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PureMathematics
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,2022,12(6),928-937
PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.126102
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PositiveSolutionsofRobinBoundary
ValueProblemsforaClassofSecond-Order
DifferenceSystem
HaiyiWu
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
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DOI:10.12677/pm.2022.126102
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Received:May3
rd
,2022;accepted:Jun.7
th
,2022;published:Jun.14
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,2022
Abstract
Inthispaper,weconsiderthecoupledgrowthofnonlineartermsforboundaryvalue
problems of systemsof differenceequations,resolve the positive solutions ofboundary
valueproblemsforaclassofnonlineardifferenceequations.AlsobyusingJensen’s
inequalityfornonnegativeconcavefunctionsandthefixedpointindextheory,we
discusstheexistenceofpositivesolutionsofRobinboundaryvalueproblemsfora
classofsecond-orderdifferencesystem
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arecontinuous.
Keywords
Jensen’sInequality,PositiveSolutions,Second-OrderDifferenceEquations,
FixedPointIndexTheory
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
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1
(
t
) = sin
πt
2
T
+1
,t
∈
[1
,T
]
Z
,
K
λ
1
T
X
t
=1
G
(
t,s
)
ϕ
1
(
t
) =
ϕ
1
(
s
)
,s
∈
[1
,T
]
Z
.
Ù
¥
,
λ
1
= 4sin
2
π
4
T
+2
.
y
²
.
Ï
•
λ
1
P
T
s
=1
G
(
t,s
)
ϕ
1
(
s
)=
ϕ
1
(
t
)
,t
∈
[1
,T
]
Z
.
5
¿
,
‚
¼
ê
é
¡
5
,
G
(
t,s
)=
G
(
s,t
)
,t,s
∈
[1
,T
]
Z
.
¤
±
λ
1
P
T
s
=1
G
(
s,t
)
ϕ
1
(
s
) =
ϕ
1
(
t
)
,t
∈
[1
,T
]
Z
.
Ï
d
(
Ø
¤
á
.
Ú
n
7
ψ
: [0
,
∞
)
→
[0
,
∞
)
ë
Y
…
3
[0
,
∞
)
þ
à
,
K
ψ
ü
N4
O
.
y
²
.
é
?
¿
x>x
2
>x
1
≥
0
,
du
ψ
þ
à
,
K
ψ
(
x
2
)
≥
ψ
(
x
1
)+
x
2
−
x
1
x
−
x
2
(
ψ
(
x
)
−
ψ
(
x
2
))
,
DOI:10.12677/pm.2022.126102932
n
Ø
ê
Æ
Ç
°
²
(
Ü
ψ
š
K
5
,lim
x
→∞
ψ
(
x
)
x
≥
0
.
l
ψ
(
x
2
)
≥
ψ
(
x
1
)
.
3.
Ì
‡
(
J
9
y
²
©
o
b
½
µ
(H1)
¼
ê
f,g
: [1
,T
]
Z
×
[0
,
∞
)
×
[0
,
∞
)
→
[0
,
∞
)
ë
Y
;
(H2)
¼
ê
ψ
1
,ψ
2
: [0
,
∞
)
→
[0
,
∞
)
ë
Y
,
ψ
1
•
þ
à
¼
ê
,
…
•
3
~
ê
α>
2(
T
+1)
2
sin
4
π
4
T
+2
,c>
0,
¦
(i)
f
(
t,x,y
)
≥
ψ
1
(
y
)
−
c,g
(
t,x,y
)
≥
ψ
2
(
x
)
−
c,t
∈
[1
,T
]
Z
,x,y
∈
[0
,
∞
);
(ii)
ψ
1
(
ψ
2
(
z
))
≥
αz
−
c,z
∈
[0
,
∞
);
(H3)
•
3
~
ê
a
1
,b
1
,c
1
,d
1
≥
0
,r>
0
,
…
r
P
×
P
(
T
1
)
<
1
,
k
f
(
t,u,v
)
≤
a
1
u
+
b
1
v, g
(
t,u,v
)
≤
c
1
u
+
d
1
v,x,y
∈
[0
,r
]
,t
∈
[1
,T
]
Z
,
Ù
¥
,
Ž
f
T
1
:
P
×
P
→
P
×
P,
T
1
(
u,v
)(
t
) := (
T
X
s
=1
G
(
t,s
)(
a
1
u
(
t
)+
b
1
v
(
t
))
,
T
X
s
=1
G
(
t,s
)(
c
1
u
(
t
)+
d
1
v
(
t
)))
.
