Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

ma hema ics
A icle
F ac ional Lange in Equa ion In ol ing Two F ac ional
O de s: Exis ence and Uniqueness Re isi ed
Hossein Fazli 1,∗, HongGuang Sun 1and Juan J. Nie o 2
1
S a e Key Labo a o y o Hyd ology-Wa e Resou ces and Hyd aulic Enginee ing, College o Mechanics and
Ma e ials, Hohai Uni e si y, Nanjing 210098, China; [email p o ec ed]
2Depa men o S a is ics, Ma hema ical Analysis and Op imiza ion, Ins i u e o Ma hema ics, Uni e si y o
San iago de Compos ela, 15782 San iago de Compos ela, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 28 Ma ch 2020; Accep ed: 5 May 2020; Published: 8 May 2020


Abs ac :
We conside he nonlinea ac ional Lange in equa ion in ol ing wo ac ional o de s
wi h ini ial condi ions. Using some basic p ope ies o P abhaka in eg al ope a o , we ind an
equi alen Vol e a in eg al equa ion wi h wo pa ame e Mi ag–Le le unc ion in he ke nel o he
men ioned equa ion. We used he con ac ion mapping heo em and Weissinge ’s ixed poin heo em
o ob ain exis ence and uniqueness o global solu ion in he spaces o Lebesgue in eg able unc ions.
The new ep esen a ion o mula o he gene al solu ion helps us o ind he ixed poin p oblem
associa ed wi h he ac ional Lange in equa ion which i s con ac i i y cons an is independen o
he ic ion coe icien . Two examples a e discussed o illus a e he easibili y o he main heo ems.
Keywo ds:
ac ional Lange in equa ion; Mi ag–Le le unc ion; P abhaka in eg al ope a o ;
exis ence; uniqueness
1. In oduc ion
Dynamical beha io o physical p ocesses a e usually ep esen ed by di e en ial equa ions. I he
model o physical sys em in some ways possesses a memo y and he edi a y p ope ies, o ins ance,
iscoelas ic de o ma ion [
1
], anomalous di usion [
2
], s ock ma ke [
3
], bac e ial chemo axis [
4
] and
complex ne wo ks [
5
], elaxa ion in illed polyme ne wo ks [
6
], elaxa ion and eac ion kine ics
o polyme s [
7
], desc ip ion o mechanical sys ems subjec o damping [
8
], Beha io o Biomedical
Ma e ials [9]; he co esponding models can be desc ibed by he ac ional di e en ial equa ions.
Lange in equa ion is a undamen al heo y o he B ownian mo ion o desc ibe he e olu ion
o physical phenomena in luc ua ing en i onmen s [
10
,
11
]. F ac ional Lange in equa ion as a
gene aliza ion o classical one gi es a ac ional Gaussian p ocess pa ame ized by wo indices,
which is mo e lexible o modeling ac al p ocesses [12–16].
The i ually simul aneous de elopmen o ac ional de i a i es, a ious gene aliza ions o he
Lange in equa ion we e p oposed and s udied by a ious esea che s du ing ecen yea s. Despi e he
widesp ead use o many o he applica ions [
17
–
22
], he ac ional Lange in equa ion is ex ensi ely
s udied in li e a u e bo h om heo e ical and nume ical poin s o iew. Au ho s in [
23
] s udied
nonlinea ac ional Lange in equa ion in ol ing wo ac ional o de s in di e en in e als as a
gene alized o m o h ee poin hi d o de nonlocal bounda y alue p oblem o nonlinea o dina y
di e en ial equa ions. In [
24
], he au ho s ha e s udied ac ional Lange in equa ions wi h nonlocal
in eg al bounda y condi ions. Recen ly, an i-pe iodic bounda y alue p oblem o Lange in equa ion
in ol ing wo ac ional o de s has been s udied in [
25
]. Exis ence and uniqueness esul s o coupled
and uncoupled sys ems o ac ional Lange in equa ions o Riemann-Liou ille and Hadama d ypes
has been discussed in [
26
]. Guo e al. [
27
] ga e an e icien nume ical me hod o sol ing he ac ional
Ma hema ics 2020,8, 743; doi:10.3390/ma h8050743 www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2020,8, 743 2 o 10
Lange in equa ion wi h o wi hou an ex e nal o ce. Some mo e ecen wo k on Lange in equa ion
can be ound in [28–37].
