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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(3),1412-1419
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.113154
á«mþ©ê‡©•§S“)
¡¡¡RRR§§§LLL
∗
§§§VVV‰‰‰jjj§§§
§ŒÆ§êƆÚOÆ§ìÀ§
ÂvFϵ2022c228F¶¹^Fϵ2022c322F¶uÙFϵ2022c329F
Á‡
©Ì‡ï Äá«mþ ©ê‡© •§§A^üNS“•{,3˜½^‡e§•§
4Š)Ú)S“S"
'…c
©ê‡©•§§S“)§Ã¡«m
IterativeSolutionsofFractional
DifferentialEquationsonInfiniteInterval
XiaodieHu,YingWang
∗
,MengqiKan,DeyangGu
SchoolofMathematicsandStatistics,LinyiUniversity,LinyiShandong
Received:Feb.28
th
,2022;accepted:Mar.22
nd
,2022;published:Mar.29
th
,2022
Abstract
Inthispaper,wemainlyinvestigatethefractionaldifferentialequationoninfinite
interval.Undercertain conditions,weestablishtheexistenceofextremalsolutionsas
∗ÏÕŠö"
©ÙÚ^:¡R,L,V‰j,.á«mþ©ê‡©•§S“)[J].A^êÆ?Ð,2022,11(3):
1412-1419.DOI:10.12677/aam.2022.113154
¡R
wellasiterativeschemesbyemployingthemonotoneiterativetechnique.
Keywords
FractionalDifferentialEquation,IterativeSolution,InfiniteInterval
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
©ïÄá«mþ©ê‡©•§È©>Š¯K(BVP):





D
α
0
+
x(t)+a(t)f(t,x(t)) = 0,0 <t<+∞,
x(0) = x
0
(0) = ···= x
(n−2)
= 0,lim
t→+∞
D
α−1
0
+
x(t) =
Z
+∞
0
h(t)x(t)dA(t),
(1.1)
Ù¥n−1 <α≤n, n≥2, D
α
0
+
´Riemann-Liouville ‡©.a∈L[0,+∞), a(t) 6≡0, 0 <
R
∞
0
a(s)ds<
+∞,h:(0,+∞)→[0,+∞)´ëY¿…h∈L
1
(0,+∞),
R
+∞
0
h(s)x(s)dA(s)L«äk2ÂÿÝ
Riemann-StieltjesÈ©,A:(0,+∞)→(−∞,+∞)´k.C¼ê,
R
+∞
0
h(t)t
α−1
dA(t)<Γ(α),
R
+∞
0
h(t)dA(t) <Γ(α).f: [0,+∞]×[0,+∞) →[0,+∞)ëY.
‡©•§Ã¡>Š¯KïÄ,©u1896cKueser[1].AgarwalÚO’Regan[2]éá«mþ
‡© •§‰•[0.ZhaoÚGe[3]$^ØÄ:½nïÄá«mþ©ê‡©•§n:
>Š¯KÃ.).LiangÚZhang[4,5]¼õ:>Š^‡e,•§õ)5.á«mþ©
ê‡©•§>Š¯KïÄ,Œõ•Ä´©êê½Âeõ:,È©>Š¯K[6–8],é
uRiemann-StieltjesÈ©½ÂeÈ©>Š¯K ïÄ,ƒé.Ïd,©¥•§ïÄ´›©
k¿Â.
2.ý•£
½Â2.1 [9,10](Riemann-Liouville)αÈ©½Â•
I
α
0+
x(t) =
1
Γ(α)
Z
t
0
(t−s)
α−1
x(s)ds,
DOI:10.12677/aam.2022.1131541413A^êÆ?Ð
¡R
Ù¥n−1 ≤α<n,n•ê.
½Â2.2 [9,10](Riemann-Liouville)αê½Â•
D
α
0+
x(t) =
1
Γ(n−α)

d
dt

n
Z
t
0
(t−s)
n−α−1
x(s)ds,
Ù¥n−1 ≤α<n,n•ê.
Ún2.1 [9,10]eα>0,x∈L(0,1),D
α
0+
x∈L(0,1),K
I
α
0+
D
α
0+
x(t) = x(t)+c
1
t
α−1
+c
2
t
α−2
+···+c
n
t
α−n
,
Ù¥c
i
∈(−∞,+∞),i= 1,2,...,n,n−1 <α≤n.
Ún2.2 by∈C(0,+∞)∩L(0,+∞),K©ê‡©•§







