设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
PureMathematics
n
Ø
ê
Æ
,2022,12(8),1370-1380
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128150
˜
a
‘
š
‚
5
>
.
^
‡
Œ
¯
K
)
•
3
5
œœœ
ZZZ
JJJ
Ü
“
‰
Œ
Æ
§
ê
Æ
†
Ú
O
Æ
§
[
‹
=
²
Â
v
F
Ï
µ
2022
c
7
19
F
¶
¹
^
F
Ï
µ
2022
c
8
19
F
¶
u
Ù
F
Ï
µ
2022
c
8
30
F
Á
‡
©
ï
Ä
Œ
¯
K
−
u
00
(
t
) =
λh
(
t
)
f
(
u
(
t
))
,t
∈
(0
,
1)
,
αu
(0)
−
βu
0
(0) = 0
,c
(
u
(1))
u
(1)+
δu
0
(1) = 0
(
P
)
)
•
3
5
§
Ù
¥
λ
•
ë
ê
§
α,δ>
0
β
≥
0
•
~
ê
§
c
∈
C
([0
,
∞
)
,
[0
,
∞
))
§
h
∈
C
([0
,
1]
,
[0
,
∞
))
§
f
∈
C
([0
,
∞
)
,
R
)
…
f>
−
M
(
M>
0)
§
f
∞
:=lim
x
→∞
f
(
x
)
x
=
∞
"
Ï
L
$
^
Krasnoselskii
Ø
Ä:½
n
y
²
•
3
~
ê
λ
0
>
0
§
0
<λ<λ
0
ž
§
¯
K
(P)
•
3
˜
‡
)
"
'
…
c
)
§
Œ
¯
K
§
š
‚
5
>
.
^
‡
§
Krasnoselskii
Ø
Ä:½
n
ExistenceofPositiveSolutionsforaClass
ofSecondOrderSemi-PositoneProblems
withNonlinearBoundaryConditions
XuanrongShi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©
Ù
Ú
^
:
œ
Z
J
.
˜
a
‘
š
‚
5
>
.
^
‡
Œ
¯
K
)
•
3
5
[J].
A^
ê
Æ
?
Ð
,2022,12(8):1370-1380.
DOI:10.12677/pm.2022.128150
œ
Z
J
Received:Jul.19
th
,2022;accepted:Aug.19
th
,2022;published:Aug.30
th
,2022
Abstract
Weareconcernedwithexistenceofpositivesolutionsforthesecondordersemi-
positoneproblem
−
u
00
(
t
) =
λh
(
t
)
f
(
u
(
t
))
,t
∈
(0
,
1)
,
αu
(0)
−
βu
0
(0) = 0
,c
(
u
(1))
u
(1)+
δu
0
(1) = 0
,
(
P
)
where
λ
isapositiveparameter,
α,δ>
0
,β
≥
0
,
c
∈
C
([0
,
∞
)
,
[0
,
∞
))
,
h
∈
C
([0
,
1]
,
[0
,
∞
))
,
f
∈
C
([0
,
∞
)
,
R
)
and
f>
−
M
(
M>
0)
,f
∞
:=lim
u
→∞
f
(
x
)
x
=
∞
. Byusingfixedpointtheorem
ofKrasnoselskii,weprovethatthereexists
λ
0
>
0
such that (P)hasapositivesolution
for
0
<λ<λ
0
.
Keywords
PositiveSolutions,Semi-PositoneProblem,NonlinearBoundaryCondition,
KrasnoselskiiFixedPointTheorem
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.
Ú
ó
‡
©•
§
Sturm-Liouville
>
Š
¯
K
´
˜
a
-
‡
¯
K
,
Ú
å
N
õ
Æ
ö
2
•
'
5
,
¿
®
²
¼
˜
•
3
5
(
J
[1–14]
'
X
,Wang
Ú
Erbe[13]
ï
Ä
X
e
Sturm-Liouville
>
Š
¯
K
u
00
(
t
)+
a
(
t
)
g
(
u
(
t
)) = 0
,t
∈
(0
,
1)
,
αu
(0)
−
βu
0
(0) = 0
,γu
(1)+
δu
0
(1) = 0
,
(1
.
