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PureMathematicsnØêÆ,2022,12(4),665-674
PublishedOnlineApril2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.124076
˜a¹ëþ[‚5‡©XÚ)•3•˜5

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Á‡
©|^ØÄ:½nïÄ˜a¹kü‡ëê[‚5‡©XÚ













−((u
0
)
p−1
)
0
= λf(t,u(t),v(t)),t∈(0,1),
−((v
0
)
q−1
)
0
= µg(t,u(t),v(t)),
u(0) = u
0
(1) = 0,
v(0) = v
0
(1) = 0
)•3•˜5§Ù¥p,q>1,f,g: [0,1]×[0,+∞)×[0,+∞) →[0,+∞)ëY"éu?¿½
λ,µ>0, f,g÷v5½^‡ž§XÚ)•3•˜5"•§Þ~`²(ØŒ15"
'…c
‡©XÚ§I§)§[‚5
ExistenceandUniquenessofPositive
SolutionsforaClassofQuasilinear
DifferentialSystemswithParameters
YangYang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:.˜a¹ëþ[‚5‡©XÚ)•3•˜5[J].nØêÆ,2022,12(4):665-674.
DOI:10.12677/pm.2022.124076

Received:Mar.17
th
,2022;accepted:Apr.20
th
,2022;published:Apr.27
th
,2022
Abstract
Inthispaper,byusinganewfixedpointtheoremtostudyexistenceanduniqueness
ofpositivesolutionsforaclassofquasilineardifferentialsystemswithparameters













−((u
0
)
p−1
)
0
= λf(t,u(t),v(t)),t∈(0,1),
−((v
0
)
q−1
)
0
= µg(t,u(t),v(t)),
u(0) = u
0
(1) = 0,
v(0) = v
0
(1) = 0,
where f,g:[0,1] ×[0,+∞) ×[0,+∞)→[0,+∞)arecontinuous, λ, and µarepositivepa-
rameters,we establish sufficient conditions forthe existence anduniqueness of positive
solutionstothissystemforanyfixedλ,µ>0.Finally,wegiveasimpleexampleto
illustrateourmainresult.
Keywords
DifferentialSystem,Cone,PositiveSolution,Quasilinear
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
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[1–8].3©z[9]¥, “<|^ØÄ:•êïÄÓž•¹p-LaplacianÚ˜ê•§



−((u
0
)
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)
0
= f(t,u,u
0
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0
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õ)•35, ÄuÉÜØªÚÙ¦Øª)•35. Ù¥p>1, f∈C
1
([0,1]×
R
+
×R
+
,R
+
)(R
+
:= [0,∞)),u∈C
2
([0,1],R)∩C
1
([0,1],R),t∈(0,1).
DOI:10.12677/pm.2022.124076666nØêÆ

3©z[10]¥,“|^ØÄ:•êïÄ‡©XÚ


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−((u
0
i
)
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)
0
= f
i
(t,u
1
,...,u
n
),t∈(0,1),
u
i
(0) = u
0
i
(1) = 0
(1.2)
õ)•35,†¯K(1.1)ƒ',(1.2)K´|^ÉÜØªÚšKÝ)•35,
n≥2,p
i
>1,f
i
∈C([0,1]×R
n
+
,R
+
)(i= 1,2,...,n,R
+
:= [0,∞)).
•C3©z[8]¥,“<|^ØÄ:•êïÄ[‚5‡©XÚ













−((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
(1.3)
)•35,†¯K(1.1),(1.2)ØÓ´, 3¯K(1.3) ¥´|^ 'ušK]¼êÚàgŽf
)•35,Ù¥p,q>1,f,g: [0,1]×[0,+∞)×[0,+∞) →[0,+∞)ëY.
¯K(1.1)-(1.3) þ™9ëê, …Ñ^ØÄ:•ênØïć©XÚ)•35, Éþã©
zéu,©|^ØÄ:½nïĹü‡ëê[‚5‡©XÚ



