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PureMathematicsnØêÆ,2021,11(6),1211-1220
PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.116134
˜aš‚5‡©XÚ)•3•˜5

Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2021c513F¶¹^Fϵ2021c615F¶uÙFϵ2021c622F
Á‡
©|^Leray-SchauderöJÚBanachØ N”nïÄ‡©XÚ















−u
00
= f(t,u(t),v(t)),t∈(0,1),
−v
00
= g(t,u(t),v(t)),
u(0) = u
0
(1) = 0,
v(0) = v
0
(1) = 0
)•3•˜5,Ù¥f,g: [0,1]×[0,+∞)×[0,+∞) →[0,+∞) ëY.
'…c
Lipschitz^‡§Ø N”n§)§Leray-SchauderöJ
ExistenceandUniquenessofPositive
SolutionsforaClassofSecond-Order
NonlinearDifferentialSystems
YangYang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:.˜aš‚5‡©XÚ)•3•˜5[J].nØêÆ,2021,11(6):1211-1220.
DOI:10.12677/pm.2021.116134

Received:May13
th
,2021;accepted:Jun.15
th
,2021;published:Jun.22
nd
,2021
Abstract
Inthispaper,byusingLeray-Schauder’salternativeandcontractionmappingprinciple
tostudythepositivesolutionsforasystemofsecond-orderboundaryvalueproblems















−u
00
= f(t,u(t),v(t)),t∈(0,1),
−v
00
= g(t,u(t),v(t)),
u(0) = u
0
(1) = 0,
v(0) = v
0
(1) = 0,
wheref,g: [0,1]×[0,+∞)×[0,+∞) →[0,+∞) arecontinuous.
Keywords
LipschitzCondition,ContractionMappingPrinciple,PositiveSolution,
LeraySchauder’sAlternative
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
š‚5~‡©•§>Š¯K)•35¯K, ´CA›c5;[ Æö¤ïÄ-‡‘K. c
Ù´~‡©•§>Š¯K, §3ÔnÆ,)ÔÆ+•kX2•A^.ÏL©z·‚•,
éuš‚5~‡©•§|ïăé[1–4].3©z[5]¥,êX|^IþØÄ:nØïÄ
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x
00
(t)+λf
1
(t,x(t),y(t)) = 0,t∈(0,1),
y
00
(t)+λf
2
(t,x(t),y(t)) = 0,t∈(0,1),
u(0) = u(1) = 0,
v(0) = v(1) = 0
DOI:10.12677/pm.2021.1161341212nØêÆ
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õ)•35¯K,Ù¥λ>0 •ëê,f
1
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2
: [0,1]×R
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ëY.
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(t,u(t),v(t)),t∈(0,1),
−v
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2
(t,u(t),v(t)),t∈(0,1),
u(0) = u(1) = 0,
v(0) = v(1) = 0
n‡)•35,Ù¥f
1
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2
: [0,1]×[0,+∞)×[0,+∞) →[0,+∞) ëY.
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−u
00
= f(x,v),t∈(0,1),
−v
00
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u(0) = u(1) = 0,
v(0) = v(1) = 0
)•35Ú)°(‡ê,Ù¥g,f: [0,1]×[0,+∞)×[0,+∞) →[0,+∞)ëY.
3©z[8]¥,“<|^ØÄ:•êïÄ[‚5‡©XÚ
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−((u
0
)
p−1
)
0
= f(t,u(t),v(t)),t∈(0,1),
−((v
0
)
q −1
)
0
= g(t,u(t),v(t)),t∈(0,1),
u(0) = u
0
(1) = 0,
v(0) = v
0
(1) = 0
)•35,Ù¥f,g: [0,1]×[0,+∞)×[0,+∞) →[0,+∞)ëY.
Éþã©zéu,©Äk|^Leray-Schauder öJy²¯K




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
−u
00
= f(t,u(t),v(t)),t∈(0,1),
−v
00
= g(t,u(t),v(t)),t∈(0,1),
u(0) = u
0
(1) = 0,
v(0) = v
0
(1) = 0
(1.1)
)•35,2ÏLBanachØ N”n¯K(1.1))•˜5.
DOI:10.12677/pm.2021.1161341213nØêÆ

