González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35
h ps://doi.o g/10.1186/s13662-020-03206-7
R E S E A R C H Open Access
Fundamen al solu ions o semidisc e e
e olu ion equa ions ia Banach algeb as
Jo ge González-Camus1, Ca los Lizama1* and Ped o J. Miana2
*Co espondence:
[email p o ec ed]
1Depa amen o de Ma emá icas y
Ciencias de la Compu ación,
Facul ad de Ciencias, Uni e sidad de
San iago de Chile, Las Sopho as 173,
Es ación Cen al, San iago, Chile
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
We gi e ep esen a ions o solu ions o ime- ac ional diffe en ial equa ions ha
in ol e ope a o s on Lebesgue spaces o sequences defined by disc e e con olu ions
in ol ing ke nels h ough he disc e e Fou ie ans o m. We conside fini e diffe ence
ope a o s o fi s and second o de s, which a e gene a o s o uni o mly con inuous
semig oups and cosine unc ions. We p esen he linea and algeb aic s uc u es (in
pa icula , ac o iza ion p ope ies) and hei no ms and spec a in he Lebesgue
space o summable sequences. We iden i y ac ional powe s o hese gene a o s and
apply o hem he subo dina ion p inciple. We also gi e some applica ions and
consequences o ou esul s.
MSC: 35R11; 35A08; 39A12
Keywo ds: Capu o ac ional de i a i e; Disc e e ac ional Laplacian; Disc e e
ac ional ope a o s; Fundamen al solu ions; W igh and Mi ag-Leffle unc ions
1 In oduc ion
In his wo k, we s udy he ollowing semidisc e e Cauchy p oblem:
⎧
⎨
⎩
∂ u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z,(1.1)
whe e Bis he con olu ion ope a o in he disc e e a iable, ha is,
Bu(n, )=
j∈Z
b(n–j)u(j, ) (1.2)
wi h bbelonging o he Banach algeb a 1(Z). A ypical example is he one-dimensional
disc e e Laplacian d, which can be ob ained by aking b=δ–1 –2δ0+δ1,whe eδi(j)
deno es he K onecke del a (o disc e e Di ac measu e). In such a case, equa ion (1.1)
co esponds o he nonhomogeneous semidisc e e diffusion equa ion (also known as he
semidisc e e hea equa ion o he la ice diffusion equa ion).
The analy ical s udy o such equa ions has ecei ed an inc easing in e es in he las
decade, mainly due o many hei applica ions in di e se a eas o knowledge. Fo ins ance,
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González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 2 o 32
in p obabili y heo y, he alue u(n, )in(1.1)wi hB=ddesc ibes he p obabili y ha a
con inuous- ime symme ic andom walk on Z isi s a poin na ime ;see[25,Sec .4].
In chemis y, (1.1) desc ibes he flow o a chemical in an infini e sys em o anks a anged
in a ow, whe e each wo neighbo s a e connec ed by pipes [42, Sec . 3], and in anspo
heo y, (1.1) desc ibes he dynamics o an infini e chain o ca s, each being coupled o i s
wo neighbo s. The alue u(n; )is hedisplacemen o ca na ime om i s equilib ium
posi ion; see [24, Example 1]. F om an analy ical poin o iew, qui e ecen ly, Sla ik [43]
s udied he asymp o ic beha io o solu ions o (1.1)whenB=d, showing ha a bounded
solu ion app oaches he a e age o he ini ial alues i he a e age exis s. No e ha choos-
ing b=δ–1 –δ0in (1.2), we ob ain he o wa d diffe ence ope a o B=, and hence (1.2)
co esponds o he semidisc e e anspo equa ion, s udied ecen ly by Abadias e al. [1].
I is in e es ing ha in [22]and[37] he au ho s s udied he undamen al solu ions o
(1.1) and he second-o de semidisc e e equa ion
⎧
⎨
⎩
∂ u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), u (n,0)=φ(n), n∈Z,(1.3)
when B=–(–d)αis he disc e e ac ional Laplacian. Pa icula ly, in [37] heau ho s
combined ope a o heo y echniques wi h he p ope ies o he Bessel unc ions o de-
elop a heo y o analy ic semig oups and cosine ope a o s gene a ed by dand –(–d)α.
Also no e ha he ac ional o wa d diffe ence ope a o B=–(–)αwas s udied in [1],
whe e he maximum and compa ison p inciples in he con ex o ha monic analysis a e
p o ed.
Howe e , o ou knowledge, o da e, he e is no a emp o in es iga e he undamen al
solu ions o he gene al equa ion (1.1)inaunifiedway.Ou goalin hispape is op opose
asolu ion o hisp oblem.
Ou key obse a ion conce ning his issue is ha he disc e e ac ional Laplacian can be
ob ained om (1.2) by allowing he ac ional powe s o b o be an elemen o he Banach
algeb a 1(Z). This o iginal app oach, which we p o ide in his pape , allows us o ob ain
new insigh s by in oducing a comple ely new me hod o analyze bo h quali a i e beha io
and undamen al solu ions o (1.1)inaunifiedway.
Mo e gene ally, o p o ide simul aneously in ou analysis he subdiffusi e and supe di-
ussi e cases associa ed wi h equa ions (1.1)and(1.3), in his pape , we include a ep e-
sen a ion o he undamen al solu ions o he ollowing semidisc e e equa ions:
⎧
⎨
⎩
Dβ
u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z,(1.4)
in case 0 < β≤1and
⎧
⎨
⎩
Dβ
u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), u (n,0)=φ(n), n∈Z,(1.5)
in case 1 < β≤2. In bo h cases, Bis he con olu ion ope a o B (n):=(b∗ )(n)onp(Z),
p∈[1,∞], b∈1(Z), and β∈(0, 2]. The symbol Dβ
deno es he Capu o ac ional de i a-
i e o o de β>0.
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 3 o 32
We obse e ha al hough he p esen s udy conside s only he cases 0 < β≤2because
hey a e he mos common in applica ions ( ha is, subdiffusion and supe diffusion) o-
ge he wi h he ac ional Capu o de i a i e, ou me hod is gene al enough o conside a
la ge o de βand o he nonlocal ope a o s in ime. Fo example, in Sec . 6.4, we conside
he new Capu o–Fab izio ac ional de i a i e o o de α∈(0,1) and gi e a ep esen a ion
o he solu ions o he co esponding equa ion (1.4).
This pape is o ganized as ollows. In Sec . 2, we conside he Banach algeb a ame-
wo k o s a e ou main esul s, which we will p esen in he o hcoming sec ions. In
pa icula , we in oduce gene alized Mi ag-Leffe unc ions on he Banach algeb a 1(Z)
and collec some basic p ope ies. Ou main esul is Theo em 2.6 conce ning he in-
a iance o con olu ion ope a o s defined on p(Z) o 1≤p≤∞.Sec ion3is de o ed
o ou fini e diffe ence ope a o s: backwa d and o wa d diffe ence ope a o s, he one-
dimensional disc e e Laplacian, and an ope a o ha o igina es in connec ion wi h c ys al
la ices [16]. Then we explici ly desc ibe hei associa ed g oups and cosine ope a o s by
means o Bessel unc ions and highligh hei main spec al p ope ies. Sec ion 4begins
wi h h ee conc e e examples o applica ion o he esul s in he p e ious sec ion: he dis-
c e e Nagumo equa ion, anspo equa ions, and a new in e es ing second-o de disc e e
equa ion, which we call he De Juhasz equa ion, appea ing in he seminal Ba eman’s pape
[16] in connec ion wi h su ges in sp ings and connec ed sys ems o sp ings. Then we s a e
he gene al undamen al solu ions o (1.4)–(1.5), fi s , in he se ing o Banach algeb as
(Theo em 5.1) and hen o con olu ion ope a o s (Co olla y 5.5). In Sec . 7,wegi eex-
plici ep esen a ions o gene alized Mi ag-Leffe unc ions in each case o he ac ional
powe s o he ou fini e diffe ence ope a o s conside ed p e iously (Theo em 7.1). This
esul , combined wi h he gene al undamen al solu ions conside ed in Theo em 5.1 and
Co olla y 5.5, gi es no only explici ep esen a ions o each o he ou diffe ence ope a-
o s conside ed in his pape – which can be conside ed as examples – bu also an efficien
me hod o ob ain ep esen a ions o solu ions in many o he cases. Besides, as a byp oduc
o ou ea men , we ob ain new Weies ass o mulae, which highligh he ole o Bessel
unc ions o fini e diffe ence ope a o s, and a subo dina ion p inciple, which connec s
he W igh and Bessel unc ions. Fo con enience o he eade , we finish his esea ch
wi h an appendix on use ul p ope ies o some special unc ions needed in his pape .
No a ion. T={eiθ:θ∈[–π,π)}is he one-dimensional o us. The Di ac measu es δ0
and δna e δn(j)=0i n=jand δn(n)=1 o n,j∈Z. Gi en a Banach space X,Xis he dual
o X,andB(X) is he se o linea bounded ope a o s on X;gi enA∈B(X), we A∈B(X)
is he adjoin o he ope a o A.Wedeno ebyχI he indica o unc ion o a se I(i.e.,
χI(n)=1i n∈Iand χI(n)=0i n/∈I). Fu he mo e, Inand Jna e he Bessel unc ions.
The usual se numbe s N,N0=N∪{0},Z,R,andCa e used. Fu he mo e, is he gamma
unc ion, βis he W igh unc ion (Sec . A.1), Eα,βis he Mi ag-Leffle unc ion, Inand
Jna e he Bessel unc ions (Sec . A.2), and he s able Lé y dis ibu ion is deno ed by ,α
(Sec . A.3).
2ABanachalgeb a amewo k
Gi en 1 ≤p≤∞, we ecall ha he Banach spaces (p(Z), ·p) a e o med by biinfini e
sequences =( (n))n∈Z⊂Csuch ha
 p:= ∞