D
α
0
+
x(t)+y(t) = 0,t∈(0,+∞),n−1 <α≤n,n≥2,
x(0) = x
0
(0) = ···= x
(n−2)
= 0,lim
t→+∞
D
α−1
0
+
x(t) =
Z
+∞
0
h(t)x(t)dA(t)
(2.1)
k)
x(t) =
Z
∞
0
G(t,s)y(s)ds,
Ù¥
G(t,s) = G
0
(t,s)+G
1
(t,s),(2.2)
G
0
(t,s) =
1
Γ(α)



t
α−1
−(t−s)
α−1
,0 ≤s≤t≤+∞,
t
α−1
,0 ≤t≤s≤+∞,
G
1
(t,s) =
t
α−1
Γ(α)−
R
+∞
0
h(t)t
α−1
dA(t)
Z
+∞
0
h(t)G
0
(t,s)dA(t).
Ún2.3d(2.2)½ÂG(t,s)ke5Ÿµ
(1)G(t,s) ≥0,(t,s) ∈[0,+∞]×[0,+∞].
(2)G(t,s)3[0,+∞]×[0,+∞]þëY.
(3)G(t,s) ≤ω,ω= max
n
1
Γ(α)
,
R
+∞
0
h(t)dA(t)
Γ(α)(Γ(α)−
R
+∞
0
h(t)t
α−1
dA(t))
o
.
DOI:10.12677/aam.2022.1131541414A^êÆ?Ð
¡R
y²ŠâG(t,s)½Â,•Iy²(3)¤á.du
G
0
(t,s) ≤
t
α−1
Γ(α)
≤
1
Γ(α)
,
G
1
(t,s) ≤
t
α−1
Γ(α)

Γ(α)−
R
+∞
0
h(t)t
α−1
dA(t)

Z
+∞
0
h(t)dA(t)
≤
1
Γ(α)

Γ(α)−
R
+∞
0
h(t)t
α−1
dA(t)

Z
+∞
0
h(t)dA(t),
¤±G(t,s) = G
0
(t,s)+G
1
(t,s) ≤ω.
3.̇(J
X= C[0,+∞),½Â
E=

x∈C[0,+∞) : sup
t∈J
|x(t)|
1+t
α−1
<+∞

.(3.1)
‰êkxk= sup
t∈J
|x(t)|
1+t
α−1
,KE´Banach˜m,P
K= {x∈E: x(t) ≥0,t∈J}.
ÏdK´X˜‡I.
©,·‚be¡^‡(H
1
)¤á.
(H
1
)f:[0,+∞) ×[0,+∞)→[0,+∞)´ëY¼ê.f(t,0)6≡0,¿…3[0,+∞)þuk.ž,
f(t,(1+t
α−1
)uk..
d(H
1
),½ÂÈ©ŽfT: K→X:
(Tx)(t) =
Z
+∞
0
G(t,s)a(s)f(s,x(s)ds,t∈[0,+∞).(3.2)
w,BVP(1.1)k)x…=x∈K´d(3.2)½ÂŽfTØÄ:.
Ún3.1[11,12]Ed(3.1)½Â,M´E¥k.8,e
n
x(t)
1+t
: x∈M
o
,{x
0
(t) : x∈M}3Jþ
?˜k.f8þÝëY,…é?¿‰½ε>0,•3N>0,t
1
,t
2
>N,¦