2.ý•£
©̇óä´Xe½n:
Ún2.1.[9](BanachØ N”n) (x,ρ) ´Ýþ˜m, T:X→X•Ø N, @
oNT3XSk…•k˜‡ØÄ:.
Ún2.2.[9](Arzela-Ascoli½n) 8ÜM⊂C(J,R
1
)ƒé;¿©7‡^‡´µ
(i)8ÜM¥¼ê˜—k.,=•3~êK>0,¦阃u=u(t)∈MÑk|u(t)|≤K,
∀t∈J;
(ii)8ÜM¥¼êÝëY,=é?‰ε>0, •3δ=δ(ε), ¦t
1
∈J,t
2
∈J,|t
1
−t
2
|<δ
ž,é?‰u= u(t) ∈M, Ñk|u(t
1
)−u(t
2
)|<ε.
Ún2.3.[10](Leray-Schauder öJ)F:E→E´˜‡ëYŽf.Kf8e(F)‡oÃ.,‡
oF–k˜‡ØÄ:,Ù¥e(F) = {x∈E: x= ΘF(x),0 <Θ <1}.
3.̇(J9Ùy²
©óŠ˜m´¢Banach ˜mE=C[0,1],‰ê•||u||=max{|u(t)|:t∈[0,1]},PI
P= {u∈C[0,1] : u(t) ≥0,t∈[0,1]},KP⊂E.
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2
,
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2
= E×E´½Â3þã‰êe¢Banach˜m…P
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2
.
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−u
00
= 0,
u(0) = u
0
(1) = 0
Green ¼ê•
G(t,s) =



s,0 ≤s≤t≤1,
t,0 ≤t≤s≤1,
Ù¥G(t,s) ≥0,…´•∀t,s∈[0,1], kG(t,s) ≤G(s,s) ¤á.
‡©XÚ(1.1) k)…=È©•§|







u(t) =
R
1
0
G(t,s)f(s,u(s),v(s))ds,
v(t) =
R
1
0
G(t,s)g(s,u(s),v(s))ds
(3.1)
k).
DOI:10.12677/pm.2021.1161341214nØêÆ

Xe½ÂŽfA
1
,A
2
,Aµ
A
1
(u,v)(t) =
Z
1
0
G(t,s)f(s,u(s),v(s))ds,
A
2
(u,v)(t) =
Z
1
0
G(t,s)g(s,u(s),v(s))ds,
A(u,v)(t) = (A
1
(u,v)(t),A
2
(u,v)(t)).
(3.2)
KA
1
,A
2
: P
2
→PÚA: P
2
→P
2
.w,‡©XÚ(1.1) Œ)5duŽf•§AkØÄ:.
©̇(J´
½n3.1.f,g:[0,1]×[0,+∞)×[0,+∞)→[0,+∞) ëY, …•3~êα
i
,γ
i
>0(i=1,2)
…α
0
,γ
0
>0,∀u
i
,v
i
∈[0,∞),i= 1,2¦
|f(t,u
1
,v
2
)|≤α
0
+α
1
|u
1
|+α
2
|u
2
|,
|g(t,u
1
,v
2
)|≤γ
0
+γ
1
|u
1
|+γ
2
|u
2
|.
K
max{ω
1
,ω
2
}<1,
Ù¥ω
1
=
1
2
(α
1
+γ
1
),ω
2
=
1
2
(α
2
+γ
2
),ω
0
=
1
2
(α
0
+γ
0
).XÚ(1.1) –k˜‡).
y²ÄkyŽfA´P
2
→P
2
ëYŽf.duf,g´ëY¼ê,lA
1
,A
2
´ëYŽf,K
A´ëYŽf.Ω ´P×P¥k.8, K∃M
3
>0, ¦é∀(u,v) ∈Ω,|f|≤M
3
.
||A
1
(u,v)(t)||= |
Z
1
0
G(t,s)f(s,u(s),v(s))ds|
≤
Z
1
0
G(s,s)M
3
ds
≤
M
3
2
.
A
1
(Ω)˜—k.. ÓnŒ
||A
2
(u,v)(t)||=



Z
1
0
G(t,s)g(s,u(s),v(s))ds



≤
Z
1
0
G(s,s)N
3
ds
≤
N
3
2
,
Ù¥N
3
>0, |g|≤N
3
.A
2
(Ω)˜—k..
DOI:10.12677/pm.2021.1161341215nØêÆ

ddŒ•∃M
3
,N
3
>0,¦
||A(u,v)(t)||= ||(A
1
(u,v)(t)||+||A
2
(u,v)(t))||≤
M
3
2
+
N
3
2
,
A(Ω) ˜—k..
Ï•Green ¼ê30≤t,s≤1 þëY, d4«mþëY¼ê˜—ëY5Œ•,Green ¼ê3
«m[0,1] þ˜—ëY,=é∀ε>0,∃δ¦é∀t
1
,t
2
∈[0,1], …|t
1
−t
1
|<δ,∀s∈[0,1],k
|G(t
2
,s)−G(t
1
,s)|<
ε
M
3
,
é∀u,v∈Ω, |t
1
−t
1
|<δ
1
ž,k
||A
1
(u,v)(t
2
)−A
1
(u,v)(t
1
)||=



