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PureMathematics
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,2021,11(5),720-730
PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.115087
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ExistenceofPositiveSolutionsfor
NonlinearFirst-OrderOrdinary
BoudaryValueProblems
RuofeiWu
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
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DOI:10.12677/pm.2021.115087
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Received:Apr.6
th
,2021;accepted:May6
th
,2021;published:May13
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,2021
Abstract
Inthispaper, weconsidertheexistenceofpositivesolutionsforthenonlinearfirst-
orderordinaryboundaryvalueproblems
u
0
(
t
)+
a
(
t
)
u
(
t
) =
λf
(
t,u
(
t
))
,
0
<t<
1
,
u
(0) =
lu
(1)
where
f
: [0
,
1]
×
[0
,
∞
)
→
[0
,
∞
)
,a
: [0
,
1]
→
[0
,
∞
)
arecontinuous functionsand
R
1
0
a
(
θ
)
dθ>
0
,
λ
isapositiveparameter,
l
isaconstant,and
0
<l
≤
e
R
1
0
a
(
θ
)
dθ
.Undertheassump-
tionthatthenonlinearterm
f
satisfiessuperlinear, sublinearandasymptoticgrowth
conditon,theexistenceofpositivesolutionsoftheproblemisobtainedbyusingthe
fixed-pointindextheory.
Keywords
PositiveSolutions,MultipleSolutions,Fixed-Point,Cone
Copyright
c
2021byauthor(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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DOI:10.12677/pm.2021.115087724
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Ø
ê
Æ
É
e
œ
d
Ú
n
2.1
Œ
i
(
T
λ
,K
p
1
,K
) = 0
.
e
f
0
= 0,
K
•
3
˜
‡
0
<r
1
<σp
1
,
0
≤
u
≤
r
1
ž
,
f
(
t,u
)
≤
ε
1
u
,
Ù
¥
ε
1
λe
R
1
0
a
(
θ
)
dθ
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
e
R
1
0
a
(
θ
)
dθ
−
l
≤
1
,
u
∈
∂K
r
1
ž
,
K
k
k
T
λ
u
k≤
λe
R
1
0
a
(
θ
)
dθ
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
h
(
s
)
e
R
s
0
a
(
θ
)
dθ
ds
≤k
u
k
.
d
Ú
n
2.1
Œ
i
(
T
λ
,K
r
1
,K
) = 1
,
¤
±
i
(
T
λ
,K
p
1
/
˚
K
r
1
,K
) =
−
1
.
e
f
∞
= 0,
K
•
3
˜
‡
R
1
>p
1
,
u
≥
R
1
ž
,
f
(
t,u
)
≤
ε
1
u
,
u
∈
∂K
R
1
,
K
k
k
T
λ
u
k≤
λe
R
1
0
a
(
θ
)
dθ
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
h
(
s
)
e
R
s
0
a
(
θ
)
dθ
ds
≤k
u
k
.
Ù
¥
R
1
λe
R
1
0
a
(
θ
)
dθ
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
e
R
1
0
a
(
θ
)
dθ
−
l
≤
1
,
d
Ú
n
2.1
Œ
i
(
T
λ
,K
R
1
,K
) = 0
,
¤
±
i
(
T
λ
,K
R
1
/
˚
K
p
1
,K
) =
−
1
,
K
T
λ
3
K
R
1
/
˚
K
p
1
½
K
p
1
/
˚
K
r
1
p
k
˜
‡
Ø
Ä:
,
=
T
λ
u
=
u
.
¯
K
(1.1)
–
•
3
˜
‡
)
.
(b).
b
(H1)-(H3)
¤
á
,
p
2
>
0,
u
∈
∂K
p
2
ž
,
Ò
k
k
T
λ
u
k≤
λe
R
1
0
a
(
θ
)
dθ
ˆ
M
p
2
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
e
R
s
0
a
(
θ
)
dθ
ds,
Ù
¥
ˆ
M
p
2
= 1+max
σp
2
≤
u
≤
p
2
{
f
(
t,u
)
}
.
