
u
(1)))
ϕ
´
¯
K
v
00
−
k
2
v
+
hf
(
v
+(
b
+
g
(
u
(1)))
ϕ
) = 0
,
0
<t<
1
,
v
(0) = 0
,v
0
(1) = 0
(6)
š
K
)
.
u
´
¯
K
(1)
)
,
K
v
=
u
−
(
b
+
g
(
u
(1)))
ϕ
´
¯
K
(6)
)
,
d
Ú
n
1
Œ
inf
t
∈
[
1
4
,
3
4
]
v
(
t
)
≥
σ
k
v
k
,
inf
t
∈
[
1
4
,
3
4
]
ϕ
(
t
)
≥
ζ
ξ
k
ϕ
k≥
σ
k
ϕ
k
,
u
´
inf
t
∈
[
1
4
,
3
4
]
v
+(
b
+
g
(
u
(1)))
ϕ
≥
σ
(
k
v
k
+(
b
+
g
(
u
(1)))
k
ϕ
k
)
≥
σ
k
v
+(
b
+
g
(
u
(1)))
ϕ
k
.
-
e
f
(
t
) = inf
t
≤
s
f
(
s
)
,
K
k
v
(
t
) =
Z
1
0
G
(
t,s
)
h
(
s
)
f
(
v
+(
b
+
g
(
u
(1)))
ϕ
(
s
))
ds
≥
Z
3
4
1
4
G
(
t,s
)
h
(
s
)
f
(
v
+(
b
+
g
(
u
(1)))
ϕ
(
s
))
ds
≥
ml
Z
3
4
1
4
f
(
v
+(
b
+
g
(
u
(1)))
ϕ
(
s
))
ds
≥
ml
e
f
(
σ
k
v
+(
b
+
g
(
u
(1)))
ϕ
k
)
,
?
e
f
(
δ
k
v
+(
b
+
g
(
u
(1)))
ϕ
k
)
k
v
+(
b
+
g
(
u
(1)))
ϕ
k
≤
e
f
(
δ
k
v
+(
b
+
g
(
u
(1)))
ϕ
k
)
k
v
k
≤
1
ml
,
DOI:10.12677/pm.2022.1271321210
n
Ø
ê
Æ
X
Ž
W
d
(H2)
9
e
f
½
Â
•
lim
s
→∞
e
f
(
s
)
s
=
∞
,
Ï
d
,
•
3
κ>
0,
¦
k
v
+(
b
+
g
(
u
(1)))
ϕ
k≤
κ
,
b
´
k
.
.
4.
þ
)
Ú
e
)
!
Ï
L
½
Â
¯
K
(1)
þ
e
)
,
‘
3
1
o
!
¥
¯
K
(1)
õ
‡
)
.
½
Â
1
α
∈
C
2
[0
,
1]
´
¯
K
(1)
þ
)
,
X
J
α
÷
v
α
00
(
t
)
−
k
2
α
(
t
)+
h
(
t
)
f
(
α
(
t
))
≤
0
,
0
<t<
1
,
α
(0)
≥
0
,α
0
(1)
−
g
(
α
(1))
≥
b.
β
∈
C
2
[0
,
1]
´
¯
K
(1)
e
)
,
X
J
β
÷
v
β
00
(
t
)
−
k
2
β
(
t
)+
h
(
t
)
f
(
β
(
t
))
≥
0
,
0
<t<
1
,
β
(0)
≤
0
,β
0
(1)
−
g
(
β
(1))
≤
b.
Ú
n
3
b
½
(H1)-(H3)
¤
á
,
α
Ú
β
©
O
´
¯
K
(1)
þ
)
Ú
e
)
,
…
k
β
(
t
)
≤
α
(
t
),
K
¯
K
(1)
–
•
3
˜
‡
)
u
÷
v
β
(
t
)
≤
u
(
t
)
≤
α
(
t
)
,t
∈
[0
,
1]
.
y
²
•
Ä
9
Ï
¯
K
u
00
(
t
)
−
k
2
u
+
h
(
t
)
f
∗
(
u
(
t
)) = 0
,
0
<t<
1
,
u
(0) = 0
,u
0
(1)
−
g
∗
(
u
(1)) =
b
(7)
Ù
¥
f
∗
(
u
(
t
)) =
f
(
α
(
t
))
,u
(
t
)
>α
(
t
)
,
f
(
u
(
t
))
,α
(
t
)
≤
u
(
t
)
≤
β
(
t
)
,
f
(
β
(
t
))
,u
(
t
)
<β
(
t
)
,
g
∗
(
u
(
t
)) =
g
(
α
(
t
))
,u
(
t
)
>α
(
t
)
,
g
(
u
(
t
))
,α
(
t
)
≤
u
(
t
)
≤
β
(
t
)
,
g
(
β
(
t
))
,u
(
t
)
<β
(
t
)
.