x(t
1
)
1+t
−
x(t
2
)
1+t



<ε,
|x
0
(t
1
)−x
0
(t
2
)|<εéx∈M˜—¤á,KM3E¥´ƒé;.
dÚn3.1´•,e¡½n3.1¤á.
½n3.1b^‡(H
1
)¤á,KT: K→K´ëYŽf.
DOI:10.12677/aam.2022.1131541415A^êÆ?Ð
¡R
½n3.2b^‡(H
1
)¤á,¿…•3~êd>0÷ve^‡:
(H
2
)f(t,u) ≤f(t,u),t∈[0,+∞),0 ≤u≤u.
(H
3
)f(t,(1+t
α−1
)u) ≤
d
%
,(t,u) ∈[0,+∞)×[0,d],
where
%= ω
Z
∞
0
a(s)ds,ωdÚn2.3½Â.
KBVP(1.1)k4Œ)Ú4)w
∗
,ν
∗
on[0,+∞),÷v
0 <sup
t∈[0,+∞)
|w
∗
(t)|
1+t
α−1
≤d,0 <sup
t∈[0,+∞)
|ν
∗
(t)|
1+t
α−1
≤d.
w
0
(t) = dt
α−1
,ν
0
(t) = 0,t∈[0,+∞),S“S{w
n
},{ν
n
}αLǥ
w
n
= ω
Z
∞
0
G(t,s)a(s)f(s,w
n−1
(s))ds,
ν
n
= ω
Z
∞
0
G(t,s)a(s)f(s,ν
n−1
(s))ds,
¿…k
lim
n→+∞
sup
t∈[0,+∞)
|w
n
(t)−w
∗
(t)|
1+t
α−1
= 0,lim
n→+∞
sup
t∈[0,+∞)
|ν
n
(t)−ν
∗
(t)|
1+t
α−1
= 0.
y²d½n3.1,T:K→K´ëYŽf.é?¿x
1
,x
2
∈K,x
1
≤x
2
,dŽfT½Â
Ú(H
2
)Υ,Tx
1
≤Tx
2
.-K
d
= {x∈K: kxk≤d}.e5,·‚Äky²T:K
d
→K
d
.é?¿
x∈K
d
,k0 ≤
x(t)
1+t
α−1
≤d,t∈[0,+∞).d(H
3
)Υ,
f(t,u) ≤ϕ
p

d
%

,(t,u) ∈[0,+∞)×[0,d].
dÚn2.3Ú(H
3
),
k(Tx)k=sup
t∈[0,+∞)
1
1+t
α−1
Z
+∞
0
G(t,s)a(s)f(s,x(s)ds
≤ω
Z
∞
0
a(s)f(s,x(s)ds≤d.
¤±,T: K
d
→K
d
.
bw
0
(t)=dt
α−1
,t∈[0,+∞),Kw
0
(t)∈K
d
.-w
1
=Tw
0
,w
2
=Tw
1
=T
2
w
0
,d½
n3.1,w
1
,w
2
∈K
d
.½Âw
n+1
=Tw
n
=T
n
w
0
, n=1,2,···.duT:K
d
→K
d
,·‚
kw
n
∈T(K
d
)⊂K
d
,w
n
∈A(K
d
)⊂K
d
.dŽfTëY5Œ•{w
n
}
∞
n=1
´E¥;8.d
DOI:10.12677/aam.2022.1131541416A^êÆ?Ð
¡R
^‡(H
3
),·‚k
w
1
(t) =
Z
+∞
0
G(t,s)a(s)f(s,w
0
(s)ds
≤ω
Z
+∞
0
a(s)f(s,w
0
(s)ds
≤ωt
α−1
Z
+∞
0
a(s)f(s,w
0
(s)ds
≤dt
α−1
= w
0
(t).
(3.3)
d(3.3)ªÚ^‡(H
2
)Œ
w
2
= Tw
1
≤Tw
0
= w
1
.(3.4)
8BŒ
w
n+1
≤w
n
,n= 1,2,···.(3.5)
Ïd,•3w
∗
∈K÷vw
n
→w
∗
,n→+∞.dTëY5Úw
n+1
= Tw
n
,kTw
∗
= w
∗
.
,˜•¡,duν
0
(t)=0,t∈[0,+∞),Kν
0
(t)∈K
d
.-ν
1
=Tν
0
,ν
2
=Tν
1
=T
2
ν
0
,
d½n3.1Œν
1
,ν
2
∈K
d
.Pν
n+1
=Tν
n
=T
n
ν
0
, n=1,2,···.duT:K
d
→K
d
,·‚
kν
n
∈T(K
d
) ⊂K
d
.dTëY5Œ•{ν
n
}
∞
n=1
´E¥;8.duν
1
= Tν
0
∈K
d
,k
ν
2
= Tν
1
≥0.
8BŒ
ν
n+1
≥ν
n
,n= 1,2,···.(3.6)
Ïd,•3ν
∗
∈K÷vν
n
→ν
∗
,n→+∞.A^TëY5Úν
n+1
= Tν
n
,·‚kTν
∗
= ν
∗
.
e¡y²w
∗
Úν
∗
´BVP(1.1)3(0,dt
α−1
]þ4Œ)Ú4).bu∈(0,dt
α−1
]´BVP(1.1)
?˜),=Tu=u.duT´š~,ν
0
(t)=0≤u(t)≤dt
α−1
=w
0
(t),Ïd,·‚kν
1
(t)=
(Tν
0
)(t) ≤u(t) ≤(Tw
0
)(t) = w
1
(t),t∈[0,+∞).8BŒ
ν
n
≤u≤w
n
,n= 1,2,3,···.(3.7)
duw
∗
= lim
n→+∞
w
n
,ν
∗
= lim
n→+∞
ν
n
,d(3.3)-(3.7)ª,Œ±
ν
0
≤ν
1
≤···ν
n
≤···≤ν
∗
≤u≤w
∗
≤···≤w
n
≤···≤w
1
≤w
0
.(3.8)
duf(t,0) 6≡0,t∈[0,+∞), 0 Ø´BVP(1.1)).¤±, d(3.8)ªŒ•w
∗
Úν
∗
´BVP(1.1)
3(0,dt
α−1
]þ4Œ)Ú4),¿…w
∗
Úν
∗
Œ±dS“Sw
n
=Tw
n−1
,ν
n
=Tν
n−1
.