In he cu en pape , we mainly ocus on he exis ence and uniqueness esul o he ac ional
Lange in equa ion in ol ing wo ac ional o de s:





Dβ(Dα+λ)x( ) = ( ,x( )), 0 < ≤1,
x(i)(0) = µi, 0 ≤i<l,
x(i+α)(0) = νi, 0 ≤i<n,
(1)
whe e
m−
1
<α≤m
,
n−
1
<β≤n
,
l=max{m
,
n}
,
m
,
n∈N
,
Dα
is he Capu o ac ional de i a i e,
x( )
is he pa icle displacemen ,
x(i+α)(
0
)
equals
DiDαx(
0
)
, in he sequen ial sense,
λ∈R
is he
ic ion coe icien and :[0, 1]×R→Ris a gi en unc ion which ep esen s a noise e m.
Based on he c i e ia speci ied in [
38
], he p oblem
(1)
is a gene al o m o anomalous sys ems
go e ned by a gene alized Lange in equa ion wi h long- ange memo y. In con as o he classical
Lange in equa ion, we use
Dβx( )
and
DβDαx( )
ins ead o he o dina y de ini ion o he eloci y and
accele a ion as he i s and second de i a i es o he displacemen o de i e a gene alized Lange in
equa ion in ol ing ic ion memo y ke nel. Fo example, i 0
<β≤
1,
α=
1, hen acco ding o he
s anda d de ini ion o he Capu o ac ional de i a e ope a o , we ha e a special case o gene alized
Lange in equa ion in ol ing ic ion memo y ke nel equal o
λ
Γ(1−β) β−1
. Based on he calcula ions
in ([
39
], Sec ion B), in his case, he esul ing mo ion is in ac subdi usi e. Fu he mo e, i is wo h
no ing ha , i
α+β>
2, hen we do no ha e any physical meaning o he main p oblem. Fo his
case, i is only a aluable p oblem in he ho y o ac ional di e en ial equa ions as a sequen ial
ac ional di e en ial equa ion wi h ini ial condi ions.
As we ha e seen in he pape s ci ed abo e abou analysis o ac ional Lange in equa ion,
using a ious classical ixed poin heo ems is a common and use ul echnique o ob aining he
exis ence and uniqueness esul s o ac ional Lange in equa ion in ol ing di e en ini ial o
bounda y condi ions. In he men ioned pape s, he con ac i i y cons an o he ixed poin p oblem
associa ed wi h he ac ional Lange in equa ion depended on he ic ion coe icien
λ
. Fo example,
in he ob ained exis ence and unique esul s in [
33
,
34
], he con ac i i y cons an s
R1
,
R2
sa is y he
ollowing condi ions
R1=sup
0≤ ≤1Z
0
( −s)α+β−1
Γ(α+β)a(s)ds +|λ|
Γ(α+1)<1, (2)
and
R2=kkakp
Γ(α+β)ds +|λ|
Γ(α)<1, (3)
whe e
k=1
1−q(1−α)1
q
and
p−1+q−1=
1, espec i ely. As s a ed in ela ions
(2)
and
(3)
,
he con ac i i y cons an s
R1
,
R2
depend on he ic ion cons an
λ
. The e o e, om
(2)
, we can
no discuss he p oblems in ol ing he ic ion cons an
|λ| ≥ Γ(α+
1
)
. Simila ly, om
(3)
, we can
no s udy he p oblems in ol ing he ic ion cons an
|λ| ≥ Γ(α)
. No e ha 0
<
1
−q(
1
−α)<
1.
