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PureMathematicsnØêÆ,2023,13(6),1769-1782
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.136181
p©ê‡©•§>Š¯K)•3•˜5
ÜÜܶ¶¶
=²nóŒÆ§nÆ§[‹=²
ÂvFϵ2023c521F¶¹^Fϵ2023c622F¶uÙFϵ2023c630F
Á‡
ïÄäk·ÜüNš‚5‘Riemann-Liouville.p©ê‡©•§>Š¯K"|
^Green¼ê5Ÿ±9·ÜüNŽfØÄ:½ny²T>Š¯K)•3•˜5§¿‰Ñ
˜‡¢~y(Ø(5"
'…c
©ê‡©•§§>Š¯K§·ÜüNŽf§•3•˜5
ExistenceandUniquenessofSolutionsto
BoundaryValueProblemsforHigher-Order
FractionalDifferentialEquations
ZhangchiWang
SchoolofScience,LanzhouUniversityofTechnology,LanzhouGansu
Received:May21
st
,2023;accepted:Jun.22
nd
,2023;published:Jun.30
th
,2023
Abstract
TheboundaryvalueproblemofRiemann-Liouvillehigherorderfractionaldifferential
©ÙÚ^:ܶ.p©ê‡©•§>Š¯K)•3•˜5[J].nØêÆ,2023,13(6):1769-1782.
DOI:10.12677/pm.2023.136181
ܶ
equationswithmixedmonotonenonlineartermsisstudied.Byusingtheproperties
ofGreen’sfunctionandthefixedpointtheoremofmixedmonotoneoperators,the
existenceanduniquenessofthesolutionoftheboundaryvalueproblemareproved,
andanexampleisgiventoverifythecorrectnessoftheconclusion.
Keywords
FractionalDifferentialEquation,BoundaryValueProblem,MixedMonotone
Operator,ExistenceandUniqueness
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
3LA›c¥,©ê‡©•§ÏÙ3>zÆ!Ê5!>^Æ!õš0Ÿ!››ˆ‡
+•A^C5-‡.k'•[&E§ žë[1–12]9Ù¥ë•©z.Cc5,'u ©
ê‡©•§ïÄÉNõÆö'5,X:ØÓ>Š^‡e)•3•˜5,)•3•˜
5,Ù̇ïÄ•{•)IþØÄ:½n,·ÜüNŽfþØÄ:½n.
©z[13]A^·ÜüNŽfØÄ:½nïÄ©ê‡©•§>Š¯K



D
α
0
+
u(t)+q(t)f(u,u
0
,···,u
(n−2)
) = 0,0 <t<1,n−1 <α≤n,
u(0) = u
0
(0) = ···= u
(n−2)
(0) = u
(n−2)
(1) = 0
(1)
)•3•˜5,Ù¥D
α
0
+
´IORiemann-Liouville.©ê‡©ê,n≥2,n∈N,¼êf
Úq÷vXeA5:
(C
1
)f(x
1
,x
2
,···,x
n−1
)=g(x
1
,x
2
,···,x
n−1
) + h(x
1
,x
2
,···,x
n−1
),Ù¥g:[0,+∞) ×R
n−2
→
[0,+∞)´ëY,h: (0,+∞)×(R\{0})
n−2
→(0,+∞)´ëY.
(C
2
)g3x
i
>0,i= 1,2,···,n−1ž´š~;d,h3x
i
>0,i= 1,2,···,n−1´šO.
(C
3
)•3β∈(0,1)¦éux
i
>0,i= 1,2,···,n−1k
g(tx
1
,···,tx
n−1
) ≥t
β
g(x
1
,···,x
n−1
), t∈(0,1),
h(t
−1
x
1
,···,t
−1
x
n−1
) ≥t
β
h(x
1
,···,x
n−1
), t∈(0,1).
(C
4
)t
r
q: [0,1] →[0,+∞)´ëY…
R
1
0
q(s)s
−β(α−1)
ds<+∞,0 ≤r<1.
DOI:10.12677/pm.2023.1361811770nØêÆ
ܶ
©z[14]ïÄ±e©ê‡©•§>Š¯K









−D
α
0
+
u(t) = f(t,u(t),u(t))+g(t,u(t),u(t))−b, t∈(0,1),n−1 <α≤n,
u
(i)
(0) = 0, i= 0,1,···,n−2,
D
β
0
+
u(1) = 0,1 ≤β≤n−2
(2)
)•3•˜5,Ù¥D
α
0
+
,D
β
0
+
´IORiemann-Liouville.©ê‡©ê,n≥3,b>0´~
ê,f,g:[0,1] ×(−∞,+∞) ×(−∞,+∞)→(−∞,+∞)´ëY¼ê.¯K(2)•)Ͷ5ù
•§Ú©z[15,16]¥•Ä©ê¯K.
•C,gGuoÚLakshmikantham[17]Ú\·ÜüNŽf±5,NõÆöÑïÄBanach˜m
¥ˆ«a .·ÜüNŽf,¿ENõƒ'½n.3©z[18]¥,BhaskarÚLakshikantham
3ŒSÝþ˜m¥ïÄ·ÜüNŽf˜Í ÜØÄ:½n.3©z[19]¥,LiÚZhao•Ä
˜aτ−φ·ÜüNŽf.d,äk6Ä·ÜüNŽf®2•ïÄ.3[20]¥,Liu<•
ÄŽf•§:
A(x,x)+B(x,x) = x
3ŒSBanach˜mþ)•3•˜5,Ù¥AÚB´ü‡·ÜüNŽf,d,Šö„‰ÑT
Žf3š‚5©ê‡©•§¥A^.
ÉþãóŠéu,©ò$^©z[14]¥·ÜüNŽfØÄ:½nïÄXeäk·ÜüN
š‚5‘p©ê‡©•§>Š¯K