n=–∞ (n)
p1
p
<∞,1≤p<∞,
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 4 o 32
 ∞:= sup
n∈Z (n)<∞.
We ecall he na u al embeddings 1(Z)→p(Z)→∞(Z) o 1≤p≤∞and ha he
dual o p(Z) is iden ified wi h p(Z), whe e 1
p+1
p=1 o 1<p<∞and p=1i p=∞.
In he case o ∈1(Z)andg∈p(Z), we define
( ∗g)(n):= ∞

j=–∞
(n–j)g(j), n∈Z.
F om Young’s inequali y i ollows ha ∗g∈p(Z). No e ha (1(Z), ∗)isacommu a i e
Banach algeb a wi h iden i y δ0:= χ{0}.Weobse e ha δ1∗δ1=δ2and, in gene al, δn∗δm=
δn+m o n,m∈Z.
The Gel and ans o m associa ed wi h (1(Z),∗) is he disc e e Fou ie ans o m F:
1(Z)→C(T) (o Fou ie se ies), whe e
ˆ
(θ):=F( )eiθ:= 
n∈Z
(n)einθ,θ∈T.
We ecall ha he spec um o ,deno edσ1(Z)( ), is defined by
σ1(Z)( ):=λ∈C:(λδ0– )–1 ∈1(Z).
In wha ollows, we conside he gene al heo y o commu a i e Banach algeb as as a
amewo k. We collec he esul s ha will be o ou in e es in he ollowing heo em.
Theo em 2.1 The ollowing p ope ies hold:
(i) The spec um Spec(1(Z)) is compac and,consequen ly,homeomo phic o he uni
complex ci cle T:= {z∈C:|z|=1}.
(ii) σ1(Z)( )⊂{z∈C;|z|< 1},and
(λδ0– )–1 =
n≥0
λ–n–1 n, 1<|λ|. (2.1)
(iii) The algeb a 1(Z)is a semisimple egula Banach algeb a,and he disc e e Fou ie
ans o m Fis injec i e.
(i ) F( ∗g)=F( )F(g),and
σ1(Z)( )=F( )(T), ∈1(Z). (2.2)
P oo Thefi s claim ollows om he ac ha healgeb a1(Z) has an iden i y; see, o
example, [35], and he second one can be ound in [35, p. 116]. The p oo o (ii) is s aigh -
o wa d. F om [35, Theo em 4.7.4] i ollows ha 1(Z)issemisimpleandFis injec i e.
By [35, Co ola y 7.2.3] 1(Z) is a egula Banach algeb a. S a emen (i ) is aken om [35,
Theo em 3.4.1.]. 
We obse e ha he ange o he Gel and ans o m is he Wiene algeb a A(T), he
poin wise algeb a o absolu ely con e gen Fou ie se ies, ha is, F(eiθ)=n∈Z (n)eiθn,
(θ∈T)wi h ∈1(Z). Fo F∈A(T), we also w i e F(z)=n∈Z (n)zn o |z|≤1.
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 5 o 32
The in e se disc e e Fou ie ans o m is gi en by he exp essions
F–1(F)(n)= 1
2ππ
–π
Feiθe–inθdθ=1
2πi|z|=1
F(z)dz
zn+1 ,n∈Z,
o F∈A(T) (and o o he unc ions in la ge se s).
The classical o mula ion o Wiene ’s lemma cha ac e izes he unc ions F∈A(T) ha
a e in e ible in A(T) as ollows. Fo F∈A(T)whe eF(eiθ)=n∈Z (n)eiθn o θ∈T,
F(eiθ)=0 o allθ∈Ti and only i 1/F∈A(T), ha is, (1/F)(eiθ)=n∈Zg(n)eiθnwi h
(g(n))n∈Z∈1(Z); in his case, ∗g=δ0[32, Theo em 5.5].
Recall he defini ion o he classical Mi ag-Leffle unc ion (see (A.3)). We now in o-
duce he ollowing defini ion.
Defini ion 2.2 Fo α,β> 0, we define he ec o - alued Mi ag-Leffle unc ion Eα,β:
1(Z)→1(Z), by
Eα,β(a):= ∞