©
Ì
‡
(
J
X
e
:
½
n
1
b
½
(H1)-(H3)
¤
á
,
•
§
|
(1)
–
•
3
˜
‡
)
.
y
²
.
P
M
1
:=
{
(
u,v
)
∈
P
×
P
: (
u,v
) =
A
(
u,v
)+
λ
(
ω
0
,ω
0
)
,λ
≥
0
}
,
Ù
¥
ω
0
(
t
) := (2
T
+1)
t
−
t
2
.
e
(
u
0
,v
0
)
∈
M
1
,
K
•
3
λ
0
≥
0
,
d
u
0
(
t
) =
T
X
s
=1
G
(
t,s
)
f
(
s,u
0
(
s
)
,v
0
(
s
))+
λ
0
ω
0
, v
0
(
t
) =
T
X
s
=1
G
(
t,s
)
g
(
s,u
0
(
s
)
,v
0
(
s
))+
λ
0
ω
0
,
(5)
(
Ü
(2),(3)
Œ
,(
u
0
,v
0
) =
A
(
u
0
,v
0
)+
λ
0
(
ω
0
,ω
0
)
.
d
(H1)
Œ
•
,∆
2
u
0
(
t
−
1)
≤
0
,
l
u
0
3
[1
,T
]
Z
þ
´
þ
à
.
Ó
n
Œ
,
v
0
3
[1
,T
]
Z
þ
•
´
þ
à
.
(
Ü
(5)
,
u
0
(
t
)
≥
T
X
s
=1
G
(
t,s
)
f
(
s,u
0
(
s
)
,v
0
(
s
))
, v
0
(
t
)
≥
T
X
s
=1
G
(
t,s
)
g
(
s,u
0
(
s
)
,v
0
(
s
))
,
d
b
½
(H2)
¥
(i)
,
u
0
(
t
)
≥
T
X
s
=1
G
(
t,s
)
ψ
1
(
v
0
(
s
))
−
c
1
, v
0
(
t
)
≥
T
X
s
=1
G
(
t,s
)
ψ
2
(
u
0
(
s
))
−
c
1
,
(6)
DOI:10.12677/pm.2022.126102933
n
Ø
ê
Æ
Ç
°
²
Ï
d
,
u
0
(
t
)
≥
T
X
s
=1
G
(
t,s
)
ψ
1
(
T
X
τ
=1
G
(
s,τ
)
ψ
2
(
u
0
(
τ
)))
−
c
2
.
(7)
du
T
X
τ
=1
G
(
s,τ
) =
s
X
τ
=1
τ
+
T
X
τ
=
s
+1
s
=
(
T
+1
−
s
)
s
2
,
-
h
(
s
) =
(
T
+1
−
s
)
s
2
,
K
h
•
Š
T
2
,
•
Œ
Š
(
T
+1)
2
8
.
5
¿
T
≥
2
,
¤
±
P
T
τ
=1
G
(
s,τ
)
≥
1
.
d
ψ
1
ü
N4
O
Ú
Ú
n
5
Œ
,
ψ
1
(
T
X
τ
=1
G
(
s,τ
)
ψ
2
(
u
0
(
τ
)))
≥
ψ
1
P
T
τ
=1
G
(
s,τ
)
ψ
2
(
u
0
(
τ
))
P
T
τ
=1
G
(
s,τ
)
≥
P
T
τ
=1
G
(
s,τ
)
ψ
1
(
ψ
2
(
u
0
(
τ
)))
P
T
τ
=1
G
(
s,τ
)
≥
8
(
T
+1)
2
T
X
τ
=1
G
(
s,τ
)
ψ
1
(
ψ
2
(
u
0
(
τ
)))
,
(
Ü
(H2)
¥
(ii)
9
(7)
Œ
,
u
0
(
t
)
≥
8
(
T
+1)
2
T
X
s
=1
G
(
t,s
)
T
X
τ
=1
G
(
s,τ
)
ψ
1
(
ψ
2
(
u
0
(
τ
)))
−
c
2
≥
8
α
(
T
+1)
2
T
X
s
=1
G
(
t,s
)
T
X
τ
=1
G
(
s,τ
)
u
0
(
τ
)
−
c
3
.