D
α
0
+
u(t)+q(t)f(u(t),u
0
(t),···,u
(n−2)
(t))−b= 0,t∈[0,1],
u(0) = u
0
(0) = ···= u
(n−2)
(0) = u
(n−2)
(1) = 0
(3)
)•3•˜5,Ù¥D
α
0
+
´IORiemann-Liouville.©ê‡©ê,b>0´˜‡~ê,
n−1 <α≤n,n≥2,f´˜‡š‚5¼ê,q∈C[0,1]÷vq(t) ≥0,q(t) 6≡0.
2.ý•£
•Bå„,Äk‰Ñ˜7‡½ÂÚÚn,•YïÄóŠJøêÆóä.
½Â2.1[2,4]¼êy∈C[0,1]α>0Riemann-Liouville©êÈ©½Â•
I
α
0
+
y(t) =
1
Γ(α)
Z
t
0
y(s)
(t−s)
1−α
ds.
½Â2.2[2,4]¼êy∈C[0,1]α>0Riemann-Liouville©êê½Â•
D
α
0
+
y(t) =
1
Γ(n−α)

d
dt

n
Z
t
0
y(s)
(t−s)
α−n−1
ds,
DOI:10.12677/pm.2023.1361811771nØêÆ
ܶ
Ù¥n= [α]+1.
Ún2.1[13]-y∈C[0,1],@o>Š¯K



D
α−n+2
0
+
x(t)+y(t) = 0,0 ≤t≤1,n−1 <α≤n,n≥2,
x(0) = x(1) = 0
(4)
k•˜)
x(t) =
Z
1
0
G(t,s)y(s)ds,
Ù¥
G(t,s) =



(t(1−s))
α−n+1
−(t−s)
α−n+1
Γ(α−n+2)
,0 ≤s≤t≤1,
(t(1−s))
α−n+1
Γ(α−n+2)
,0 ≤t≤s≤1.
(5)
Ún2.2[13]½Â(5)¥¼êG(t,s)÷vXe^‡:
(1)G(t,s) ≥0,G(t,s) ≤t
α−n+1
/Γ(α−n+2),G(t,s) ≤G(s,s)§0 ≤t,s≤1;
(2)•3˜‡¼êρ∈C(0,1)¦min
γ≤t≤δ
G(t,s) ≥ρ(s)G(s,s),s∈(0,1),Ù¥0 <γ<δ<1.
Ún2.3-u(t)=I
n−2
0
+
x(t),x(t)∈C[0,1],@oD
n−2
0
+
u(t)=x(t).¯K(3)Œ±=C¤Xe¯
K(6):



D
α−n+2
0
+
x(t)+q(t)f(I
n−2
0
+
x(t),I
n−3
0
+
x(t),...,x(t))−b= 0,0 ≤t≤1,n−1 <α≤n,n≥2,
x(0) = x(1) = 0.
(6)
XJx∈C[0,1]´¯K(6)),@ou(t) = I
n−2
0
+
x(t)´¯K(3)).
y.Ty²†©z[13]¥Ún2.7aq,d?ŽÑ.
©¥,E´D‰¢Banach˜m,θ´E¥".˜‡š˜4à8P∈EXJ÷
v(1)x∈P,λ≥0⇒λx∈P;(2)x∈P,−x∈P ⇒x=θ,K¡P´Eþ˜‡I.E
¥ŒS'X•x.y…=y−x∈P.d,XJ•3˜‡~êN >0¦é¤k
x,y∈E,θ.x.ykkxk≤Nkyk,K¡P´˜‡5I,Ù¥•N¡•´P5~
ê.‰½h>θ(i.e.,θ.h…h6= θ),·‚½Â8ÜC
h
•
C
h
= {x∈E|•3λ>0Úµ>0 ¦λh.x.µh}.
e∈P,…θ.e.h.½Â
C
h,e
= {x∈E|x+e∈C
h
}.
½Â2.3[21]A:C
h,e
×C
h,e
→E,XJA(x,y)'uxš~'uyšO,K¡A´·ÜüN,=
XJx
i
,y
i
∈C
h,e
(i= 1,2),x
1
≤x
2
,y
1
≥y
2
,KA(x
1
,y
1
) ≤A(x
2
,y
2
).
Ún2.4[14]A,B: C
h,e
×C
h,e
→E´ü‡·ÜüNŽf…÷vXe^‡:
DOI:10.12677/pm.2023.1361811772nØêÆ
ܶ
(i)éu¤kt∈(0,1),•3ψ(t) ∈(t,1)¦
A(tx+(t−1)e,t
−1
y+(t
−1
−1)e) ≥ψ(t)A(x,y)+(ψ(t)−1)e,∀x,y∈C
h,e
;
(ii)éu¤kt∈(0,1)Úx,y∈C
h,e
,
B(tx+(t−1)e,t
−1
y+(t
−1
−1)e) ≥tB(x,y)+(t−1)e;
(iii)A(h,h) ∈C
h,e
…B(h,h) ∈C
h,e
;
(iv)•3˜‡~êδ>0,¦éu¤kx,y∈C
h,e
,k
A(x,y) ≥δB(x,y)+(δ−1)e.
@oŽf•§A(x,x)+ B(x,x)+e=x3C
h,e
þk•˜)x
∗
,¿…éu?ÛЊx
0
,y
0
∈C
h,e
,
ES
x
n
=A(x
n−1
,y
n−1
)+B(x
n−1
,y
n−1
)+e,
y
n
=A(y
n−1
,x
n−1
)+B(y
n−1
,x
n−1
)+e,n= 1,2,···,
n→∞ž,3E¥kx
n
→x
∗
,y
n
→x
∗
.
3.̇(J
•Bå„,·‚½ÂXeÎÒ.
éut∈[0,1],
e(t) =
b
Γ(α−n+3)