j=0
aj
(αj+β),a∈1(Z).
No e ha
E1,1(a)= ∞

j=0
aj
j!=ea;E2,1(a)= ∞

j=0
aj
(2j)!.
The se exp(1(Z)) := {ea;a∈1(Z)}is he connec ed componen o δ0in he se o egula
elemen s in 1(Z)[35, Theo em 6.4.1].
We ollow he usual e minology in semig oup heo y: he elemen ais called he gen-
e a o o he en i e g oup (eza)z∈C; he cosine and sine unc ions a e defined as Cos(z,a):=
E2,1(z2a)andSin(z,a):=zE2,2(z2a). We ha e
Sin(z,a)=[0,z]
Cos(s,a)ds,z∈C,
o a∈1(Z); see [10, Sec s. 3.1 and 3.14]. Mo eo e , he Laplace ans o m o an en i e
g oup o a cosine unc ion is connec ed wi h he esol en o i s gene a o as ollows:
(λ–a)–1 =∞
0
e–λseas ds,λ>a1,
λλ2–a–1 =∞
0
e–λsCos(s,a)ds,λ>a1; (2.3)
see, o example, [10, p. 213].
Example 2.3 Fo α,β>0,weha e
Eα,β(zδ0)=Eα,β(z)δ0;Eα,β(zδ1)= ∞

j=0
zjδj
(αj+β).
In pa icula , ezδ1=∞
j=0
zjδj
j!and Cos(z,δ1)=∞
j=0
z2jδj
(2j)! a e gene a ed by δ1.

González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 6 o 32
Conside ing gene alized e sions o he Mi ag-Leffle unc ion, as well as o o he hy-
pe geome ic se ies, as p esen ed, o example, in [3–5], mo e examples can be easily de-
i ed.
In he nex p oposi ion, we collec some basic p ope ies o hese ec o - alued Mi ag-
Leffle unc ions. As usual, we conside Bochne ec o - alued in eg a ion in he Banach
space 1(Z); see, o example, [41, Sec . 1.2]. Fo he defini ion o he W igh unc ion γ,
see he Appendix, o mula (A.1).
P oposi ion 2.4 Fo α,β>0and a ∈1(Z), we ha e:
(i) Eα,β(a)1≤Eα,β(a1).
(ii) F(Eα,β(a)) = Eα,β(F(a));in pa icula ,F(eaz)=ezF(a)and
F(Cos(z,a)) = Cos(F(z),a) o z∈C.
(iii) σ1(Z)(Eα,β(a)) = Eα,β(σ1(Z)(a)).
(i ) The ollowing Laplace ans o m o mula holds:
∞
0
e–λ αk+β–1E(k)
α,β αad =k!λα–βλα–a–1(k+1),(λ)>a1/α
1, (2.4)
o k∈N∪{0}.
( ) Fo 0<γ<1,Eγ,1(a)=∞
0γ( )e a d .
P oo P oo s o pa s (i) and (ii) a e s aigh o wa d. Pa (iii) is he spec al mapping heo-
emshownin[35, Theo em 6.2.1]. Since he algeb a 1(Z)issemisimple(seeTheo em2.1),
o mulae in (i ) and ( ) a e di ec consequences o he scala iden i ies [39, o mula (180),
p. 21]. 
Gi en a∈1(Z), he modified Mi ag-Leffle unc ion Sα,β:(0,∞)→1(Z), which we
define by
Sα,β( ,a):= β–1Eα,β αa, > 0, (2.5)
is a (gα,gβ)- egula ized esol en amily gene a ed by ain he algeb a 1(Z). Fo he defini-
ion o (gα,gβ)- egula ized esol en amilies and mo e de ail in he gene al case o linea
and bounded ope a o s in a Banach space, we e e he eade o [2,Sec .4]and hesu ey
[36].
We in oduce he unc ions
ψα,β( ,s):= β–1 ∞

