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PureMathematicsnØêÆ,2021,11(6),979-989
PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.116112
˜aš‚5nÛÄ>Š¯K)•35Ú
ìCO
ÜÜÜaaaÿÿÿ
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2021c426F¶¹^Fϵ2021c527F¶uÙFϵ2021c63F
Á‡
©ïĘan:>Š^‡eš‚5nÛÄ>Š¯K





x
000
(t) = f(t,x(t),x
0
(t),x
00
(t)),0 ≤t≤1,0 <1,
x(0,) = x
0
(0,) = 0,x
0
(1,)−ξx
0
(η,) = 0
)•35ÚìCO,Ù¥0<η<1,0<ξη<1.ÏLE˜‡T2Âþe)éÚ$^
Nagumo^‡Ú>.¼ê,·‚þã¯K)•35,¿…‰Ñ)˜—kìCO.
'…c
n§ÛÉħ•35§ìCO
ExistenceandAsymptoticEstimatesof
SolutionsforaClassofNonlinear
Third-OrderSingularlyPerturbed
BoundaryValueProblems
RuiyanZhang
©ÙÚ^:Üaÿ.˜aš‚5nÛÄ>Š¯K)•35ÚìCO[J].nØêÆ,2021,11(6):979-989.
DOI:10.12677/pm.2021.116112
Üaÿ
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Apr.26
th
,2021;accepted:May27
th
,2021;published:Jun.3
rd
,2021
Abstract
Thispaperisdevotedtostudytheexistenceandasymptoticestimatesofsolutionsfor
aclassofnonlinearthird-ordersingularlyperturbedboundaryvalueproblemswith
three-pointboundaryvalueconditions





x
000
(t) = f(t,x(t),x
0
(t),x
00
(t)),0 ≤t≤1,0 <1,
x(0,) = x
0
(0,) = 0,x
0
(1,)−ξx
0
(η,) = 0,
where0<η<1,0<ξη<1.Byconstructinganappropriategeneralizedupper-and
lower-solutionpairandemployingtheNagumoconditionsandboundarylayerfunc-
tions,weobtaintheexistenceofsolutionstotheaboveproblemandgiveuniformly
validasymptoticestimatesofthesolutions.
Keywords
Third-Order,SingularlyPerturbed,Existence,AsymptoticEstimates
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
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(J.~X,3©z[1]¥, GuoïÄnn:>Š¯K



u
000
(t)+a(t)f(u(t)) = 0,t∈(0,1),
u(0) = u
0
(0) = 0,u
0
(1) = αu
0
(η),
(P)
DOI:10.12677/pm.2021.116112980nØêÆ
Üaÿ
Ù¥0 <η<1,1 <α<
1
η
,¿$^I.†Ø ØÄ:½nXe(J:
½nAf∈C([0,∞),[0,∞)), a∈C([0,1],[0,∞)) …3t∈[
η
α
,η] þØð•".ef÷v
(i)f
0
= 0,f
∞
= ∞,
½
(ii)f
0
= ∞,f
∞
= 0,
K¯K(P) –•3˜‡),Ù¥
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u→∞
f(u)
u
.
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/ž§nn:>Š¯K



u
000
(t)+a(t)f(t,u(t)) = 0,0 <t<1,
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1
η
),f∈C([0,1]×[0,∞),[0,∞)), a∈L
1
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þØð•".nn:>Š¯KÙ¦ƒ'(J,ë„©z[3–7].
,, 3þã©z[1–7]¥, ïÄ¯Kþ•´Ä•3)?•3A‡)?éš‚5nÛÉÄ
n:>Š¯KïÄé. ¤±,©$^2Âþ e)•{Ú>.¼ê, ïıeš‚5nÛÉ
Än:>Š¯K
x
000
(t) = f(t,x(t),x
0
(t),x
00
(t)),0 ≤t≤1,0 <1,(1)
x(0,) = x
0
(0,) = 0,x
0
(1,)−ξx
0
(η,) = 0(2)
)•35(JÚìCO,Ù¥0 <η<1,0 <ξη<1.
•ïÄ>Š¯K(1)-(2),·‚I‡?رeÛÄ>Š¯K
y
00
(t) = f(t,
Z
t
0
y(s)ds,y(t),y
0
(t)),0 ≤t≤1,0 <1,(3)
y(0,) = 0,y(1,)−ξy(η,) = 0.(4)
Äk?رe¯K
y
00
(t) = f(t,
Z
t
0
y(s)ds,y(t),y
0
(t)),0 ≤t≤1,(5)
y(0) = 0,y(1)−ξy(η) = 0.(6)
DOI:10.12677/pm.2021.116112981nØêÆ
Üaÿ
2.ý•£
½Â1eα(t) ∈C
2
[0,1]÷v



