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Maximal $ ell_p$- egula i y o disc e e ime ol e a equa ions wi h delay
A icleinJou nal o Di e ence Equa ions and Applica ions · Ap il 2019
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MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA
EQUATIONS WITH DELAY
CARLOS LIZAMA AND MARINA MURILLO-ARCILA
Abs ac . In his pape we in es iga e he exis ence and uniqueness o solu ions
belonging o he ec o - alued space `p(Z, X) by using Blunck’s heo em on he
equi alence be ween ope a o - alued `p-mul iplie s and he no ion o R-boundedness
o he disc e e ime ol e a equa ion wi h delay gi en by
u(n) =
n
X
j=−∞
b(n−j)Au(j) +
k
X
j=1
βju(n−τj) + (n), n ∈Z,
whe e Ais a closed linea ope a o wi h domain D(A) de ined on a Banach space
Xand b∈`1(Z) e i ies sui able condi ions such as 1- egula i y. We cha ac e ize
maximal `p- egula i y o solu ions o such p oblems in e ms o he da a and an
spec al condi ion and we p o ide op imal es ima es. Mo eo e , we illus a e ou
esul s p o iding di e en models ha label in o ou gene al scheme such as he
disc e e ime wa e and Kuzne so equa ions.
Keywo ds: ol e a equa ions, maximal `p- egula i y, R-bounded, disc e e wa e
equa ion, disc e e kuzne so equa ion
Ma hema ics Subjec Classi ica ion (2010): 45D05, 35R09, 65Q10, 39A06.
1. In oduc ion
In his pape we analyze he exis ence and uniqueness o solu ions in ec o alued
`p(Z, X) spaces o disc e e ime o mula ions o he ollowing in eg o pa ial di e en ial
equa ion wi h delay,
u( ) = Z
−∞
a( −s)Au(s)ds +βu( −τ) + ( , u( )), ∈R,
whe e Ais a closed linea ope a o de ined on a Banach space X. This equa ion models
iscoelas ic luids, hea conduc ion wi h memo y and elec odynamics p ocesses wi h
memo y [4, 28]. Typical models ha a e included in his a icle co espond o di e en
disc e e e sions o he mul idimensional wa e and Kuzne so equa ions
(1) u −c2∆u−ν∆u = ( ), ∈R.
The analysis o quali a i e p ope ies o Vol e a ype equa ions has been conside ed
by a ious au ho s, see [10] and all he e e ences he ein. Mo eo e , nume ical me hods
o he esolu ion o Vol e a equa ions ha e been s udied among o he s in [6, 7, 9, 19,
26, 27]. On he o he hand, he s udy o maximal egula i y o disc e e sys ems ha
belong o he Lebesgue space o ec o - alued sequences since he pionee wo k o S.
Blunck [5] has expe imen ed a g ea de elopmen as i can be seen in he ecen pape s
The i s au ho is pa ially suppo ed by FONDECYT g an numbe 1180041.
The second au ho is suppo ed by MEC, g an MTM2016-75963-P and GVA, G an 18I264.01/1.
1
2 C. LIZAMA AND M. MURILLO
[13, 21, 20, 22]. This s udy is s ongly connec ed wi h he necessi y o op imal `p−`q
ime-space es ima es o he co esponding linea ized p oblem [2, 11, 13, 16, 14, 18,
17, 24, 25].
In ou wo k, we succeed cha ac e izing maximal `p- egula i y o he ollowing ab-
s ac model
(2) u(n) =
n
X
j=−∞
b(n−j)Au(j) +
k
X
j=1
βju(n−τj) + (n), n ∈Z,
whe e ∈`p(Z, X), A is a closed linea ope a o wi h domain D(A) de ined on Xand
b∈`1(Z). I is wo hwhile o obse e ha , o ins ance, model (2) includes among
o he s he disc e e Kuzne so equa ion (1) aking A=−∆d,N , he mul idimensional
disc e e Laplacian, b(n) = −(c2+ν)δ0(n) + ν δ0(n−1), β1= 2 , τ1= 1 and β2=
− 2, τ2= 2.