The e o e, we canno discuss he exis ence and uniqueness o solu ions o he p oblems in ol ing
la ge ic ion coe icien
λ
. In his pape , we s i e o o e come his majo limi a ion. Fi s we p opose
a new cons uc ion o he gene al solu ion o he Equa ion
(1)
using wo pa ame e Mi ag–Le le
unc ions and some o he basic p ope ies o P abhaka ope a o . This is done in Sec ion 2. Then we
ob ain a new exis ence and uniqueness esul s unde some weak condi ions by using con ac i e
mapping heo em and Weissinge ’s ixed poin heo em. This is con en o Sec ion 3. Two examples
a e gi en in Sec ion 4 o illus a e ou esul s.
Ma hema ics 2020,8, 743 3 o 10
2. P elimina ies and Auxilia y Resul s
In he ollowing sec ion, we apply some echnical calcula ions ela ed o ac ional calculus o
build a new gene al solu ion co esponding o ini ial alue p oblem (1) which p o ides an ex emely
powe ul ool o he p oo o he main esul . Fu he mo e, we p esen some p elimina ies and
no a ions ega ding ac ional calculus o he eade ’s con enience. Fo de ails, see [40–46].
De ini ion 1.
The Riemann-Liou ille ac ional in eg al o o de
α>
0 o he unc ion
x:[
0, 1
]→R
,
x∈L1[0, 1]is de ined as
Iαx( ) = 1
Γ(α)Z
0( −s)α−1x(s)ds.
De ini ion 2. The Capu o ac ional de i a i e o o de α>0o a unc ion x :[0, 1]→Ris de ined as
Dαx( ) = 1
Γ(n−α)Z
0( −s)n−α−1x(n)(s)ds,
whe e n −1<α≤n and n ∈N, p o ided ha he igh -hand-side in eg al exis s and is ini e.
De ini ion 3 ([46]).Le α,β>0,λ∈Rand x ∈L1[0, 1]. The P abhaka in eg al can be w i en as
E[α,β,λ]x( ) = Z
0( −s)β−1Eα,β(λ( −s)α)x(s)ds,
whe e Eα,β(·)is he so-called wo pa ame e Mi ag-Le le unc ion, de ined by
Eα,β(z) =
∞
∑
n=0
zn
Γ(nα+β),
and Eα(·) = Eα,1(·). Like he Mi ag–Le le unc ion Eα(z), Eα,β(z)is an en i e unc ion o o de 1
α.
Lemma 1 ([46]).Le α,β,γ≥0and x ∈L1[0, 1]. Then
IγE[α,β,λ]x( ) = E[α,β,λ]Iγx( ) = E[α,β+γ,λ]x( ),
holds almos e e ywhe e on [0, 1]. Fu he mo e, E[α,β,λ] γ=Γ(γ+1) γ+βEα,β(λ α).
Lemma 2. The gene al solu ion o (1)is gi en by
x( ) =
m−1
∑
j=0
µj jEα+j(−λ α)+
n−1
∑
i=0
νi α+iEα,α(−λ α) +
n−1
∑
i=0
µi i1
Γ(i+1)−Eα(−λ α)
+Z
0( −s)α+β−1Eα,α+β(−λ( −s)α) (s,x(s)) ds.
(4)
P oo . Le x( )be a solu ion o he p oblem (1), we ha e
(Dα+λ)x( ) =
n−1
∑
i=0
ai i+Z
0
( −s)β−1
Γ(β) (s,x(s))ds.
Ma hema ics 2020,8, 743 4 o 10
By using he ini ial condi ions o he ini ial p oblem
(1)
, we ind ha
ai=νi+λµi
Γ(i+1)
,
i=
0, 1,
···
,
n−
1.