α
00
(t) ≤f(t,
R
t
0
α(s)ds,α(t),α
0
(t)),0 ≤t≤1,
α(0) ≥0,α(1)−ξα(η) ≥0,
K¡α(t)•>Š¯K(5)-(6) ˜‡þ).
eβ(t) ∈C
2
[0,1]÷v



β
00
(t) ≥f(t,
R
t
0
β(s)ds,β(t),β
0
(t)),0 ≤t≤1,
β(0) ≤0,β(1)−ξβ(η) ≤0,
K¡β(t) •>Š¯K(5)-(6) ˜‡e).
½Â2f(t,x,y,z) 3[0,1]×R
3
þ÷vNagumo ^‡, Ò´`, fëY…é∀a>0, •3¼
êΦ : [0,+∞) →[a,+∞), é∀(t,x,y,z)∈[0,1]×R
3
,Ñk|f(t,x,y,z)|≤Φ(|z|), …
R
+∞
0
s
Φ(s)
ds=
+∞.
½Â3½Â
f
∗
(t,
Z
t
0
y(s)ds,y(t),y
0
(t)) =







f(t,
R
t
0
α(s)ds,α(t),α
0
(t)),y(t) >α(t),
f(t,
R
t
0
y(s)ds,y(t),y
0
(t)),β(t) ≤y(t) ≤α(t),
f(t,
R
t
0
β(s)ds,β(t),β
0
(t)),y(t) <β(t),
K¼êf
∗
(t,
R
t
0
y(s)ds,y(t),y
0
(t)) ∈C([0,1]×R
3
,R)•f(t,
R
t
0
y(s)ds,y(t),y
0
(t))'u(β(t),α(t))
?¼ê.
51:eα(t),β(t)3[0,1]þëY,f3[0,1]×R
3
þëY,K?¼êf
∗
3[0,1]×R
3
þëY…
k..d,α(t) ≤β(t) ž,f
∗
= f.
Ún1[8]ef(t,
R
t
0
y(s)ds,y(t),y
0
(t)) 3[0,1]×R
3
þëY…k., K>Š¯K(5)-(6) •3˜
‡)y(t) ∈C
2
([0,1],R).
Ún2b½
(i)f(t,x,y,z) 3[0,1]×R
3
þëY,
(ii)é∀(t,x,y,z) ∈[0,1]×R
3
,f(t,x,y,z) 'uz÷vNagumo ^‡,
(iii)>Š¯K(5)-(6) •3þ)α(t) Úe)β(t), ÷v
β(0) ≤0 ≤α(0),(7)
DOI:10.12677/pm.2021.116112982nØêÆ
Üaÿ
β(1)−ξβ(η) ≤0 ≤α(1)−ξα(η),(8)
K>Š¯K(5)-(6)•3˜‡)y(t) ∈C
2
([0,1],R),¦
β(t) ≤y(t) ≤α(t),|y
0
(t)|≤D,t∈[0,1],
Ù¥D>0 ´˜‡~ê.
y²-
λ=max
t∈[0,1]
α(t)−min
t∈[0,1]
β(t).
d^‡(ii),•3M>0, ¦
R
M
λ
s
Φ(s)
ds>λ.
½Â
N= max{|α
0
(t)|,|β
0
(t)|,M,2λ},
KN>0.d51 ÚÚn1, Œ±?¯K