n=0
(–s –α)n
n!(–αn+β),s, >0,
o 0 < α<1andβ>0.No e ha ψα,1–α(s, )= –αα(s α) o 0<α<1.
A di ec consequence o [2, Theo em 12] is he ollowing subo dina ion heo em.
Theo em 2.5 Le 0<η1,0<η2,and a ∈1(Z), and le Sη1,η2be defined in (2.5). Then
Sαη1,αη2+β( ,a)=∞
0
ψα,β( ,s)Sη1,η2(s,a)ds, >0,
o 0<α<1and β≥0.
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 7 o 32
No e ha in he case o η1=2,η2=1,andα=β=1
2in Theo em 2.5,weob ain he
well-known ela ion be ween cosine and semig oup ope a o s gene a ed by a,knownas
he Weie s ass o mula:
ea =1
√π ∞
0
e–s2
4 Cos(s,a)ds, > 0, (2.6)
o a∈1(Z); see, o example, [10, Theo em 3.14.17].
A nice applica ion o he classical Wiene lemma is he in a iance o spec um o con-
olu ion ope a o s defined on p(Z) o 1≤p≤∞. This issue is con ained in he ollowing
heo em ha is hekeyabs ac esul in hispape .
Theo em 2.6 Fo a ∈1(Z), we define
A(b)(n):=(a∗b)(n), n∈Z,b∈p(Z). (2.7)
Then A ∈B(p(Z)) o all 1≤p≤∞.Mo eo e ,A=a1,and o all 1≤p≤∞,we
ha e he ollowing iden i ies:
σB(p(Z))(A)=σ1(Z)(a)=F(a)(T). (2.8)
Fo all a ∈1(Z), we ha e ha eza is an en i e g oup in p(Z)wi h gene a o a,and o all
1≤p≤∞,we ha e he ollowing iden i ies:
σB(p(Z))eza=σ1(Z)eza=ezF(a)(T),z∈C. (2.9)
P oo F om Young’s inequali y i ollows ha A∈B(p(Z)). Since he algeb a p(Z)has he
iden i y δ0, he p ope y o he no m ollows. Fo iden i ies (2.8),we e e o[32, Co ol-
la y 5.20]. Finally, o he spec al mapping heo em (2.9), we use (2.8)and[35,Theo-
em 6.2.1]. 
The elemen ain he heo em is also called he symbol o he ope a o A.
Rema k 2.7 I is also s aigh o wa d o check ha he adjoin ope a o o Ais again a
con olu ion ope a o gi en by A(g)(n):=(˜
a∗g)(n), whe e
˜
a(n)=a(–n), n∈Z.
3 Some fini e di e ence ope a o s in 1(Z)
An impo an case o fini e diffe ence ope a o s is gi en by sequences in he se
cc(Z):=a∈1(Z):∃m∈Z+:a(n)=0,∀|n|>m).
In such a case, he disc e e Fou ie ans o m o a∈cc(Z) is he igonome ic polynomial
F(a)eiθ=
m

j=–m
a(j)eijθ. (3.1)
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 8 o 32
I is in e es ing o obse e ha i m
j=–ma(j)=0, hen0∈σ1(Z)(a). This immedia ely ol-
lows om (2.8).
In his pape , we concen a e ou s udy on he ope a o s ha appea in he seminal pape
o Ba eman [16].
Defini ion 3.1 Fo ∈p(Z)wi h1≤p≤∞, we define he ollowing ope a o s:
(1) – (n):= (n)– (n+1)=((δ0–δ–1)∗ )(n);
(2) ∇ (n):= (n)– (n–1)=((δ0–δ1)∗ )(n);
(3) d (n):= (n+1)–2 (n)+ (n–1)=((δ–1 –2δ0+δ1)∗ )(n);and
(4) dd (n):= (n+2)–2 (n)+ (n–2)=((δ–2 –2δ0+δ2)∗ )(n)
o n∈Z.
We ema k ha when conside ing he abo e-defined ope a o s in he con ex o nume -
ical analysis, he ope a o s –and ∇a e ela ed o he Eule scheme o app oxima ion,
and he ope a o dco esponds o he second-o de cen al diffe ence app oxima ion
o he second-o de de i a i e. The ope a o dd appea s in Ba eman’s pape [16, p. 506]
in connec ion wi h he equa ions o Bo n and Ka man on c ys al la ices in ib a ion.
3.1 The ope a o –
The o wa d diffe ence ope a o  (n):= (n+1)– (n)isaclassicalope a o usedin
app oxima ion heo y and in he heo y o diffe ence equa ions. Conside ing i as an op-
e a o om p(Z) op(Z), ou main esul is as ollows.
Theo em 3.2 The ope a o – =a∗ ,whe e a := δ0–δ–1,possesses he ollowing p op-
e ies:
(1) The no m is gi en by =2;
(2) The Fou ie ans o m is F(a)(z)=1–z,|z|=1;
(3) Fo all 1≤p≤∞, he spec um is gi en by σB(p(Z))(–)={z∈T:|z–1|=1};
(4) Fo |λ+1|>1,
(λδ0+a)–1 =
j≥0
δ–j
(1 + λ)j+1 .
(5) The associa ed g oup is e–za(n)=e–zz–n
(–n)! χ–N0(n),z∈C,n∈Z,and i s gene a o is
–a.
(6) The no m o he g oup is gi en by e– a1=1, >0;
(7) The associa ed cosine unc ion is Cos(z,–a)(n)= √π
(–n)! (z
2)–n+1
2J–n–1
2(z)χ–N0(n) o
z∈Cand n∈Z.
P oo (1) The Minkowski inequali y shows ha ≤2. Then obse e ha δ0∈p(Z)
wi h δ0p=1sa isfiesδ0p= 2, p o ing he claim. (2) Follows immedia ely om he
defini ion o he disc e e Fou ie ans o m. (3) Follows om o mula (2.8)inTheo em2.6
and (2).
To p o e (4), we apply (2.1) oge
(λδ0+a)–1 =(λ+1)δ0–δ–1–1 =
j≥0
δ–j
(1 + λ)j+1
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o |λ+1|> 1. We show (5) di ec ly:
e–za(n)=ezδ–1 ∗e–zδ0(n)=ezδ–1 ∗e–zδ0(n)=e–zz–n
(–n)!χ–N0(n)
o z∈Cand n∈Z.Theno me– a1=1 o >0iss aigh o wa d om(5).Finally, o
show (7), we apply he Laplace ans o m and o mula (A.10) oge
√π
(–n)! ∞
0
e–λ 
2–n+1
2
J–n–1
2( )d =λ
(λ2+1)
–n+1 ,λ>1,
o n≤0. By (4) we ha e ha
λ
(λ2+1)
–n+1 =λλ2+a–1(n), n≤0,
and we apply (2.3) o conclude he claimed equali y and iden i y he gene a o o he cosine
unc ion wi h –a.
We ema k ha g oups gene a ed by a e ea ed in [1, Sec . 2] and cosine unc ions
in [16, In oduc ion].
3.2 The ope a o ∇
This ope a o co esponds o he classical backwa d diffe ence ope a o .
Theo em 3.3 The ope a o ∇ =a∗ ,whe e a := δ0–δ1,possesses he ollowing p ope -
ies:
(1) ∇=2;
(2) F(a)(z)=1–1
z;
(3) Fo all 1≤p≤∞,we ha e σB(p(Z))(∇)={z∈T:|z–1|=1};
(4) Fo |λ+1|>1,
(λδ0+a)–1 =
j≥0
δj
(1 + λ)j+1 .
(5) e–za(n)=e–zzn
n!χN0(n),z∈C,n∈Z;
(6) e– a1=1, >0;
(7) Cos(z,–a)=√π
n!(z
2)n+1
2Jn–1
2(z)χN0(n),z∈C,n∈Z.
P oo The p oo s o s a emen s (1), (2), (3), and (4) ollow he lines o Theo em 3.2.Fo
s a emen (5), we ha e
e–za(n)=ezδ1∗e–zδ0(n)=e–zzn
n!χN0(n)
o z∈Cand n∈Z. Claim (6) ollows om (5). Finally, we check (7) as ollows. We apply
Laplace ans o m and o mula (A.10) oge
√π
n!∞
0
e–λ 
2n+1
2
Jn–1
2( )d =λ
(λ2+1)
n+1 ,λ>1,
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Now we apply he Lé y subo dina ion p inciple (4.2) oa=δ–1 –δ0o a=δ1–δ0and
by Theo em 4.4 ob ain he ollowing esul . No e ha his o mula is also ob ained om
(A.12).
Co olla y 4.5 Le 0<α<1,and le s,αbe he Lé y s able p ocess defined by (A.11). Then
∞

