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PureMathematics
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,2021,11(1),115-125
PublishedOnlineJanuary2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.111017
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ExistenceandMultiplicityofPositive
PeriodicSolutionsforSystems
withParameters
WeiYang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
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DOI:10.12677/pm.2021.111017
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Received:Dec.17
th
,2020;accepted:Jan.19
th
,2021;published:Jan.26
th
,2021
Abstract
Inthispaper, weconsidersystemswithparameters
u
0
(
t
)+
a
(
t
)
u
(
t
) =
λf
(
t,u
(
t
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,v
(
t
))
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1
,
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0
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t
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(
t
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,v
(
t
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0
≤
t
≤
1
,
u
(0) =
u
(1)
,v
(0) =
v
(1)
,
where
λ
isapositiveparameter,
a,b
:[0
,
1]
→
[0
,
∞
)
arecontinuousfunctionanddo
notvanishidenticallyonanysubintervalof
[0
,
1]
,f,g
: [0
,
1]
×
[0
,
∞
)
×
[0
,
∞
)
→
[0
,
∞
)
are
continuousfunction.Inthispaper,basedontheKrasnoselskillfixedpointtheorem,
aninfinitenumberofpositiveperiodicsolutionsforsystemswithparameters.
Keywords
PositiveSolutions,MultipleSolutions,Parametric
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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ª
ü
à
Ó
ž
¦
±
e
R
t
0
a
(
θ
)
dθ
§
n
Œ
(
e
R
t
0
a
(
θ
)
dθ
u
(
t
))
0
=
λe
R
t
0
a
(
θ
)
dθ
f
(
t,z
(
t
))
,
(2
.
4)
ò
(2.4)
ª
'
u
0
t
È
©
Œ
u
(0) =
u
(
t
)
e
R
t
0
a
(
θ
)
dθ
−
λ
Z
t
0
f
(
s,z
(
s
))
e
R
s
0
a
(
θ
)
dθ
ds,
ò
(2.4)
ª
'
u
t
1
È
©
Œ
u
(1) =
u
(
t
)
e
R
t
0
a
(
θ
)
dθ
+
λ
R
1
t
f
(
s,z
(
s
))
e
R
s
0
a
(
θ
)
dθ
ds
e
R
1
0
a
(
θ
)
dθ
,
DOI:10.12677/pm.2021.111017119
n
Ø
ê
Æ
•
d
u
(0) =
u
(1)
Œ
u
(
t
) =
λ
Z
t
0
e
R
s
t
a
(
θ
)
dθ
1
−
e
−
R
1
0
a
(
θ
)
dθ
f
(
s,z
(
s
))
ds
+
λ
Z
1
t
e
R
s
t
a
(
θ
)
dθ
e
R
1
0
a
(
θ
)
dθ
−
1
f
(
s,z
(
s
))
ds,
(2.5)
l
G
u
(
t,s
) =
e
R
s
t
a
(
θ
)
dθ
1
−
e
−
R
1
0
a
(
θ
)
dθ
,
0
≤
s
≤
t
≤
1
,
e
R
s
t
a
(
θ
)
dθ
e
R
1
0
a
(
θ
)
dθ
−
1
,
0
≤
t
≤
s
≤
1
.
Ó
n
Œ
v
(
t
) =
λ
Z
t
0
e
R
s
t
b
(
θ
)
dθ
1
−
e
−
R
1
0
b
(
θ
)
dθ
g
(
s,z
(
s
))
ds
+
λ
Z
1
t
e
R
s
t
b
(
θ
)
dθ
e
R
1
0
b
(
θ
)
dθ
−
1
g
(
s,z
(
s
))
ds,
(2.6)
G
v
(
t,s
) =
e
R
s
t
b
(
θ
)
dθ
1
−
e
−
R
1
0
b
(
θ
)
dθ
,
0
≤
s
≤
t
≤
1
,
e
R
s
t
b
(
θ
)
dθ
e
R
1
0
b
(
θ
)
dθ
−
1
,
0
≤
t
≤
s
≤
1
.
´
„
σ
u
1
−
σ
u
≤
G
u
(
t,s
)
≤
1
1
−
σ
u
,
Ú
σ
v
1
−
σ
v
≤
G
v
(
t,s
)
≤
1
1
−
σ
v
,
2
(
Ü
σ
= min
{
σ
u
,σ
v
}
Œ
σ
1
−
σ
≤
G
u
(
t,s
)
≤
1
1
−
σ
,
(2
.
7)
Ú
σ
1
−
σ
≤
G
v
(
t,s
)
≤
1
1
−
σ
.
(2
.
8)
½
Â
Ž
f
T
λ
:
K
→
X
,
K
T
λ
z
(
t
) := (
T
u
λ
z
(
t
)
,T
v
λ
z
(
t
))
,
(2
.
9)
Ù
¥
T
u
λ
z
(
t
) :=
λ
Z
1
0
G
u
(
t,s
)
f
(
s,z
(
s
))
ds,
(2
.
