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PureMathematics
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,2021,11(6),979-989
PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.116112
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ExistenceandAsymptoticEstimatesof
SolutionsforaClassofNonlinear
Third-OrderSingularlyPerturbed
BoundaryValueProblems
RuiyanZhang
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[J].
n
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,2021,11(6):979-989.
DOI:10.12677/pm.2021.116112
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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
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) =
x
0
(0
,
) = 0
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0
(1
,
)
−
ξx
0
(
η,
) = 0
,
where
0
<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/
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(
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
,
g
ñ
.
Ï
d
β
(
t
)
≤
y
(
t
)
,t
∈
[0
,
1]
.
Ó
n
Œ
y
y
(
t
)
≤
α
(
t
)
,t
∈
[0
,
1].
¤
±
,
β
(
t
)
≤
y
(
t
)
≤
α
(
t
)
,t
∈
[0
,
1]
.
(2)
y
²
|
y
0
(
t
)
|≤
N,t
∈
[0
,
1]
.
‡
þ
¡
(
Ø
Ø
¤
á
,
K
•
3
t
1
∈
[0
,
1],
¦
y
0
(
t
1
)
>N.
-
d
= max
{
y
0
(
t
)
−
N,t
∈
[0
,
1]
}
>
0
,
K
d
¥Š
½
n
Ú
β
(
t
)
≤
y
(
t
)
≤
α
(
t
)
,t
∈
[0
,
1]
Œ
•
,
•
3
θ
∈
(0
,
1),
¦
|
y
0
(
θ
)
|
=
|
y
(1)
−
y
(0)
|≤
λ<N.
du
y
0
(
t
)
∈
C
[0
,
1],
K
•
3
«
m
[
t
2
,t
3
]
⊆
[0
,
1] (
½
ö
[
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
)
,
Ï
d
,
|
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)
g
ñ
,
b
Ø
¤
á
,
¤
±
|
y
0
(
t
)
|≤
N,t
∈
[0
,
1]
.
3.
Ì
‡
½
n
9
Ù
y
²
½
n
1[9]
b
½
(i)
>
Š
¯
K
(3)-(4)
ò
z
¯
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.2021.116112984
n
Ø
ê
Æ
Ü
a
ÿ
ò
z
)
y
0
(
t
)
∈
C
2
([0
,
1]
,
R
),
÷
v
y
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
)
'
u
z
÷
v
Nagumo
^
‡
,
(iii)
•
3
~
ê
m
=
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
•
š
K
¼
ê
,
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)
y
²
E
¼
ê
α
(
t,
)
,β
(
t,
)
X
e
:
α
(
t,
) =
y
0
(
t
)+
ω
(
t,
)+
r,
β
(
t,
) =
y
0
(
t
)
−
r,
¯¢
þ
,
ω
(
t,
)
´
t
=
η
?
>
.
¼
ê
,
÷
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
¡
y
²
α
(
t,
)
,β
(
t,
)
©
O
•¯
K
(3)
þ
)
Ú
e
)
.
é
∀
>
0,
Ø
J
y
²
•
3
¿
©
Œ
~
ê
r
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.2021.116112985
n
Ø
ê
Æ
Ü
a
ÿ
=
α
(
t,
)
,β
(
t,
)
÷
v
Ø
ª
(7)-(8),
d
¥Š
½
n
9
(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,
))
.
d
y
0
(
t
)
∈
C
2
([0
,
1])
,y
0
0
(
t
)
>
0
,
0
≤
t
≤
1
Œ
•
,
y
0
0
(
t
)
,y
00
0
(
t
)
3
[0
,
1]
þ
k
.
.
Ï
d
,
•
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
−
mry
0
0
−
mrω
0
≤
y
00
0
−
mry
0
0
≤
(
n
1
−
mrn
2
)
,
r
≥
r
2
=
n
1
mn
2
ž
,
Ò
k
α
00
(
t
)
≤
f
t,
Z
t
0
α
(
s
)
ds,α
(
t
)
,α
0
(
t
)
,
0
≤
t
≤
1
.
DOI:10.12677/pm.2021.116112986
n
Ø
ê
Æ
Ü
a
ÿ
a
q
,
Œ
±
f
t,
Z
t
0
β
(
s
)
ds,β
(
t
)
,β
0
(
t
)
≤−
mry
0
0
,
@
o
,
β
00
(
t
)
−
f
t,
Z
t
0
β
(
s
)
ds,β
(
t
)
,β
0
(
t
)
≥
y
00
0
+
mry
0
0
≥
(
mrn
2
−
n
1
)
,
r
≥
r
2
=
n
1
mn
2
ž
,
Ò
k
β
00
(
t
)
≥
f
t,
Z
t
0
β
(
s
)
ds,β
(
t
)
,β
0
(
t
)
,
0
≤
t
≤
1
.
Ï
d
,
r
≥
max
{
r
1
,r
2
}
ž
,
α
(
t,
)
,β
(
t,
)
Ò
©
O
•¯
K
(3)
þ
)
Ú
e
)
.
d
Ú
n
2
Œ
±
,
>
Š
¯
K
(3)-(4)
•
3
˜
‡
)
y
(
t
)
∈
C
2
([0
,
1]
,
R
),
¦
α
(
t,
)
≤
y
(
t,
)
≤
β
(
t,
)
,
0
≤
t
≤
1
.
½
n
2
b
½
(i)
>
Š
¯
K
(1)-(2)
ò
z
¯
K
f
(
t,x
(
t
)
,x
0
(
t
)
,x
00
(
t
)) = 0
,
0
≤
t
≤
1
,
x
(0) =
x
0
(0) = 0
,
k
˜
‡
ò
z
)
x
0
(
t
)
∈
C
3
([0
,
1]
,
R
),
÷
v
x
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
)
'
u
z
÷
v
Nagumo
^
‡
,
(iii)
•
3
~
ê
m
=
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
´
š
K
¼
ê
,
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)
y
²
-
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)
ò
z
¯
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.2021.116112987
n
Ø
ê
Æ
Ü
a
ÿ
k
˜
‡
ò
z
)
u
0
(
t
)
∈
C
2
([0
,
1]
,
R
),
÷
v
u
0
0
(
t
)
>
0
,C
=:
u
(1)
−
ξu
(
η
)
>
0
.
½
n
1
¥
^
‡
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)
¥
¤
½
Â
¼
ê
,
d
(15)-(18)
Ø
J
y
²
>
Š
¯
K
(1)-(2)
•
3
˜
‡
)
x
(
t,
)
∈
C
3
([0
,
1]
,
R
),
¿
…
÷
v
(14).
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