¯¢
þ
,
‡
y
²
¯
K
(1)
•
3
˜
‡
)
u
,
…
k
β
(
t
)
≤
u
(
t
)
≤
α
(
t
)
,t
∈
[0
,
1]
,
•
I
y
²
¯
K
(7)
•
3
)
u
…
÷
v
T
^
‡
.
d
ª
(5)
Œ
,
¯
K
(7)
d
u
È
©•
§
u
(
t
) =
ϕ
(
t
)(
b
+
g
∗
(
u
(1)))+
Z
1
0
G
(
t,s
)
h
(
s
)
f
∗
(
u
(
s
))
ds,t
∈
[0
,
1]
,
DOI:10.12677/pm.2022.1271321211
n
Ø
ê
Æ
X
Ž
W
½
Â
Ž
f
T
∗
:
X
→
X
T
∗
u
(
t
) =
ϕ
(
t
)(
b
+
g
∗
(
u
(1)))+
Z
1
0
G
(
t,s
)
h
(
s
)
f
∗
(
u
(
s
))
ds,t
∈
[0
,
1]
,
du
f
∗
,g
∗
´
ë
Y
¼
ê
,
N
´
y
,
T
∗
´
ë
Y
Ž
f
,
B
=
{
u
∈
X
:
β
(
t
)
≤
u
(
t
)
≤
α
(
t
)
, t
∈
[0
,
1]
}
,
w
,
B
´
X
¥
k
.
4
8
,
K
|
f
∗
(
u
(
t
))
|≤
max
u
(
t
)
∈
B
|
f
(
u
(
t
))
|
,
Ï
d
f
∗
k
.
,
Ó
n
,
g
∗
k
.
,
l
T
∗
´
k
.
Ž
f
,
Š
â
Schauder
Ø
Ä:½
n
,
T
∗
k
Ø
Ä:
u
,
=
u
´
¯
K
(7)
)
.
e
y
u
(
t
)
≤
α
(
t
).
‡
é
,
t
0
∈
[0
,
1],
k
u
(
t
0
)
>α
(
t
0
),
-
ω
(
t
)=
α
(
t
)
−
u
(
t
),
e
¡
©
o
«
œ
/
?
Ø
.
(i)
é
?
¿
t
∈
[0
,
1],
b
Ñ
k
ω
(
t
)
<
0,
d
ž
,
f
∗
(
u
(
t
)) =
f
(
α
(
t
))
,g
∗
(
u
(
t
)) =
g
(
α
(
t
))
,
Ï
d
ω
00
(
t
) =
α
00
(
t
)
−
u
00
(
t
)
≤
k
2
α
(
t
)
−
h
(
t
)
f
(
α
(
t
))
−
k
2
u
(
t
)
−
h
(
t
)
f
∗
(
u
(
t
))
≤
0
,
,
˜
•
¡
ω
(0) =
α
(0)
−
u
(0)
≥
0
,
ω
0
(1) =
α
0
(1)
−
u
0
(1)
≥
b
+
g
(
α
(1))
−
b
−
g
∗
(
u
(1)) = 0
,
Š
â
4
Œ
Š
n
,
ω
(
t
0
)
≥
0
,t
0
∈
[0
,
1],
ù
†
b
g
ñ
.
(ii)
0
<a<
1
÷
v
ω
(
a
) = 0,
b
ω
(
t
)
<
0
,t
∈
[
a,
1],
K
k
ω
00
(
t
)
≤
0
,ω
(
a
) = 0
, ω
0
(1)
≥
0
,
ù
†
(i)
a
q
Œ
g
ñ
,
l
k
ω
(
t
)
≥
0.
(iii)
0
<a<
1
÷
v
ω
(
a
) = 0,
b
ω
(
t
)
<
0
,t
∈
[0
,a
),
K
k
ω
00
(
t
)
≤
0
,ω
(0)
≥
0
,ω
0
(
a
)
≥
0
,
Ó
n
,
Œ
g
ñ
.
(iv)
0
<a,b<
1
÷
v
ω
(
a
) = 0
,ω
(
b
) = 0,
ω
(
t
)
<
0
,t
∈
[
a,b
],
K
k
ω
00
(
t
)
≤
0
,ω
(
a
) = 0
, ω
0
(
b
)
≥
0
,
DOI:10.12677/pm.2022.1271321212
n
Ø
ê
Æ
X
Ž
W
a
q
/
,
d
«
œ
/
e
½
k
ω
(
t
)
≥
0.
é
u
u
(
t
)
≥
β
(
t
)
,t
∈
[0
,
1],
^
Ó
•{
Œ
y
,
ù
p
Ø
3
K
ã
.
u
´
¯
K
(7)
)
u
÷
v
β
(
t
)
≤
u
(
t
)
≤
α
(
t
)
,t
∈
[0
,
1]
,
Š
â
f
∗
,g
∗
½
´
•
u
´
¯
K
(1)
)
.
5.