10)
T
v
λ
z
(
t
) :=
λ
Z
1
0
G
v
(
t,s
)
g
(
s,z
(
s
))
ds.
(2
.
11)
Ú
n
2.3
b
(H1)-(H2)
^
‡
¤
á
§
K
T
λ
(
K
)
⊂
K
"
DOI:10.12677/pm.2021.111017120
n
Ø
ê
Æ
•
y
²
b
z
= (
u,v
)
∈
K
§
(
Ü
min
0
6
t
6
1
{|
u
|
+
|
v
|}≥
σ
k
z
k
§
K
T
u
λ
z
(
t
)
≥
λ
σ
u
1
−
σ
u
Z
1
0
f
(
s,z
(
s
))
ds
≥
λσ
u
1
1
−
σ
u
Z
1
0
f
(
s,z
(
s
))
ds
≥
σ
u
sup
t
∈
[0
,
1]
|
T
u
λ
z
(
t
)
|
≥
σ
u
k
u
k
∞
,
(2.12)
Ó
n
Œ
T
v
λ
z
(
t
)
≥
σ
v
k
v
k
∞
,
(2.13)
l
min
0
6
t
6
1
T
λ
z
(
t
)
≥
σ
k
z
k
.
(2
.
14)
=
T
λ
(
K
)
⊂
K.
Ú
n
2.4
b
(H1)-(H2)
^
‡
¤
á
§
K
Ž
f
T
λ
:
K
→
K
´
˜
‡
;
Ž
f
"
y
²
Ä
k
•
Ä
(2.10)
ª
"
S
⊂
C
[0
,
1]
´
˜
‡
k
.
8
§
@
o
•
3
~
ê
B>
0
§
¦
é
?
¿
z
∈
S,
k
z
k≤
B.
2
(
Ü
f
ë
Y5
§
f
3
[0
,
1]
×
[0
,B
]
´
˜
—
ë
Y
"
l
§
•
3
A>
0
§
¦
|
F
(
t,z
(
t
))
|≤
A,z
∈
[0
,B
]
§
K
|
T
u
λ
z
(
t
)
|
=
|
λ
Z
1
0
G
u
(
t,s
)
f
(
s,z
(
s
))
ds
|
≤
λ
Z
1
0
|
G
u
(
t,s
)
f
(
s,z
(
s
))
|
ds
≤
λ
1
−
σ
Z
1
0
|
f
(
s,z
(
s
))
|
ds
≤
λ
1
−
σ
A,
(2.15)
Ï
d
§
Ž
f
T
u
λ
:
K
→
K
´
˜
—
k
.
"
DOI:10.12677/pm.2021.111017121
n
Ø
ê
Æ
•
é
?
¿
t
1
,t
2
∈
[0
,
1](
t
1
<t
2
)
,z
∈
S
§
¿
…
•
3
ξ
∈
[0
,
1]
§
K
|
T
u
λ
z
(
t
1
)
−
T
u
λ
z
(
t
2
)
|
=
λ
|
Z
1
0
G
u
(
t
1
,s
)
f
(
s,z
(
s
))
ds
−
Z
1
0
G
u
(
t
2
,s
)
f
(
s,z
(
s
))
ds
|
≤
λA
|
Z
1
0
G
u
(
t
1
,s
)
ds
−
Z
1
0
G
u
(
t
2
,s
)
ds
|
≤
λA
|
G
0
u
(
ξ,s
)
||
t
1
−
t
2
|
≤
λA
|
1+
a
(
ξ
)
u
(
ξ
)
||
t
1
−
t
2
|
≤
D
|
t
1
−
t
2
|
,
(2.16)
Ù
¥
§
D
:=max
0
6
t
6
1
{
λA
|
1+
a
(
t
)
u
(
t
)
|}
.
l
§
é
?
¿
ε>
0
§
δ
=
1
D
§
@
o
é
?
¿
t
1
,t
2
∈
[0
,
1]
§
e
|
t
1
−
t
2
|
<δ
§
é
?
¿
z
∈
C
[0
,
1]
§
Ñ
k
|
T
u
λ
z
(
t
1
)
−
T
u
λ
z
(
t
2
)
|
<ε.
(2.17)
Ï
d
§
Ž
f
T
u
λ
:
K
→
K
´
Ý
ë
Y
"
Š
â
Arzela-Ascoli
½
n
[15]
§
Ž
f
T
u
λ
:
K
→
K
´
;
"
Ó
n
Œ
§
Ž
f
T
v
λ
:
K
→
K
•
´
;
"
n
þ
¤
ã
§
Ž
f
T
λ
:
K
→
K
´
;
"
3.
Ì
‡
(
J
y
²
½
n
1.1
y
²
Ä
k
•
Ä
m
8
K
¥
ü
‡
ê
{
Ω
1
,k
}
m
k
=1
Ú
{
Ω
2
,k
}
m
k
=1
,
Ù
½
Â
X
e
{
Ω
1
,k
}
=
{
z
∈
K
:
k
z
k
<r
k
}
,k
= 1
,
2
,...,m,
Ú
{
Ω
2
,k
}
=
{
z
∈
K
:
k
z
k
<R
k
}
,k
= 1
,
2
,...,m.