j=1
k–αj(n)(– )j
j!=∞
0
,α(s)e–ssn
n!ds, >0,n≥1.
In pa icula ,when α=1
2,we ha e
∞

j=1
k–j
2(n)(– )j
j!=∞
0
√4πs3e– 2
4se–ssnds,n≥1.
4.2 The ope a o (–d)α
The ope a o (–d)α o 0 < α≤1, called he ac ional disc e e Laplacian, has been deeply
ea ed in [22,23,26,37]. In [37, Sec . 3] he sequence (–δ–1 +2δ0–δ1)αis deno ed by Kα
d.
To ollow he no a ion in ha pape , we w i e
Kα
d(n):= 1
2ππ
–π4sin2(θ/2)αe–inθdθ=(–1)n(2α+1)
(1 + α+n)(1 + α–n)
o n∈Zand α>0[37, Fo mula (22)]. In he case 1 + α+n∈–N0,Kα
d(n)=0.Then
|Kα
d(n)|∼(2α+1)
π|n|–2α–1 as n→±∞.
We summa ize he main p ope ies o he ke nel Kα
din he ollowing esul .
Theo em 4.6 Fo 0<α<1,we ha e
FKα
d(z)=2–z+1
zα
=4sin2θ
2α
,z=eiθ∈T, (4.7)
and
Kα
d=Kα
+∗Kα
–. (4.8)
In pa icula ,
Kα
d=∞

j=0 k–α
–∗k–α(j)δj, (4.9)
whe e k–α
–(n):=k–α(–n). Mo eo e ,σ(Kα
d)=[0,4
α], and

Kα
d
1=2(1 + 2α)
(1 + α)2.
P oo Iden i y (4.7) ollows om (4.1). To show (4.8), we apply he disc e e Fou ie ans-
o m o ob ain ha
FKα
+∗Kα
–(z)=1–1
zα
(1 – z)α=2–z+1
zα
=FKα
d(z)

González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 17 o 32
o z∈T. Since he disc e e Fou ie ans o m is one- o-one, we ob ain he equali y. To
p o e (4.9), we no e ha he igh -hand side e alua ed a n∈Zis equal o (k–α
–∗kα)(n),
and we ha e
k–α
–∗kα(n)=
n

j=0
k–α–(n–j)k–α(j)=
n

j=0
Kα
+(n–j)Kα
–(j)=Kα
+∗Kα
–(n),
and he esul ollows om (4.8). The spec um is gi en in [37, Theo em 1.3 (iii)], and he
no m o Kαis calcula ed in [37, Lemma 3.2]. 
An in e es ing consequence is he ollowing co olla y, which seems o be a new o mula
o binomials o nonin ege en ies.
Co olla y 4.7 Le α∈(0, 1) and n ∈N0.We ha e he ollowing equali y:
2α
α+n=∞

j=0 α
j+nα
j.
P oo The combina o ial equali y is a s aigh o wa d consequence o he explici exp es-
sion o he ke nel con olu ions Kα
+,Kα
–,andKα
d.
The ollowing esul collec s he main esul s on he ac ional disc e e semig oup. Fo
o he esul s, see also [37].
Theo em 4.8 Fo any 0<α<1,we ha e ha he ac ional disc e e semig oup gene a ed
by –Kα
dis gi en by
e–zKα
d(n)=(–1)
n∞

k=1
(–1)kzk
k!
(2kα+1)
(1 + kα+n)(1 + kα–n)+δ0(n)
o n ∈Zand z ∈C.Mo eo e :
(i) The disc e e Fou ie ans o m o e–zKα
dis gi en by
Fe–zKα
deiθ=e–z(4sin2(θ
2))α,θ∈[–π,π),z∈C.
(ii) e– Kα
d(n)≥0,and e– Kα
d1=1 o n∈Zand ≥0, ha is,i is a Ma ko ian
semig oup.
(iii) σ(e–zKα
d)={e–z(4sin2(θ
2))α:θ∈[–π,π)}.
P oo The ac ional disc e e semig oup gene a ed by –Kα
dis gi en in [37, Theo em 1.3].
The e he en i e g oup (e–zKα
d)z∈Cis w i en as Lα
z. S a emen (i) ollows om P oposi-
ion 2.4(ii) combined wi h (4.7)inTheo em4.6. The p oo o (ii) is con ained in [37,The-
o em 1.3( )]. Finally, o p o e (iii), we use (2.9)inTheo em2.6 and (4.7)inTheo em4.6.
We also apply he Lé y subo dina ion p inciple (4.2) o he semig oup gene a ed by a=
δ–1 –2δ0+δ1 o ob ain he ollowing esul .
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Co olla y 4.9 Le 0<α<1,and le s,αbe he Lé y s able p ocess defined by (A.11). Fo
n∈Zand 0< <1,we ha e
∞

