h p://www.aimsp ess.com/jou nal/Ma h
AIMS Ma hema ics, 8(4): 8633–8649.
DOI:10.3934/ma h.2023433
Recei ed: 07 No embe 2022
Re ised: 28 Janua y 2023
Accep ed: 01 Feb ua y 2023
Published: 06 Feb ua y 2023
Resea ch a icle
Fixed poin app oach o he Mi ag-Le le ke nel- ela ed ac ional
di e en ial equa ions
Hasanen A. Hammad1,2,∗, H¨
useyin Is¸ık3, Hassen Aydi4,5,6,∗and Manuel De la Sen7
1Depa men o Ma hema ics, Unaizah College o Sciences and A s, Qassim Uni e si y, Bu aydah
52571, Saudi A abia
2Depa men o Ma hema ics, Facul y o Science, Sohag Uni e si y, Sohag 82524, Egyp
3Depa men o Enginee ing Science, Bandı ma Onyedi Eyl¨
ul Uni e si y, 10200 Bandı ma,
Balıkesi , Tu key
4Ins i u Sup´
e ieu d’In o ma ique e des Techniques de Communica ion, Uni e si ´
e de Sousse, H.
Sousse 4000, Tunisia
5China Medical Uni e si y Hospi al, China Medical Uni e si y, Taichung 40402, Taiwan
6Depa men o Ma hema ics and Applied Ma hema ics, Se ako Makga ho Heal h Sciences
Uni e si y, Ga-Rankuwa, Sou h A ica
7Ins i u e o Resea ch and De elopmen o P ocesses, Depa men o Elec ici y and Elec onics,
Facul y o Science and Technology, Uni e si y o he Basque Coun y, 48940-Leioa ( Bizkaia),
Spain
*Co espondence: Email: h.abdelw[email p o ec ed], [email p o ec ed].
Abs ac : The goal o his pape is o p esen a new class o con ac ion mappings, so-called η`
θ-
con ac ions. Also, in he con ex o pa ially o de ed me ic spaces, some coupled ixed-poin esul s
o η`
θ-con ac ion mappings a e in oduced. Fu he mo e, o suppo ou esul s, wo examples a e
p o ided. Finally, he heo e ical esul s a e applied o ob ain he exis ence o solu ions o coupled
ac ional di e en ial equa ions wi h a Mi ag-Le le ke nel.
Keywo ds: ac ional di e en ial equa ion; A angana-Baleanu ac ional ope a o ; ixed poin
me hodology; Riemann-Liou ille ac ional in eg al
Ma hema ics Subjec Classi ica ion: 34A08, 34A12, 47H10, 54H25
8634
1. In oduc ion and basic ac s
F ac ional di e en ial equa ions a e hough o be he mos e ec i e models o a a ie y o pe inen
e en s. This makes i possible o in es iga e he exis ence, uniqueness, con ollabili y, s abili y, and
o he p ope ies o analy ical solu ions. Fo example, applying conse a ion laws o he ac ional
Black-Scholes equa ion in Lie symme y analysis, inding exis ence solu ions o some con o mable
di e en ial equa ions, and inding exis ence solu ions o some classical and ac ional di e en ial
equa ions on he basis o disc e e symme y analysis, o mo e de ails, see [1–5].
A angana and Baleanu uni ied and ex ended he de ini ion o Capu o-Fab izio [5] by in oducing
exci ing de i a i es wi hou singula ke nel. Also, he same au ho s p esen ed he de i a i e con aining
Mi ag-Le le unc ion as a nonlocal and nonsingula ke nel. Many esea che s showed hei in e es in
his de ini ion because i opens many and sobe di ec ions and ca ies Riemann-Liou ille and Capu o
de i a i es [6–13].
A a ie y o p oblems in economic heo y, con ol heo y, global analysis, ac ional analysis, and
nonlinea analysis ha e been ea ed by ixed poin (FP) heo y. The FP me hod con ibu es g ea ly
o he ac ional di e en ial/in eg al equa ions, h ough which i is possible o s udy he exis ence and
uniqueness o he solu ion o such equa ions [14–17]. Also, his opic has been densely s udied and
se e al signi ican esul s ha e been eco ded in [18–21].