1)
Ù
¥
α,β,γ,δ
≥
0
…
γβ
+
αγ
+
αδ>
0,
g
∈
C
([0
,
∞
)
,
[0
,
∞
)).
a
∈
C
([0
,
1]
,
[0
,
∞
))
…
3
[0
,
1]
?
¿
f
«
m
þ
Ø
ð
•
"
.Wang
Ú
Erbe
$
^
I
þ
Ø
Ä:½
n
,
ï
á
X
e
(
J
:
DOI:10.12677/pm.2022.1281501371
n
Ø
ê
Æ
œ
Z
J
½
n
A
([13],
½
n
1)
e
g
÷
v
e
^
‡
:
lim
u
→
0
max
t
∈
[0
,
1]
g
(
t,u
)
u
= 0
,
lim
u
→∞
min
t
∈
[0
,
1]
g
(
t,u
)
u
= +
∞
½
lim
u
→
0
min
t
∈
[0
,
1]
g
(
t,u
)
u
= +
∞
,
lim
u
→∞
max
t
∈
[0
,
1]
g
(
t,u
)
u
= 0
,
K
¯
K
(1
.
1)
–
•
3
˜
‡
)
.
3#
N
f
•
K
Š
œ
/
e
,
é
u
Œ
œ
/
, Anuradha
[14]
$
^
I
þ
Ø
Ä:½
n
ï
Ä
Sturm-Liouville
>
Š
¯
K
−
u
00
(
t
) =
λf
(
t,u
(
t
))
,t
∈
(
r,R
)
,
au
(
r
)
−
bu
0
(
r
) = 0
,cu
0
(
R
)+
du
0
(
R
) = 0
,
(1
.
2)
Ù
¥
λ
•
ë
ê
,
¿
…
:
(A1)
a,b,c,d
≥
0
…
ac
+
ad
+
bc>
0;
(A2)
f
: [
r,R
]
×
[0
,
∞
)
→
R
ë
Y
…
f
(
t,u
)
>
−
M
(
M>
0)
,t
∈
[
r,R
];
(A3)
f
∞
=lim
x
→∞
f
(
x
)
x
=
∞
.
¦
‚
ï
á
X
e
(
J
:
½
n
B
([14],
½
n
2.1)
e
(A1)–(A3)
¤
á
,
λ>
0
…
¿
©
ž
,
¯
K
(1
.
2)
–
•
3
˜
‡
)
.
Š
5
¿
´
,
©
z
[13]
¤
ï
Ä
¯
K
¥
š
‚
5
‘
f
š
K
,
©
z
[14]
•
,
ï
Ä
Œ
¯
K
•
Î
´
‚
5
>
.
^
‡
.
g
,
‡
¯
:
š
‚
5
‘
f
•
Œ
œ
/
…
>
.
^
‡
•
š
‚
5
ž
¯
K
(P)
´
Ä
•
3
)
?
Ï
d
,
©
•
Ä
−
u
00
(
t
) =
λh
(
t
)
f
(
u
(
t
))
,t
∈
(0
,
1)
,
αu
(0)
−
βu
0
(0) = 0
,c
(
u
(1))
u
(1)+
δu
0
(1) = 0
(1
.
3)
)
•
3
5
.
·
‚
o
b
½
:(H1)
λ>
0
•
ë
ê
,
α,δ>
0
,β
≥
0
•
~
ê
;(H2)
c
: [0
,
∞
)
→
[0
,
∞
)
ë
Y
,
h
: [0
,
1]
→
[0
,
∞
)
ë
Y
,
…
3
[0
,
1]
?
¿
f
«
m
Ø
ð
•
"
;(H3)
f
:[0
,
∞
)
→
R
ë
Y
…
f
÷
v
f
(
s
)
>
−
M
(
M>
0);
©
Ì
‡
(
J
X
e
:
½
n
1.1
b
(H1)–(H3)
…
(A3)
¤
á
,
K
•
3
~
ê
λ
0
>
0
¦
0
<λ<λ
0
ž
,
¯
K
(1
.