j=0
Kαj
d(n)(– )j
j!=∞
0
,α(s)e–2sIn(2s)ds;
in pa icula , o α=1
2,
∞

j=0
K
j
2
d(n)(– )j
j!=∞
0
√4πs3e– 2
4se–2sIn(2s)ds.
4.3 The ope a o (–dd)α
Since he elemen (δ2–2δ0+δ–2) gene a es a uni o mly bounded C0-semig oup, we con-
side he ac ional powe (δ2–2δ0+δ–2)α o 0 < α<1.Fo simplici y,wew i eKα
dd ins ead
o (δ2–2δ0+δ–2)α.
Theo em 4.10 Le 0<α<1.
(i) We ha e
Kα
dd(n)= (2α+1)
(1 + α+n
2)(1 + α–n
2)cosn
2π,n∈Z.
(ii) Kα
dd(2n)=Kα
d(n)and Kα
dd(2n–1)=0 o n∈Z.
(iii) Kα
dd1=2(1+2α)
(1+α)2.
(i ) F(Kα
dd)(eiθ)=(4sin2(θ))α o θ∈[–π,π),and σ(Kα
dd)=[0,4
α].
P oo (i) Fo n∈Z,weha e ha
Kα
dd(n)= 1
2ππ
–π4sin2(θ)αe–inθdθ=4α
ππ
0
sin2α(θ)cos(nθ)dθ
=(2α+1)
(1 + α+n
2)(1 + α–n
2)cosn
2π,
whe eweha eapplied[31, Fo mula 3.631 (8)]. Pa s (ii), (iii), and (i ) a e s aigh o wa d
om pa (i). 
Now we conside he en i e g oup (e–zKα
dd )z∈Cgene a ed by –Kα
dd.
Theo em 4.11 Fo 0<α<1,we ha e:
(i) e–zKα
dd (n)=cos(n
2π)∞
k=1(–1)kzk
k!
(2kα+1)
(1+kα+n
2)(1+kα–n
2)+δ0(n),n∈Z.
(ii) e–zKα
dd (2n)=e–zKα
d(n)and e–zKα
dd (2n–1)=0 o n∈Z.
(iii) e– Kα
dd (n)≥0and e– Kα
dd 1=1 o n∈Zand ≥0, ha is,i is a Ma ko ian
semig oup.
(i ) F(e–zKα
dd )(eiθ)=e–z(4sin2(θ))α o θ∈[–π,π),and
σe–zKα
dd =σe–zKα
dd =e–zuα|u∈[0,4].
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 19 o 32
P oo By [47, Theo em 1, p. 263] we ha e ha
e– Kα
dd (n)=∞
0
,α(s)e–2sIn
2(2s)dsχ2Z(n)=e– Kα
dn
2χ2Z(n),
and we conclude equali ies (i) and (ii). Pa s (iii) and (i ) a e p o ed om simila p ope ies
o e–zKα
d.
Rema k 4.12 The spec um o he disc e e ac ional Laplacian –(–d)αis de e mined by
σ(–(–d)α)=[–4
α,0], eco e ing he esul o Lizama and Roncal [37], Theo em 1.3(iii).
A simila esul is ob ained o he ope a o –(–dd)αgi en by σ(–(–dd)α)=[–4
α,0].Fo
he disc e e ac ional diffe ence ope a o s –and ∇,weha eσ((–)α) = [–(1+eiT)α]and
σ(∇α) = [–(1 + eiT)α], espec i ely.
5 Fundamen al solu ions o semidisc e e e olu ion equa ions
In his sec ion, we conside he ope a o B (n):=(b∗ )(n)wi hb∈1(Z), ∈p(Z), p∈
[1,∞], and n∈Z. Ou objec i e is ob aining a undamen al ep esen a ion o solu ions
o he ollowing semidisc e e ac ional e olu ion equa ion:
⎧
⎨
⎩
Dβ
u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), u (n,0)=φ(n), n∈Z,
whe e β∈(0,2]. Fo a sufficien ly egula unc ion ,wedeno ebyDβ
he Capu o de i a-
i e o o de βgi en by
Dβ
( )= 1
(1 – β)
0
( –s)–β (s)ds =g1–β∗ ( ), >0,
o 0 < β<1and
Dβ
( )= 1
(2 – β)
0
( –s)1–β (s)ds =g2–β∗ ( ), >0,
o 1 < β<2.Fo β=1andβ= 2, we conside he usual fi s - and second-o de de i a i es.
No e ha
lim
β→1–Dβ
( )= ( ), lim
β→2–Dβ
( )= ( ), >0;
howe e ,
lim
β→0+Dβ
( )= ( )– (0), lim
β→1+Dβ
( )= ( )– (0), > 0; (5.1)
see, o example, [17,30].
To begin wi h, we conside he semidisc e e Cauchy p oblem (1.1) gi en in he in o-
duc ion,
⎧
⎨
⎩
∂ u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z,(5.2)
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 20 o 32
and i s undamen al solu ion, which is ob iously gi en by Duhamel’s o mula
u(n, )=eB ϕ(n)+
0
eB( –s)g(n,s)ds,n∈Z, ≥0.
Analogously, in he case o he second-o de semidisc e e Cauchy p oblem
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∂ u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z,
u(n,0)=ψ(n), n∈Z,
(5.3)
we ha e ha he undamen al solu ion is gi en by D’Alembe o mula
u(n, )=Cos( ,B)ϕ(n)+Sin( ,B)ψ(n)+
0
Sin( –s,B) (s)ds,
whe e Cos( ,B)andSin( ,B)a egene a edbyB.
We now conside ac ional in ime gene aliza ions. Gi en 0 < β≤1, we fi s conside
he equa ion
⎧
⎨
⎩
Dβ
u(n, )=Bu(n, )+g(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z.(5.4)
We ecall ha Eα,β(b)(wi hb∈1(Z)) is he ec o - alued Mi ag-Leffle unc ion gi en
in Defini ion 2.2. The main esul is he ollowing heo em.
Theo em 5.1 Le ϕ,φ∈p(Z), and le g :Z×R+→Cbe such ha o each ∈R+,g(·, )∈
p(Z), and sups∈[0, ]g(·,s)p<∞wi h 1≤p≤∞.
(i) Fo 0<β<1, he unc ion
u(n, )=Eβ,1 βb∗ϕ(n)
+
0
( –s)β–1Eβ,β( –s)βb∗g(·,s)(n)ds,n∈Z,
is he unique solu ion o he ini ial alue p oblem (5.4). Mo eo e ,u(·, )belong o
p(Z) o >0.
(ii) Fo 1<β<2, he unc ion
u(n, )=Eβ,1 βb∗ϕ(n)+ Eβ,2 βb∗φ(n)
+
0
( –s)β–1Eβ,β( –s)βb∗g(·,s)(n)ds,n∈Z,
is he unique solu ion o he ini ial alue p oblem (1.5). Mo eo e ,u(·, )belong o
p(Z) o >0.
P oo Since he algeb a 1(Z) is semisimple (see Theo em 2.1), he o mulae in (i) and (ii)
a e di ec consequences o he scala iden i ies, which in case 0 < α<1canbe oundin
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 21 o 32
[36, Sec . 3.3, o mula (8)] combined wi h [36, Sec . 1.2]. The case 1 < α< 2 ollows om
[36, Sec . 3.3, o mula (11)]. See also he e e ences he ein. 
Rema k 5.2 Now we conside he beha io o he solu ion as β ends o he in ege pa am-
e e , ha is, β= 1, 2. Fo simplici y, we conside he homogeneous case g=0.Asβ→1–,
he solu ion o equa ion (1.4) con e ges o semig oup amily ope a o s E1,1( b), and as
β→2–, he solu ion o equa ion (1.5),
u(·, )=Eβ,1 βb∗ϕ+ Eβ,2 βb∗φ, >0,
con e ges o unique mild solu ion o second-o de Cauchy p oblem, ha is, he sum o a
cosine unc ion and a sine unc ion gene a ed by b;see[10, Co olla y 3.14.8].
Howe e , as in he scala case, as β→1+, he solu ion o equa ion (1.5)con e ges o
u(·, )=E1,1(b )+ E1,2( b), >0.
No e ha his unc ion is a solu ion o he ollowing fi s -o de modified Cauchy p oblem:
⎧
⎨
⎩
(n, )=B (n, )+φ(n), n∈Z, >0,
(n,0)=ϕ(n), n∈Z,
o φ,ϕ∈p(Z). This ac is in acco dance wi h he in e pola ion p ope y o he Capu o
ac ional de i a i e; see (5.1).
The undamen al solu ions uβ,1 o sys ems (1.4)and(1.5) a e ob ained by equi ing
ha he ini ial alue ψand he ini ial eloci y φbe he sequences ψ=δ0and φ=0.In
he case 1 < β≤2 (including he wa e equa ion), a second undamen al solu ion uβ,2 is
gi en by ψ=0andφ=δ0;see[26, Rema k 3.2]. A consequence o Theo ems 5.1 and
2.5 is he ollowing subo dina ion heo em o undamen al solu ions, which ex ends [26,
Co olla y 3.5].
Co olla y 5.3 Le uβ,1 and uβ,2 be he undamen al solu ions o p oblems (1.4)and (1.5),
and le αbe he W igh unc ion defined by (A.1).
(i) Le 0<β<1.Then
uβ,1(n, )=∞
0
β(τ)u1,1n,τ βdτ,n∈Z, >0.
(ii) Le 1<β<2.Then
uβ,1(n, )=∞
0
β
2(τ)u2,1n,τ β
2dτ,
uβ,2(n, )=
0
( –u)–β
2
(1 – β
2)∞
0
β
2(τ)u2,2n,τuβ
2dτdu
o n∈Zand >0.