The concep s o mixed mono one p ope y (MMP) and a coupled ixed poin (CFP) o a con ac i e
mapping Ξ:χ×χ→χ, whe e χis a pa ially o de ed me ic space (POMS) ha e been ini ia ed by
Bhaska and Lakshmikan ham [22]. To suppo hese ideas, hey p esen ed some CFP heo ems and
de e mined he exis ence and uniqueness o he solu ion o a pe iodic bounda y alue p oblem [23–25].
Many au ho s wo ked in his di ec ion and ob ained some nice esul s conce ned wi h CFPs in a ious
spaces [26–28].
De ini ion 1.1. [22] Conside a se χ,∅.A pai (a,b)∈χ×χis called a CFP o he mapping
Ξ:χ×χ→χi a= Ξ(a,b) and b= Ξ(b,a).
De ini ion 1.2. [22] Assume ha (χ, ≤) is a pa ially o de ed se and Ξ:χ×χ→χis a gi en mapping.
We say ha Ξhas a MMP i o any a,b∈χ,
a1,a2∈χ, a1≤a2⇒Ξ(a1,b)≤Ξ(a2,b),
and
b1,b2∈χ, b1≤b2⇒Ξ(a,b1)≥Ξ(a,b2).
Theo em 1.1. [22] Le (χ, ≤,d)be a comple e POMS and Ξ:χ×χ→χbe a con inuous mapping
ha ing he MMP on χ. Assume ha he e is a τ∈[0,1) so ha
d(Ξ(a,b),Ξ(k,l))≤τ
2(d(a,k)+d(b,l)),
o all a ≥k and b ≤l.I he e a e a0,b0∈χso ha a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0), hen Ξhas a
CFP, ha is, he e exis a0,b0∈χsuch ha a = Ξ(a,b)and b = Ξ(b,a).
The same au ho s p o ed ha Theo em 1.1 is s ill alid i we eplace he hypo hesis o con inui y
wi h he ollowing: Assume χhas he p ope y below:
AIMS Ma hema ics Volume 8, Issue 4, 8633–8649.
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(†) i a non-dec easing sequence {am} → a, hen am≤a o all m;
(‡) i a non-inc easing sequence {bm} → b, hen b≤bm o all m.
The ollowing auxilia y esul s a e aken om [29,30], which a e used e icien ly in he nex sec ion.
Le Θ ep esen a amily o non-dec easing unc ions θ: [0,∞)→[0,∞) so ha P∞
m=1θm(τ)<∞
o all τ > 0,whe e θnis he n- h i e a e o θjus i ying:
(i) θ(τ)=0⇔τ=0;
(ii) o all τ > 0, θ(τ)< τ;
(iii) o all τ > 0,lims→τ+θ(s)< τ.
Lemma 1.1. [30] I θ: [0,∞)→[0,∞)is igh con inuous and non-dec easing, hen limm→∞ θm(τ)=
0 o all τ≥0i θ(τ)< τ o all τ > 0.
Le e
Lbe he se o all unc ions e
`: [0,∞)→[0,1) which e i y he condi ion:
lim
m→∞ e
`(τm)=1 implies lim
m→∞ τm=0.
Recen ly, Same e al. [29] epo ed exci ing FP esul s by p esen ing he concep o α-θ-con ac i e
mappings.
De ini ion 1.3. [29] Le χbe a non emp y-se , Ξ:χ→χbe a map and α:χ×χ→Rbe a gi en
unc ion. Then, Ξis called α-admissible i
α(a,b)≥1⇒α(Ξa,Ξb))≥1,∀a,b∈χ.
De ini ion 1.4. [29] Le (χ, d) be a me ic space. Ξ:χ→χis called an α-θ-con ac i e mapping, i
he e exis wo unc ions α:χ×χ→[0,+∞) and θ∈Θsuch ha
α(a,b)d(Ξ(a,b))≤θ(d(a,b)),
o all a,b∈χ.