The e o e, we ha e
(Dα+λ)x( ) =
n−1
∑
i=0
νi+λµi
Γ(i+1) i+Z
0
( −s)β−1
Γ(β) (s,x(s))ds. (5)
Now, using he app oach o Kilbas e al. ([
40
], Sec ion 3.1), he solu ion o he Equa ion
(5)
is
gi en by he ollowing exp ession
x( ) =
m−1
∑
j=0
µj jEα,j+1(−λ α)+Z
0( −s)α−1Eα,α(−λ( −s)α) n−1
∑
i=0
νi+λµi
Γ(i+1)si+Iβ (·,x(·))(s)!ds.(6)
No e
Eα,α(z) = αE0
α(z)
and so
( −s)α−1Eα,α(−λ( −s)α)=d
ds 1
λEα(−λ( −s)α)
. This yields ha
R
0( −s)α−1Eα,α(−λ( −s)α)ds =1
λ(1−Eα(−λ α))
. On he o he hand, an in eg a ion by pa s e eals
Z
0( −s)α−1Eα,α(−λ( −s)α)sids =1
λsiEα(−λ( −s)α)
0−iZ
0Eα(−λ( −s)α)si−1ds, (7)
o each i∈N. Applying Lemma 1 o he second e m in he igh -hand side o (7), we conclude
Z
0( −s)α−1Eα,α(−λ( −s)α)sids =1
λ i−Γ(i+1) iEα(−λ α),
o each i∈N. The e o e
x( ) =
m−1
∑
j=0
µj jEα,j+1(−λ α)+
n−1
∑
i=0
νi
Γ(i+1)Z
0( −s)α−1Eα,α(−λ( −s)α)sids
+
n−1
∑
i=0
λµi
Γ(i+1)Z
0( −s)α−1Eα,α(−λ( −s)α)sids +Z
0( −s)α−1Eα,α(−λ( −s)α)Iβ (·,x(·))(s)ds
=
m−1
∑
j=0
µj jEα,j+1(−λ α)+
n−1
∑
i=0
νi α+iEα,α(−λ α) +
n−1
∑
i=0
λµi
Γ(i+1)1
λ i−Γ(i+1) iEα(−λ α)
+Z
0( −s)α+β−1Eα,α+β(−λ( −s)α) (s,x(s)) ds
=
m−1
∑
j=0
µj jEα,j+1(−λ α)+
n−1
∑
i=0
νi α+iEα,α(−λ α) +
n−1
∑
i=0
µi i1
Γ(i+1)−Eα(−λ α)
+Z
0( −s)α+β−1Eα,α+β(−λ( −s)α) (s,x(s)) ds,
which is he desi ed esul .
Now, we s a e Weissinge ’s ixed poin heo em ([
41
], Theo em D.7) as a gene aliza ion o he
so-called con ac ion mapping heo em which is needed o p o e Theo em 3.
Theo em 1.
Le
X
o be a Banach space and le
θn≥
0 o e e y
n∈N∪{
0
}
such ha
∑∞
n=0θn
con e ges.
Fu he mo e, assume
T:X→X
is a nonlinea mapping which sa is ies he inequali y
kTnx−Tnyk ≤
θnkx−yk
o e e y
n∈N
and e e y
x
,
y∈X
. Then,
T
has a unique ixed poin
x∗
. Mo eo e , he sequence
{Tnx0}∞
n=0con e ges o his ixed poin x∗, o any x0∈X.
3. Exis ence and Uniqueness
Ou aim in he ollowing sec ion is o deeply in es iga e he exis ence and uniqueness esul s o
he main p oblem (1) in he Lebesgue space.
Theo em 2. Le max{1, 1
α+β} ≤ p≤∞, p−1+q−1=1and he ollowing hypo heses 1–3 hold:
Ma hema ics 2020,8, 743 5 o 10
Hypo hesis 1. ( , 0)∈Lq[0, 1].
Hypo hesis 2.
The e exis s nonnega i e
a∈Lp[
0, 1
]
such ha
| (
,
x2)− (
,
x1)| ≤ a( )|x2−x1|
, o each
∈[0, 1]and x1,x2∈R.
Hypo hesis 3. R:=M1kakp
(1−q+q(α+β)) 1
q
<1whe e M1=sup ∈[0,1]Eα,α+β(−λ α).
Then he in eg al Equa ion (4)has a unique solu ion in Lq[0, 1].
