Maximal lp-regularity for discrete time Volterra equations with delay

See discussions, s a s, and au ho p o iles o his publica ion a : h ps://www. esea chga e.ne /publica ion/330779516 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 DOI: 10.1080/10236198.2019.1638916 CITATIONS 0 READS 30 2 au ho s: Some o he au ho s o his publica ion a e also wo king on hese ela ed p ojec s: Dynamics o Ope a o s View p ojec Special Issue on: “Mode n ac ional dynamic sys ems and applica ions” View p ojec Ca los Lizama Uni e si y o San iago, Chile 173 PUBLICATIONS2,134 CITATIONS SEE PROFILE Ma ina Mu illo Uni e si a Jaume I 38 PUBLICATIONS144 CITATIONS SEE PROFILE All con en ollowing his page was uploaded by Ma ina Mu illo on 02 Sep embe 2019. The use has eques ed enhancemen o he downloaded ile. 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. Re e ences [1] R. P. Aga wal, C. Cue as and C. Lizama. Regula i y o Di e ence Equa ions on Banach Spaces, Sp inge -Ve lag, Cham, 2014. [2] G. Ak i is, B. Li and C. Lubich. Combining maximal egula i y and ene gy es ima es o ime disc e iza ions o quasilinea pa abolic equa ions. Ma h. 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Sp inge , Basel Heidelbe g, 1993. (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] View publica ion s a sView publica ion s a s