õ
)
5
9
Ì
‡
(
J
y
²
•
¯
K
(1)
)
,
•
B
å
„
,
!
o
b
(H4)
f
(
u
) = 0
,g
(
u
) = 0
,u<
0
.
Ú
n
4
b
½
(H1)-(H3)
¤
á
,
I
⊂
(0
,
∞
)
•
;
f
«
m
,
e
b
∈
I
,
K
•
3
~
ê
e
m>
0,
¦
¯
K
(1)
¤
k
)
u
÷
v
k
u
k≤
e
m
.
y
²
:
{
u
n
}
´
¯
K
(1)
Ã
.)
S
,
†
Ù
é
A
{
b
n
}∈
I
.
d
Ú
n
1
•
,
u
n
∈
K
,
Ï
•
f
∞
=
∞
,
K
•
3
~
ê
q>
0
,
u
≥
q
ž
,
f
(
u
)
≥
ηu
,
Ù
¥
η>
0
÷
v
ησ
Z
3
4
1
4
G
(
1
2
,s
)
h
(
s
)
ds
≥
2
,
q
Ï
n
→∞
ž
,
k
u
n
k→∞
,
K
•
3
N,
n>N
,
ž
k
min
t
∈
[
1
4
,
3
4
]
u
n
(
t
)
≥
σ
k
u
n
k≥
q,
l
u
n
(
1
2
)
≥
Z
3
4
1
4
G
(
1
2
,s
)
h
(
s
)
f
(
u
n
(
s
))
ds
≥
η
Z
3
4
1
4
G
(
1
2
,s
)
h
(
s
)
u
n
(
s
)
ds
≥
ησ
Z
3
4
1
4
G
(
1
2
,s
)
h
(
s
)
k
u
n
k
≥
2
k
u
n
k
,
w
,
ù
´
˜
‡
g
ñ
,
·
K
y
.
Ú
n
5
P
Γ =
{
b>
0
|
¯
K
(1)
–
k
˜
‡
)
}
,
supΓ =
b
∗
,
K
Γ
k
.
,
¿
…
b
∗
∈
Γ
.
y
²
d
½
n
2
Œ
•
,Γ
´
k
.
.
{
b
n
}∈
Γ
…
÷
v
b
n
→
b
∗
,n
→∞
,
DOI:10.12677/pm.2022.1271321213
n
Ø
ê
Æ
X
Ž
W
w
,
{
b
n
}
´
k
.
,
d
Ú
n
4
•
b
n
é
Au
¯
K
(1)
)
u
n
k
.
,
(
Ü
Ž
f
T
;
5
´
•
b
∗
∈
Γ
.
é
ε>
0,
u
∗
´
é
Au
b
∗
¯
K
(1)
)
.
-
e
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,
e
g
(
u
(
t
)) =
g
(
u
∗
(
t
)+
ε
)
,u
(
t
)
>u
∗
(
t
)+
ε,
g
(
u
(
t
))
,
−
ε
≤
u
(
t
)
≤
u
∗
(
t
)+
ε,
g
(
−
ε
)
,u
(
t
)
<
−
ε,
½
Â
e
Tu
(
t
) =
ϕ
(
t
)(
b
+
e
g
(
u
(1)))+
Z
1
0
G
(
t,s
)
h
(
s
)
e
f
(
u
(
s
))
ds,t
∈
[0
,
1]
,
(8)
Ω =
{
u
∈
X
:
−
ε<u
(
t
)
<u
∗
(
t
)+
ε,t
∈
[0
,
1]
}
.
Ú
n
6
b
½
(H1)-(H4)
¤
á
,
¿
©
ê
ε
,
¦
é
?
¿
u
∈
C
[0
,
1]
…
÷
v
e
Tu
=
u
,
0
<b<b
∗
ž
,
k
u
∈
Ω
.
y
²
:
d
ª
(8)
Œ
•
,
u
≥
0.
e
¡
y
²
u
≤
u
∗
+
ε
.
Š
â
f
˜
—
ë
Y5
,
0
<ε<ε
0
ž
,
•
3
ê
c
,
¦
cL
≤
k
2
,
K
k
|
f
(
u
∗
+
ε
)
−
f
(
u
∗
)
|
<cε,
u
´
(
u
∗
+
ε
)
00
= (
u
∗
)
00
=
k
2
u
∗
−
hf
(
u
∗
)
=
k
2
(
u
∗
+
ε
)
−
hf
(
u
∗
+
ε
)+
h
(
f
(
u
∗
+
ε
)
−
f
(
u
∗
))
−
k
2
ε
≤
k
2
(
u
∗
+
ε
)
−
hf
(
u
∗
+
ε
)+(
cL
−
k
2
)
ε
≤
k
2
(
u
∗
+
ε
)
−
hf
(
u
∗
+
ε
)
,
,
˜
•
¡
,
(
u
∗
+
ε
)
0
(1) =
b
∗
+
g
(
u
∗
(1))
≥
b
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