Z
1
0
G(t
2
,s)f(s,u(s),v(s))ds−
Z
1
0
G(t
1
,s)f(s,u(s),v(s))ds



≤
Z
1
0
M
3



G(t
2
,s)−G(t
1
,s)



ds
≤ε,
A
1
(Ω)ÝëY.
Ón∀u,v∈Ω ,|t
1
−t
1
|<δ
2
ž,k
||A
2
(u,v)(t
2
)−A
2
(u,v)(t
1
)||=



Z
1
0
G(t
2
,s)g(s,u(s),v(s))ds−
Z
1
0
G(t
1
,s)g(s,u(s),v(s))ds



≤
Z
1
0
N
3



G(t
2
,s)−G(t
1
,s)



ds
≤ε,
A
2
(Ω)ÝëY.
é∀u,v∈Ω, δ= min(δ
1
,δ
2
),|t
1
−t
1
|<δ,k
||A(u,v)(t
2
)−A(u,v)(t
1
)||=



(A
1
(u,v)(t
2
),A
2
(u,v)(t
2
))−(A
1
(u,v)(t
1
),A
2
(u,v)(t
1
)



=



(A
1
(u,v)(t
2
)−A
1
(u,v)(t
1
))



+



(A
2
(u,v)(t
2
)−A
2
(u,v)(t
1
))



≤2ε,
A(Ω) ÝëY.ÏddArzela-Ascoli’s½n•A´ëYŽf.
½ÂΘ(A)={(u,v)∈P×P:(u,v)=ϑA(u,v);0≤ϑ≤1}.é(u,v)∈ΘA.k(u,v)=
ϑA(u,v).é∀t∈[0,1], ku(t) = ϑA
1
(u,v)(t),v(t) = ϑA
2
(u,v)(t)
DOI:10.12677/pm.2021.1161341216nØêÆ

|u(t)|= |ϑA
1
(u,v)(t)|
≤|A
1
(u,v)(t)|
≤|
Z
1
0
G(s,s)f(s,u(s),v(s))ds|
≤
1
2
(α
0
+α
1
|u
1
|+α
2
|u
2
|),
Šâ‰ê½Âk
||u(t)||≤
1
2
(α
0
+α
1
||u
1
||+α
2
||u
2
||).
Ó·‚k
||v(t)||≤
1
2
(γ
0
+γ
1
||u
1
||+γ
2
||u
2
||).
d‰ê||(u,v)||= ||u||+||v||, •
||u||+||v||=
1
2
(α
0
+α
1
||u
1
||+α
2
||u
2
||)+
1
2
(γ
0
+γ
1
||u
1
||+γ
2
||u
2
||)
≤
1
2
(α
0
+γ
0
)+max{
1
2
(α
1
+γ
1
),
1
2
(α
2
+γ
2
)}(||u||+||v||)
Ké∀t∈[0,1]
||(u,v)||≤
ω
0
1−max{ω
1
,ω
2
}
l8ÜΘ(A) k..dÚn(3.2)•ŽfA–k˜‡ØÄ:, =¯K(1.1)k).
½n3.2f,g:[0,1] ×[0,+∞) ×[0,+∞)→[0,+∞) ëY,¿…•3K
i
,L
i
>0, é∀t∈
[0,1],u
i
,v
i
∈[0,∞),i= 1,2, ÷vLipschitz ^‡
|f(t,u
1
,v
1
)−f(t,u
2
,v
2
) ≤K
1
|u
1
−u
2
|+L
1
|v
1
−v
2
|,
|g(t,u
1
,v
1
)−g(t,u
2
,v
2
) ≤K
2
|u
1
−u
2
|+L
2
|v
1
−v
2
|.
XJ
M
2
+
N
2
<1,
(3.3)
Ù¥M= max{K
1
,L
1
},N= max{K
2
,L
2
}.KXÚ(1.1) 3B
r
þk•˜).
y²PM
1
=sup
t∈[0,1]
|f(t,0,0)|,M
2
=sup
t∈[0,1]
|g(t,0,0)|,B
r
= {(u,v) ∈P×P: ||(u,v)||≤r},
Ù¥
r≥
M
1
2
+
M
2
2
1−
M
2
−
N
2
.
DOI:10.12677/pm.2021.1161341217nØêÆ