-
0
<λ
0
≤
p
2
(
e
R
1
0
a
(
θ
)
dθ
−
l
)
e
R
1
0
a
(
θ
)
dθ
ˆ
M
p
2
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
,
λ<λ
0
ž
,
k
T
λ
u
k≤k
u
k
,
DOI:10.12677/pm.2021.115087725
n
Ø
ê
Æ
É
e
œ
d
Ú
n
2.1
Œ
i
(
T
λ
,K
p
2
,K
) = 1
.
e
f
0
=
∞
,
K
•
3
0
<r
2
<σp
2
,
¦
f
(
t,u
)
≥
M
1
u
,0
≤
u
≤
r
2
,
Ù
¥
M
1
λl
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
e
R
1
0
a
(
θ
)
dθ
−
l
≥
1
,
u
∈
∂K
r
2
ž
,
K
k
k
T
λ
u
k≥
λl
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
h
(
s
)
e
R
s
0
a
(
θ
)
dθ
ds
≥k
u
k
.
d
Ú
n
2.1
Œ
i
(
T
λ
,K
r
2
,K
) = 0
,
¤
±
i
(
T
λ
,K
p
2
/
˚
K
r
2
,K
) =
−
1
.
e
f
∞
=
∞
,
K
•
3
ˆ
R
2
>p
2
¦
f
(
t,u
)
≥
M
1
u
,
u
≥
ˆ
R
2
,
R
2
= max
{
2
p
2
,
ˆ
R
2
/σ
}
,
u
∈
∂K
R
2
ž
min
0
≤
t
≤
1
u
(
t
)
≥
σ
k
u
k≥
ˆ
R
2
,
¤
±
k
T
λ
u
k≥k
u
k
,u
∈
∂K
R
2
.
d
Ú
n
2.1
Œ
i
(
T
λ
,K
R
2
,K
) = 0
,
¤
±
i
(
T
λ
,K
R
2
/
˚
K
p
2
,K
) =
−
1
,
K
T
λ
3
K
R
2
/
˚
K
p
2
½
K
p
2
/
˚
K
r
2
p
k
˜
‡
Ø
Ä:
,
=
T
λ
u
=
u
.
¯
K
(1.1)
–
•
3
˜
‡
)
.
(c).
b
(H1)-(H3)
¤
á
,
p
3
>
0,
u
∈
∂K
p
3
ž
,
Ò
k
k
T
λ
u
k≥
ˆ
m
p
3
λl
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
e
R
s
0
a
(
θ
)
dθ
ds,
Ù
¥
ˆ
m
p
3
=min
σp
3
≤
u
≤
p
3
{
f
(
u
)
}
,
λ
0
≥
p
3
(
e
R
1
0
a
(
θ
)
dθ
−
l
)
ˆ
m
p
3
l
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
.
λ>λ
0
ž
,
k
T
λ
u
k≥k
u
k
,
DOI:10.12677/pm.2021.115087726
n
Ø
ê
Æ
É
e
œ
d
Ú
n
2.1
Œ
i
(
T
λ
,K
p
3
,K
) = 0
.
e
f
0
= 0,
K
•
3
0
<r
3
<σp
3
,
¦
f
(
t,u
)
≤
ε
2
u
,0
≤
u
≤
r
3
,
Ù
¥
ε
2
λe
R
1
0
a
(
θ
)
dθ
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
e
R
1
0
a
(
θ
)
dθ
−
l
≤
1
,
u
∈
∂K
r
3
ž
,
K
k
k
T
λ
u
k≤
λe
R
1
0
a
(
θ
)
dθ
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
h
(
s
)
e
R
s
0
a
(
θ
)
dθ
ds
≤k
u
k
.
d
Ú
n
2.1
Œ
i
(
T
λ
,K
r
3
,K
) = 1
,
¤
±
i
(
T
λ
,K
p
3
/
˚
K
r
3
,K
) =
−
1
.
e
f
∞
= 0,
K
•
3
R
3
¦
f
(
t,u
)
≤
ε
2
u
,
u
≥
R
3
,
u
∈
∂K
R
3
,
K
k
k
T
λ
u
k≤k
u
k
.
d
Ú
n
2.1
Œ
i
(
T
λ
,K
R
3
,K
) = 1
,
¤
±
i
(
T
λ
,K
R
3
/
˚
K
p
3
,K
) = 1
,
K
T
λ
3
K
p
3
/
˚
K
r
3
,K
R
3
/
˚
K
p
3
p
ˆ
k
˜
‡
Ø
Ä:
,
=
T
λ
u
=
u
.