y
00
(t) = f
∗
(t,
R
t
0
y(s)ds,y(t),y
0
(t)),0 ≤t≤1,
y(0) = 0,y(1)−ξy(η) = 0,
(9)
•3˜‡)y(t) ∈C
2
([0,1],R).
df
∗
½Â,N´y²¯K(9) )÷v
β(t) ≤y(t) ≤α(t),|y
0
(t)|≤N,t∈[0,1].(10)
e¡©üÚy²(10)¤á.
(1)y²β(t) ≤y(t) ≤α(t),t∈[0,1].
kyβ(t) ≤y(t),t∈[0,1].‡β(t) ≤y(t)),t∈[0,1] ؤá,K•3t
0
∈[0,1], ¦
β(t
0
) >y(t
0
).
-p(t) =: β(t)−y(t),Kp(c) = max{β(t)−y(t),t∈[0,1]}>0.
ec= 0,Kβ(0)−y(0) >0, d(7) Œ±β(0) ≤0 = y(0), gñ.
ec= 1,Kβ(1)−y(1) >0,d(8)Œ±β(1)−ξβ(η) ≤0 = y(1)−ξy(η),=β(1)−y(1) ≤
ξ(β(η)−y(η)) <β(η)−y(η), gñ.
ec∈(0,1), Kβ(c)−y(c) >0,β
00
(c)−y
00
(c) <0.
,˜•¡,
DOI:10.12677/pm.2021.116112983nØêÆ
Üaÿ
β
00
(c)−y
00
(c) ≥f(c,
Z
c
0
β(s)ds,β(c),β
0
(c))−f
∗
(c,
Z
c
0
y(s)ds,y(c),y
0
(c))
= f(c,
Z
c
0
β(s)ds,β(c),β
0
(c))−f(c,
Z
c
0
β(s)ds,β(c),β
0
(c))
= 0,
gñ.Ïdβ(t) ≤y(t),t∈[0,1].ÓnŒyy(t) ≤α(t),t∈[0,1].
¤±,
β(t) ≤y(t) ≤α(t),t∈[0,1].
(2)y²|y
0
(t)|≤N,t∈[0,1].
‡þ¡(Øؤá,K•3t
1
∈[0,1], ¦y
0
(t
1
) >N.
-
d= max{y
0
(t)−N,t∈[0,1]}>0,
Kd¥Š½nÚβ(t) ≤y(t) ≤α(t),t∈[0,1] Œ•,•3θ∈(0,1), ¦
|y
0
(θ)|= |y(1)−y(0)|≤λ<N.
duy
0
(t) ∈C[0,1], K•3«m[t
2
,t
3
] ⊆[0,1] (½ö[t
3
,t
2
] ⊆[0,1] ),¦
y
0
(t
2
) = λ,y
0
(t
3
) = N,λ<y
0
(t) <N,t∈(t
2
,t
3
),
Ïd,
|y
00
(t)|= |f
∗
(t,
Z
t
0
y(s)ds,y(t),y
0
(t))|
= |f(t,
Z
t
0
y(s)ds,y(t),y
0
(t))|
≤Φ(|y
0
(t)|),t∈(t
2
,t
3
).
@o,
|
Z
t
3
t
2
y
0
(t)y
00
(t)
Φ(y
0
(t))
dt|≤|
Z
t
3
t
2
y
0
(t)dt|≤λ,(11)
,˜•¡,
|
Z
t
3
t
2
y
0
(t)y
00
(t)
Φ(y
0
(t))
dt|= |
Z
N
λ
s
Φ(s))
ds|>λ,(12)
(11)Ú(12) gñ,bؤá, ¤±|y
0
(t)|≤N,t∈[0,1].
3.̇½n9Ùy²
½n1[9]b½
(i) >Š¯K(3)-(4) òz¯K(=f(t,
R
t
0
y(s)ds,y(t),y
0
(t))=0,0≤t≤1,y(0)=0)k˜ ‡
DOI:10.12677/pm.2021.116112984nØêÆ
Üaÿ
òz)y
0
(t) ∈C
2
([0,1],R),÷vy
0
0
(t) >0,0 ≤t≤1,…C=: y
0
(1)−ξy
0
(η) >0,
(ii)é∀(t,x,y,z) ∈[0,1]×R
3
,f(t,x,y,z) 'uz÷vNagumo ^‡,
(iii)•3~êm=
2ξ
C(1−η)(ξ+1)
,¦f
yz
=
∂
2
f(t,x,y,z)
∂y∂z
≥m>0,f
x
=
∂f(t,x,y,z)
∂x
,f
y
=
∂f(t,x,y,z)
∂y
,f
z
=
∂f(t,x,y,z)
∂z
þ•šK¼ê,
K>0 ¿©ž,>Š¯K(3)-(4) •3˜‡)y(t,) ÷v
|y(t,)−y
0
(t)|≤ω(t,)+r,0 ≤t≤1,0 <<<1,
Ù¥,r´˜‡–½vŒ~ê,…
ω(t,) =
2C
2+mC(η−t)
.(13)
y²E¼êα(t,),β(t,) Xe:



