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PureMathematicsnØêÆ,2021,11(1),115-125
PublishedOnlineJanuary2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.111017
¹ëXÚ±Ï)•359õ)5
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

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
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




u
0
(t)+a(t)u(t) = λf(t,u(t),v(t)),0 ≤t≤1,
v
0
(t)+b(t)v(t) = λg(t,u(t),v(t)),0 ≤t≤1,
u(0) = u(1),v(0) = v(1),
Ù¥§λ´˜‡ëê§a,b: [0,1] →[0,∞) ´ëY¼ê…a,b3[0,1] ?¿f«mþØð•"§
f,g:[0,1]×[0,∞) ×[0,∞)→[0,∞)´ëY¼ê"©ÄuKrasnoselskill ØÄ:½n
˜~‡©XÚáõ‡±Ï)"
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ExistenceandMultiplicityofPositive
PeriodicSolutionsforSystems
withParameters
WeiYang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:•.¹ëXÚ±Ï)•359õ)5[J].nØêÆ,2021,11(1):115-125.
DOI:10.12677/pm.2021.111017
•
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),v(t)),0 ≤t≤1,
v
0
(t)+b(t)v(t) = λg(t,u(t),v(t)),0 ≤t≤1,
u(0) = u(1),v(0) = v(1),
whereλisapositiveparameter,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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DOI:10.12677/pm.2021.111017116nØêÆ
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v
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DOI:10.12677/pm.2021.111017117nØêÆ
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DOI:10.12677/pm.2021.111017118nØêÆ
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R
s
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DOI:10.12677/pm.2021.111017119nØêÆ
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DOI:10.12677/pm.2021.111017120nØêÆ
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DOI:10.12677/pm.2021.111017121nØêÆ
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λ
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Z
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2
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1
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1
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1
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2
∈[0,1]§e|t
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λ
z(t
1
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u
λ
z(t
2
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Ïd§ŽfT
u
λ
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u
λ
: K→K´;
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λ
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3.̇(Jy²
½n1.1y²Äk•Äm8K¥ü‡ê{Ω
1,k
}
m
k=1
Ú{Ω
2,k
}
m
k=1
,Ù½ÂXe
{Ω
1,k
}= {z∈K: kzk<r
k
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Ú
{Ω
2,k
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k
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éz˜‡½~êk§z∈K∩∂Ω
1,k
ž§k
σr
k
= σkr
k
k≤min
t∈[0,1]
|z(t)|≤z(s) ≤kzk= r
k
,
DOI:10.12677/pm.2021.111017122nØêÆ
•
é?¿s∈[0,1],(Ü^‡(C
1
)§k
kT
λ
zk= kT
u
λ
zk+kT
v
λ
zk
= 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
2σ
1−σ
≥r
k
= kzk.(3.1)
=^‡(C
1
)÷vž§k
kT
λ
zk≥kzk.(3.2)
,˜•¡§z∈K∩∂Ω
2,k
ž§k
z(s) ≤kzk,(3.3)
é?¿s∈[0,1]§(Ü^‡(C
2
)§k
kT
λ
zk= kT
u
λ
zk+kT
v
λ
zk
≤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
= kzk.(3.4)
=^‡(C
2
)÷vž§k
kT
λ
zk≤kzk.(3.5)
ŠâÚn2.1^‡(i)§ŒŽfT
λ
km‡ØÄ:"ØÄ:^α
k
5L«§KÙ÷v
r
k
≤kα
k
k≤R
k
,k= 1,2,...,m.
Ùg§•Äm8K¥ê{Ω
3,k
}
m
k=1
§Ù½ÂXe
{Ω
3,k
}= {z∈K: kzk<R
k+1
},k= 1,2,...,m−1.
DOI:10.12677/pm.2021.111017123nØêÆ
•
éz˜‡½~êk§z∈K∩∂Ω
3,k
ž§k
z(s) ≤kzk,
é?¿s∈[0,1],(Ü^‡(C
2
)§k
kT
λ
zk= kT
u
λ
zk+kT
v
λ
zk
≤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
= kzk.(3.6)
=^‡(C
2
)÷vž§k
kT
λ
zk≤kzk.
,˜•¡§(Ü(3.2) ª9Ún2.1 ^‡(ii)§ŒŽfT
λ
km−1 ‡ØÄ:"ØÄ:^
β
k
5L«§KÙ÷v
R
k+1
≤kβ
k
k≤r
k
,k= 1,2,...,m−1.
nþŒ§¯K(1.5)k2m−1‡)"(Üm∈N∪{+∞}§K¯K(1.5)káõ‡)"
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11671322§11361054)"
ë•©z
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