¯
K
(1.1)
–
•
3
ü
‡
)
.
(d).
b
(H1)-(H3)
¤
á
,
p
4
>
0,
u
∈
∂K
p
4
ž
,
Ò
k
k
T
λ
u
k≤
λe
R
1
0
a
(
θ
)
dθ
ˆ
M
p
4
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
e
R
s
0
a
(
θ
)
dθ
ds,
Ù
¥
ˆ
M
p
4
= 1+max
σp
4
≤
u
≤
p
4
{
f
(
u
)
}
,
-
0
<λ
0
≤
p
4
(
e
R
1
0
a
(
θ
)
dθ
−
l
)
e
R
1
0
a
(
θ
)
dθ
ˆ
M
p
4
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
.
λ<λ
0
ž
,
k
T
λ
u
k≤k
u
k
,
d
Ú
n
2.1
Œ
i
(
T
λ
,K
p
4
,K
) = 1
.
DOI:10.12677/pm.2021.115087727
n
Ø
ê
Æ
É
e
œ
e
f
0
=
∞
,
K
•
3
0
<r
4
<σp
4
,
¦
f
(
t,u
)
≥
M
2
u
,0
≤
u
≤
r
4
,
Ù
¥
M
2
λl
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
e
R
1
0
a
(
θ
)
dθ
−
l
≥
1
,
u
∈
∂K
r
4
ž
,
K
k
k
T
λ
u
k≥
λl
e
R
1
0
a
(
θ
)
dθ
−
l
Z
1
0
h
(
s
)
e
R
s
0
a
(
θ
)
dθ
ds
≥k
u
k
,
d
Ú
n
2.1
Œ
i
(
T
λ
,K
r
4
,K
) = 0
.
e
f
∞
=
∞
,
K
•
3
ˆ
R
4
>p
4
,
¦
f
(
t,u
)
≥
M
2
u
,
u
≥
R
4
.
R
4
=
max
{
2
p
2
,
ˆ
R
4
/σ
}
,
u
∈
∂K
R
4
ž
min
0
≤
t
≤
1
u
(
t
)
≥
σ
k
u
k≥
ˆ
R
4
,
¤
±
k
T
λ
u
k≥k
u
k
,u
∈
∂K
R
4
,
d
Ú
n
2.1
Œ
i
(
T
λ
,K
R
4
,K
) = 0
,
¤
±
i
(
T
λ
,K
p
4
/
˚
K
r
4
,K
) = 1
,i
(
T
λ
,K
R
4
/
˚
K
p
4
,K
) =
−
1
,
K
T
λ
3
K
p
4
/
˚
K
r
4
,K
R
4
/
˚
K
p
4
p
ˆ
k
˜
‡
Ø
Ä:
,
=
T
λ
u
=
u
.
¯
K
(1.1)
–
•
3
ü
‡
)
.
(e)
e
f
0
<
∞
…
f
∞
<
∞
,
@
o
•
3
ü
‡
ê
ε
∗
1
,ε
∗
2
,
±
9
r
∗
1
<R
∗
1
,
¦
f
(
t,u
)
≤
ε
∗
1
u,u
∈
[0
,r
∗
1
]
,
f
(
t,u
)
≤
ε
∗
2
u,u
∈
[
R
∗
1
,
∞
)
.
-
ε
∗
3
=
{
max
ε
∗
1
,ε
∗
2
,
max
r
∗
1
≤
u
≤
R
∗
1
f
(
t,u
)
u
}
,
¤
±
f
(
t,u
)
≤
ε
∗
3
uu
∈
[0
,
∞
)
.
b
v
(
t
)
´
¯
K
(1.1)
˜
‡
)
.