é
z
˜
‡
½
~
ê
k
§
z
∈
K
∩
∂
Ω
1
,k
ž
§
k
σr
k
=
σ
k
r
k
k≤
min
t
∈
[0
,
1]
|
z
(
t
)
|≤
z
(
s
)
≤k
z
k
=
r
k
,
DOI:10.12677/pm.2021.111017122
n
Ø
ê
Æ
•
é
?
¿
s
∈
[0
,
1]
,
(
Ü
^
‡
(
C
1
)
§
k
k
T
λ
z
k
=
k
T
u
λ
z
k
+
k
T
v
λ
z
k
=
λ
Z
1
0
G
u
(
t,s
)
f
(
s,z
(
s
))
ds
+
λ
Z
1
0
G
v
(
t,s
)
G
(
s,z
(
s
))
ds
≥
λ
σ
1
−
σ
Z
1
0
(
f
(
t,z
(
t
))+
g
(
t,z
(
t
)))
ds
≥
λ
σ
1
−
σ
2min
{
f
(
t,z
(
t
))
,g
(
t,z
(
t
))
}
≥
Br
k
λ
2
σ
1
−
σ
≥
r
k
=
k
z
k
.
(3.1)
=
^
‡
(
C
1
)
÷
v
ž
§
k
k
T
λ
z
k≥k
z
k
.
(3
.
2)
,
˜
•
¡
§
z
∈
K
∩
∂
Ω
2
,k
ž
§
k
z
(
s
)
≤k
z
k
,
(3
.
3)
é
?
¿
s
∈
[0
,
1]
§
(
Ü
^
‡
(
C
2
)
§
k
k
T
λ
z
k
=
k
T
u
λ
z
k
+
k
T
v
λ
z
k
≤
λ
1
1
−
σ
Z
1
0
(
f
(
s,z
(
s
))+
g
(
s,z
(
s
))
ds
≤
λ
1
1
−
σ
2max
{
f
(
s,z
(
s
))
,g
(
s,z
(
s
))
}
≤
AR
k
λ
2
1
−
σ
≤
R
k
=
k
z
k
.
(3.4)
=
^
‡
(
C
2
)
÷
v
ž
§
k
k
T
λ
z
k≤k
z
k
.
(3
.
5)
Š
â
Ú
n
2.1
^
‡
(i)
§
Œ
Ž
f
T
λ
k
m
‡
Ø
Ä:
"
Ø
Ä:
^
α
k
5
L
«
§
K
Ù
÷
v
r
k
≤k
α
k
k≤
R
k
,k
= 1
,
2
,...,m.
Ù
g
§
•
Ä
m
8
K
¥
ê
{
Ω
3
,k
}
m
k
=1
§
Ù
½
Â
X
e
{
Ω
3
,k
}
=
{
z
∈
K
:
k
z
k
<R
k
+1
}
,k
= 1
,
2
,...,m
−
1
.
DOI:10.12677/pm.2021.111017123
n
Ø
ê
Æ
•
é
z
˜
‡
½
~
ê
k
§
z
∈
K
∩
∂
Ω
3
,k
ž
§
k
z
(
s
)
≤k
z
k
,
é
?
¿
s
∈
[0
,
1]
,
(
Ü
^
‡
(
C
2
)
§
k
k
T
λ
z
k
=
k
T
u
λ
z
k
+
k
T
v
λ
z
k
≤
λ
1
1
−
σ
Z
1
0
(
f
(
s,z
(
s
))+
G
(
s,z
(
s
)))
ds
≤
λ
1
1
−
σ
2max
{
f
(
s,z
(
s
))
,g
(
s,z
(
s
))
}
≤
AR
k
+1
λ
2
1
−
σ
≤
R
k
+1
=
k
z
k
.
(3.6)
=
^
‡
(
C
2
)
÷
v
ž
§
k
k
T
λ
z
k≤k
z
k
.
,
˜
•
¡
§
(
Ü
(3.2)
ª
9
Ú
n
2.1
^
‡
(ii)
§
Œ
Ž
f
T
λ
k
m
−
1
‡
Ø
Ä:
"
Ø
Ä:
^
β
k
5
L
«
§
K
Ù
÷
v
R
k
+1
≤k
β
k
k≤
r
k
,k
= 1
,
2
,...,m
−
1
.
n
þ
Œ
§
¯
K
(1.5)
k
2
m
−
1
‡
)
"
(
Ü
m
∈
N
∪{
+
∞}
§
K
¯
K
(1.5)
k
Ã
¡
õ
‡
)
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7
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8
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[1]Joseph,W., So, H.and Yu, J.S. (1994)Global Attractivity andUniformPersistence in Nichol-
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[2]Zhang,G.andCheng,S.S.(2002)PositivePeriodicSolutionofNon-AutonomousFunctional
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[9]Torres,P.J.(2003)ExistenceofOne-SignedPeriodicSolutionsofSomeSecond-OrderDiffer-
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