α(t,) = y
0
(t)+ω(t,)+r,
β(t,) = y
0
(t)−r,
¯¢þ,ω(t,) ´t= η?>.¼ê,÷v



ω
00
(t,)−mω(t,)ω
0
(t,) = 0,0 ≤t≤1,
ω(1,)−ξω(η,) = C,
w,,ω(t,) >0,ω
00
(t,) >0,0 ≤t≤1.
e¡y²α(t,),β(t,) ©O•¯K(3) þ)Úe).
é∀>0, ØJy²•3¿©Œ~êr
1
,r>r
1
ž,
α(t,),β(t,) ∈C
2
([0,1]),α(t,) ≥β(t,),0 ≤t≤1,
α(0,) = y
0
(0)+ω(0,)+r= ω(0,)+r≥0,
β(0,) = y
0
(0)−r= −r≤0,
α(1,)−ξα(η,) = y
0
(1)+ω(1,)+r−ξ(y
0
(η)+ω(η,)+r)
= y
0
(1)−ξy
0
(η)+ω(1,)−ξω(η,)+(1−ξ)r≥0,
β(1,)−ξβ(η,) = y
0
(1)−r−ξ(y
0
(η)−r)
= y
0
(1)−ξy
0
(η)−(1−ξ)r,
= C−(1−ξ)r≤0,
DOI:10.12677/pm.2021.116112985nØêÆ
Üaÿ
=α(t,),β(t,) ÷vØª(7)-(8), d¥Š½n9(i) Ú(iii), Œ±
f

t,
Z
t
0
α(s)ds,α,α
0

= f

t,
Z
t
0
α(s)ds,α,α
0

−f

t,
Z
t
0
α(s)ds,α,y
0
0

+f

t,
Z
t
0
α(s)ds,α,y
0
0

−f

t,
Z
t
0
α(s)ds,y
0
,y
0
0

+f

t,
Z
t
0
α(s)ds,y
0
,y
0
0

−f

t,
Z
t
0
y
0
(s)ds,y
0
,y
0
0

+f

t,
Z
t
0
y
0
(s)ds,y
0
,y
0
0

= ω
0
(t,)
Z
1
0
f
z

t,
Z
t
0
α(s)ds,α,y
0
0
+θ(α
0
−y
0
0
)

dθ
+(ω(t,)+r)
Z
1
0
f
y

t,
Z
t
0
α(s)ds,y
0
+θ(α−y
0
),y
0
0

dθ
+
Z
t
0
(ω(s,)+r)ds
Z
1
0
f
x

t,
Z
t
0
[y
0
(s)+θ(α(s)−y
0
(s))]ds,y
0
,y
0
0

dθ
≥(ω(t,)+r)
Z
1
0
f
y

t,
Z
t
0
α(s)ds,y
0
+θ(α−y
0
),y
0
0

dθ
−(ω(t,)+r)
Z
1
0
f
y

t,
Z
t
0
α(s)ds,y
0
+θ(α−y
0
),y
0
0
+α
0

dθ
+(ω(t,)+r)
Z
1
0
f
y

t,
Z
t
0
α(s)ds,y
0
+θ(α−y
0
),y
0
0
+α
0

dθ
≥(ω(t,)+r)(y
0
0
+ω
0
(t,))
Z
1
0
Z
1
0
f
yz

t,
Z
t
0
α(s)ds,y
0
+θ(α−y
0
),y
0
0
+sα
0

dθds
≥m(ω(t,)+r)(y
0
0
+ω
0
(t,)).
dy
0
(t)∈C
2
([0,1]),y
0
0
(t)>0,0≤t≤1Œ•,y
0
0
(t),y
00
0
(t)3[0,1]þk..Ïd,•3~ê
n
1
,n
2
,¦
|y
00
0
(t)|≤n
1
,|y
0
0
(t)|≥n
2
,t∈[0,1].
@o,
α
00
(t)−f

t,
Z
t
0
α(s)ds,α(t),α
0
(t)