¤
±
t
∈
[0
,
1]
ž
,
Ò
k
v
(
t
) =
T
λ
v
(
t
),
-
λ
0
=
e
R
1
0
a
(
θ
)
dθ
−
l
ε
∗
3
e
R
1
0
a
(
θ
)
dθ
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
,
λ<λ
0
ž
,
k
v
k
=
k
T
λ
v
k≤
λe
R
0
1
a
(
θ
)
dθ
ε
∗
3
e
R
0
1
a
(
θ
)
dθ
−
l
Z
1
0
e
R
0
sa
(
θ
)
dθ
ds
k
v
k
<
k
v
k
.
DOI:10.12677/pm.2021.115087728
n
Ø
ê
Æ
É
e
œ
ù
´
˜
‡
g
ñ
,
Ï
d•
3
˜
‡
λ
0
>
0,
¦
0
<λ<λ
0
ž
,
¯
K
(1.1)
Ã
)
.
(f)
e
f
0
>
0
…
f
∞
>
0,
@
o
•
3
ü
‡
ê
η
∗
1
,η
∗
2
,
±
9
r
∗
2
<R
∗
2
,
¦
f
(
t,u
)
≥
η
∗
1
uu
∈
[0
,r
∗
2
]
,
f
(
t,u
)
≥
η
∗
2
uu
∈
[
R
∗
2
,
∞
)
,
-
η
∗
3
= max
{
η
∗
1
,η
∗
2
,
max
r
∗
2
≤
u
≤
R
∗
2
f
(
t,u
)
u
}
,
¤
±
f
(
t,u
)
≥
η
∗
3
uu
∈
[0
,
∞
)
.
b
v
(
t
)
´
¯
K
(1.1)
˜
‡
)
,
¤
±
t
∈
[0
,
1]
ž
,
Ò
k
v
(
t
) =
T
λ
v
(
t
),
-
λ
0
=
e
R
1
0
a
(
θ
)
dθ
−
l
η
∗
3
l
R
1
0
e
R
s
0
a
(
θ
)
dθ
ds
,
λ>λ
0
ž
,
k
v
k
=
k
T
λ
v
k≥
λlη
∗
3
e
R
0
1
a
(
θ
)
dθ
−
l
Z
1
0
e
R
0
sa
(
θ
)
dθ
ds
k
v
k
>
k
v
k
.
ù
´
˜
‡
g
ñ
,
Ï
d•
3
˜
‡
λ
0
>
0,
¦
λ>λ
0
ž
,
¯
K
(1.1)
Ã
)
,
y
.
.
~
1
-
¯
K
(1.3)
¥
~
ê
l
=
1
2
,
‚
5
‘
a
(
t
)
≡
1,
‚
5
‘
a
(
t
)
≡
1,
š
‚
5
‘
f
(
t,u
(
t
))=
t
(
u
2
+
u
1
2
),
Ù
¥
λ
´
˜
‡
ë
ê
,
=
u
0
(
t
)+
u
(
t
) =
λt
(
u
2
+
u
1
2
)
,
0
≤
t
≤
1
,
u
(0) =
1
2
eu
(1)
y
²
é
u
f
(
t,u
(
t
))=
t
(
u
2
+
u
1
2
),
f
∈
C
([0
,
1]
×
R
+
,
R
+
),
^
‡
(H1)
¤
á
.
w
,
,
a
(
t
)
≡
1
…
l
=
1
2
e
÷
v
^
‡
(H2)
†
(H3),
´
f
0
=
∞
,f
∞
=
∞
.
λ
0
=
p
2
2
ˆ
M
p
2
(
e
−
1)
,
d
½
n
1.1
,
0
<λ
≤
λ
0
ž
,
¯
K
(1.3)
–
•
3
ü
‡
)
.
4.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
]
Ï
‘
8
(12061064).
DOI:10.12677/pm.2021.115087729
n
Ø
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