t
α−n+1
−t
α−n+2

,
E
∗
= max{I
n−2
0
+
e(t),I
n−3
0
+
e(t),...,e(t)}.
½n3.1XJ±e^‡¤á:
(H
1
) éu?¿x
i
∈[−E
∗
,+∞)(i= 0,1,2,···,n−2),kf(x
0
,x
1
,···,x
n−2
) = g(x
0
,x
1
,···,x
n−2
,
x
0
,
x
1
,...,x
n−2
)+φ(x
0
,x
1
,···,x
n−2
,x
0
,x
1
,···,x
n−2
)Ù¥g∈C([−E
∗
,+∞)
2(n−1)
,(−∞,+∞)),
φ∈C([−E
∗
,+∞)
2(n−1)
,(−∞,+∞)).
(H
2
)éu?¿½y
i
∈[−E
∗
,+∞)(i= 0,1,2,···,n−2),g(x
0
,x
1
,...,x
n−2
,y
0
,y
1
,···,y
n−2
)Ú
φ(x
0
,x
1
,···,x
n−2
,y
0
,y
1
,···,y
n−2
)3x
i
∈[−E
∗
,+∞)š~;éu?¿½x
i
∈[−E
∗
,+∞)
(i= 0,1,2,...,n−2),g(x
0
,x
1
,···,x
n−2
,y
0
,y
1
,···,y
n−2
)Úφ(x
0
,x
1
,···,x
n−2
,y
0
,y
1
,···,y
n−2
) 3y
i
∈[−E
∗
,+∞)šO.
(H
3
)éu?¿τ∈(0,1),•3ϕ(τ)∈(τ,1)¦éu¤kx
i
,y
i
∈[−E
∗
,+∞)(i=0,1,···,n−
DOI:10.12677/pm.2023.1361811773nØêÆ
ܶ
2),k
g(τx
0
+(τ−1)ρ
0
,τx
1
+(τ−1)ρ
0
,···,τx
n−2
+(τ−1)ρ
0
,τ
−1
y
0
+(τ
−1
−1)ρ
0
,
τ
−1
y
1
+(τ
−1
−1)ρ
0
,···,τ
−1
y
n−2
+(τ
−1
−1)ρ
0
)≥ϕ(τ)g(x
0
,x
1
,···,x
n−2
,y
0
,y
1
,···,y
n−2
),ρ
0
∈[0,E
∗
],
φ(τx
0
+(τ−1)ρ
0
,τx
1
+(τ−1)ρ
0
,...,τx
n−2
+(τ−1)ρ
0
,τ
−1
y
0
+(τ
−1
−1)ρ
0
,
τ
−1
y
1
+(τ
−1
−1)e,...,τ
−1
y
n−2
+(τ
−1
−1)e)≥τh(x
0
,x
1
,...,x
n−2
,y
0
,y
1
,...,y
n−2
),ρ
0
∈[0,E
∗
].
(H
4
)éu?¿½x
i
,y
i
∈[−E
∗
,+∞)(i= 0,1,2,...,n−2),•3˜‡~êδ
0
>0¦
g(x
0
,x
1
,...,x
n−2
,y
0
,y
1
,...,y
n−2
) ≥δ
0
φ(x
0
,x
1
,...,x
n−2
,y
0
,y
1
,...,y
n−2
).
(H
5
)•3˜‡~êM=
b
(α−n+1)Γ(α−n+3)
,¦¼êgÚ¼êφ÷v
g(M,M,...,M,0,0,...,0) <+∞,
φ(M,M,...,M,0,0,...,0) <+∞,
φ(0,0,...,0,M,M,...,M) >0.
@o¯K(6)3C
h,e
k•˜² …)x
∗
,Ù¥h(t)= Mt
α−n+1
,t∈[0,1].d,Œ±ï±eü‡
S:
ω
n
(t) =
Z
1
0
G(t,s)g