González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 22 o 32
Rema k 5.4 The W igh unc ion 1
3can be exp essed in e ms o he Ai y unc ion Ai(z),
ha is,
1
3(z)=32
3Aiz
31
3,z∈C;
see, o example, [28]. A in eg al ep esen a ion o he Ai y unc ion is gi en by he im-
p ope Riemann in eg al:
Ai(x)= 1
π∞
0
cos 3
3+x d ,x∈R.
This unc ion appea s in se e al applied p oblems, in pa icula , in he Sch ödinge equa-
ion o quan um physics and in op ics (s udy o caus ics); see mo e de ails in [44]. By Co ol-
la y 5.3(i) we conclude ha
E1
3,1 1
3b(n)=32
3∞
0
Aiτ
31
3eτ 1
3b(n)dτ,n∈Z, >0,
o b∈1(Z).
The pa icula case o Theo em 5.1 wi h B=–(–A)α,whe eAis he infini esimal gene -
a o o an uni o mly bounded C0-semig oup in B(p(Z)), has ecei ed a special a en ion.
In [34, Theo em 3.3] and [26, Theo em 3.1] he ime/space ac ional e olu ion equa-
ions (1.4)and(1.5)o o de s0<β≤1and1<β≤2, espec i ely, a e sol ed, whe e
B=–(–d)α,anddis he disc e e Laplacian ope a o . Bo h p oo s a e based on he ex-
plici exp essions o ec o - alued Mi ag-Leffle unc ions Eβ,1(– βKα
d), Eβ,2(– βKα
d), and
Eβ,β(– βKα
d). As a consequence o he esul s in Sec . 4, we can easily gi e a gene al e sion,
which ex ends bo h esul s.
Co olla y 5.5 Le ϕ,φ∈p(Z), and le g :Z×R+→Cbe such ha o each ∈R+,
g(·, )∈p(Z)and sups∈[0, ]g(·,s)p<∞wi h 1≤p≤∞.Fo a ∈1(Z)gene a ing a uni-
o mly con inuous semig oup in 1(Z), we w i e (–a)α o he ac ional powe s gi en in
Defini ion 4.1 and B( ):=–(–a)α∗ o ∈p(Z)and 0<α<1.Then hesame ep esen-
a ion o he undamen al solu ions gi en in Theo em 5.1 wi h b =–(–a)αholds.
6 Applica ions
We s udy some conc e e examples ha appea in a ious applied fields.
6.1 The disc e e Nagumo equa ion
Le us conside he linea pa o he disc e e Nagumo equa ion, which can be w i en as
ollows:
⎧
⎨
⎩
∂ u(n, )=du(n, )–ku(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z,(6.1)
whe e 0 < k< 1/2. The disc e e Nagumo equa ion is used as a model o he sp ead o
gene ic ai s and o he p opaga ion o ne e pulses in a ne e axon, neglec ing eco e y;
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 23 o 32
see [48] and e e ences he ein. Using Theo em 3.4(3), we ob ain
σe (d–kI)=e σ(d–kI)=e s : ≥0,–4 – k≤s≤–k.
This implies ha he unique solu ion o equa ion (6.1) is uni o mly asymp o ically s able,
ha is,
u(n, )=e (d–kI)ϕ(n)→0as →∞.
Mo eo e , using Theo em 3.4(4) and he semig oup p ope y, we can ob ain he ollowing
ep esen a ion o he undamen al solu ion:
u(n, )=e– kIe dϕ(n):=e– kI ∗e d∗ϕ(n)=
n