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







−((u
0
)
p−1
)
0
= λf(t,u(t),v(t)),t∈(0,1),
−((v
0
)
q−1
)
0
= µg(t,u(t),v(t)),
u(0) = u
0
(1) = 0,
v(0) = v
0
(1) = 0
(1.4)
)•3•˜5,Ù¥p,q>1,λ,µ>0,f,g: [0,1]×[0,+∞)×[0,+∞) →[0,+∞)ëY.
2.ý•£
©̇½ÂÚÚn:
½Â2.1. [11] (E,||.||) ´¢Banach˜m,P∈E´˜‡I,XJP÷v
(i)∀p∈E, Úλ≥0,Ñkλp∈P;
(ii)e−x∈P,Kx= Θ
E
,Ù¥Θ
E
´Banach˜mE¥"ƒ.
½Â2.2.[11](E,||.||)´¢Banach˜m,P ∈E´˜‡I,∀x,y∈E,y≥xž,K
y−x∈P.
½Â2.3. [12] XJ÷v
∀x,y∈P,Θ
E
≤x≤y,∃N>0,||x||≤N||y||,
K¡P∈E´˜‡5I.
DOI:10.12677/pm.2022.124076667nØêÆ

½Â2.4. [12] é∀x,y∈E,x≤y, kAx≤Ay, K¡A´OŽf.
½Âd'Xx∼y,=•3~êα,β>0,¦αy≤x≤βy.P
h
={x∈E,x∼h}, Ù¥
h>Θ
E
.´
•P
h
⊂P.é?¿h
1
,h
2
∈P,h
1
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2
6= Θ
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2
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h
= P×P,XJP´5I,K
¯
P
h
= (P,P) ´5I.
Φ = {ϕ(r) ∈(0,1) : ϕ(r) >r,r∈(0,1)}.
Ún2.5. [13]
¯
P
h
= {(u,v) : u∈P
h
1
,v∈P
h
2
}= P
h
1
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h
2
.
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1
,h
2
)∈P×P,Ù¥
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1
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2
6= Θ.Žf
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(C
1
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1
,ϕ
2
∈Φ ¦
A(ru,rv) ≥ϕ
1
(r)A(u,v),B(ru,rv) ≥ϕ
2
(r)B(u,v),r∈(0,1),u,v∈P;
(C
2
)•3(c
1
,c
2
) ∈
¯
P
h
,¦A(c
1
,c
2
) ∈P
h
1
,B(c
1
,c
2
) ∈P
h
2
.
K
(a)A: P
h
1
×P
h
2
→P
h
1
,B: P
h
1
×P
h
2
→P
h
2
,…•3u
1
,v
1
∈P
h
1
,u
2
,v
2
∈P
h
2
,r∈(0,1)¦

r(v
1
,v
2
) ≤(u
1
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2
) ≤(v
1
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2
),u
1
≤A((u
1
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2
)) ≤v
1
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2
≤B((u
1
,u
2
)) ≤v
2
;
(b)é?¿½λ,µ,Žf•§(u,v) = (λA(u,v),µB(u,v))k•˜ØÄ:(u
ˆ
∗
λ,µ
,v
ˆ
∗
λ,µ
) ∈
¯
P
h
,,éu
?¿Щ:(u
0
,v
0
) ∈
¯
P
h
,kS
(u
n
,v
n
) = (λA(u
n−1
,v
n−1
),µB(u
n−1
,v
n−1
)),n= 1,2,...
…÷v||u
n
−u
ˆ
∗
λ,µ
||→0,||v
n
−v
ˆ
∗
λ,µ
||→0, n→∞.
3.̇(J9Ùy²
©óŠ˜m´¢Banach˜mE=C[0,1],‰ê•||u||=max{|u(t)|:t∈[0,1]},PI
P= {u∈E: u(t) ≥0,t∈[0,1]}, KP⊂E.…´•5~êN= 1ž, P´5I.
½Â
||(u,v)||= ||u||+||v||,(u,v) ∈E
2
,
Ù¥E
2
= E×E´½Â3þã‰êe¢Banach˜m,…P
2
⊂E
2
.
¯
P
h
= {(u,v) ∈E×E: u(t) ≥0,v(t) ≥0,t∈[0,1]},
DOI:10.12677/pm.2022.124076668nØêÆ