3)
–
•
3
˜
‡
)
u
λ
,
…
λ
→
0
+
ž
,
u
λ
→∞
.
2.
ý
•
£
-
˜
m
X
:=
C
[0
,
1],
Ù
3
‰
ê
k
u
k
∞
=max
t
∈
[0
,
1]
|
u
(
t
)
|
e
¤
Banach
˜
m
,
L
1
(0
,
1)
3
‰
ê
k
y
k
1
=
R
1
0
|
y
(
t
)
|
dt
e
¤
Banach
˜
m
.
Ú
n
2.1
[15]
-
E
•
Banach
˜
m
,
P
•
E
¥
I
,
T
:
P
→
P
•
ë
Y
Ž
f
.
b
h
∈
E,h
6
= 0
DOI:10.12677/pm.2022.1281501372
n
Ø
ê
Æ
œ
Z
J
…
•
3
ê
r,R,r
6
=
R
¦
(a)
e
y
∈
P
÷
v
y
=
θTy,θ
∈
[0
,
1],
K
k
y
k
∞
6
=
r
;
(b)
e
y
∈
P
÷
v
y
=
Ty
+
ξh,ξ
≥
0,
K
k
y
k
∞
6
=
R
.
K
T
•
3
Ø
Ä:
y
,
…
min
{
r,R
}
<
k
y
k
∞
<
max
{
r,R
}
.
Ú
n
2.2
-
ω
÷
v
−
ω
00
(
t
) =
m
(
t
)
,t
∈
(0
,
1)
,
αω
(0)
−
βω
0
(0) = 0
,γω
(1)+
δω
0
(1) = 0
,
Ù
¥
γ>
0
•
~
ê
,
m
∈
L
1
[0
,
1]
,m
(
t
)
≥
0
,t
∈
(0
,
1),
…
ρ
:=
γβ
+
γα
+
αδ>
0
.
K
ω
(
t
)
≥k
ω
k
∞
σ,t
∈
[0
,
1]
,
Ù
¥
σ
= min
{
δ
δ
+
γ
,
β
α
+
β
}
.
y
²
N
´
y
²
ω
(
t
) =
Z
1
0
G
(
t,s
)
ω
(
s
)
ds
:=
Lω
(
t
)
,
Ù
¥
G
(
t,s
) =
(
γ
+
δ
−
γt
)(
β
+
αs
)
ρ
,
0
≤
s
≤
t
≤
1
,
(
β
+
αt
)(
γ
+
δ
−
γs
)
ρ
,
0
≤
t
≤
s
≤
1
.
w
,
,
G
(
t,s
)
≥
0
,t
∈
[0
,
1].
-
k
ω
k
∞
=
ω
(
τ
)
,τ
∈
[0
,
1].
¯¢
þ
G
(
t,s
)
G
(
τ,s
)
≥
G
(
t,s
)
G
(
s,s
)
≥
δ
δ
+
γ
,
0
≤
s
≤
t
≤
1
,
β
α
+
β
,
0
≤
t
≤
s
≤
1
.
Ï
d
ω
(
t
) =
Z
1
0
G
(
t,s
)
G
(
τ,s
)
G
(
τ,s
)
ω
(
s
)
ds
≥k
ω
k
∞
σ.
Ú
n
2.3
-
k
∈
L
1
(0
,
1),
…
k
≥
0,
-
u
∈
C
1
[0
,
1]
∩
C
2
(0
,
1)
÷
v
−
u
00
(
t
)
≥−
k,t
∈
(0
,
1)
,
αu
(0)
−
βu
0
(0)
≥
0
,γu
(1)+
δu
0
(1)
≥
0
.
-
ϕ
:=
β
+
α
α
,
b
k
u
k
∞
>
2
ϕ
(1+
1
σ
)
k
k
k
1
,
K
u
(
t
)
≥
0
…
u
(
t
)
≥
σ
2
k
u
k
∞
,t
∈
[0
,
1]
.