Äky²AB
r
⊂B
r
,é?¿(u,v) ∈B
r
,t∈[0,1], k
|f(t,u(t),v(t))|≤|f(t,u(t),v(t))−f(t,0,0)+f(t,0,0)|
≤K
1
|u(t)|+L
1
|v(t)|+M
1
≤M(||u(t)||+||v(t)||)+M
1
≤M||(u,v)||+M
1
≤Mr+M
1
.
(3.4)
ÓnŒy
|g(t,u(t),v(t))|≤Nr+M
2
.
(3.5)
d(3.4) Ú(3.5)
||A
1
(u,v)||=max
t∈[0,1]
|A
1
(u,v)(t)|
≤
Z
1
0
G(s,s)|f(s,u(s),v(s))|ds
≤
Z
1
0
G(s,s)(Mr+M
1
)ds
≤
1
2
(Mr+M
1
).
(3.6)
ÓŒ
||A
2
(u,v)||≤
1
2
(Nr+M
2
).
(3.7)
d(3.6) Ú(3.7)
||A(u,v)||= ||A
1
(u,v)||+||A
2
(u,v)||≤r,
=AB
r
⊂B
r
.
y3y²A´Ø Žf, é?¿(u
1
,v
1
),(u
2
,v
2
) ∈B
r
,t∈[0,1],k
||A
1
(u
2
,v
2
)(t)−A
1
(u
1
,v
1
)(t)||=



Z
1
0
G(t,s)f(s,u
2
(s),v
2
(s))ds−
Z
1
0
G(t,s)f(s,u
1
(s),v
1
(s))ds



≤
Z
1
0
G(s,s)



f(s,u
2
(s),v
2
(s))−f(s,u
1
(s),v
1
(s))



ds
≤
Z
1
0
G(s,s)(K
1
|u
2
(s)−u
1
(s)|+L
1
|v
2
(s)−v
1
(s)|)ds
≤
Z
1
0
G(s,s)(K
1
||u
2
−u
1
||+L
1
||v
2
−v
1
||)ds
≤M
Z
1
0
G(s,s)(||u
2
−u
1
||+||v
2
−v
1
||)ds
=
M
2
(||u
2
−u
1
||+||v
2
−v
1
||).
DOI:10.12677/pm.2021.1161341218nØêÆ

Ón
||A
2
(u
2
,v
2
(t))−A
2
(u
1
,v
1
(t))||=



Z
1
0
G(t,s)g(s,u
2
(s),v
2
(s))ds−
Z
1
0
G(t,s)g(s,u
1
(s),v
1
(s))ds



≤
Z
1
0
G(s,s)



g(s,u
2
(s),v
2
(s))−g(s,u
1
(s),v
1
(s))



ds
≤
Z
1
0
G(s,s)(K
2
|u
2
(s)−u
1
(s)|+L
2
|v
2
(s)−v
1
(s)|ds
≤
Z
1
0
G(s,s)(K
2
||u
2
−u
1
||+L
2
||v
2
−v
1
||)ds
≤N
Z
1
0
G(s,s)(||u
2
−u
1
||+||v
2
−v
1
||)ds
=
N
2
(||u
2
−u
1
||+||v
2
−v
1
||).
Ïd
||A(u
2
,v
2
)(t)−A(u
1
,v
1
)(t)||=



(A
1
(u
2
,v
2
)(t),A
2
(u
2
,v
2
)(t))−(A
1
(u
1
,v
1
)(t),A
2
(u
1
,v
1
)(t))



=



(A
1
(u
2
,v
2
)(t)−A
1
(u
1
,v
1
)(t)),(A
2
(u
2
,v
2
)(t)−A
2
(u
1
,v
1
)(t))



≤(
M
2
+
N
2
)(||u
2
−u
1
||+||v
2
−v
1
||).
d(3.3) •,A(u,v)(t)•B
r
→B
r
Ø Žf. ÏddBanachØ N”n•Žf•§(3.2) k•
˜ØÄ:(u,v), =¯K(1.1)k•˜)(u,v),¦A(u,v) = (u,v).
Ä7‘8
I[g,䮀7(11561063).
ë•©z
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DifferentialEquations.Monatsheftef¨urMathematik,195,107-117.
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