I
n−2
0
+
ω
n−1
(s),I
n−3
0
+
ω
n−1
(s),...,ω
n−1
(s),I
n−2
0
+
τ
n−1
(s),I
n−3
0
+
τ
n−1
(s),
...,τ
n−1
(s)

ds+
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
ω
n−1
(s),I
n−3
0
+
ω
n−1
(s),...,ω
n−1
(s),
I
n−2
0
+
τ
n−1
(s),I
n−3
0
+
τ
n−1
(s),...,τ
n−1
(s)

ds−
b
Γ(α−n+3)

t
α−n+1
−t
α−n+2

,
n= 1,2,···,
τ
n
(t) =
Z
1
0
G(t,s)g

I
n−2
0
+
τ
n−1
(s),I
n−3
0
+
τ
n−1
(s),...,τ
n−1
(s),I
n−2
0
+
ω
n−1
(s),I
n−3
0
+
ω
n−1
(s),
...,ω
n−1
(s)

ds+
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
τ
n−1
(s),I
n−3
0
+
τ
n−1
(s),...,τ
n−1
(s),
I
n−2
0
+
ω
n−1
(s),I
n−3
0
+
ω
n−1
,...,ω
n−1

ds−
b
Γ(α−n+3)

t
α−n+1
−t
α−n+2

,
n= 1,2,···,
éu?¿‰½ω
0
,τ
0
∈C
h,e
,t∈[0,1],k{ω
n
(t)}Ú{τ
n
(t)}ј—Âñux
∗
.
yyy.éut∈[0,1],
e(t) =
b
Γ(α−n+3)

t
α−n+1
−t
α−n+2

≥0,
E
∗
= max{I
n−2
0
+
e(t),I
n−3
0
+
e(t),...,e(t)}.
=e,E
∗
∈P.d,e(t) ≤E
∗
≤
b
(α−n+1)Γ(α−n+3)
t
α−n+1
= h(t),C
h,e
= {x∈C[0,1]|x+e∈C
h
}.
DOI:10.12677/pm.2023.1361811774nØêÆ
ܶ
y²ƒc,Äk=†x(t).dÚn2.1Ú(H
1
),¯K(6)È©úª•
x(t) =
Z
1
0
G(t,s)[q(s)f(I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s))−b]ds
=
Z
1
0
G(t,s)q(s)f

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s)

ds−b
Z
1
0
G(t,s)ds
=
Z
1
0
G(t,s)q(s)[g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s)

+
φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s)

]ds−
b(t
α−n+1
−t
α−n+2
)
Γ(α−n+3)
=
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s)

ds+
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s)

ds−e(t)
=
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s)

ds−e(t)+
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s)

ds−e(t)+e(t),
Ù¥G(t,s)3(5)‰Ñ.éuz‡t∈[0,1],x,y∈C
h,e
,½ÂXeŽf:
A(x,y)(t) =
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t),
B(x,y)(t) =
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t).
éN´y²x´¯K(6))…=x= A(x,x)+B(x,x)+e.e5,©oÚy²AÚB÷vÚ
n2.4¥^‡.
(1)Äk,y²A,B:C
h,e
×C
h,e
→E´·ÜüNŽf.¯¢þ,éu¤kx
i
,y
i
∈C
h,e
(i=
1,2)…x
1
≥x
2
,y
1
≤y
2
,d(H
2
),Œ
A(x
1
,y
1
)(t) =
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x
1
(s),I
n−3
0
+
x
1
(s),...,x
1
(s),I
n−2
0
+
y
1
(s),I
n−3
0
+
y
1
(s),...,y
1
(s)

ds−e(t)
≥
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x
2
(s),I
n−3
0
+
x
2
(s),...,x
2
(s),I
n−2
0
+
y
2
(s),I
n−3
0
+
y
2
(s),...,y
2
(s)

ds−e(t)
= A(x
2
,y
2
)(t).
=A(x
1
,y
1
) ≥A(x
2
,y
2
).^Ó•{Œ±ÑB(x
1
,y
1
) ≥B(x
2
,y
2
).
(2)Ùg, yÚn2.4¥(i)Ú(ii)¤á.d(H
3
),éz‡τ∈(0,1),t∈[0,1], •3ϕ(τ) ∈(τ,1)
¦éz‡x,y∈C
h,e
,ρ
0
= E
∗
,k
A

