Maximal lp-regularity for discrete time Volterra equations with delay

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/330779516 Maximal $\ell_p$-regularity for discrete time volterra equations with delay ArticleinJournal of Difference Equations and Applications · April 2019 DOI: 10.1080/10236198.2019.1638916 CITATIONS 0 READS 30 2 authors: Some of the authors of this publication are also working on these related projects: Dynamics of Operators View project Special Issue on: “Modern fractional dynamic systems and applications” View project Carlos Lizama University of Santiago, Chile 173 PUBLICATIONS2,134 CITATIONS SEE PROFILE Marina Murillo Universitat Jaume I 38 PUBLICATIONS144 CITATIONS SEE PROFILE All content following this page was uploaded by Marina Murillo on 02 September 2019. The user has requested enhancement of the downloaded file. MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY CARLOS LIZAMA AND MARINA MURILLO-ARCILA Abstract. In this paper we investigate the existence and uniqueness of solutions belonging to the vector-valued space `p(Z, X) by using Blunck’s theorem on the equivalence between operator-valued `p-multipliers and the notion of R-boundedness for the discrete time volterra equation with delay given by u(n) = n X j=−∞ b(n−j)Au(j) + k X j=1 βju(n−τj) + f(n), n ∈Z, where Ais a closed linear operator with domain D(A) defined on a Banach space Xand b∈`1(Z) verifies suitable conditions such as 1-regularity. We characterize maximal `p-regularity of solutions of such problems in terms of the data and an spectral condition and we provide optimal estimates. Moreover, we illustrate our results providing different models that label into our general scheme such as the discrete time wave and Kuznetsov equations. Keywords: volterra equations, maximal `p-regularity, R-bounded, discrete wave equation, discrete kuznetsov equation Mathematics Subject Classification (2010): 45D05, 35R09, 65Q10, 39A06. 1. Introduction In this paper we analyze the existence and uniqueness of solutions in vector valued `p(Z, X) spaces for discrete time formulations of the following integro partial differential equation with delay, u(t) = Zt −∞ a(t−s)Au(s)ds +βu(t−τ) + f(t, u(t)), t ∈R, where Ais a closed linear operator defined on a Banach space X. This equation models viscoelastic fluids, heat conduction with memory and electrodynamics processes with memory [4, 28]. Typical models that are included in this article correspond to different discrete versions of the multidimensional wave and Kuznetsov equations (1) utt −c2∆u−ν∆ut=f(t), t ∈R. The analysis of qualitative properties of Volterra type equations has been considered by various authors, see [10] and all the references therein. Moreover, numerical methods for the resolution of Volterra equations have been studied among others in [6, 7, 9, 19, 26, 27]. On the other hand, the study of maximal regularity for discrete systems that belong to the Lebesgue space of vector-valued sequences since the pioneer work of S. Blunck [5] has experimented a great development as it can be seen in the recent papers The first author is partially supported by FONDECYT grant number 1180041. The second author is supported by MEC, grant MTM2016-75963-P and GVA, Grant 18I264.01/1. 