P oo . We de ine he ope a o Tas ollows:
Tx( ) = Z
0( −s)α+β−1Eα,α+β(−λ( −s)α) (s,x(s))ds +φ( ), (8)
whe e
φ( ) =
m−1
∑
j=0
µj jEα,j+1(−λ α)+
n−1
∑
i=0
νi α+iEα,α(−λ α) +
n−1
∑
i=0
µi i1
Γ(i+1)−Eα(−λ α). (9)
Le
M( ) = α+β−1Eα,α+β(−λ α)
,
M1=sup ∈[0,1]Eα,α+β(−λ α)
and
M2=sup ∈[0,1]|φ( )|
.
No e ha he gene alized Mi ag–Le le unc ions a e en i e unc ions [
43
,
44
]. Fo each
x∈Lq[
0, 1
]
,
we ha e
|Tx( )| ≤ Z
0M( −s) (s,x(s))ds
+M2
≤Z
0|M( −s)|| (s, 0)|+|M( −s)|| (s,x(s)) − (s, 0)|ds +M2
≤Z
0|M( −s)|1
q| (s, 0)||M( −s)|1
pds +Z
0|M( −s)||x(s)||a(s)|ds +M2
≤Z
0|M( −s)|| (s, 0)|qds1
qZ
0|M( −s)|ds1
p
+M1Z
0|x(s)|q
( −s)q−q(α+β)ds1
qZ
0|a(s)|pds1
p+M2
≤M
1
q
1Z
0| (s, 0)|q
( −s)1−α+βds1
q
·M
1
p
1
(α+β)1
p
+M1kakpZ
0|x(s)|q
( −s)q−q(α+β)ds1
q
+M2
=M1
(α+β)1
pZ
0| (s, 0)|q
( −s)1−α+βds1
q
+M1kakpZ
0|x(s)|q
( −s)q−q(α+β)ds1
q
+M2.
The e o e, we ha e
kTxkq≤M1
(α+β)1
pZ1
0Z
0| (s, 0)|q
( −s)1−α+βdsd 1
q
+M1kakpZ1
0Z
0|x(s)|q
( −s)q−q(α+β)dsd 1
q
+M2
=M1
(α+β)1
pZ1
0Z1
s| (s, 0)|q
( −s)1−α+βd ds1
q
+M1kakpZ1
0Z1
s|x(s)|q
( −s)q−q(α+β)d ds1
q
+M2
=M1
α+βk (s, 0)kq+M1
(1−q+q(α+β))1
qkakpkxkq+M2,

Ma hema ics 2020,8, 743 6 o 10
which yields T:Lq[0, 1]→Lq[0, 1]. Now, o x,y∈Lq[0, 1], we ob ain
|Tx( )−Ty( )| ≤ Z
0|M( −s)|| (s,x(s)) − (s,y(s))|ds
≤Z
0|M( −s)||x(s)−y(s)||a(s)|ds
≤M1Z
0|x(s)−y(s)|q
( −s)q−q(α+β)ds1
qZ
0|a(s)|pds1
p
=M1kakpZ
0|x(s)−y(s)|q
( −s)q−q(α+β)ds1
q
,
which implies ha
kTx −Tykq=M1kakpZ1
0Z
0|x(s)−y(s)|q
( −s)q−q(α+β)dsd 1
q
=M1kakpZ1
0Z1
s|x(s)−y(s)|q
( −s)q−q(α+β)d ds1
q
=M1kakp Z1
0
(1−s)1−q+q(α+β)
1−q+q(α+β)|x(s)−y(s)|qds!1
q
≤M1kakp
(1−q+q(α+β))1
qkx−ykq,
=Rkx−ykq.
No e ha 1
−q+q(α+β)≥
0 because o
p≥1
α+β
. The e o e,
T
is a con ac ion since
R<
1.
By he Banach con ac ion p inciple,
T
has a unique ixed poin , which is he unique solu ion o he
ini ial p oblem (1).
Rema k 1.