This pape is o ganized as ollows: in Sec ion 2, we i s ecall he no ions o UMD-
spaces, R-boundedness, `p-mul iplie s, sec o ial ope a o s and he disc e e ime Fou ie
ans o m de ined on he space o dis ibu ions. Mo eo e , we ecall he well-known
Blunck’s Fou ie mul iplie heo em [5] o ope a o - alued symbols on UMD-spaces
ha es ablishes he equi alence be ween `p-mul iplie s and R-boundedness.
In Sec ion 3, we p o e ou main esul , namely, i b∈`1(Z) is 1- egula , ˆ
b( )6= 0 o
all ∈Tand (1−Pk
j=1 βje−i τj
ˆ
b( )) ∈T⊂ρ(A).
hen he ollowing asse ions a e equi alen :
(i) Fo all ∈`p(Z, X) equa ion
u(n) =
n
X
j=−∞
b(n−j)Au(j) +
k
X
j=1
βju(n−τj) + (n), n ∈Z,
has a unique solu ion in `p(Z,[D(A)]);
(ii) M( ) := (1 −Pk
j=1 βje−i τj−ˆ
b( )A)−1is an `p-mul iplie om X o [D(A)];
(iii) The se {M( ) : ∈T}is R-bounded.
Obse e, ha ou esul demands 1- egula i y o he ke nel sequence b(n). We in o-
duce his concep o he i s ime in de ini ion 3.2 and i co esponds o he disc e e
coun e pa o he no ion o 1- egula i y in oduced in [15]. Fu he mo e when Xis
Hilbe we simpli y he p e ious esul by eplacing he condi ion (iii) abo e by an
easie compu able condi ion
sup
∈TkM( )k<∞.
We also ensu e op imal es ima es o model (2) unde any o he abo e condi ions, ha
is, he ollowing es ima e also holds
kuk`p(Z;X)+kb∗Auk`p(Z;X)≤Ck k`p(Z;X).
Finally, in sec ion 4, we p o e, as an applica ion o ou cha ac e iza ion, he exis ence
and uniqueness o `p(Z;`q(ZN)) solu ions o ime disc e iza ions o ms o he wa e and
MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY 3
Kuzne so equa ions in e ms o he da a o he p oblem as i can be seen in heo ems
4.2,4.3 and 4.4. In addi ion, we ob ain maximal `p−`q- es ima es o such models.
2. Analy ical amewo k and no a ion
In his sec ion, we p esen some esul s ha will be needed h oughou he pape .
Le Xbe a Banach space. We deno e by S(Z;X) he space o all ec o - alued
sequences :Z→Xsuch ha o each k∈N0 he e exis s a cons an Ck>0
sa is ying pk( ) := supn∈Z|n|kk (n)k< Ckand when X=Rwe deno e S(Z).
We w i e as Cn
pe (R;X), n ∈N0, he space o all 2π-pe iodic X- alued and n- imes
con inuously di e en iable unc ions de ined in R.In wha ollows, we will deno e
T:= (−π, π) and T0:= (−π, π) {0}.The space o es unc ions is he space
C∞
pe (T;X) := Tn∈N0Cn
pe (R;X).When X=Rwe simply w i e C∞
pe (T).
Fo each ∈`p(Z;X) we can de ine he map
(3) T (ψ) := hT , ψi:= X
n∈Z
(n)ψ(n), ψ ∈ S(Z),
and we ha e T ∈ S0(Z, X) = {T:S(Z)→X:Tis linea and con inuous}.
Rema k 2.1.By his mapping we iden i y `p(Z;X) wi h a subspace o S0(Z;X).When
con enien and con usion seems unlikely, a unc ion ∈`p(Z;X) is iden i ied wi h
T ∈ S0(Z, X).