K
,
u
0
(
t
)
≥
8
α
(
T
+1)
2
T
X
s
=1
G
(
t,s
)
T
X
τ
=1
G
(
s,τ
)
u
0
(
τ
)
−
c
3
.
(8)
P
µ
2
=
P
T
t
=1
sin
πt
2
T
+1
>
0.
é
(8)
ü
à
Ó
ž
¦
±
sin
πt
2
T
+1
,
2
l
t
=1
T
þ
¦
Ú
,
¿
d
Ú
n
6
Œ
,
T
X
t
=1
u
0
(
t
)sin
πt
2
T
+1
≥
8
α
(
T
+1)
2
T
X
t
=1
T
X
s
=1
G
(
t,s
)
T
X
τ
=1
G
(
s,τ
)
u
0
(
τ
)sin
πt
2
T
+1
−
µ
2
c
3
=
8
α
(
T
+1)
2
T
X
s
=1
T
X
t
=1
G
(
t,s
)sin
πt
2
T
+1
T
X
τ
=1
G
(
s,τ
)
u
0
(
τ
)
−
µ
2
c
3
=
8
α
(
T
+1)
2
λ
1
T
X
s
=1
T
X
τ
=1
G
(
s,τ
)sin
πs
2
T
+1
u
0
(
τ
)
−
µ
2
c
3
=
8
α
(
T
+1)
2
λ
1
T
X
τ
=1
T
X
s
=1
G
(
s,τ
)sin
πs
2
T
+1
u
0
(
τ
)
−
µ
2
c
3
=
8
α
(
T
+1)
2
λ
2
1
T
X
t
=1
sin
πt
2
T
+1
u
0
(
t
)
−
µ
2
c
3
.
DOI:10.12677/pm.2022.126102934
n
Ø
ê
Æ
Ç
°
²
l
T
X
t
=1
u
0
(
t
)sin
πt
2
T
+1
≤
(
T
+1)
2
λ
2
1
µ
2
c
3
8
α
−
(
T
+1)
2
λ
2
1
.
(9)
Ú
n
4
,
k
u
0
k
<
r
µ
1
T
X
t
=1
u
0
(
t
)sin
πt
2
T
+1
≤
(
T
+1)
2
rλ
2
1
µ
2
c
3
µ
1
[8
α
−
(
T
+1)
2
λ
2
1
]
.
u
0
k
.
,
e
y
v
0
k
.
.
d
Ú
n
4
Œ
•
,
k
v
0
k
<
r
µ
1
T
X
t
=1
v
0
(
t
)sin
πt
2
T
+1
.
d
(6)
Ú
(9)
(
Ü
Ú
n
6
9
Ú
n
4
Œ
í
•
,
(
T
+1)
2
λ
2
1
µ
2
c
3
8
α
−
(
T
+1)
2
λ
2
1
≥
T
X
t
=1
u
0
(
t
)sin
πt
2
T
+1
≥
T
X
t
=1
T
X
s
=1
G
(
t,s
)
ψ
1
(
v
0
(
s
))sin
πt
2
T
+1
−
µ
2
c
1
=
T
X
s
=1
T
X
t
=1
G
(
t,s
)
ψ
1
(
v
0
(
s
))sin
πt
2
T
+1
−
µ
2
c
1
=
1
λ
1
T
X
t
=1
ψ
1
(
v
0
(
t
))sin
πt
2
T
+1
−
µ
2
c
1
=
1
λ
1
T
X
t
=1
ψ
1
v
0
(
t
)
k
v
0
k
·k
v
0
k
sin
πt
2
T
+1
−
µ
2
c
1
≥
1
λ
1
ψ
1
(
k
v
0
k
)
k
v
0
k
T
X
t
=1
v
0
(
t
)sin
πt
2
T
+1
−
µ
2
c
1
>
µ
1
rλ
1
ψ
1
(
k
v
0
k
)
−
µ
2
c
1
.