τx+(τ−1)e,τ
−1
y+(τ
−1
−1)e

(t)
=
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
[τx(s)+(τ−1)e(s)],I
n−3
0
+
[τx(s)+(τ−1)e(s)],...,[τx(s)+(τ−1)e(s)],
DOI:10.12677/pm.2023.1361811775nØêÆ
ܶ
I
n−2
0
+
[τ
−1
y(s)+(τ
−1
−1)e(s)],I
n−3
0
+
[τ
−1
y(s)+(τ
−1
−1)e(s)],...,[τ
−1
y(s)+(τ
−1
−1)e(s)]

ds−e(t)
=
Z
1
0
G(t,s)q(s)g

τI
n−2
0
+
x(s)+(τ−1)I
n−2
0
+
e(s),τI
n−3
0
+
x(s)+(τ−1)I
n−3
0
+
e(s),...,τx(s)+(τ−1)e(s),
τ
−1
I
n−2
0
+
y(s)+(τ
−1
−1)I
n−2
0
+
e(s),τ
−1
I
n−3
0
+
y(s)+(τ
−1
−1)I
n−3
0
+
e(s),...,τ
−1
y+(τ
−1
−1)e(s)

ds−e(t)
≥
Z
1
0
G(t,s)q(s)g

τI
n−2
0
+
x(s)+(τ−1)E
∗
,τI
n−3
0
+
x(s)+(τ−1)E
∗
,...,τx(s)+(τ−1)E
∗
,
τ
−1
I
n−2
0
+
y(s)+(τ
−1
−1)E
∗
,τ
−1
I
n−3
0
+
y(s)+(τ
−1
−1)E
∗
,...,τ
−1
y+(τ
−1
−1)E
∗

ds−e(t)
=
Z
1
0
G(t,s)q(s)g

τI
n−2
0
+
x(s)+(τ−1)ρ
0
,τI
n−3
0
+
x(s)+(τ−1)ρ
0
,...,τx(s)+(τ−1)ρ
0
,
τ
−1
I
n−2
0
+
y(s)+(τ
−1
−1)ρ
0
,τ
−1
I
n−3
0
+
y(s)+(τ
−1
−1)ρ
0
,...,τ
−1
y+(τ
−1
−1)ρ
0

ds−e(t)
≥ϕ(τ)
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)
=ϕ(τ)
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)+ϕ(τ)e(t)−ϕ(τ)e(t)
=ϕ(τ)[
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)]+(ϕ(τ)−1)e(t)
=ϕ(τ)A(x,y)(t)+(ϕ(τ)−1)e(t),
B

τx+(τ−1)e,τ
−1
y+(τ
−1
−1)e

(t)
=
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
[τx(s)+(τ−1)e(s)],I
n−3
0
+
[τx(s)+(τ−1)e(s)],...,τx(s)+(τ−1)e(s),
I
n−2
0
+
[τ
−1
y(s)+(τ
−1
−1)e(s)],I
n−3
0
+
[τ
−1
y(s)+(τ
−1
−1)e(s)],...,τ
−1
y(s)+(τ
−1
−1)e(s)

ds−e(t)
=
Z
1
0
G(t,s)q(s)φ

τI
n−2
0
+
x(s)+(τ−1)I
n−2
0
+
e(s),τI
n−3
0
+
x(s)+(τ−1)I
n−3
0
+
e(s),...,τx(s)+(τ−1)e(s),
τ
−1
I
n−2
0
+
y(s)+(τ
−1
−1)I
n−2
0
+
e(s),τ
−1
I
n−3
0
+
y(s)+(τ
−1
−1)I
n−3
0
+
e(s),...,τ
−1
y+(τ
−1
−1)e(s)

ds−e(t)
≥
Z
1
0
G(t,s)q(s)φ

τI
n−2
0
+
x(s)+(τ−1)E
∗
,τI
n−3
0
+
x(s)+(τ−1)E
∗
,...,τx(s)+(τ−1)E
∗
,
τ
−1
I
n−2
0
+
y(s)+(τ
−1
−1)E
∗
,τ
−1
I
n−3
0
+
y(s)+(τ
−1
−1)E
∗
,...,τ
−1
y+(τ
−1
−1)E
∗

ds−e(t)
=
Z
1
0
G(t,s)q(s)φ

τI
n−2
0
+
x(s)+(τ−1)ρ
0
,τI
n−3
0
+
x(s)+(τ−1)ρ
0
,...,τx(s)+(τ−1)ρ
0
,
τ
−1
I
n−2
0
+
y(s)+(τ
−1
−1)ρ
0
,τ
−1
I
n−3
0
+
y(s)+(τ
−1
−1)ρ
0
,...,τ
−1
y+(τ
−1
−1)ρ
0

ds−e(t)
≥τ
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)
=τ
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)+τe(t)−τe(t)
=τ[
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)]+(τ−1)e(t)
=τB(x,y)(t)+(τ−1)e(t).
DOI:10.12677/pm.2023.1361811776nØêÆ
ܶ
(3)1n,y²A(h,h)∈C
h,e
,B(h,h)∈C
h,e
.=yA(h,h) + e∈C
h
,B(h,h) +e∈C
h
.dÚ
n2.2,(H
4
),(H
5
)Œ±Ñ
A(h,h)(t)+e(t) =
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
h(s),I
n−3
0
+
h(s),...,h(s),I
n−2
0
+
h(s),I
n−3
0
+
h(s),...,h(s)

ds
=
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
Ms
α−n+1
,I
n−3
0
+
Ms
α−n+1
,
...,Ms
α−n+1
,I
n−2
0
+
Ms
α−n+1
,I
n−3
0
+
Ms
α−n+1
,...,Ms
α−n+1