1 2 C. LIZAMA AND M. MURILLO [13, 21, 20, 22]. This study is strongly connected with the necessity of optimal `p−`q time-space estimates for the corresponding linearized problem [2, 11, 13, 16, 14, 18, 17, 24, 25]. In our work, we succeed characterizing maximal `p-regularity for the following abstract model (2) u(n) = n X j=−∞ b(n−j)Au(j) + k X j=1 βju(n−τj) + f(n), n ∈Z, where f∈`p(Z, X), A is a closed linear operator with domain D(A) defined on Xand b∈`1(Z). It is worthwhile to observe that, for instance, model (2) includes among others the discrete Kuznetsov equation (1) taking A=−∆d,N ,the multidimensional discrete Laplacian, b(n) = −(c2+ν)δ0(n) + νrδ0(n−1), β1= 2r, τ1= 1 and β2= −r2, τ2= 2. This paper is organized as follows: in Section 2, we first recall the notions of UMDspaces, R-boundedness, `p-multipliers, sectorial operators and the discrete time Fourier transform defined on the space of distributions. Moreover, we recall the well-known Blunck’s Fourier multiplier theorem [5] for operator-valued symbols on UMD-spaces that establishes the equivalence between `p-multipliers and R-boundedness. In Section 3, we prove our main result, namely, if b∈`1(Z) is 1-regular, ˆ b(t)6= 0 for all t∈Tand (1−Pk j=1 βje−itτj ˆ b(t))t∈T⊂ρ(A). then the following assertions are equivalent: (i) For all f∈`p(Z, X) equation u(n) = n X j=−∞ b(n−j)Au(j) + k X j=1 βju(n−τj) + f(n), n ∈Z, has a unique solution in `p(Z,[D(A)]); (ii) M(t) := (1 −Pk j=1 βje−itτj−ˆ b(t)A)−1is an `p-multiplier from Xto [D(A)]; (iii) The set {M(t) : t∈T}is R-bounded. Observe, that our result demands 1-regularity of the kernel sequence b(n). We introduce this concept for the first time in definition 3.2 and it corresponds to the discrete counterpart of the notion of 1-regularity introduced in [15]. Furthermore when Xis Hilbert we simplify the previous result by replacing the condition (iii) above by an easier computable condition sup t∈TkM(t)k<∞. We also ensure optimal estimates for model (2) under any of the above conditions, that is, the following estimate also holds kuk`p(Z;X)+kb∗Auk`p(Z;X)≤Ckfk`p(Z;X). Finally, in section 4, we prove, as an application of our characterization, the existence and uniqueness of `p(Z;`q(ZN)) solutions for time discretizations forms of the wave and MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY 3 Kuznetsov equations in terms of the data of the problem as it can be seen in theorems 4.2,4.3 and 4.4. In addition, we obtain maximal `p−`qestimates for such models. 