Theo em 1.2. [29] Le (χ, d)be a me ic space, Ξ:χ→χbe an α-ψ-con ac i e mapping jus i ying
he hypo heses below:
(i) Ξis α-admissible;
(ii) he e is a0∈χso ha α(a0,Ξa0)≥1;
(iii) Ξis con inuous.
Then Ξhas a FP.
Mo eo e , he au ho s in [29] showed ha Theo em 1.2 is also ue i we use he ollowing condi ion
ins ead o he con inui y o he mapping Ξ.
•I {am}is a sequence o χso ha α(am,am+1)≥1 o all mand limm→+∞am=a∈χ, hen o all
m,α(am,a)≥1.
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The idea o an α-admissible mapping has sp ead widely, and he FPs ob ained unde his idea a e
no small, o example, see [31–34].
Fu he mo e, one o he in e es ing di ec ions o ob aining FPs is o in oduce he idea o Ge agh y
con ac ions [30]. The au ho [30] gene alized he Banach con ac ion p inciple and ob ained some
pi o al esul s in a comple e me ic space. I is wo h no ing ha a good numbe o esea che s ha e
ocused hei a en ion on his idea, o example, see [35–37]. In espec o comple eness, we s a e
Ge agh y’s heo em.
Theo em 1.3. [30] Le Ξ:χ→χbe an ope a o on a comple e me ic space (χ, d). Then Ξhas a
unique FP i Ξsa is ies he ollowing inequali y:
d(Ξa,Ξb)≤e
`(d(a,b))d(a,b), o any a,b∈χ,
whe e e
`∈e
L.
We need he ollowing esul s in he las pa .
De ini ion 1.5. [5] Le σ∈H1(s, ),s< ,and ν∈[0,1).The A angana–Baleanu ac ional de i a i e
in he Capu o sense o σo o de νis desc ibed by
ABC
sDνσ(ζ)=Q(ν)
1−ν
ζ
Zs
σ0(ϑ)Mν −ν(ζ−ϑ)ν
1−ν!dϑ,
whe e Mνis he Mi ag-Le le unc ion gi en by Mν( )=
∞
P
m=0
m
Γ(mν+1) and Q(ν) is a no malizing posi i e
unc ion ul illing Q(0) =Q(1) =1 (see [4]). The ela ed ac ional in eg al is desc ibed as
AB
sIνσ(ζ)=1−ν
Q(ν)σ(ζ)+ν
Q(ν)(sIνσ)(ζ),(1.1)
whe e sIνis he le Riemann-Liou ille ac ional in eg al de ined by
(sIνσ)(ζ)=1
Γ(ν)
ζ
Zs
(ζ−ϑ)ν−1σ(ϑ)dϑ. (1.2)
Lemma 1.2. [38] Fo ν∈(0,1),we ha e
AB
sIνABC Dνσ(ζ)=σ(ζ)−σ(s).
The ou line o his pape is as ollows: In Sec ion 1, we p esen ed some known consequences abou
α-admissible mappings and some use ul de ini ions and heo ems ha will be used in he sequel. In
Sec ion 2, we in oduce an η`
θ-con ac ion ype mapping and ob ain some ela ed CFP esul s in he
con ex o POMSs. Also, we suppo ou heo e ical esul s wi h some examples. In Sec ion 5, an
applica ion o ind he exis ence o a solu ion o he A angana-Baleanu coupled ac ional di e en ial
equa ion (CFDE) in he Capu o sense is p esen ed.
AIMS Ma hema ics Volume 8, Issue 4, 8633–8649.
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2. Main esul s
Le Lbe he se o all unc ions `: [0,∞)→[0,1) sa is ying he ollowing condi ion:
lim
m→∞ `(τn)=1 implies lim
m→∞ τn=1.
We begin his pa wi h he ollowing de ini ions:
De ini ion 2.1. Suppose ha Ξ:χ×χ→χand η:χ2×χ2→[0,∞) a e wo mappings. The mapping
Ξis called η-admissible i
η((a,b),(k,l))≥1⇒η((Ξ(a,b),Ξ(b,a)),(Ξ(k,l),Ξ(l,k))) ≥1,∀a,b,k,l∈χ.