The e also exis s a na u al mapping ha iden i ies C∞
pe (T;X) wi h a subspace o
D0(T;X) = {T:C∞
pe (T)→X:Tis linea and con inuous}which assigns o each
S∈C∞
pe (T;X) he linea map
LS(ϕ) := hLS, ϕi:= 1
2πZπ
−π
ϕ( )S( )d , ϕ ∈C∞
pe (T),
and we ha e LS∈ D0(T;X).
De ini ion 2.2. The disc e e ime Fou ie ans o m F:S(Z;X)→C∞
pe (T;X) is
de ined by
Fϕ( )≡bϕ( ) :=
∞
X
j=−∞
e−ij ϕ(j), ∈(−π, π]
and he co esponding in e se ans o m is gi en by
(4) F−1ϕ(n)≡ˇϕ(n) := 1
2πZπ
−π
ϕ( )ein d , n ∈Z,
whe e ϕ∈C∞
pe (T;X).
This isomo phism, allows us o de ine he disc e e ime Fou ie ans o m (DTFT)
be ween he spaces o dis ibu ions S0(Z;X) and D0(T;X) as ollows:
(5) hFT, ψi≡F(T)(ψ) := b
T(ψ)≡ hT, ˇ
ψi, T ∈ S0(Z;X), ψ ∈C∞
pe (T),
whose in e se F−1:D0(T;X)→ S0(Z;X) is gi en by
hF−1L, ψi ≡ F−1(L)(ψ) := ˇ
L(ψ)≡ hL, b
ψi, L ∈ D0(T;X), ψ ∈ S(Z).
4 C. LIZAMA AND M. MURILLO
We inally p esen a echnical lemma in oduced in [22] which will be necessa y
h ougou he pape . We i s need he ollowing de ini ion.
De ini ion 2.3. Gi en u∈`p(Z;X) and ∈`1(Z) he con olu ion p oduc be ween
uand is de ined as
(u∗ )(n) :=
n
X
j=−∞
u(n−j) (j) =
∞
X
j=0
u(j) (n−j), n ∈Z.
Mo eo e , he con olu ion o a dis ibu ion T∈ S0(Z, X) wi h a unc ion a∈`1(Z+)
is de ined by
(6) hT∗a, ϕi:= hT, a ◦ϕi, ϕ ∈ S(Z),
whe e
(a◦ϕ)(n) :=
∞
X
j=0
a(j)ϕ(j+n).
Lemma 2.4. Le u, ∈`p(Z;X)be gi en and a∈`1(Z+)which is de ined by 0 o
nega i e alues o n. The ollowing asse ions a e equi alen :
(i) a∗ ∈`p(Z, X)and (a∗ )(n) = u(n) o all n∈Z.
(ii) hu, ˇϕi=h , (ϕ·ba−ˇ
)i o all ϕ∈C∞
pe (T),
whe e
(ϕ·ba−ˇ
)(n) := 1
2πZπ
−πba(− )ϕ( )ein d , n ∈Z.
We ecall he no ion o R-bounded se s and `p-mul iplie s in he space B(X, Y ) o
bounded linea ope a o s om Xin o Yendowed wi h he uni o m ope a o opology.
De ini ion 2.5. Le Xand Ybe Banach spaces. A subse To B(X, Y ) is called
R-bounded i he e is a cons an c > 0 such ha
(7) k(T1x1, ..., Tnxn)kR≤ck(x1, ..., xn)kR,
o all T1, ..., Tn∈ T, x1, ..., xn∈X, n ∈N,whe e
k(x1, ..., xn)kR:= 1
2nX
j∈{−1,1}n


n
X
j=1
jxj

,
o x1, ..., xn∈X.
Fo mo e in o ma ion abou R-bounded se s and hei p ope ies see [1, Sec ion 2.2]
and [8]. We nex ecall he ollowing no ion.