•
¯
P
h
⊂E×E, duP´5I,K
¯
P
h
= P×P´5I. 3E×EþkeS'X,
eu
1
(t) ≤u
2
(t),v
1
(t) ≤v
2
(t),t∈[0,1], K(u
1
,v
1
) ≤(u
2
,v
2
).
‡©XÚ(1.4) k)…=È©•§|



u(t) =
R
t
0
(
R
1
s
λf(τ,u(τ),v(τ))dτ)
p−1
ds,
v(t) =
R
t
0
(
R
1
s
µg(τ,u(τ),v(τ))dτ)
q−1
ds
(3.1)
k).
½ÂXeŽfA
1
,A
2
: P
2
→P, A: P
2
→P
2
A
1
(u,v)(t) = λ
p−1
Z
t
0
(
Z
1
s
f(τ,u(τ),v(τ))dτ)
p−1
ds,
A
2
(u,v)(t) = µ
q−1
Z
t
0
(
Z
1
s
g(τ,u(τ),v(τ))dτ)
q−1
ds,
A(u,v)(t) = (λ
p−1
A
1
(u,v)(t),µ
q−1
A
2
(u,v)(t)).
(3.2)
KA
1
,A
2
: P
2
→PÚA: P
2
→P
2
.w,‡©XÚ(1.4) Œ)5duŽf•§AkØÄ:.
P
h
1
(t) =
Z
t
0
(1−s)
p−1
ds, h
2
(t) =
Z
t
0
(1−s)
q−1
ds,t∈[0,1],
´•h
1
(t),h
2
(t) ≥0,t∈[0,1], Kh
1
,h
2
∈P.
P
l
1
=min
t∈[0,1]
Z
t
0
(1−s)
p−1
ds,l
2
=min
t∈[0,1]
Z
t
0
(1−s)
q−1
ds,
L
1
=max
t∈[0,1]
Z
t
0
(1−s)
p−1
ds,L
2
=max
t∈[0,1]
Z
t
0
(1−s)
q−1
ds.
w,
l
1
≤h
1
(t) ≤L
1
,l
2
≤h
2
(t) ≤L
2
.
©̇(J´
½n3.1.h
1
,h
2
∈P, b
(H
1
)f,g∈C([0,1]×R
+
×R
+
,R
+
),…f(t,l
1
,l
2
) >0,g(t,l
1
,l
2
) >0,t∈[0,1];
(H
2
)f,g'u1Ú1nCþ4O, =é?¿0 ≤u
1
≤u
2
,0 ≤v
1
≤v
2
,t∈[0,1],
f(t,u
1
,v
1
) ≤f(t,u
2
,v
2
),g(t,u
1
,v
1
) ≤g(t,u
2
,v
2
);
(H
3
)•3ϕ
1
,ϕ
2
∈Φ,∀u,v∈R
+
,t∈[0,1],r∈(0,1)¦
f(t,ru,rv) ≥ϕ
p−1
1
f(t,u,v),g(t,ru,rv) ≥ϕ
q−1
2
g(t,u,v).
DOI:10.12677/pm.2022.124076669nØêÆ

K
(a)•3u
1
,v
1
∈P
h
1
,u
2
,v
2
∈P
h
2
,r∈(0,1),¦r(v
1
,v
2
) ≤(u
1
,u
2
) ≤(v
1
,v
2
),…
u
1
≤
Z
t
0
(
Z
1
s
f(τ,u(τ),v(τ))dτ)
p−1
ds≤v
1
,t∈[0,1],
u
2
≤
Z
t
0
(
Z
1
s
g(τ,u(τ),v(τ))dτ)
q−1
ds≤v
2
,t∈[0,1].
(b)é?¿½λ,µ>0,t∈[0,1], XÚ(1.4)k•˜)(u
∗
λ,µ
,v
∗
λ,µ
) ∈
¯
P
h
;
(c)�Щ:(u
0
,v
0
) ∈
¯
P
h
,÷v
u
n+1
=
Z
t
0
(
Z
1
s
λf(τ,u
n
(τ),v
n
(τ))dτ)
p−1
ds,n= 1,2,...
v
n+1
=
Z
t
0
(
Z
1
s
µg(τ,u
n
(τ),v
n
(τ))dτ)
q−1
ds,n= 1,2,...
n→∞,u
n
→u
∗
λ,µ
,v
n
→v
∗
λ,µ
.
y²ÄkyA
1
,A
2
´OŽf.é?¿u
i
,v
i
∈P,i=1,2,u
1
≤u
2
,v
1
≤v
2
,=u
1
(t)≤
u
2
(t),v
1
(t) ≤v
2
(t),d(H
2
)•
A
1
(u
1
,v
1
)(t) =
Z
t
0
(
Z
1
s
f(τ,u
1
(τ),v
1
(τ))dτ)
p−1
≤
Z
t
0
(
Z
1
s
f(τ,u
2
(τ),v
2
(τ))dτ)
p−1
= A
1
(u
2
,v
2
)(t),
(3.3)
A
2
(u
1
,v
1
)(t) =
Z
t
0
(
Z
1
s
g(τ,u
1
(τ),v
1
(τ))dτ)
q−1
≤
Z
t
0
(
Z
1
s
g(τ,u
2
(τ),v
2
(τ))dτ)
q−1
= A
2
(u
2
,v
2
)(t).
(3.4)
d(3.3),(3.4) •A
1
(u
1
,v
1
) ≤A
1
(u
2
,v
2
),A
2
(u
1
,v
1
) ≤A
2
(u
2
,v
2
).
é?¿u,v∈P,r∈(0,1),d(H
3
)Œ
A
1
(ru,rv)(t) =
Z
t
0
(
Z
1
s
f(τ,ru(τ),rv(τ))dτ)
p−1
≥ϕ
1
(r)
Z
t
0
(
Z
1
s
f(τ,u(τ),v(τ))dτ)
p−1
= ϕ
1
A
1
(u,v)(t),
DOI:10.12677/pm.2022.124076670nØêÆ