ds
=
Z
1
0
G(t,s)q(s)g

MI
n−2
0
+
s
α−n+1
,MI
n−3
0
+
s
α−n+1
,
...,Ms
α−n+1
,MI
n−2
0
+
s
α−n+1
,MI
n−3
0
+
s
α−n+1
,...,Ms
α−n+1

ds
≤
Z
1
0
t
α−n+1
Γ(α−n+2)
q(s)g(M,M,...,M,0,0,...,0)ds
=
1
Γ(α−n+2)
g(M,M,...,M,0,0,...,0)
Z
1
0
q(s)ds·t
α−n+1
=
1
MΓ(α−n+2)
g(M,M,...,M,0,0,...,0)
Z
1
0
q(s)ds·h(t)
Ú
A(h,h)(t)+e(t) ≥
Z
1
0
ρ(s)G(s,s)q(s)g(0,0,...,0,M,M,...,M)ds
≥
Z
1
0
ρ(s)G(s,s)q(s)δ
0
φ(0,0,...,0,M,M,...,M)ds
≥δ
0
φ(0,0,...,0,M,M,...,M)
Z
1
0
ρ(s)G(s,s)q(s)ds·t
α−n+1
=
1
M
δ
0
φ(0,0,...,0,M,M,...,M)
Z
1
0
ρ(s)G(s,s)q(s)ds·h(t).
-
l
1
=
1
M
δ
0
φ(0,0,...,0,M,M,...,M)
Z
1
0
ρ(s)G(s,s)q(s)ds,
L
1
=
1
MΓ(α−n+2)
g(M,M,...,M,0,0,...,0)
Z
1
0
q(s)ds.
=0 <l
1
≤L
1
<+∞,l
1
h(t) ≤A(h,h)(t)+e(t)≤L
1
h(t),t∈[0,1],ÏdkA(h,h)∈C
h,e
.Ó
•{k
B(h,h)(t)+e(t) =
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
h(s),I
n−3
0
+
h(s),...,h(s),I
n−2
0
+
h(s),I
n−3
0
+
h(s),...,h(s)

ds
DOI:10.12677/pm.2023.1361811777nØêÆ
ܶ
≤
Z
1
0
t
α−n+1
Γ(α−n+2)
q(s)φ(M,M,...,M,0,0,...,0)ds
=
1
Γ(α−n+2)
φ(M,M,...,M,0,0,...,0)
Z
1
0
q(s)ds·t
α−n+1
=
1
MΓ(α−n+2)
φ(M,M,...,M,0,0,...,0)
Z
1
0
q(s)ds·h(t)
Ú
B(x,x)(t)+e(t) ≥
Z
1
0
ρ(s)G(s,s)q(s)φ(0,0,...,0,M,M,...,M)ds
≥φ(0,0,...,0,M,M,...,M)
Z
1
0
ρ(s)G(s,s)q(s)ds·t
α−n+1
=
1
M
φ(0,0,...,0,M,M,...,M)
Z
1
0
ρ(s)G(s,s)q(s)ds·h(t).
-
l
2
=
1
M
φ(0,0,...,0,M,M,...,M)
Z
1
0
ρ(s)G(s,s)q(s)ds,
L
2
=
1
MΓ(α−n+2)
φ(M,M,...,M,0,0,...,0)
Z
1
0
q(s)ds.
=0 <l
2
≤L
2
<+∞,l
2
h(t) ≤B(h,h)(t)+e(t) ≤L
2
h(t),t∈[0,1],ÏdkB(h,h) ∈C
h,e
.
(4)•,y²Ún2.4¥(iv)¤á.éz‡x,y∈C
h,e
,t∈[0,1],d(H
4
)Ñ
A(x,y)(t) =
Z
1
0
G(t,s)q(s)g

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)
≥δ
0
Z
1
0
G(t,s)q(s)φ

I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s)

ds−e(t)−
δ
0
e(t)+δ
0
e(t)
=δ
0
[
Z
1
0
G(t,s)q(s)φ(I
n−2
0
+
x(s),I
n−3
0
+
x(s),...,x(s),I
n−2
0
+
y(s),I
n−3
0
+
y(s),...,y(s))ds−e(t)]+
(δ
0
−1)e(t)
=δ
0
B(x,y)(t)+(δ
0
−1)e(t),
=A(x,y)≥δ
0
B(x,y)+(δ
0
−1)e.ÏdÚn2.4¥¤k^‡Ñ÷v.Ïd½n3.1¥(ؤá.