2. Analytical framework and notation In this section, we present some results that will be needed throughout the paper. Let Xbe a Banach space. We denote by S(Z;X) the space of all vector-valued sequences f:Z→Xsuch that for each k∈N0there exists a constant Ck>0 satisfying pk(f) := supn∈Z|n|kkf(n)k< Ckand when X=Rwe denote S(Z). We write as Cn per(R;X), n ∈N0,the space of all 2π-periodic X-valued and n-times continuously differentiable functions defined in R.In what follows, we will denote T:= (−π, π) and T0:= (−π, π)\ {0}.The space of test functions is the space C∞ per(T;X) := Tn∈N0Cn per(R;X).When X=Rwe simply write C∞ per(T). For each f∈`p(Z;X) we can define the map (3) Tf(ψ) := hTf, ψi:= X n∈Z f(n)ψ(n), ψ ∈ S(Z), and we have Tf∈ S0(Z, X) = {T:S(Z)→X:Tis linear and continuous}. Remark 2.1.By this mapping we identify `p(Z;X) with a subspace of S0(Z;X).When convenient and confusion seems unlikely, a function f∈`p(Z;X) is identified with Tf∈ S0(Z, X). There also exists a natural mapping that identifies C∞ per(T;X) with a subspace of D0(T;X) = {T:C∞ per(T)→X:Tis linear and continuous}which assigns to each S∈C∞ per(T;X) the linear map LS(ϕ) := hLS, ϕi:= 1 2πZπ −π ϕ(t)S(t)dt, ϕ ∈C∞ per(T), and we have LS∈ D0(T;X). Definition 2.2. The discrete time Fourier transform F:S(Z;X)→C∞ per(T;X) is defined by Fϕ(t)≡bϕ(t) := ∞ X j=−∞ e−ijtϕ(j), t ∈(−π, π] and the corresponding inverse transform is given by (4) F−1ϕ(n)≡ˇϕ(n) := 1 2πZπ −π ϕ(t)eintdt, n ∈Z, where ϕ∈C∞ per(T;X). This isomorphism, allows us to define the discrete time Fourier transform (DTFT) between the spaces of distributions S0(Z;X) and D0(T;X) as follows: (5) hFT, ψi≡F(T)(ψ) := b T(ψ)≡ hT, ˇ ψi, T ∈ S0(Z;X), ψ ∈C∞ per(T), whose inverse F−1:D0(T;X)→ S0(Z;X) is given 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 finally present a technical lemma introduced in [22] which will be necessary througout the paper. We first need the following definition. Definition 2.3. Given u∈`p(Z;X) and v∈`1(Z) the convolution product between uand vis defined as (u∗v)(n) := n X j=−∞ u(n−j)v(j) = ∞ X j=0 u(j)v(n−j), n ∈Z. Moreover, the convolution of a distribution T∈ S0(Z, X) with a function a∈`1(Z+) is defined by (6) hT∗a, ϕi:= hT, a ◦ϕi, ϕ ∈ S(Z), where (a◦ϕ)(n) := ∞ X j=0 a(j)ϕ(j+n). Lemma 2.4. Let u, v ∈`p(Z;X)be given and a∈`1(Z+)which is defined by 0for negative values of n. The following assertions are equivalent: (i) a∗v∈`p(Z, X)and (a∗v)(n) = u(n)for all n∈Z. (ii) hu, ˇϕi=hv, (ϕ·ba−ˇ )ifor all ϕ∈C∞ per(T), where (ϕ·ba−ˇ )(n) := 1 2πZπ −πba(−t)ϕ(t)eintdt, n ∈Z. We recall the notion of R-bounded sets and `p-multipliers in the space B(X, Y ) of bounded linear operators from Xinto Yendowed with the uniform operator topology. Definition 2.5. Let Xand Ybe Banach spaces. A subset Tof B(X, Y ) is called R-bounded if there is a constant c > 0 such that (7) k(T1x1, ..., Tnxn)kR≤ck(x1, ..., xn)kR, for all T1, ..., Tn∈ T, x1, ..., xn∈X, n ∈N,where k(x1, ..., xn)kR:= 1 2nX j∈{−1,1}n   n X j=1 jxj  , for x1, ..., xn∈X. For more information about R-bounded sets and their properties see [1, Section 2.2] and [8]. We next recall the following notion. Definition 2.6. [22] Let X,Ybe Banach spaces, 1 < p < ∞.A function M∈ C∞ per(T,B(X, Y )) is an `p-multiplier (from Xto Y) if there exists a bounded operator T:`p(Z;X)→`p(Z;Y) such that (8) X n∈Z (Tf)(n) ˇϕ(n) = X n∈Z (ϕ·M−ˇ )(n)f(n) for all f∈`p(Z;X) and all ϕ∈C∞ per(T).Here (ϕ·M−ˇ )(n) := 1 2πZπ −π eintϕ(t)M(−t)dt, n ∈Z. MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY 5 We now recall the following Fourier