j=0 e– kI ∗e d(n–j)ϕ(j)
=e–2
n

j=0
n–j

l=0
(–k )l
l!In–j–l(2 )ϕ(j).
Since σ(–(–d)α)=[–4
α,0](seeRema k4.12), we ha e ha he same asymp o ic beha io
also holds o he undamen al solu ion o he ac ional Laplacian e sion o he disc e e
Nagumo equa ion [37,Sec .7]:
⎧
⎨
⎩
∂ u(n, )=–(–d)αu(n, )–ku(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z.
6.2 The semidisc e e anspo equa ion associa ed wi h he -di e ence
ope a o
Le us conside he semidisc e e anspo equa ion
⎧
⎨
⎩
∂ u(n, )= u(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z,(6.2)
whe e >0,and is he - o wa d diffe ence ope a o defined by  (n):= (n+1)–
(n); see [7, Sec . 5.5]. Obse e ha =+(1– )I,whe eIis he iden i y ope a o .
Then by pe u ba ion semig oup heo y he unique solu ion o (6.2)has he o mu(n, )=
e (+(1– )I)ϕ(n), n∈Z. By he spec al mapping (2.9)inTheo em2.6 we ob ain ha
σe (+(1– )I)=e σ(+(1– )I)).
Hence by Theo em 3.2(3)wededuce ha σ(+(1– )I)={z∈T:|z+ |=1}. The e o e
o any > 1, we ha e ha he uppe bound o he spec um o B=+(1– )Iis nega i e,
ha is,
ωσ+(1– )I:= supz:z∈σ+(1– )I<0,
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 24 o 32
and, consequen ly, we ob ain ha o any > 1, he unique solu ion o equa ion (6.2)is
uni o mly asymp o ically s able, ha is,

e  ϕ
→0as →∞,
uni o mly wi h espec o ϕ≤1. O cou se, his esul can be also di ec ly deduced om
Theo em 3.2(5). Analogously, using he ac ha
 –ei > 0 implies  –ei α>0
o any 0 < α<1and > 1, we can deduce om Theo em 4.4 ha he same p ope y o
asymp o ic s abili y emains ue o he unique solu ion o he ac ional semidisc e e
anspo equa ion
⎧
⎨
⎩
∂ u(n, )=–(–
)αu(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z.(6.3)
6.3 The De Juhasz equa ion
We conside he ollowing semidisc e e equa ion:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∂ u(n, )=du(n, )–2ku(n, ), n∈Z, >0,
u(n,0)=ϕ(n), n∈Z,
u (n,0)=ψ(n), n∈Z,
(6.4)
whe e k> 0. This equa ion can be ound in he seminal pape o Ba eman [16]incon-
nec ion wi h su ges in sp ings and connec ed sys ems o sp ings. We call i he De Juhasz
equa ion because, acco ding Ba eman’s pape , De Juhasz deduced o he fi s ime he
modeling o such equa ion in mechanical heo y. Following Ba eman’s pape , his semidis-
c e e equa ion is ob ained when he concen a ed masses on a ligh s ing a e moun ed on
sp ings a anged ei he along a s aigh line o on he ci cum e ence o a ci cle o helix [16,
Sec . 5, o mula (5.1)]. Applying Theo em 3.4 and conside ing he ope a o B=d–2kI,
we ob ain
σ(d–2kI)=[–4–2k,–2k],
and he e o e
σCos( ,B)=cos( √s):s∈[2k,4+2k].
In pa icula , his implies ha on he Hilbe space 2(Z), we ha e Cos( )≤1, and, con-
sequen ly, he unique solu ion o (6.4)whenψ≡0mus bebounded.Thisex ends he
p e ious esul o Ba eman [16,Sec .5],whos udied(6.4) wi h he ini ial condi ions ψ≡0
and ϕ(n)=δ0(n).
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 25 o 32
6.4 The Capu o–Fab izio de i a i e
We ecall ha gi en a sufficien ly egula unc ion uand 0 < α<1, heCapu o–Fab izio
de i a i e o o de αis defined as [19]
CFDαu( )= 1
1–α
0
e–α
1–α( –s)u(s)ds.
No e ha he Capu o–Fab izio de i a i e has been e y ecen ly used o p opose a new
ma hema ical modeling o human li e [12], HIV [13], pa allel RCL ci cui s [8], he Rubella
disease model [14], epidemic childhood diseases [11], and COVID-19 [15]. Howe e , wi h
he excep ion o he implici solu ion o he linea model in he scala case, p oposed by
Losada and Nie o in [38], so a no explici o mulas ha e been p oposed o he solu ion
o he ac ional Cauchy p oblem in he con ex o Banach algeb as.
We conside he equa ion
⎧
⎨
⎩
CFDα
u(n, )=Bu(n, )+g(n, ), n∈Z, ≥0,
u(n,0)=ϕ(n), n∈Z,(6.5)
whe e we ecall ha B (n):=(b∗ )(n)wi hb∈1(Z), ∈p(Z), p∈[1,∞], and n∈Z.
Since Bis bounded, assuming ha 1
1–α∈ρ(B), we ob ain he ollowing ep esen a ion o
he solu ion o (6.5):
u(n, )=T( )ϕ(n)+
0
T( –s)h(n,s)ds,n∈Z, > 0, (6.6)
whe e
T( ):=eα
1–α(1–(1–α)B)–1 e–α
1–α , ≥0, (6.7)
and
h(n, ):=I–(1–α)B–1(1 – α)g (n, )+αg(n, ). (6.8)
Indeed, since Bis bounded, om [38, P oposi ion 2] and aking in o accoun [38, o mula
(8) and he explici o mula o M(α) gi en in Rema k p. 89] we know ha he unique
solu ion o p oblem (6.5)isgi enby heuniquesolu iono hep oblem
u (n, )=αBI–(1–α)B–1u(n, )
+I–(1–α)B–1(1 – α)g (n, )+αg(n, )
(no e ha he e is a small bu impo an misp in in [38, p. 90, l. 16], whe e we mus ead
˜σ( ) ins ead o ˜σ( )). Using he iden i y
(1 – α)B1–(1–α)B–1 =I–(1–α)B–1 –I,
González-Camus e al. Ad ances in Diffe ence Equa ions ( 2021) 2021:35 Page 32 o 32
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