De ini ion 2.6. [22] Le X,Ybe Banach spaces, 1 < p < ∞.A unc ion M∈
C∞
pe (T,B(X, Y )) is an `p-mul iplie ( om X o Y) i he e exis s a bounded ope a o
T:`p(Z;X)→`p(Z;Y) such ha
(8) X
n∈Z
(T )(n) ˇϕ(n) = X
n∈Z
(ϕ·M−ˇ
)(n) (n)
o all ∈`p(Z;X) and all ϕ∈C∞
pe (T).He e
(ϕ·M−ˇ
)(n) := 1
2πZπ
−π
ein ϕ( )M(− )d , n ∈Z.

MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY 5
We now ecall he ollowing Fou ie mul iplie heo em o ope a o - alued symbols
gi en by S. Blunck [5, 1]. This heo em p o ides su icien condi ions o ensu e when an
ope a o - alued symbol is a mul iplie , and allows o es ablish an equi alence be ween
`p-mul iplie s and he no ion o R- boundedness o he UMD class o Banach spaces.
Fo mo e in o ma ion abou hese spaces see [3, Sec ion III.4.3-III.4.5].
Theo em 2.7. [5, Theo em 1.3] and [22] Le p∈(1,∞)and le X, Y be UMD spaces.
Le M∈C∞
pe (T0,B(X;Y)) such ha he se s
{M( ) : ∈T0}and (1 −ei )(1 + ei )M0( ) : ∈T0,
a e bo h R-bounded. Then Mis an `p-mul iplie ( om X o Y) o 1<p<∞.
The con e se o Blunck’s heo em also holds wi hou any es ic ion on he Banach
spaces X, Y as ollows:
Theo em 2.8. [5, P oposi ion 1.4] Le p∈(1,∞)and le X, Y be Banach spaces. Le
M:T→ B(X;Y)be an ope a o alued unc ion. Suppose ha he e is a bounded
ope a o TM:lp(Z;X)→lp(Z;Y)such ha (8) holds. Then he se
{M( ) : ∈T}
is R-bounded.
We now p o ide some no ions conce ning sec o ial ope a o s. Le Σφ⊂Cdeno e
he open sec o
Σφ={λ∈C {0}:|a g λ|< φ},0< φ ≤π.
We deno e by
H(Σφ) = { : Σφ→Cholomo phic}.
and
H∞(Σφ) = { : Σφ→Cholomo phic and bounded}.
H∞(Σφ) is equipped wi h he no m
|| ||φ
∞= sup
|a g λ|<φ | (λ)|.
We u he de ine he subspace H0(Σφ) o H(Σφ) as ollows
H0(Σφ) = [
α,β<0{ ∈ H(Σφ) : || ||φ
α,β <∞},
whe e
|| ||φ
α,β = sup
|λ|≤1|λα (λ)|+ sup
|λ|≥1|λ−β (λ)|.
De ini ion 2.9. A closed linea ope a o Ain Xis called sec o ial i he ollowing
condi ions hold:
(i) D(A) = X, R(A) = X, (−∞,0) ⊂ρ(A);
(ii)|| ( +A)−1|| ≤ M o all > 0 and some M > 0.
Ais called R-sec o ial i he se { ( +A)−1} >0is R-bounded.
6 C. LIZAMA AND M. MURILLO
The class o sec o ial ( esp. R-sec o ial) ope a o s in Xwill be deno ed by S(X)
( esp. RS(X).Se RA(φ) = R(λ(λ+A)−1:|a g λ| ≤ φ}.I A∈ S(X) hen Σφ⊂
ρ(−A) o some φ > 0 and
sup
|a g λ|<φ ||λ(λ+A)−1|| <∞.
We deno e he spec al angle o A∈ S(X) by
φA= in {φ: Σπ−φ⊂ρ(−A),sup
λ∈Σπ−φ||λ(λ+A)−1|| <∞}.