De ini ion 2.2. Le (χ, $) be a POMS and Ξ:χ×χ→χbe a gi en mapping. Ξis e med as an
η`
θ-coupled con ac ion mapping i he e a e wo unc ions η:χ2×χ2→[0,∞) and θ∈Θso ha
η((a,b),(k,l))$(Ξ(a,b),Ξ(k,l))≤` θ $(a,k)+$(b,l)
2!!θ $(a,k)+$(b,l)
2!,(2.1)
o all a,b,k,l∈χwi h a≥kand b≤l,whe e `∈L.
Rema k 2.1. No ice ha since `: [0,∞)→[0,1),we ha e
η((a,b),(k,l))$(Ξ(a,b),Ξ(k,l))
≤` θ $(a,k)+$(b,l)
2!!×θ $(a,k)+$(b,l)
2!
< θ $(a,k)+$(b,l)
2!, o any a,b,k,l∈χwi h a,b,k,l.
Theo em 2.1. Le (χ, ≤, $)be a comple e POMS and Ξbe an η`
θ-coupled con ac ion which has he
mixed mono one p ope y so ha
(i) Ξis η-admissible;
(ii) he e a e a0,b0∈χso ha
η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1and η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0))) ≥1;
(iii) Ξis con inuous.
I he e a e a0,b0∈χso ha a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0), hen Ξhas a CFP.
P oo . Le a0,b0∈χbe such ha η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1,
η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0))) ≥1,a0≤Ξ(a0,b0)=a1(say) and b0≥Ξ(b0,a0)=b1(say).
Conside a2,b2∈χso ha Ξ(a1,b1)=a2and Ξ(b1,a1)=b2.Simila o his app oach, we ex ac wo
sequences {am}and {bm}in χso ha
am+1= Ξ (am,bm)and bm+1= Ξ (bm,am), o all m≥0.
Now, we shall show ha
am≤am+1and bm≥bm+1, o all m≥0.(2.2)
By a ma hema ical induc ion, we ha e
AIMS Ma hema ics Volume 8, Issue 4, 8633–8649.
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(1) A m=0,because a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0)and since Ξ(a0,b0)=a1and Ξ(b0,a0)=b1,
we ob ain a0≤a1and b0≥b1, hus (2.2) holds o m=0.
(2) Suppose ha (2.2) holds o some ixed m≥0.
(3) A emp ing o p o e he alidi y o (2.2) o any m,by assump ion (2) and he mixed mono one
p ope y o Ξ,we ge
am+2= Ξ (am+1,bm+1)≥Ξ(am,bm+1)≥Ξ(am,bm)=am+1,
and
bm+2= Ξ (bm+1,am+1)≤Ξ(bm,am+1)≤Ξ(bm,am)=bm+1.
This implies ha
am+2≥am+1and bm+2≤bm+1.
Thus, we conclude ha (2.2) is alid o all n≥0.
Nex , i o some m≥0,(am+1,bm+1)=(am,bm), hen am= Ξ (am,bm)and bm= Ξ (bm,am),i.e., Ξ
has a CFP. So, le (am+1,bm+1),(am,bm) o all m≥0.As Ξis η-admissible, we ge
η((a0,b0),(a1,b1)) =η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1,
implies
η((Ξ(a0,b0),Ξ(b0,a0)) ,(Ξ(a1,b1),Ξ(b1,a1))) =η((a1,b1),(a2,b2)) ≥1.
Thus, by induc ion, one can w i e
η((am,bm),(am+1,bm+1)) ≥1 and η((bm,am),(bm+1,am+1)) ≥1 o all m≥0.(2.3)
Using (2.1) and (2.3) and he de ini ion o `, we ha e
$(am,am+1)=$(Ξ(am−1,bm−1),Ξ(am,bm))
≤η((am−1,bm−1),(am,bm))$(Ξ(am−1,bm−1),Ξ(am,bm))
≤` θ $(am−1,am)+$(bm−1,bm)
2!!θ $(am−1,am)+$(bm−1,bm)
2!