multiplier theorem for operator-valued symbols given by S. Blunck [5, 1]. This theorem provides sufficient conditions to ensure when an operator-valued symbol is a multiplier, and allows to establish an equivalence between `p-multipliers and the notion of Rboundedness for the UMD class of Banach spaces. For more information about these spaces see [3, Section III.4.3-III.4.5]. Theorem 2.7. [5, Theorem 1.3] and [22] Let p∈(1,∞)and let X, Y be UMD spaces. Let M∈C∞ per(T0,B(X;Y)) such that the sets {M(t) : t∈T0}and (1 −eit)(1 + eit)M0(t) : t∈T0, are both R-bounded. Then Mis an `p-multiplier (from Xto Y) for 1<p<∞. The converse of Blunck’s theorem also holds without any restriction on the Banach spaces X, Y as follows: Theorem 2.8. [5, Proposition 1.4] Let p∈(1,∞)and let X, Y be Banach spaces. Let M:T→ B(X;Y)be an operator valued function. Suppose that there is a bounded operator TM:lp(Z;X)→lp(Z;Y)such that (8) holds. Then the set {M(t) : t∈T} is R-bounded. We now provide some notions concerning sectorial operators. Let Σφ⊂Cdenote the open sector Σφ={λ∈C\{0}:|arg λ|< φ},0< φ ≤π. We denote by H(Σφ) = {f: Σφ→Cholomorphic}. and H∞(Σφ) = {f: Σφ→Cholomorphic and bounded}. H∞(Σφ) is equipped with the norm ||f||φ ∞= sup |arg λ|<φ |f(λ)|. We further define the subspace H0(Σφ) of H(Σφ) as follows H0(Σφ) = [ α,β<0{f∈ H(Σφ) : ||f||φ α,β <∞}, where ||f||φ α,β = sup |λ|≤1|λαf(λ)|+ sup |λ|≥1|λ−βf(λ)|. Definition 2.9. A closed linear operator Ain Xis called sectorial if the following conditions hold: (i) D(A) = X, R(A) = X, (−∞,0) ⊂ρ(A); (ii)||t(t+A)−1|| ≤ Mfor all t > 0 and some M > 0. Ais called R-sectorial if the set {t(t+A)−1}t>0is R-bounded. 6 C. LIZAMA AND M. MURILLO The class of sectorial (resp. R-sectorial) operators in Xwill be denoted by S(X) (resp. RS(X).Set RA(φ) = R(λ(λ+A)−1:|arg λ| ≤ φ}.If A∈ S(X) then Σφ⊂ ρ(−A) for some φ > 0 and sup |arg λ|<φ ||λ(λ+A)−1|| <∞. We denote the spectral angle of A∈ S(X) by φA= inf{φ: Σπ−φ⊂ρ(−A),sup λ∈Σπ−φ||λ(λ+A)−1|| <∞}. Definition 2.10. A sectorial operator Ais said to admit a bounded H∞−calculus if there are φ > φAand a constant Kφ>0 such that (9) ||f(A)|| ≤ Kφ||f||φ ∞for all f∈ H0(Σφ). The class of sectorial operators Awhich admit a bounded H∞−calculus is denoted by H∞(X). Moreover, the H∞−angle is defined by φ∞ A= inf{φ>φA: (9) holds } When A∈ H∞(X) we say that Aadmits an R-bounded H∞−calculus if the set {h(A) : h∈ H∞(Σθ),||f||θ ∞≤1} is R-bounded for some θ > 0.We denote the class of such operators by RH∞(X). The corresponding angle is defined in an obvious way and denoted by θR∞ A. Remark 2.11.If Ais a sectorial operator on a Hilbert space, Lebesgue spaces Lp(Ω),1< p < ∞,Sobolev spaces Ws,p(Ω),1<p<∞, s ∈Ror Besov spaces Bs p,q(Ω),1< p, q < ∞, s ∈Rand Aadmits a bounded H∞calculus of angle β, then Aalready admits and RH∞calculus on the same angle βon each of the above described spaces (see Kalton and Weis [12]). More generally, this property is true whenever Xis a UMD space with the so called property (α) (see [12]). Example 2.12. Well known examples for general classes of closed linear operators with a bounded H∞calculus are: normal sectorial operators in a Hilbert space; maccretive operators in a Hilbert space; generators of bounded C0-groups on Lp-spaces and negative generators of positive contraction semigroups on Lp-spaces. The following result will be necessary for establishing `p−`qestimates in section 4. It can be found in [8, Proposition 4.10]. Proposition 2.13. Let A∈ RH∞(X)and suppose that {hλ}λ∈Λ⊂ H∞(Σθ)is uniformly bounded for some θ > θR∞ A,where Λis an arbitrary index set. Then the set {hλ(A)}λ∈Λis R-bounded. 3. Abstract setting: A characterization of maximal `p-regularity Let β∈R, τj∈Z,b∈`1(Z) and Xbe a Banach space. For a given vector-valued sequence f:Z→Xwe consider the abstract discrete equation (10) u(n) = n X j=−∞ b(n−j)Au(j) + k X j=1 βju(n−τj) + f(n), n ∈Z, MAXIMAL `p-REGULARITY FOR DISCRETE TIME VOLTERRA EQUATIONS WITH DELAY 7 where Ais a closed linear operator with domain D(A) defined in a Banach space X. Recall that by [D(A)] we denote the domain of Aendowed with the graph norm. Definition 3.1. Let 1 <p<∞be given. We say that equation (10) has maximal `p-regularity if for each f∈`p(Z;X) there exists a unique solution u∈`p(Z; [D(A]) of (10). In this section, our purpose is to provide a characterization of maximal `p-regularity of equation (10). For the sake of simplicity, we will first obtain this characterization for the following equation (11) u(n) = n X j=−∞ b(n−j)Au(j) + βu(n−τ) + f(n), n ∈Z. As a corollary, we will have a full characterization of maximal `p-regularity for the more general equation (10). We first introduce the following definition. Observe that, in some sense, it corresponds to the discrete counterpart of the notion of k-regularity introduced in the paper [15]. See also [28]. Definition 3.2. Let k∈N0be given. A sequence b∈`1(Z) is called k-regular if there exists a constant c > 0 such that |((1+eit)(1−eit))n[ˆ b(t)](n)| ≤ c|ˆ b(t)|for all 1 ≤n≤k and all t∈T0. Remark 3.3.A simple example of a k-regular sequence is given by b(n) = 1 2n, n ∈ N0and 0 otherwise. The 1-regularity follows easily since ˆ b(t) = 2(2 −e−it)−1and (1 + eit)(1 −eit)[ˆ b(t)]0 ˆ b(t)=i(e2it−1) (2eit−1) ≤2. The case k > 1 follows analogously. Let τ∈Zbe given. In what follows we denote by δτ:Z→Rthe sequence defined by δτ(n) =    1n=τ, 0otherwise. We are now ready to prove our main theorem. Theorem 3.4. Let Xbe a UMD space, 1<p<∞,β∈R,b∈`1(Z)such that b(n) = 0 for all n∈Z−and τ∈Z.Suppose that bis 1-regular, ˆ b(t)6= 0 for all t∈T and ((1 −βe−itτ ) ˆ b(t))t∈T⊂ρ(A), The following assertions are equivalent: (i) Equation (11) has maximal `p-regularity; (ii) M(t) := (1 −βe−itτ −ˆ b(t)A)−1is an `p-multiplier from Xto [D(A)]; (iii) The set {M(t) : t∈T}is R-bounded. 