We ecall om [
43
,
44
] ha
Eα,β(−z)
is comple ely mono onic unc ion o 0
<α≤
1and
β≥α
,
ha is,
Eα,β(−z)
possesses de i a i es
dn
dznEα,β(−z)
o all
n=
0, 1, 2,
···
and
(−
1
)ndn
dznEα,β(−z)≥
0
o all z >0. The e o e, Eα,α+β(−λ α)≤Eα,α+β(0) = 1
Γ(α+β) o λ≥0,0<α≤1and 0≤ ≤1.
Theo em 3. Le 1≤q≤∞and he ollowing hypo heses 4 and 5 hold:
Hypo hesis 4. ( , 0)∈Lq[0, 1].
Hypo hesis 5.
The e exis s
L>
0such ha
| (
,
x2)− (
,
x1)| ≤ L|x2−x1|
, o almos e e y
∈[
0, 1
]
and x1,x2∈R.
Then he in eg al Equa ion (4)has a unique solu ion in Lq[0, 1].
P oo . Wi h no a ions as in he p oo o The oem 2, and using he same a gumen s, we ob ain
|Tx( )| ≤ M1
(α+β)1
pZ
0| (s, 0)|q
( −s)1−α−βds1
q
+M1L
(α+β)1
pZ
0|x(s)|q
( −s)1−α−βds1
q
+M2,
Ma hema ics 2020,8, 743 7 o 10
and he e o e,
kTxkq≤M1
(α+β)1
pZ1
0Z
0| (s, 0)|q
( −s)1−α−βdsd 1
q
+M1L
(α+β)1
pZ1
0Z
0|x(s)|q
( −s)1−α−βdsd 1
q
+M2
=M1
(α+β)1
pZ1
0Z1
s| (s, 0)|q
( −s)1−α−βd ds1
q
+M1L
(α+β)1
pZ1
0Z1
s|x(s)|q
( −s)1−α−βd ds1
q
+M2
≤M1
α+βk (0, s)kq+Lkxkq+M2,
which yields
T:Lq[
0, 1
]→Lq[
0, 1
]
. On he o he hand, o e e y
n∈N
and o each
∈[
0, 1
]
, we ha e
|Tnx( )−Tny( )| ≤ Z
0|M( −s1)| (s1,Tn−1x(s1)) − (s1,Tn−1y(s1))ds1
≤M1LZ
0( −s1)α+β−1Tn−1x(s1)−Tn−1y(s1)ds1
≤(M1L)2Z
0( −s1)α+β−1Zs1
0(s1−s2)α+β−1Tn−2x(s2)−Tn−2y(s2)ds2ds1
=(M1L)2Z
0Z
s2
( −s1)α+β−1(s1−s2)α+β−1Tn−2x(s2)−Tn−2y(s2)ds1ds2
=(M1L)2Z
0Z
s2
( −s1)α+β−1(s1−s2)α+β−1ds1Tn−2x(s2)−Tn−2y(s2)ds2
=(Γ(α+β)M1L)2
Γ(2α+2β)Z
0( −s2)2α+2β−1Tn−2x(s2)−Tn−2y(s2)ds2
.
.
.
≤(Γ(α+β)M1L)n
Γ(nα+nβ)Z
0( −sn)nα+nβ−1|x(sn)−y(sn)|dsn
=(Γ(α+β)M1L)n
Γ(nα+nβ)Z
0( −sn)nα+nβ−1
q|x(sn)−y(sn)|( −sn)nα+nβ−1
pdsn
≤(Γ(α+β)M1L)n
Γ(nα+nβ)Z
0( −sn)nα+nβ−1|x(sn)−y(sn)|qdsn1
qZ
0( −sn)nα+nβ−1dsn1
p
≤(Γ(α+β)M1L)n
(nα+nβ)1
pΓ(nα+nβ)Z
0( −sn)nα+nβ−1|x(sn)−y(sn)|qdsn1
q
.