DOI:10.12677/pm.2022.1281501373
n
Ø
ê
Æ
œ
Z
J
y
²
-
ν
0
(
t
)
•¯
K
−
ν
00
(
t
) =
−
k,t
∈
(0
,
1)
,
αν
(0)
−
βν
0
(0) = 0
,γν
(1)+
δν
0
(1) = 0
•
˜
)
,
K
−
ν
0
(
t
) =
Z
1
0
G
(
t,s
)
k
(
s
)
ds
=
Z
t
0
(
γ
+
δ
−
γt
)(
β
+
αs
)
ρ
k
(
s
)
ds
+
Z
1
t
(
β
+
αt
)(
γ
+
δ
−
γs
)
ρ
k
(
s
)
ds
≤
Z
1
0
(
γ
+
δ
−
γt
)(
β
+
αt
)
ρ
ds
k
k
k
1
≤
(
γ
+
δ
)(
β
+
α
)
γβ
+
γα
+
αδ
k
k
k
1
≤
(
β
+
α
)
α
k
k
k
1
.
Ï
d
−
ν
0
(
t
)
≤
ϕ
k
k
k
1
.
-
y
(
t
) =
u
(
t
)
−
ν
0
(
t
),
k
−
y
00
(
t
)
≥
0
,t
∈
(0
,
1)
,
αy
(0)
−
βy
0
(0)
≥
0
,γy
(1)+
δy
0
(1)
≥
0
.
d
Ú
n
2.2
Œ
•
y
(
t
)
≥k
y
k
∞
σ,t
∈
[0
,
1]
,
¤
±
u
(
t
) =
y
(
t
)+
ν
0
(
t
)
≥k
y
k
∞
σ
−
ϕ
k
k
k
1
=
k
u
−
ν
0
k
∞
σ
−
ϕ
k
k
k
1
≥
(
k
u
k
∞
−k−
ν
0
k
∞
)
σ
−
ϕ
k
k
k
1
≥
(
k
u
k
∞
−
ϕ
(1+
1
σ
)
k
k
k
1
)
σ
≥
σ
2
k
u
k
∞
,t
∈
(0
,
1)
.
DOI:10.12677/pm.2022.1281501374
n
Ø
ê
Æ
œ
Z
J
3.
Ì
‡
(
J
y
²
½
n
1.1
y
²
-
λ>
0,
é
v
∈
X
k
L
λ
v
=
u
,
Ù
¥
u
•
−
u
00
(
t
) =
λh
(
t
)
f
(˜
v
(
t
))
,t
∈
(0
,
1)
,
αu
(0)
−
βu
0
(0) = 0
,γ
v
u
(1)+
δu
0
(1) = 0
)
.
…
-
γ
v
=
c
(
|
v
(1)
|
),
σ
(
t
) = min
{
δ
δ
+
γ
v
β
α
+
β
}
˜
v
(
t
) = max
{
v
(
t
)
,σ
(
t
)
}
.
d
Ú
n
2.2
Œ
•
u
(
t
) =
λ
Z
1
0
G
v
(
t,s
)
h
(
s
)
f
(˜
v
(
s
))
ds,
Ù
¥
G
v
(
t,s
) =
(
γ
v
+
δ
−
γ
v
t
)(
β
+
αs
)
αδ
+
γ
v
β
+
γ
v
α
,
0
≤
s
≤
t
≤
1
,
(
β
+
αt
)(
γ
v
+
δ
−
γ
v
s
)
αδ
+
γ
v
β
+
γ
v
α
,
0
≤
t
≤
s
≤
1
.