8 C. LIZAMA AND M. MURILLO In addition, if any of the hypothesis holds true, then u, b ∗Au ∈`p(Z;X)and there exists a constant C > 0( independent of f∈`p(Z;X)) such that (12) kuk`p(Z;X)+kb∗Auk`p(Z;X)≤Ckfk`p(Z;X). Proof. We first show (i) implies (ii). Let f∈`p(Z;X) be given. By hypothesis there exists a unique sequence u:Z→[D(A)] such that u∈`p(Z; [D(A)]) satisfies: (13) u(n) = n X j=−∞ b(n−j)Au(j) + βu(n−τ) + f(n), n ∈Z. Let Tα:`p(Z;X)→`p(Z; [D(A)]) be defined by Tα(f) = u. It can be easily shown using the closed graph theorem that Tαis bounded. Since b∈`1(Z), we obtain the following identities: (b◦ˇ S)(n) = ∞ X j=0 b(j)ˇ S(j+n) = ∞ X j=0 b(j)1 2πZπ −π ei(n+j)tS(t)dt =1 2πZπ −π eint∞ X j=0 eijtb(j)S(t)dt =1 2πZπ −π eintbb(−t)S(t)dt =: (bb−·Sˇ )(n),(14) valid for any S∈C∞ per(T,B(X, Y )).Therefore, using the hypothesis, the fact that M∈C∞ per(T,B(X, [D(A)]),and the identity I=M(−t)−βeitτ M(−t)−b(−t)AM(−t) we get hTαf, ˇϕi=hu, ˇϕi=X n∈Z ˇϕ(n)u(n) = X n∈Z 1 2πZπ −π eintϕ(t)u(n)dt =X n∈Z 1 2πZπ −π eintϕ(t)(1 −βeitτ −ˆ b(−t)A)−1u(n)dt −βX n∈Z 1 2πZπ −π eitτ (1 −βeitτ −ˆ b(−t)A)−1u(n)eintϕ(t)dt −X n∈Z 1 2πZπ −π (1 −βeitτ −ˆ b(−t)A)−1ˆ b(−t)Au(n)eintϕ(t)dt =X n∈Z 1 2πZπ −π eintϕ(t)M(−t)u(n)dt −βX n∈Z 1 2πZπ −π eintb δτ(t)ϕ(t)M(−t)u(n)dt −X n∈Z 1 2πZπ −π eintϕ(t)M(−t)ˆ b(−t)Au(n)dt =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 fact, a simple computation shows that <(r2−2reit+e2it (c2−νr)+νreit )>0 if and only if P(t) := <((r2−2reit +e2it)((c2−νr) + νre−it)) >0.Now, we observe that, in view of the hypothesis (35) we have P(t)=(c2−νr) cos 2t+ (ν −2rc2+ 2νr2+r2ν) cos t+ (r2c2−νr3−2rν) ≥ −(c2−νr)−(ν −2rc2+ 2νr2+r2ν)+(r2c2−νr3−2rν) = [(2c2−ν)r−(c2+ν)] + r2[(c2−3ν)−νr]>0. This proves the claim. We now define the following complex valued function ht(z) := r2−2reit +e2it (c2−νr) + νreit +z−1 . Then, as a consequence of the above computation, we obtain the following estimate |ht(z)| ≤ 1 [(2c2−ν)r−(c2+ν)] + r2[(c2−3ν)−νr] proving that the set {ht}t∈T⊂ H∞(Σπ/2) is uniformly bounded and then the set {hλ(∆d,N )}t∈T is R-bounded and the conclusion follows as before. We arrive at the following result. Theorem 4.4. Let f∈`p(Z;`q(ZN)),1<p<∞be given and suppose that (36) c2+ν 2c2−ν < r < c2−3ν ν , c2>3ν. Then the numerical solution (u(n, m))n∈Z,m∈ZNof (34), obtained by the forward Euler r-method exists, belongs to u∈`p(Z;`q(ZN)) and satisfies the discrete maximal `p−`q regularity estimate X n∈Zku(n)kp `q(ZN)1/p +X n∈Zkc2∆d,N u(n) + ν∆d,N ∆ru(n)kp `q(ZN)1/p ≤CX n∈Zkf(n)kp `q(ZN)1/p, where the constant C > 0is independent of f. Remark 4.5.It is interesting to observe that the case r= 1 can be reached under the hypothesis: c2>4ν which shows that insofar as the damping term in (34) is not too small, the possibility of discretizing the temporal derivative by means of the usual backward difference operator increases. This reveals that in order to have `p−`qestimates for the model (25), the difference operator that will be used in the temporal discretization of the equation will depend on the structure of the equation, i.e. on the parameters βand b. 16 C. LIZAMA AND M. MURILLO Acknowledgments The first author was partially supported by FONDECYT, Grant No 1180041. The second author was supported by was supported by MEC, grant MTM2016-75963-P and GVA/2018/110. References [1] R. P. Agarwal, C. Cuevas and C. Lizama. Regularity of Difference Equations on Banach Spaces, Springer-Verlag, Cham, 2014. [2] G. Akrivis, B. Li and C. Lubich. 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