De ini ion 2.10. A sec o ial ope a o Ais said o admi a bounded H∞−calculus i
he e a e φ > φAand a cons an Kφ>0 such ha
(9) || (A)|| ≤ Kφ|| ||φ
∞ o all ∈ H0(Σφ).
The class o sec o ial ope a o s Awhich admi a bounded H∞−calculus is deno ed
by H∞(X). Mo eo e , he H∞−angle is de ined by
φ∞
A= in {φ>φA: (9) holds }
When A∈ H∞(X) we say ha Aadmi s an R-bounded H∞−calculus i he se
{h(A) : h∈ H∞(Σθ),|| ||θ
∞≤1}
is R-bounded o some θ > 0.We deno e he class o such ope a o s by RH∞(X). The
co esponding angle is de ined in an ob ious way and deno ed by θR∞
A.
Rema k 2.11.I Ais a sec o ial ope a o on a Hilbe space, Lebesgue spaces Lp(Ω),1<
p < ∞,Sobole spaces Ws,p(Ω),1<p<∞, s ∈Ro Beso spaces Bs
p,q(Ω),1< p, q <
∞, s ∈Rand Aadmi s a bounded H∞calculus o angle β, hen Aal eady admi s
and RH∞calculus on he same angle βon each o he abo e desc ibed spaces (see
Kal on and Weis [12]). Mo e gene ally, his p ope y is ue whene e Xis a UMD
space wi h he so called p ope y (α) (see [12]).
Example 2.12. Well known examples o gene al classes o closed linea ope a o s
wi h a bounded H∞calculus a e: no mal sec o ial ope a o s in a Hilbe space; m-
acc e i e ope a o s in a Hilbe space; gene a o s o bounded C0-g oups on Lp-spaces
and nega i e gene a o s o posi i e con ac ion semig oups on Lp-spaces.
The ollowing esul will be necessa y o es ablishing `p−`qes ima es in sec ion 4.
I can be ound in [8, P oposi ion 4.10].
P oposi ion 2.13. Le A∈ RH∞(X)and suppose ha {hλ}λ∈Λ⊂ H∞(Σθ)is uni-
o mly bounded o some θ > θR∞
A,whe e Λis an a bi a y index se . Then he se
{hλ(A)}λ∈Λis R-bounded.
3. Abs ac se ing: A cha ac e iza ion o maximal `p- egula i y
Le β∈R, τj∈Z,b∈`1(Z) and Xbe a Banach space. Fo a gi en ec o - alued
sequence :Z→Xwe conside he abs ac disc e e equa ion
(10) u(n) =
n
X
j=−∞
b(n−j)Au(j) +
k
X
j=1
βju(n−τj) + (n), n ∈Z,
MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY 7
whe e Ais a closed linea ope a o wi h domain D(A) de ined in a Banach space X.
Recall ha by [D(A)] we deno e he domain o Aendowed wi h he g aph no m.
De ini ion 3.1. Le 1 <p<∞be gi en. We say ha equa ion (10) has maximal
`p- egula i y i o each ∈`p(Z;X) he e exis s a unique solu ion u∈`p(Z; [D(A]) o
(10).
In his sec ion, ou pu pose is o p o ide a cha ac e iza ion o maximal `p- egula i y
o equa ion (10). Fo he sake o simplici y, we will i s ob ain his cha ac e iza ion
o he ollowing equa ion
(11) u(n) =
n
X
j=−∞
b(n−j)Au(j) + βu(n−τ) + (n), n ∈Z.
As a co olla y, we will ha e a ull cha ac e iza ion o maximal `p- egula i y o he
mo e gene al equa ion (10).
We i s in oduce he ollowing de ini ion. Obse e ha , in some sense, i co e-
sponds o he disc e e coun e pa o he no ion o k- egula i y in oduced in he pape
[15]. See also [28].
De ini ion 3.2. Le k∈N0be gi en. A sequence b∈`1(Z) is called k- egula i he e
exis s a cons an c > 0 such ha |((1+ei )(1−ei ))n[ˆ
b( )](n)| ≤ c|ˆ
b( )| o all 1 ≤n≤k
and all ∈T0.