½
Â
X
¥
I
P
=
{
u
∈
X
|
u
(
t
)
≥
σ
2
||
u
||
∞
,t
∈
[0
,
1]
}
.
e
u
∈
P
,
(
Ü
Ú
n
2.3
Œ
•
L
λ
u
(
t
) =
λ
Z
1
0
G
(
t,s
)
h
(
s
)
f
(˜
v
(
s
))
ds
≥
λ
Z
1
0
G
(
t,s
)
G
(
s,s
)
G
(
s,s
)
h
(
s
)
f
(˜
v
(
s
))
ds
≥
σ
||
L
λ
u
||
∞
≥
σ
2
||
L
λ
u
||
∞
,t
∈
[0
,
1]
.
Ï
d
L
λ
(
P
)
⊂
P
.
e
y
L
λ
ë
Y
.
Ä
k
,
S
⊂
C
[0
,
1]
•
k
.
8
,
K
•
3
ê
B
,
¦
é
?
¿
v
∈
S
k
||
v
||
∞
≤
B
.
d
f
ë
Y
5
•
,
•
3
D>
0,
k
f
(
v
)
≤
D,v
∈
S.
-
v
n
∈
S
…
v
n
→
v
,
K
u
n
=
L
λ
v
n
,u
=
L
λ
v.
d
.
‚
K
F
¥Š
½
n
Œ
•
,
•
3
~
ê
N>
0,
k
|
G
v
n
(
t,s
)
−
G
v
(
t,s
)
|≤
N
|
γ
v
n
−
γ
v
|
.
DOI:10.12677/pm.2022.1281501375
n
Ø
ê
Æ
œ
Z
J
Ï
•
|
f
(˜
v
n
(
s
))
−
f
(˜
v
(
s
))
|→
0
,n
→∞
,s
∈
[0
,
1]
,
|
L
λ
v
n
−
L
λ
v
|
=
|
λ
Z
1
0
G
v
n
(
t,s
)
h
(
s
)
f
(˜
v
n
(
s
))
ds
−
λ
Z
1
0
G
v
(
t,s
)
h
(
s
)
f
(˜
v
(
s
))
ds
|
≤
λ
(
Z
1
0
|
G
v
n
(
t,s
)
−
G
v
(
t,s
)
|
h
(
s
)
|
f
(˜
v
n
(
s
))
|
ds
+
Z
1
0
G
v
(
t,s
)
h
(
s
)
|
f
(˜
v
n
(
s
))
−
f
(˜
v
(
s
))
|
ds
)
<λ
(
Z
1
0
N
|
γ
v
n
−
γ
v
|
h
(
s
)
|
f
(˜
v
n
(
s
))
|
ds
+
Z
1
0
G
v
(
t,s
)
h
(
s
)
|
f
(˜
v
n
(
s
))
−
f
(˜
v
(
s
))
|
ds
)
,
¤
±
|
L
λ
v
n
−
L
λ
v
|→
0
,n
→∞
,
L
λ
v
n
→
L
λ
v
.
¤
±
L
λ
ë
Y5
y
.
e
y
L
λ
´
;
Ž
f
.
é
u
v
∈
S
k
||
L
λ
v
||≤
D
Z
1
0
G
v
(
s,s
)
h
(
s
)
ds,
Ï
d
,
L
λ
:
C
[0
,
1]
→
C
[0
,
1]
˜
—
k
.
.
é
?
¿
t
1
,t
2
∈
[0
,
1] (
t
1
<t
2
)
k
|
L
λ
v
(
t
1
)
−
L
λ
v
(
t
2
)
|
=
|
λ
Z
1
0
G
v
(
t
1
,s
)
h
(
s
)
f
(˜
v
(
s
))
ds
−
λ
Z
1
0
G
v
(
t
2
,s
)
h
(
s
)
f
(˜
v
(
s
))
ds
|
≤
λD
Z
1
0
|
G
v
(
t
1
,s
)
−
G
v
(
t
2
,s
)
|
h
(
s
)
ds.
d
G
(
t,s
)
ë
Y5
Œ
•
,
L
λ
Ý
ë
Y
.
d
Arz`ela-Ascoli
½
n
•
,
L
λ
:
C
[0
,
1]
→
C
[0
,
1]
•
ë
Y
Ž
f
.