DOI:10.12677/aam.2022.1131541417A^êÆ?Ð
¡R
Ä7‘8
©É§ŒÆŒÆ)M#M’ÔöOy‘8(X202110452130)Ü©]Ï"
ë•©z
[1]Kueser,A. (1996)Untersuchung und asymptotische darstellung der intergrale genwisserdiffer-
entialgleichungenbeigrossenwerthendesarguments.Journalf¨urdieReineundAngewandte
Mathematik,116,178-212.
[2]Agarwal,P.R.andO’Regan,D.(2001)InfiniteIntervalProblemsforDifferential,Difference
andIntegralEquations.KluwerAcademicPublishers,Dordrecht.
https://doi.org/10.1007/978-94-010-0718-4
[3]Zhao, X. and Ge, W.(2010) Unbounded SolutionsforaFractional DifferentialBoundary Value
ProblemsontheInfiniteInterval.ActaApplicandaeMathematicae,109,495-505.
https://doi.org/10.1007/s10440-008-9329-9
[4]Liang,S.andZhang,J.(2011)ExistenceofThreePositiveSolutionsofm-PointBoundary
ValueProblemsforSomeNonlinearFractionalDifferentialEquationsonanInfiniteInterval.
ComputersandMathematicswithApplications,61,3343-3354.
https://doi.org/10.1016/j.camwa.2011.04.018
[5]Liang,S.and Zhang,J.(2011)Existence of MultiplePositive Solutionsform-PointFractional
BoundaryValueProblemsonan InfiniteInterval.MathematicalandComputerModelling,54,
1334-1346.https://doi.org/10.1016/j.mcm.2011.04.004
[6]Zhang,X.andZhong,Q.(2018)TriplePositiveSolutionsforNonlocalFractionalDifferential
Equations with Singularities Bothon Timeand SpaceVariables.AppliedMathematicsLetters,
80,12-19.https://doi.org/10.1016/j.aml.2017.12.022
[7]Wang,F.,Liu,L.andWu,Y.(2019)IterativeUniquePositiveSolutionsforaNewClassof
Nonlinear SingularHigher OrderFractionalDifferential EquationswithMixed-Type Boundary
ValueConditions.JournalofInequalitiesandApplications,2019,ArticleNo.210.
https://doi.org/10.1186/s13660-019-2164-x
[8]Tan,J.,Zhang,X.,Liu,L.andWu,Y.(2021)AnIterativeAlgorithmforSolvingn-Order
FractionalDifferentialEquationwithMixedIntegralandMultipointBoundaryConditions.
Complexity,2021,ArticleID:8898859.https://doi.org/10.1155/2021/8898859
[9]Podlubny,I.(1999)FractionalDifferentialEquations,Vol.198.AcademicPress,SanDiego,
CA.
[10]Miller,K.S.andRoss,B.(1993)AnIntroductiontotheFractionalCalculusandFractional
DifferentialEquations.Wiley,NewYork.
DOI:10.12677/aam.2022.1131541418A^êÆ?Ð
¡R
[11]Liu, Y.(2002)BoundaryValueProblemforSecondOrderDifferentialEquationsonUnbounded
Domain.ActaAnalysisFunctionalisApplicata,4,211-216.
[12]Corduneanu, C.(1973)Integral Equations andStability of Feedback Systems.AcademicPress,
NewYork.
DOI:10.12677/aam.2022.1131541419A^êÆ?Ð

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