Rema k 3.3.A simple example o a k- egula sequence is gi en by b(n) = 1
2n, n ∈
N0and 0 o he wise. The 1- egula i y ollows easily since ˆ
b( ) = 2(2 −e−i )−1and
(1 + ei )(1 −ei )[ˆ
b( )]0
ˆ
b( )=i(e2i −1)
(2ei −1) ≤2. The case k > 1 ollows analogously.
Le τ∈Zbe gi en. In wha ollows we deno e by δτ:Z→R he sequence de ined
by
δτ(n) = 


1n=τ,
0o he wise.
We a e now eady o p o e ou main heo em.
Theo em 3.4. Le Xbe a UMD space, 1<p<∞,β∈R,b∈`1(Z)such ha
b(n) = 0 o all n∈Z−and τ∈Z.Suppose ha bis 1- egula , ˆ
b( )6= 0 o all ∈T
and ((1 −βe−i τ )
ˆ
b( )) ∈T⊂ρ(A),
The ollowing asse ions a e equi alen :
(i) Equa ion (11) has maximal `p- egula i y;
(ii) M( ) := (1 −βe−i τ −ˆ
b( )A)−1is an `p-mul iplie om X o [D(A)];
(iii) The se {M( ) : ∈T}is R-bounded.
8 C. LIZAMA AND M. MURILLO
In addi ion, i any o he hypo hesis holds ue, hen u, b ∗Au ∈`p(Z;X)and he e
exis s a cons an C > 0( independen o ∈`p(Z;X)) such ha
(12) kuk`p(Z;X)+kb∗Auk`p(Z;X)≤Ck k`p(Z;X).
P oo . We i s show (i) implies (ii). Le ∈`p(Z;X) be gi en. By hypo hesis he e
exis s a unique sequence u:Z→[D(A)] such ha u∈`p(Z; [D(A)]) sa is ies:
(13) u(n) =
n
X
j=−∞
b(n−j)Au(j) + βu(n−τ) + (n), n ∈Z.
Le Tα:`p(Z;X)→`p(Z; [D(A)]) be de ined by Tα( ) = u. I can be easily shown
using he closed g aph heo em ha Tαis bounded. Since b∈`1(Z), we ob ain he
ollowing iden i ies:
(b◦ˇ
S)(n) =
∞
X
j=0
b(j)ˇ
S(j+n) =
∞
X
j=0
b(j)1
2πZπ
−π
ei(n+j) S( )d
=1
2πZπ
−π
ein ∞
X
j=0
eij b(j)S( )d
=1
2πZπ
−π
ein bb(− )S( )d =: (bb−·Sˇ
)(n),(14)
alid o any S∈C∞
pe (T,B(X, Y )).The e o e, using he hypo hesis, he ac ha
M∈C∞
pe (T,B(X, [D(A)]),and he iden i y I=M(− )−βei τ M(− )−b(− )AM(− )
we ge
hTα , ˇϕi=hu, ˇϕi=X
n∈Z
ˇϕ(n)u(n) = X
n∈Z
1
2πZπ
−π
ein ϕ( )u(n)d
=X
n∈Z
1
2πZπ
−π
ein ϕ( )(1 −βei τ −ˆ
b(− )A)−1u(n)d
−βX
n∈Z
1
2πZπ
−π
ei τ (1 −βei τ −ˆ
b(− )A)−1u(n)ein ϕ( )d
−X
n∈Z
1
2πZπ
−π
(1 −βei τ −ˆ
b(− )A)−1ˆ
b(− )Au(n)ein ϕ( )d
=X
n∈Z
1
2πZπ
−π
ein ϕ( )M(− )u(n)d
−βX
n∈Z
1
2πZπ
−π
ein b
δτ( )ϕ( )M(− )u(n)d
−X
n∈Z
1
2πZπ
−π
ein ϕ( )M(− )ˆ
b(− )Au(n)d
=hu, (ϕ·M−ˇ
)i−βhu, (b
δτ−·ϕ·M−ˇ
)i−hAu, (ˆ
b−·ϕ·M−ˇ
)i
=hu, (ϕ·M−ˇ
)i−βhu, δτ◦(ϕ·M−ˇ
)i−hAu, b ◦(ϕ·M−ˇ
)i,
MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY 15
In ac , a simple compu a ion shows ha <( 2−2 ei +e2i
(c2−ν )+ν ei )>0 i and only i P( ) :=
<(( 2−2 ei +e2i )((c2−ν ) + ν e−i )) >0.Now, we obse e ha , in iew o he
hypo hesis (35) we ha e
P( )=(c2−ν ) cos 2 + (ν −2 c2+ 2ν 2+ 2ν) cos + ( 2c2−ν 3−2 ν)
≥ −(c2−ν )−(ν −2 c2+ 2ν 2+ 2ν)+( 2c2−ν 3−2 ν)
= [(2c2−ν) −(c2+ν)] + 2[(c2−3ν)−ν ]>0.