-
a>
1
k
f
(
z
)
>
0
,z
≥
a.
d
f
ë
Y5
Œ
•
|
f
(
z
)
|≤
M,z
∈
(0
,a
)
.
(3
.
1)
Ï
d
|
f
(
z
)
|≤
(
M
+
ˆ
f
(max
{
z,a
}
)
,z>
0
,
(3
.
2)
Ù
¥
ˆ
f
(
x
) =sup
a
≤
z
≤
x
f
(
z
)
,x
≥
a.
b
λ<
a
2(
c
1
+
c
2
f
(
a
))
,
Ù
¥
c
1
=
(
α
+
β
)
M
α
Z
1
0
h
(
s
)
ds,c
2
=
(
α
+
β
)
α
Z
1
0
h
(
s
)
ds.
DOI:10.12677/pm.2022.1281501376
n
Ø
ê
Æ
œ
Z
J
e
y
Ú
n
2.2
b
é
L
λ
¤
á
.
(a)
•
3
r
λ
>
0,
e
u
∈
P
÷
v
u
=
θL
λ
u,θ
∈
[0
,
1]
,
K
||
u
||
∞
6
=
r
λ
.
¯¢
þ
,
-
u
∈
P
÷
v
u
=
θL
λ
u,θ
∈
[0
,
1],
K
u
(
t
) =
θλ
Z
1
0
G
u
(
t,s
)
h
(
s
)
f
(˜
u
(
s
))
ds,t
∈
[0
,
1]
.
Ï
•
a>
1,
σ<
1,
d
(3.2)
Œ
•
|
f
(
u
)
|≤
M
+
ˆ
f
(max
{
u,a
}
)
,
…
G
u
(
t,s
)
≤
α
+
β
α
,t
∈
(0
,
1)
.
Ï
d
u
(
t
)
≤
λ
α
+
β
α
(
M
+
ˆ
f
(max
{
u,a
}
)
Z
1
0
h
(
s
)
ds
≤
λ
(
c
1
+
c
2
ˆ
f
(max
{||
u
||
∞
,a
}
))
,t
∈
[0
,
1]
.
=
||
u
||
∞
c
1
+
c
2
ˆ
f
(max
{||
u
||
∞
,a
}
≤
λ.
(3
.
3)
Ï
•
a
(
c
1
+
c
2
f
(
a
))
>
2
λ,
(
Ü
(A4),
K
•
3
r
λ
>a
¦
r
λ
c
1
+
c
2
ˆ
f
(
r
λ
)
= 2
λ.
(3
.
4)
d
(3.3),(3.4)
||
u
||
∞
6
=
r
λ
…
r
λ
→∞
,λ
→
0
.
(b)
•
3
R
λ
>
0,
é
u
u
∈
P
k
u
=
L
λ
u
+
ξ,ξ
≥
0
.
K
||
u
||
∞
6
=
R
λ
.
¯¢
þ
,
-
u
∈
P
÷
v
u
=
L
λ
u
+
ξ,ξ
≥
0,
K
u
(
t
)
−
ξ
=
λ
Z
1
0
G
u
(
t,s
)
h
(
s
)
f
(˜
u
(
s
))
ds.
Ï
•
u
(
t
)
−
ξ
÷
v
−
u
00
(
t
) =
λh
(
t
)
f
(
u
(
t
))
,t
∈
(0
,
1)
,
u
(0)
−
βu
0
(0) =
ξ
≥
0
,γ
v
u
(1)+
δu
0
(1) =
γ
v
ξ
≥
0
.