≤θ $(am−1,am)+$(bm−1,bm)
2!.(2.4)
Analogously, we ge
$(bm,bm+1)=$(Ξ(bm−1,am−1),Ξ(bm,am))
≤η((bm−1,am−1),(bm,am))$(Ξ(bm−1,am−1),Ξ(bm,am))
≤θ $(bm−1,bm)+$(am−1,am)
2!.(2.5)
Adding (2.4) and (2.5) we ha e
$(am,am+1)+$(bm,bm+1)
2≤θ $(am−1,am)+$(bm−1,bm)
2!.
AIMS Ma hema ics Volume 8, Issue 4, 8633–8649.
8639
Con inuing in he same way, we ge
$(am,am+1)+$(bm,bm+1)
2≤θm $(a0,a1)+$(b0,b1)
2!, o all m∈N.
Fo > 0, he e exis s m()∈Nso ha
X
m≥m()
θm $(a0,a1)+$(b0,b1)
2!<
2,
o some θ∈Θ.Le m,j∈Nbe so ha j>m>m().Then based on he iangle inequali y, we ob ain
$am,aj+$bm,bj
2≤
j−1
X
i=m
$(ai,ai+1)+$(bi,bi+1)
2
≤
j−1
X
i=m
θi $(a0,a1)+$(b0,b1)
2!
≤X
m≥m()
θm $(a0,a1)+$(b0,b1)
2!<
2,
his leads o $am,aj+$bm,bj< . Because
$am,aj≤$am,aj+$bm,bj< ,
and
$bm,bj≤$am,aj+$bm,bj< ,
hence {am}and {bm}a e Cauchy sequences in χ. The comple eness o χimplies ha he sequences {am}
and {bm}a e con e gen in χ, ha is, he e a e a,b∈χso ha
lim
m→∞ am=aand lim
m→∞ bm=b.
Since Ξis con inuous, am+1= Ξ (am,bm)and bm+1= Ξ (bm,am),we ob ain a e aking he limi as
m→ ∞ ha
a=lim
m→∞ am=lim
m→∞ Ξ(am−1,bm−1)= Ξ(a,b),
and
b=lim
m→∞ bm=lim
m→∞ Ξ(bm−1,am−1)= Ξ(b,a).
The e o e, Ξhas a CFP and his ends he p oo .
In he abo e heo em, when omi ing he con inui y assump ion on Ξ,we de i e he ollowing
heo em.
Theo em 2.2. Le (χ, ≤, $)be a comple e POMS and Ξbe an η`
θ-coupled con ac ion and ha ing he
mixed mono one p ope y so ha
(a) Ξis η-admissible;
AIMS Ma hema ics Volume 8, Issue 4, 8633–8649.
8640
(b) he e a e a0,b0∈χso ha
η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1and η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0))) ≥1;
(c) i {am}and {bm}a e sequences in χsuch ha
η((am,bm),(am+1,bm+1)) ≥1, η ((bm,am),(bm+1,am+1)) ≥1
o all m ≥0,limm→∞ am=a∈χand limm→∞ bm=b∈χ, hen
η((am,bm),(a,b)) ≥1and η((bm,am),(b,a)) ≥1.
I a0,b0∈χa e ha a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0), hen Ξhas a CFP.
P oo . Wi h he same app oach as o he p oo o Theo em 2.1, he sequences {am}and {bm}a e Cauchy
sequences in χ. The comple eness o χimplies ha he e a e a,b∈χso ha
lim
m→∞ am=aand lim
m→∞ bm=b.