This p o es he claim. We now de ine he ollowing complex alued unc ion
h (z) :=  2−2 ei +e2i
(c2−ν ) + ν ei +z−1
.
Then, as a consequence o he abo e compu a ion, we ob ain he ollowing es ima e
|h (z)| ≤ 1
[(2c2−ν) −(c2+ν)] + 2[(c2−3ν)−ν ]
p o ing ha he se {h } ∈T⊂ H∞(Σπ/2) is uni o mly bounded and hen he se
{hλ(∆d,N )} ∈T
is R-bounded and he conclusion ollows as be o e. We a i e a he ollowing esul .
Theo em 4.4. Le ∈`p(Z;`q(ZN)),1<p<∞be gi en and suppose ha
(36) c2+ν
2c2−ν < < c2−3ν
ν , c2>3ν.
Then he nume ical solu ion (u(n, m))n∈Z,m∈ZNo (34), ob ained by he o wa d Eule
-me hod exis s, belongs o u∈`p(Z;`q(ZN)) and sa is ies he disc e e maximal `p−`q
egula i y es ima e
X
n∈Zku(n)kp
`q(ZN)1/p +X
n∈Zkc2∆d,N u(n) + ν∆d,N ∆ u(n)kp
`q(ZN)1/p
≤CX
n∈Zk (n)kp
`q(ZN)1/p,
whe e he cons an C > 0is independen o .
Rema k 4.5.I is in e es ing o obse e ha he case = 1 can be eached unde he
hypo hesis:
c2>4ν
which shows ha inso a as he damping e m in (34) is no oo small, he possibili y o
disc e izing he empo al de i a i e by means o he usual backwa d di e ence ope a o
inc eases. This e eals ha in o de o ha e `p−`qes ima es o he model (25), he
di e ence ope a o ha will be used in he empo al disc e iza ion o he equa ion will
depend on he s uc u e o he equa ion, i.e. on he pa ame e s βand b.

16 C. LIZAMA AND M. MURILLO
Acknowledgmen s
The i s au ho was pa ially suppo ed by FONDECYT, G an No 1180041. The
second au ho was suppo ed by was suppo ed by MEC, g an MTM2016-75963-P
and GVA/2018/110.
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(C. Lizama) Depa amen o de Ma em´
a ica y Ciencia de la Compu aci´
on, Facul ad de
Ciencias, Uni e sidad de San iago de Chile, Casilla 307, Co eo 2, San iago, Chile
E-mail add ess:[email p o ec ed]
(M. Mu illo) Ins i u de Ma em`
a iques i Aplicacions de Cas ell´
o (IMAC), Uni e si a
Jaume I, Campus del Riu Sec s/n, 12071 Cas ell´
o, Spain
E-mail add ess:[email p o ec ed]
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