DOI:10.12677/pm.2022.1281501377
n
Ø
ê
Æ
œ
Z
J
-
k
(
t
) =
Mh
(
t
)
,t
∈
[0
,
1]
.
d
(3.1)
Œ
•
h
(
t
)
f
(˜
u
(
t
))
≥−
Mh
(
t
) =
−
k
(
t
)
,t
∈
[0
,
1]
.
d
Ú
n
2.3
Œ
•
u
(
t
)
≥
(
||
u
||
∞
−
ϕ
(1+
1
σ
)
||
k
||
1
)
σ,t
∈
[0
,
1]
.
b
||
u
||
∞
>
max
{
2
ϕ
(1+
1
σ
)
,
2
σ
}
,
K
u
(
t
)
≥
(
||
u
||
∞
−
||
u
||
∞
2
)
σ
=
σ
||
u
||
∞
2
,t
∈
[0
,
1]
.
s
∈
Λ = [
1
4
,
3
4
]
k
G
u
(
t,s
)
≥
(
δ
+
1
4
γ
)(
1
4
α
+
β
)
(
δ
+
γ
)(
α
+
β
)
≥
(4+
γ
)(1+4
β
)
16(1+
γ
)(1+
β
)
≥
1
16
.
Ï
d
u
(
t
) =
λ
Z
1
0
G
u
(
t,s
)
h
(
s
)
f
(˜
u
(
s
))
ds
≥
λ
Z
Λ
G
u
(
t,s
)
h
(
s
)
f
(˜
u
(
s
))
ds
+
λ
Z
Λ
c
G
u
(
t,s
)
h
(
s
)
f
(˜
u
(
s
))
ds
≥
λ
(
1
16
ˇ
f
(
||
u
||
∞
2
σ
)
Z
Λ
h
(
s
)
ds
−
(
α
+
β
)
α
M
||
k
||
1
)
,
Ù
¥
ˇ
f
(
x
) =inf
z
≥
x
f
(
z
).
¯¢
þ
1
16
ˇ
f
(
||
u
||
∞
2
σ
)
R
Λ
h
(
s
)
ds
−
(
α
+
β
)
α
K
||
k
||
1
||
u
||
∞
≤
1
λ
.
(3
.
5)
(
Ü
(A3),(3.5)
Œ
•
,
•
3
R
λ
>
1
…
||
u
||
∞
<R
λ
,
Ï
d
(b)
¤
á
.
Ï
d
,
d
Ú
n
2.1
Œ
•
,
¯
K
(1.3)
•
3
˜
‡
)
u
λ
(
t
),
λ
→
0
ž
u
λ
(
t
)
→∞
,t
∈
[0
,
1]
.
4.
A^
~
•
Ä
¯
K
−
u
00
(
t
) =
λt
2
(
u
2
(
t
)
−
1)
,t
∈
(0
,
1)
,
u
(0)
−
u
0
(0) = 0
,u
3
(1)+
u
0
(1) = 0
(4
.
1)
DOI:10.12677/pm.2022.1281501378
n
Ø
ê
Æ
œ
Z
J
)
•
3
5
,
Ù
¥
λ>
0
.
)
ù
p
f
(
u
) =
u
2
(
t
)
−
1
,α
=
β
=
δ
= 1
,h
(
t
) =
t
2
,c
(
u
(1)) =
u
2
(1)
.
é
u
¯
K
(4.1)
ó
, (H1), (H2)
w
,
¤
á
,
Ï
•
lim
u
→∞
f
(
u
)
u
=
∞
…
f
(
u
)
>
−
1
,u
∈
[0
,
∞
),
K
f
÷
v
(H3),(A3).
Š
â
½
n
1.1,
•
3
~
ê
λ
∗
>
0,
¦
0
<λ
∗
<λ
0
ž
,
¯
K
(4.1)
•
3
˜
‡
)
u
λ
,
λ
→
0
+
ž
,
u
λ
→∞
.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
]
Ï
‘
8
(
1
O
Ò
:12061064).
ë
•
©
z
[1]Castro,A. andShivaji,R.(1988) Nonnegative Solutionsfora Classof NonpositoneProblems.
ProceedingsoftheRoyalSocietyofEdinburghSectionA
,
108
,291-302.
https://doi.org/10.1017/S0308210500014670
[2]Rabinowitz,P.H.(1970)NonlinearSturm-LiouvilleProblemsforSecondOrderOrdinaryDif-
ferentialEquations.