≤(y
00
0
+ω
00
)−m(ω+r)(y
0
0
+ω
0
)
= y
00
0
+ω
00
−mωy
0
0
−mωω
0
−mry
0
0
−mrω
0
≤y
00
0
−mry
0
0
≤(n
1
−mrn
2
),
r≥r
2
=
n
1
mn
2
ž,Òk
α
00
(t) ≤f

t,
Z
t
0
α(s)ds,α(t),α
0
(t)

,0 ≤t≤1.
DOI:10.12677/pm.2021.116112986nØêÆ
Üaÿ
aq,Œ±
f

t,
Z
t
0
β(s)ds,β(t),β
0
(t)

≤−mry
0
0
,
@o,
β
00
(t)−f

t,
Z
t
0
β(s)ds,β(t),β
0
(t)

≥y
00
0
+mry
0
0
≥(mrn
2
−n
1
),
r≥r
2
=
n
1
mn
2
ž,Òk
β
00
(t) ≥f

t,
Z
t
0
β(s)ds,β(t),β
0
(t)

,0 ≤t≤1.
Ïd, r≥max{r
1
,r
2
}ž, α(t,),β(t,) Ò©O•¯K(3) þ)Úe). dÚn2 Œ±, >
Š¯K(3)-(4)•3˜‡)y(t) ∈C
2
([0,1],R),¦α(t,) ≤y(t,) ≤β(t,),0 ≤t≤1.
½n2b½
(i)>Š¯K(1)-(2) òz¯K



f(t,x(t),x
0
(t),x
00
(t)) = 0,0 ≤t≤1,
x(0) = x
0
(0) = 0,
k˜‡òz)x
0
(t) ∈C
3
([0,1],R),÷vx
00
0
(t) >0,0 ≤t≤1,…C
∗
=: x
0
0
(1)−ξx
0
0
(η) >0,
(ii)é∀(t,x,y,z) ∈[0,1]×R
3
,f(t,x,y,z) 'uz÷vNagumo ^‡,
(iii) •3~êm=
2ξ
C(1−η)(ξ+1)
, ¦
∂
2
f(t,x,y,z)
∂y∂z
≥m>0,…
∂f(t,x,y,z)
∂x
,
∂f(t,x,y,z)
∂y
,
∂f(t,x,y,z)
∂z
´šK¼ê,
K>0 ¿©ž,>Š¯K(1)-(2) •3˜‡)x(t,) ∈C
3
([0,1],R)÷v
|x(t,)−x
0
(t)|≤ω(t,)+r.(14)
y²-
x
0
(t) = u(t),(15)
K>Š¯K(1)-(2)Œ±=z•



u
00
(t) = f(t,
R
t
0
u(s),u(s),u
0
(s)),0 ≤t≤1,0 <1,
u(0,) = 0,u(1,)−ξu(η,) = 0,
(16)
(16)òz¯K



f(t,
R
t
0
u(s),u(s),u
0
(s)) = 0,0 ≤t≤1,
u(0) = 0,
(17)
DOI:10.12677/pm.2021.116112987nØêÆ
Üaÿ
k˜‡òz)u
0
(t) ∈C
2
([0,1],R),÷v
u
0
0
(t) >0,C=: u(1)−ξu(η) >0.
½n1¥^‡þ÷v, @o>Š¯K(16) •3˜‡)u(t,) ∈C
2
([0,1],R),¦
|u(t,)−u
0
(t)|≤ω(t,)+r,0 ≤t≤1,0 <<<1,(18)
Ù¥,ω(t,)•(13)¥¤½Â¼ê,d(15)-(18)ØJy²>Š¯K(1)-(2)•3˜‡)x(t,)∈
C
3
([0,1],R),¿…÷v(14).
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(12061064).
ë•©z
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OrderThree-PointBoundaryValueProblem.NonlinearAnalysis:Theory,MethodsandAp-
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[2]Torres, F.J.(2013)PositiveSolutionsforaThird-OrderThreePointBoundaryValueProblems.
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[6]Henderson,J.andKosmatov,N.(2014)Three-PointThird-OrderProblemswithaSign-
ChangingNonlinearTerm.ElectronicJournalofDifferentialEquations,2014,1-10.
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