Acco ding o he assump ion (c) and (2.3), one can w i e
η((am,bm),(a,b)) ≥1 and η((bm,am),(b,a)) ≥1, o all m∈N.(2.6)
I ollows by (2.3), he de ini ion o `and he p ope y o θ(τ)< τ o all τ > 0, ha
$(Ξ(a,b),a)≤$(Ξ(a,b),Ξ(am,bm))+$(Ξ(am,bm),a)
≤η((am,bm),(a,b)) $(Ξ(am,bm),Ξ(a,b))+$(am+1,a)
≤` θ $(am,a)+$(bm,b)
2!!θ $(am,a)+$(bm,b)
2!+$(am+1,a)
≤θ $(am,a)+$(bm,b)
2!+$(am+1,a)
<$(am,a)+$(bm,b)
2+$(am+1,a).(2.7)
Simila ly, we ind ha
$(Ξ(b,a),b)≤$(Ξ(b,a),Ξ(bm,am))+$(Ξ(bm,am),b)
≤η((bm,am),(b,a)) $(Ξ(bm,am),Ξ(b,a))+$(bm+1,b)
≤` θ $(bm,b)+$(am,a)
2!!θ $(bm,b)+$(am,a)
2!+$(bm+1,b)
≤θ $(bm,b)+$(am,a)
2!+$(bm+1,b)
<$(bm,b)+$(am,a)
2+$(bm+1,b).(2.8)
As m→ ∞ in (2.7) and (2.8), we ha e
$(Ξ(a,b),a)=0 and $(Ξ(b,a),b)=0.
Hence, a= Ξ(a,b) and b= Ξ(b,a).Thus, Ξhas a CFP and his comple es he p oo .
AIMS Ma hema ics Volume 8, Issue 4, 8633–8649.
8641
In o de o show he uniqueness o a CFP, we gi e he heo em below. I (χ, ≤) is a pa ially o de ed
se , we de ine a pa ial o de ela ion ≤on he p oduc χ×χas ollows:
(a,b)≤(k,l)⇔a≤kand b≥l, o all (a,b),(k,l)∈χ×χ.
Theo em 2.3. In addi ion o he asse ions o Theo em 2.1, assume ha o each (a,b),(y,z)in χ×χ,
he e is (k,l)∈χ×χso ha
η((a,b),(k,l)) ≥1and η((y,z),(k,l)) ≥1.
Suppose also (k,l)is compa able o (a,b)and (y,z).Then Ξhas a unique CFP.
P oo . Theo em 2.1 asse s ha he se o CFPs is non-emp y. Le (a,b) and (y,z) be CFPs o he
mapping Ξ, ha is, a= Ξ(a,b),b= Ξ(b,a) and y= Ξ(y,z),z= Ξ(z,y).By hypo hesis, he e is
(k,l)∈χ×χso ha (k,l) is compa able o (a,b) and (y,z).Le (a,b)≤(k,l),k=k0and l=l0. Choose
k1,l1∈χ×χso ha k1= Ξ(k1,l1),l1= Ξ(l1,k1).Thus, we can cons uc wo sequences {km}and {lm}as
km+1= Ξ(km,lm) and lm+1= Ξ(lm,km).
Since (k,l) is compa able o (a,b),in an easy way we can p o e ha a≤k1and b≥l1.Hence, o
m≥1,we ha e a≤kmand b≥lm.Because o e e y (a,b),(y,z)∈χ×χ, he e is (k,l)∈χ×χso ha
η((a,b),(k,l)) ≥1 and η((y,z),(k,l)) ≥1.(2.9)
Because Ξis η-admissible, hen by (2.9), we ge
η((a,b),(k,l)) ≥1 implies η((Ξ(a,b),Ξ(b,a)),(Ξ(k,l),Ξ(l,k))) ≥1.
Since k=k0and l=l0,we ob ain
η((a,b),(k,l)) ≥1 implies η((Ξ(a,b),Ξ(b,a)),(Ξ(k0,l0),Ξ(l0,k0))) ≥1.
Hence,
η((a,b),(k,l)) ≥1 implies η((a,b),(k1,l1)) ≥1.
So, by induc ion, we conclude ha
η((a,b),(km,lm)) ≥1,(2.10)
o all m∈N.Analogously, one can ob ain ha η((b,a),(lm,km)) ≥1.The e o e, he ob ained esul s
hold i (a,b)≤(k,l). Based on (2.9) and (2.10), we can w i e
$(a,km+1)=$(Ξ(a,b),Ξ(km,lm))
≤η((a,b),(km,lm)) $(Ξ(a,b),Ξ(km,lm))
≤` θ $(a,km)+$(b,lm)
2!!θ $(a,km)+$(b,lm)
2!
≤θ $(a,km)+$(b,lm)
2!.(2.11)
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