The e o e, we conclude
kTnx−Tnykq≤(Γ(α+β)M1L)n
Γ(n(α+β) + 1)kx−ykq,
o e e y
n∈N
and all
x
,
y∈Lq[
0, 1
]
. Now le
θn=(Γ(α+β)M1L)n
Γ(n(α+β)+1)
. F om he de ini ion o he
gene alized Mi ag–Le le unc ions, we ha e
∑∞
n=0θn=Eα+β(Γ(α+β)M1L)
and hence he se ies
∑∞
n=0θn
con e ges. The e o e, he exis ence o he unique ixed poin o
T
ollows om Weissinge ’s
ixed poin Theo em.
4. Illus a i e Examples
In his sec ion, some examples a e p o ided o show he applicabili y o he analy ical
achie emen s o he pape .
Example 1. Conside he ini ial alue p oblem



D4
5D1
2+λx( ) = 1+ 2+sin +a c an x( )
2e 3
√ 0< ≤1,
x(0) = 1, D1
2x(0) = 1.
(10)
Ma hema ics 2020,8, 743 8 o 10
He e ( ,x) = 1+ 2+sin +a c an x
2e 3
√ ,α=1
2,β=4
5and he ic ion cons an λ≥0.
Le
p=q=
2. Clea ly,
(
, 0
) =
1
+ 2+sin
2e 3
√
and
(
, 0
)∈L2[
0, 1
]
. In ac , i is easily seen
ha
k (
, 0
)k2≤
1
+1
51
2+1
2Γ(1
3)
3
√21
2
. On he o he hand,
| (
,
x)− (
,
y)| ≤ 1
2e 3
√ |x−y|
wi h
a( ) = 1
2e 3
√
. Simila ly, we see ha
a∈L2[
0, 1
]
and
kak2≤1
2Γ(1
3)
3
√21
2
. Fu he , om Rema k 1i ollows
ha M1=sup ∈[0,1]E1
2,13
10 (−λ 1
2)≤1
Γ(13
10 ). The e o e,
R=M1kakp
(1−q+q(α+β))1
q
<
1
2 Γ(1
3)
3
√2
√1.6Γ13
10 =0.64226 <1.
No e ha he con ac ion cons an
R
is independen o ic ion cons an
λ
. Thus, by Theo em 2, he ini ial
alue p oblem (10)has a unique solu ion in L2[0, 1].
Example 2. Conside he ini ial alue p oblem



D1
3D5
4+λx( ) = g( )|x( )|
1+|x( )|0< ≤1,
x(0) = 1, x0(0) = −1, D5
4x(0) = 1,
(11)
whe e g ∈L∞[0, 1]and he ic ion cons an λ∈R.
Obse e ha
(
, 0
) =
0and
| (
,
x)− (
,
y)| ≤ L|x−y|
o almos e e y
∈[
0, 1
]
wi h
L=kgk∞
.
Thus, by Theo em 3, he ini ial alue p oblem (11)has a unique solu ion in L∞[0, 1].
5. Conclusions
In his a icle, we ha e conside ed ini ial alue p oblem o nonlinea ac ional Lange in equa ion
in ol ing wo ac ional o de s. As a i s s ep, by applying he ools o ac ional calculus and
using some basic p ope ies o P abhaka in eg al ope a o , we build a gene al s uc u e o solu ions
associa ed wi h ou p oposed model. Once he ixed poin ope a o equa ion is a ailable, he exis ence
esul s a e es ablished by means o con ac ion mapping heo em and Weissinge ’s ixed poin heo em.
Finally, wo examples we e p esen ed o suppo he esul .
Au ho Con ibu ions:
Supe ision, H.S.; W i ing— e iew and edi ing, H.F. and J.J.N. All au ho s ha e ead
and ag eed o he published e sion o he manusc ip .
Funding:
The au ho s a e hank ul o he Edi o (s) and e iewe s o he manusc ip o hei help ul commen s.
The wo k o H. Fazli and H. Sun was suppo ed by he Na ional Key R&D P og am o China (2017YFC0405203),
he Na ional Na u al Science Founda ion o China unde G an No. 11972148. The ese ach o J. J. Nie o was
pa ially suppo ed by Xun a de Galicia, ED431C 2019/02, and by p ojec MTM2016-75140-P o AEI/FEDER (Spain).
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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