A
2
(ru,rv)(t) =
Z
t
0
(
Z
1
s
g(τ,ru(τ),rv(τ))dτ)
q−1
≥ϕ
2
(r)
Z
t
0
(
Z
1
s
g(τ,u(τ),v(τ))dτ)
q−1
= ϕ
2
(t)A
2
(u,v)(t),
=∀u,v∈P,r∈(0,1),A
1
(ru,rv)(t) ≥ϕ
1
A
1
(u,v)(t),A
2
(ru,rv)(t) ≥ϕ
2
(t)A
2
(u,v)(t).
2yA
1
(h
1
,h
2
) ∈P
h
1
,A
2
(h
1
,h
2
) ∈P
h
2
,
r
1
=min
t∈[0,1]
{f(t,l
1
,l
2
)},R
1
=max
t∈[0,1]
{f(t,L
1
,L
2
)},
r
2
=min
t∈[0,1]
{g(t,l
1
,l
2
)},R
2
=max
t∈[0,1]
{g(t,L
1
,L
2
)},
d(H
1
),(H
2
)
A
1
(u,v)(t) =
Z
t
0
(
Z
1
s
f(τ,u
1
(τ),v
1
(τ))dτ)
p−1
ds
≥
Z
t
0
(
Z
1
s
f(τ,l
1
,l
2
)dτ)
p−1
ds
= r
p−1
1
Z
t
0
(1−s)
p−1
ds
= r
p−1
1
h
1
,
(3.5)
A
1
(u,v)(t) =
Z
t
0
(
Z
1
s
f(τ,u
1
(τ),v
1
(τ))dτ)
p−1
ds
≤
Z
t
0
(
Z
1
s
f(τ,L
1
,L
2
)dτ)
p−1
ds
= R
p−1
1
Z
t
0
(1−s)
p−1
ds
= R
p−1
1
h
1
,
(3.6)
d(3.5),(3.6) •r
p−1
1
h
1
≤A
1
(u,v)(t) ≤R
p−1
1
h
1
,=A
1
(u,v) ∈P
h
1
.ÓnŒ±A
2
(u,v) ∈P
h
2
.
•dÚn2.6 ŒXe(Ø:
(1)∃u
1
,v
1
∈P
h
1
,u
2
,v
2
∈P
h
2
,r∈(0,1),¦r(v
1
,v
2
) ≤(u
1
,u
2
) ≤(v
1
,v
2
),…
u
1
≤A
1
(u
1
,v
1
) ≤v
1
,u
2
≤A
2
(u
1
,v
1
) ≤v
2
,
u
1
(t) ≤
Z
t
0
(
Z
1
s
f(τ,u(τ),v(τ))dτ)
p−1
ds≤v
1
(t),t∈[0,1],
DOI:10.12677/pm.2022.124076671nØêÆ