CommunicationsonPureandAppliedMathematics
,
23
,939-961.
https://doi.org/10.1002/cpa.3160230606
[3]Jiang,J.Q.,Liu,L.S.andWu,Y.H.(2009)Second-OrderNonlinearSingularSturm-Liouville
ProblemswithIntegralBoundaryConditions.
AppliedMathematicsandComputation
,
215
,
1573-1582.https://doi.org/10.1016/j.amc.2009.07.024
[4]Ge,W.G.andRen,J.L.(2004)NewExistenceTheoremsofPositiveSolutionsforSturm-
LiouvilleBoundaryValueProblems.
AppliedMathematicsandComputation
,
148
,631-644.
https://doi.org/10.1016/S0096-3003(02)00921-9
[5]Tian, Y.andGe,W.G.(2007)MultipleSolutionsforaSecond-OrderSturm-LiouvilleBoundary
ValueProblem.
TaiwaneseJournalofMathematics
,
11
,975-988.
https://doi.org/10.11650/twjm/1500404796
[6]Rynne,B.P.(2012)LinearSecond-OrderProblemswithSturm-Liouville-TypeMulti-Point
BoundaryConditions.
ElectronicJournalofDifferentialEquations
,
146
,1-21.
[7]Lv,H.Y.andShi,Y.M.(2009)ErrorEstimateofEigenvaluesofPerturbedSecond-Order
DiscreteSturm-LiouvilleProblems.
LinearAlgebraanditsApplications
,
430
,2389-2415.
https://doi.org/10.1016/j.laa.2008.12.016
[8]Du,Z.J.andYin,J.(2014)ASecondOrderDifferentialEquationwithGeneralizedSturm-
LiouvilleIntegralBoundaryConditionsatResonance.
Filomat
,
28
,1437-1444.
https://doi.org/10.2298/FIL1407437D
DOI:10.12677/pm.2022.1281501379
n
Ø
ê
Æ
œ
Z
J
[9]Liu,Y.J.(2008)OnSturm-LiouvilleBoundaryValueProblemsforSecond-OrderNonlinear
FunctionalFiniteDifferenceEquations.
JournalofComputationalandAppliedMathematics
,
216
,523-533.https://doi.org/10.1016/j.cam.2007.06.003
[10]Xie,S.L.(2001)Sturm-LiouvilleBoundaryValueProblemsforNonlinearSecondOrderIm-
pulsiveDifferentialEquations.
MathematicaApplicata(Wuhan)
,
14
,93-98.
[11]Yang,J.B.andWei,Z.L.(2010)ExistenceofPositiveSolutionsofSturm-LiouvilleBoundary
ValueProblemsforaNonlinearSingularSecondOrderDifferentialSystemwithaParameter.
JournalofAppliedMathematicsandComputing
,
34
,129-145.
https://doi.org/10.1007/s12190-009-0312-z
[12]Drame,A.K.andCosta,D.G.(2012)OnPositiveSolutionsofOne-dimensionalSemipositone
EquationswithNonlinearBoundaryConditions.
AppliedMathematicsLetters
,
25
,2411-2416.
https://doi.org/10.1016/j.aml.2012.07.015
[13]Erbe,L.H.andWang,H.Y.(1994)OntheExistenceofPositiveSolutionsofOrdinaryDiffer-
entialEquations.
ProceedingsoftheAmericanMathematicalSociety
,
120
,743-748.
https://doi.org/10.1090/S0002-9939-1994-1204373-9
[14]Anuradha, V.,Hai, D.D.and Shivaji, R.(1996) ExistenceResults forSuperlinear Semipositone
BVP’s.
ProceedingsoftheAmericanMathematicalSociety
,
124
,757-763.
https://doi.org/10.1090/S0002-9939-96-03256-X
[15]Castro, A.andShivaji, R.(1988)Non-NegativeSolutionsforaClassofNon-PositoneProblems.
ProceedingsoftheRoyalSocietyofEdinburghSectionA
,
108
,291-302.
https://doi.org/10.1017/S0308210500014670
DOI:10.12677/pm.2022.1281501380
n
Ø
ê
Æ