~3.2•ÄXeBVP:



D
5
2
0
+
u(t)+q(t)f(u(t),u
0
(t))−1 = 0,t∈[0,1],
u(0) = u
0
(0) = u
0
(1) = 0,
(7)
Ù¥q(t) = t
1
2
,f(u(t),u
0
(t)) = (1+t)[(
e(t)
E
∗
u(t)+e(t))
1
2
+(
e(t)
E
∗
u(t)+e(t))
1
3
+(
e(t)
E
∗
u
0
(t)+e(t))
−
1
5
+
(
e(t)
E
∗
u
0
(t)+e(t))
−
1
6
],E
∗
= max{e(t),I
1
0
+
e(t) : t∈[0,1]}=
472Γ(
7
2
)
2539
.
DOI:10.12677/pm.2023.1361811778nØêÆ
ܶ
-u(t) = I
1
0
+
x(t),@o•§(7)Œ=z•Xe/ª:



D
3
2
0
+
x(t)+q(t)f(I
1
0
+
x(t),x(t))−1 = 0,t∈[0,1],
x(0) = x(1) = 0,
(8)
Ù¥
f(I
1
0
+
x(t),x(t))=f(x,y)=(1+t)[(
e(t)
E
∗
x+e(t))
1
2
+(
e(t)
E
∗
x+e(t))
1
3
+(
e(t)
E
∗
y+e(t))
−
1
5
+(
e(t)
E
∗
y+e(t))
−
1
6
],
g(x,x,y,y)=(
e(t)
E
∗
x+e(t))
1
2
+(
e(t)
E
∗
x+e(t))
1
3
+(
e(t)
E
∗
y+e(t))
−
1
5
+(
e(t)
E
∗
y+e(t))
−
1
6
,
φ(x,x,y,y)=t(
e(t)
E
∗
x+e(t))
1
2
+t(
e(t)
E
∗
x+e(t))
1
3
+t(
e(t)
E
∗
y+e(t))
−
1
5
+t(
e(t)
E
∗
y+e(t))
−
1
6
.
éu?¿x,y>−E
∗
, kf(x,y) = g(x,x,y,y)+φ(x,x,y,y),e(t) =
t
3
2
−t
5
2
Γ(
7
2
)
,E
∗
= max{e(t),I
1
0
+
e(t) :
t∈[0,1]}=
472Γ(
7
2
)
2539
.e5,u½n3.1¥¤k^‡´ÄÑ÷v.
(i)w,,¼êg,φ: [−E
∗
,+∞)
4
→(−∞,+∞)´ëY.
(ii)éu½y
i
∈[−E
∗
,+∞)(i= 1,2),g(x,x,y,y),φ(x,x,y,y)3x
i
∈[−E
∗
,+∞)(i= 1,2)
þ´š~;éu½x
i
∈[−E
∗
,+∞)(i=1,2),g(x,x,y,y),φ(x,x,y,y)3y
i
∈[−E
∗
,+∞)(i=
1,2)þ´šO.
(iii)éτ∈(0,1),x
i
,y
i
∈[−E
∗
,+∞)(i= 1,2),ρ
0
= e,ϕ(τ) = τ
1
2
,k
g(τx+(τ−1)ρ
0
,τx+(τ−1)ρ
0
,τ
−1
y+(τ
−1
−1)ρ
0
,τ
−1
y+(τ
−1
−1)ρ
0
)
=g(τx+(τ−1)e,τx+(τ−1)e,τ
−1
y+(τ
−1
−1)e,τ
−1
y+(τ
−1
−1)e)
=

τ
e(t)
E
∗
x+(τ−1)e(t)+e(t)

1
2
+(τ
e(t)
E
∗
x+(τ−1)e(t)+e(t))
1
3
+

τ
−1
e(t)
E
∗
y+
(τ
−1
−1)e(t)+e(t)

−
1
5
+

τ
−1
e(t)
E
∗
y+(τ
−1
−1)e(t)+e(t)

−
1
6
=(τ
e(t)
E
∗
x+τe(t))
1
2
+(τ
e(t)
E
∗
x+τe(t))
1
3
+(τ
−1
e(t)
E
∗
y+τ
−1
e(t))
−
1
5
+(τ
−1
e(t)
E
∗
y+τ
−1
e(t))
−
1
6
=τ
1
2
(
e(t)
E
∗
x+e(t))
1
2
+τ
1
3
(
e(t)
E
∗
x+e(t))
1
3
+τ
1
5
(
e(t)
E
∗
y+e(t))
−
1
5
+τ
1
6
(
e(t)
E
∗
y+e(t))
−
1
6
>τ
1
2
[(
e(t)
E
∗
x+e(t))
1
2
+(
e(t)
E
∗
x+e(t))
1
3
+(
e(t)
E
∗
y+e(t))
−
1
5
+(
e(t)
E
∗
y+e(t))
−
1
6
]
=ϕ(τ)g(x,x,y,y).
éuτ∈(0,1)Úx
i
,y
i
∈[−E
∗
,+∞),(i= 1,2),ρ
0
= e,k
φ(τx+(τ−1)ρ
0
,τx+(τ−1)ρ
0
,τ
−1
y+(τ
−1
−1)ρ
0
,τ
−1
y+(τ
−1
−1)ρ
0
)
=φ(τx+(τ−1)e,τx+(τ−1)e,τ
−1
y+(τ
−1
−1)e,τ
−1
y+(τ
−1
−1)e)
=t[(τ
e(t)
E
∗
x+(τ−1)e(t)+e(t))
1
2
+(τ
e(t)
E
∗
x+(τ−1)e(t)+e(t))
1
3
+

τ
−1
e(t)
E
∗
y+
(τ
−1
−1)e(t)+e(t)