u
2
(t) ≤
Z
t
0
(
Z
1
s
g(τ,u(τ),v(τ))dτ)
p−1
ds≤v
2
(t),t∈[0,1].
(2)é?¿½λ,µ>0,Žf•§(u,v)=(λ
p−1
A
1
(u,v),µ
q−1
A
2
(u,v))k•˜)
(u
∗
λ,µ
,v
∗
λ,µ
) ∈
¯
P
h
,¦(u
∗
λ,µ
,v
∗
λ,µ
) = A(u
∗
λ,µ
,v
∗
λ,µ
).ÏdXÚ(1.6) k•˜)(u
∗
λ,µ
,v
∗
λ,µ
) ∈
¯
P
h
.
(3)�Щ:(u
0
,v
0
) ∈
¯
P
h
,½Â
u
n+1
= λ
p−1
A
1
(u
n
,v
n
)(t) = λ
p−1
Z
t
0
(
Z
1
s
f(τ,u
n
(τ),v
n
(τ))dτ)
p−1
ds,n= 1,2,...
v
n+1
= µ
q−1
A
2
(u
n
,v
n
)(t) = µ
q−1
Z
t
0
(
Z
1
s
g(τ,u
n
(τ),v
n
(τ))dτ)
q−1
ds,n= 1,2,...
n→∞, u
n
(t) →u
∗
λ,µ
,v
n
(t) →v
∗
λ,µ
.
4.Þ~
~4.1.•Äe¡‡©XÚ:



−((u
0
)
p−1
)
0
= 2(u
1
3
+v
1
4
)+2a,t∈(0,1),
−((v
0
)
q−1
)
0
= 3(u
1
5
+v
1
6
)+3b
(4.1)
Ù¥a,b>0 ,0 <p−1 <1,0 <q−1 <1, 
f(t,u,v) = 2(u
1
3
+v
1
4
)+2a,g(t,u,v) = 3(u
1
5
+v
1
6
)+3b.
w,f,g:[0,1] ×[0,+∞) ×[0,+∞)→[0,+∞)ëY,…f,g'u1Ú1nCþ´4O.-
ϕ
1
(r)=r
1
3
,ϕ
2
(r)=r
1
5
,r∈(0,1),´•ϕ
1
(r)=r
1
3
>r,ϕ
2
(r)=r
1
5
>r.é?¿u,v∈R
+
,t∈
[0,1],r∈(0,1),÷v
f(t,ru,rv) = 2((ru)
1
3
+(rv)
1
4
)+2a
≥r
1
3
[2(u
1
3
+v
1
4
)+2a]
≥r
1
3(p−1)
[2(u
1
3
+v
1
4
)+2a]
= ϕ
p−1
1
(r)f(t,u,v),
g(t,ru,rv) = 3((ru)
1
5
+(rv)
1
6
)+3b
≥r
1
5
[3(u
1
5
+v
1
6
)+3b]
≥r
1
5(q−1)
[3(u
1
5
+v
1
6
)+3b]
= ϕ
q−1
2
(r)f(t,u,v).
,
h
1
(t) =
Z
t
0
(1−s)
p−1
ds, h
2
(t) =
Z
t
0
(1−s)
q−1
ds,t∈[0,1],
DOI:10.12677/pm.2022.124076672nØêÆ

f(t,l
1
,l
2
)≥f(t,0,0)=2a>0 ,g(t,l
1
,l
2
)≥g(t,0,0)=3b>0. Ù¥l
1
,l
2
d1nÜ©Œ•, ´•
½n3.1 ¥^‡Ñ÷v. d½n3.1 ŒXÚ4.1 k•˜)(u
∗
λ,µ
,v
∗
λ,µ
)∈
¯
P
h
, �Щ:
(u
0
,v
0
) ∈
¯
P
h
,½Â
u
n+1
=
Z
t
0
(
Z
1
s
[2(u
1
3
n−1
(τ)+v
1
4
n−1
(τ))+2a]dτ)
p−1
ds,n= 1,2,...
v
n+1
=
Z
t
0
(
Z
1
s
[3(u
1
5
n−1
(τ)+v
1
6
n−1
(τ))+3b]dτ)
q−1
ds,n= 1,2,...
n→∞, u
n
(t) →u
∗
λ,µ
,v
n
(t) →v
∗
λ,µ
.
Ä7‘8
I[g,䮀7(11561063)"
ë•©z
[1]Lopushanska,G.P.andChmir,O.Y.(2007)OnSolutionsofGeneralizedNormalBoundary
ValueProblemsforQuasilinearParabolicSystemsofEquationswithLinearPrincipalParts.
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
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