Ï
d
,
ψ
1
(
k
v
0
k
)
<
(
T
+1)
2
rλ
3
1
µ
2
c
3
µ
1
(8
α
−
(
T
+1)
2
λ
2
1
)
+
rλ
1
µ
2
c
1
µ
1
.
d
(H2)
Œ
•
,lim
z
→∞
ψ
1
(
z
) =
∞
,
K
•
3
c
4
>
0
,
¦
k
v
0
k
<c
4
,
M
1
k
.
.
-
R>
sup
{
(
u,v
):(
u,v
)
∈
M
1
}
,
K
•
3
ω
0
6
=0
,
¦
(
u,v
)
6
=
A
(
u,v
)+
λ
(
ω
0
,ω
0
)
.
d
Ú
n
1 ,
?
¿
(
u,v
)
∈
∂B
R
∩
P
×
P,λ
6
= 0
,
i
(
A,B
R
∩
P
×
P,P
×
P
) = 0
.
(10)
P
M
2
=
{
(
u,v
)
∈
¯
B
r
∩
P
×
P
:(
u,v
)=
λA
(
u,v
)
,λ
∈
[0
,
1]
}
,
e
y
M
2
=
{
0
}
.
e
(
u
0
,v
0
)
∈
DOI:10.12677/pm.2022.126102935
n
Ø
ê
Æ
Ç
°
²
M
2
,
(
u
0
,v
0
)
∈
¯
B
r
∩
P
×
P,
K
(
u
0
,v
0
)=
λ
0
A
(
u
0
,v
0
)
.
du
¯
K
(1)
d
u
Ž
f
•
§
(
u
0
,v
0
)=
A
(
u
0
,v
0
)
,
(
Ü
(2),
é
,
‡
λ
0
∈
[0
,
1]
,
u
0
(
t
)
≤
T
X
s
=1
G
(
t,s
)
f
(
s,u
0
(
s
)
,v
0
(
s
))
, v
0
(
t
)
≤
T
X
s
=1
G
(
t,s
)
g
(
s,u
0
(
s
)
,v
0
(
s
))
,
d
(H3)
•
,
u
0
(
t
)
≤
T
X
s
=1
G
(
t,s
)(
a
1
u
0
+
b
1
v
0
)
, v
0
(
t
)
≤
T
X
s
=1
G
(
t,s
)(
c
1
u
0
+
d
1
v
0
)
,
l
(
u
0
,v
0
)
≤
T
1
(
u
0
,v
0
)
.
(11)
d
í
Ø
1
Ú
(10)
Œ
•
,
u
0
=
v
0
= 0
,
M
2
=
{
0
}
.
d
Ú
n
2
Œ
•
,
é
u
?
¿
(
u,v
)
∈
B
r
∩
P
×
P,λ
∈
[0
,
1]
,
(
u,v
)
6
=
λA
(
u,v
)
.
K
i
(
A,B
r
∩
P
×
P,P
×
P
) = 1
.
(12)
(
Ü
(10),(12)
,
i
(
A,
(
B
R
/B
r
)
∩
P
×
P,P
×
P
) =
−
1
.
K
A
3
(
B
R
/B
r
)
∩
P
×
P
–
k
˜
‡
Ø
Ä:
.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
“
c
Ä
7
]
Ï
‘
8
(11801453,11901464),
[
‹
Ž
“
c
‰
E
Ä
7
O
y
]
Ï
(20JR10RA100).
ë
•
©
z
[1]Chen,T.L.,Ma,R.Y.andLiang,Y.W.(2019)MultiplePositiveSolutionsofSecond-Order
NonlinearDifferenceEquationswithDiscreteSingular
φ
-Laplacian.
JournalofDifferenceE-
quationsandApplications
,
25
,38-55.https://doi.org/10.1080/10236198.2018.1554064
[2]Gao,C.H. andMa,R.Y.(2016) EigenvaluesofDiscrete Sturm-Liouville Problemswith Eigen-
parameterDependentBoundaryConditions.
LinearAlgebraandItsApplications
,
503
,100-
119.https://doi.org/10.1016/j.laa.2016.03.043
[3]Ma,R.Y.andAn,Y.L.(2009)GlobalStructureofPositiveSolutionsforSuperlinearSecond
Order
m
-PointBoundaryValueProblems.
TopologicalMethodsinNonlinearAnalysis
,
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