−
1
5
+(τ
−1
e(t)
E
∗
y+(τ
−1
−1)e(t)+e(t))
−
1
6
]
DOI:10.12677/pm.2023.1361811779nØêÆ
ܶ
=t[(τ
e(t)
E
∗
x+τe(t))
1
2
+(τ
e(t)
E
∗
x+τe(t))
1
3
+(τ
−1
e(t)
E
∗
y+τ
−1
e(t))
−
1
5
+(τ
−1
e(t)
E
∗
y+τ
−1
e(t))
−
1
6
]
=t[τ
1
2
(
e(t)
E
∗
x+e(t))
1
2
+τ
1
3
(
e(t)
E
∗
x+e(t))
1
3
+τ
1
5
(
e(t)
E
∗
y+e(t))
−
1
5
+τ
1
6
(
e(t)
E
∗
y+e(t))
−
1
6
]
>t{τ
1
6
[(
e(t)
E
∗
x+e(t))
1
2
+(
e(t)
E
∗
x+e(t))
1
3
+(
e(t)
E
∗
y+e(t))
−
1
5
+(
e(t)
E
∗
y+e(t))
−
1
6
]}
>τ{t[(
e(t)
E
∗
x+e(t))
1
2
+(
e(t)
E
∗
x+e(t))
1
3
+(
e(t)
E
∗
y+e(t))
−
1
5
+(
e(t)
E
∗
y+e(t))
−
1
6
]}
=τφ(x,x,y,y).
(iv)ρ
0
= e,δ
0
= 1 >0.@o
g(x,x,y,y) =(
e(t)
E
∗
x+ρ
0
)
1
2
+(
e(t)
E
∗
x+ρ
0
)
1
3
+(
e(t)
E
∗
y+ρ
0
)
−
1
5
+(
e(t)
E
∗
y+ρ
0
)
−
1
6
=(
e(t)
E
∗
x+e(t))
1
2
+(
e(t)
E
∗
x+e(t))
1
3
+(
e(t)
E
∗
y+e(t))
−
1
5
+(
e(t)
E
∗
y+e(t))
−
1
6
>1{t[(
e(t)
E
∗
x+e(t))
1
2
+(
e(t)
E
∗
x+e(t))
1
3
+(
e(t)
E
∗
y+e(t))
−
1
5
+(
e(t)
E
∗
y+e(t))
−
1
6
]}
=δ
0
φ(x,x,y,y).
(v)M= 1.51 >
1
(
5
2
−3+1)Γ(
5
2
−3+3)
,
g(M,M,0,0) <(1.51+1)
1
2
+(1.51+1)
1
3
+1+1 = 4.9433 <+∞,
φ(M,M,0,0) <(1.51+1)
1
2
+(1.51+1)
1
3
+1+1 = 4.9433 <+∞,
φ(0,0,M,M) >0.
Ïd,½n3.1¥bÜ÷v.(ÜÚn2.3,BVP7•3•˜š²…).d,˜±eS
:
ω
n
(t) =
Z
1
0
G(t,s)

(
e(s)
E
∗
ω
n−1
(s)+e(s))
1
2
+(
e(s)
E
∗
ω
n−1
(s)+e(s))
1
3
+(
e(s)
E
∗
τ
n−1
(s)+
e(s))
−
1
5
+(
e(s)
E
∗
τ
n−1
(s)+e(s))
−
1
6

ds+
Z
1
0
G(t,s)

t(
e(s)
E
∗
ω
n−1
(s)+e(s))
1
2
+
t(
e(s)
E
∗
ω
n−1
(s)+e(s))
1
3
+t(
e(s)
E
∗
τ
n−1
(s)+e(s))
−
1
5
+s(
e(s)
E
∗
τ
n−1
(s)+
e(s))
−
1
6

ds+
t
3
2
−t
5
2
Γ(
7
2
)
,n= 1,2,···
Ú
τ
n
(t) =
Z
1
0
G(t,s)

(
e(s)
E
∗
τ
n−1
(s)+e(s))
1
2
+(
e(s)
E
∗
τ
n−1
(s)+e(s))
1
3
+(
e(s)
E
∗
ω
n−1
(s)+
e(s))
−
1
5
+(
e(s)
E
∗
ω
n−1
(s)+e(s))
−
1
6

ds+
Z
1
0
G(t,s)

t(
e(s)
E
∗
τ
n−1
(s)+e(s))
1
2
+
DOI:10.12677/pm.2023.1361811780nØêÆ
ܶ
t(
e(s)
E
∗
τ
n−1
(s)+e(s))
1
3
+t(
e(s)
E
∗
ω
n−1
(s)+e(s))
−
1
5
+s(
e(s)
E
∗
ω
n−1
(s)+
e(s))
−
1
6

ds+
t
3
2
−t
5
2
Γ(
7
2
)
,n= 1,2,···,
éu?¿‰½Њω
0
,τ
0
∈C
h,e
,…t∈[0,1],·‚k{ω
n
(t)}Ú{τ
n
(t)}ј—Âñux
∗
.Ïd,½
n3.1¥b¤á.(ÜÚn2.3,BVP7•3•˜š²…).
ë•©z
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