Fundamental solutions for semidiscrete evolution equations via Banach algebras

González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 https://doi.org/10.1186/s13662-020-03206-7 R E S E A R C H Open Access Fundamental solutions for semidiscrete evolution equations via Banach algebras Jorge González-Camus1, Carlos Lizama1* and Pedro J. Miana2 *Correspondence: [email protected] 1Departamento de Matemáticas y Ciencias de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile Full list of author information is available at the end of the article Abstract We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. We identify fractional powers of these generators and apply to them the subordination principle. We also give some applications and consequences of our results. MSC: 35R11; 35A08; 39A12 Keywords: Caputo fractional derivative; Discrete fractional Laplacian; Discrete fractional operators; Fundamental solutions; Wright and Mittag-Leffler functions 1 Introduction In this work, we study the following semidiscrete Cauchy problem: ⎧ ⎨ ⎩ ∂tu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z,(1.1) where Bis the convolution operator in the discrete variable, that is, Bu(n,t)= j∈Z b(n–j)u(j,t) (1.2) with bbelonging to the Banach algebra 1(Z). A typical example is the one-dimensional discrete Laplacian d, which can be obtained by taking b=δ–1 –2δ0+δ1,whereδi(j) denotes the Kronecker delta (or discrete Dirac measure). In such a case, equation (1.1) corresponds to the nonhomogeneous semidiscrete diffusion equation (also known as the semidiscrete heat equation or the lattice diffusion equation). The analytical study of such equations has received an increasing interest in the last decade, mainly due to many their applications in diverse areas of knowledge. For instance, ©The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 2 of 32 in probability theory, the value u(n,t)in(1.1)withB=ddescribes the probability that a continuous-time symmetric random walk on Zvisits a point nat time t;see[25,Sect.4]. In chemistry, (1.1) describes the flow of a chemical in an infinite system of tanks arranged in a row, where each two neighbors are connected by pipes [42, Sect. 3], and in transport theory, (1.1) describes the dynamics of an infinite chain of cars, each being coupled to its two neighbors. The value u(n;t)isthedisplacementofcarnat time tfrom its equilibrium position; see [24, Example 1]. From an analytical point of view, quite recently, Slavik [43] studied the asymptotic behavior of solutions of (1.1)whenB=d, showing that a bounded solution approaches the average of the initial values if the average exists. Note that choosing b=δ–1 –δ0in (1.2), we obtain the forward difference operator B=, and hence (1.2) corresponds to the semidiscrete transport equation, studied recently by Abadias et al. [1]. It is interesting that in [22]and[37] the authors studied the fundamental solutions of (1.1) and the second-order semidiscrete equation ⎧ ⎨ ⎩ ∂ttu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), ut(n,0)=φ(n), n∈Z,(1.3) when B=–(–d)αis the discrete fractional Laplacian. Particularly, in [37]theauthors combined operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by dand –(–d)α. Also note that the fractional forward difference operator B=–(–)αwas studied in [1], where the maximum and comparison principles in the context of harmonic analysis are proved. However, to our knowledge, to date, there is no attempt to investigate the fundamental solutions of the general equation (1.1)inaunifiedway.Ourgoalinthispaperistopropose asolutiontothisproblem. Our key observation concerning this issue is that the discrete fractional Laplacian can be obtained from (1.2) by allowing the fractional powers of bto be an element of the Banach algebra 1(Z). This original approach, which we provide in this paper, allows us to obtain new insights by introducing a completely new method to analyze both qualitative behavior and fundamental solutions of (1.1)inaunifiedway. More generally, to provide simultaneously in our analysis the subdiffusive and superdifussive cases associated with equations (1.1)and(1.3), in this paper, we include a representation of the fundamental solutions for the following semidiscrete equations: ⎧ ⎨ ⎩ Dβ tu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z,(1.4) in case 0 < β≤1and ⎧ ⎨ ⎩ Dβ tu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), ut(n,0)=φ(n), n∈Z,(1.5) in case 1 < β≤2. In both cases, Bis the convolution operator Bf (n):=(b∗f)(n)onp(Z), p∈[1,∞], b∈1(Z), and β∈(0, 2]. The symbol Dβ tdenotes the Caputo fractional derivative of order β>0. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 3 of 32 We observe that although the present study considers only the cases 0 < β≤2because they are the most common in applications (that is, subdiffusion and superdiffusion) together with the fractional Caputo derivative, our method is general enough to consider a larger order βand other nonlocal operators in time. For example, in Sect. 6.4, we consider the new Caputo–Fabrizio fractional derivative of order α∈(0,1) and give a representation of the solutions for the corresponding equation (1.4). This paper is organized as follows. In Sect. 2, we consider the Banach algebra framework to state our main results, which we will present in the forthcoming sections. In particular, we introduce generalized Mittag-Leffer functions on the Banach algebra 1(Z) and collect some basic properties. Our main result is Theorem 2.6 concerning the invariance for convolution operators defined on p(Z)for1≤p≤∞.Section3is devoted to four finite difference operators: backward and forward difference operators, the onedimensional discrete Laplacian, and an operator that originates in connection with crystal lattices [16]. Then we explicitly describe their associated groups and cosine operators by means of Bessel functions and highlight their main spectral properties. Section 4begins with three concrete examples of application of the results in the previous section: the discrete Nagumo equation, transport equations, and a new interesting second-order discrete equation, which we call the De Juhasz equation, appearing in the seminal Bateman’s paper [16] in connection with surges in springs and connected systems of springs. Then we state the general fundamental solutions for (1.4)–(1.5), first, in the setting of Banach algebras (Theorem 5.1) and then for convolution operators (Corollary 5.5). In Sect. 7,wegiveexplicit representations of generalized Mittag-Leffer functions in each case of the fractional powers of the four finite difference operators considered previously (Theorem 7.1). This result, combined with the general fundamental solutions considered in Theorem 5.1 and Corollary 5.5, gives not only explicit representations of each of the four difference operators considered in this paper – which can be considered as examples – but also an efficient method to obtain representations of solutions in many other cases. Besides, as a byproduct of our treatment, we obtain new Weiestrass formulae, which highlight the role of Bessel functions for finite difference operators, and a subordination principle, which connects the Wright and Bessel functions. For convenience of the reader, we finish this research with an appendix on useful properties of some special functions needed in this paper. Notation. T={eiθ:θ∈[–π,π)}is the one-dimensional torus. The Dirac measures δ0 and δnare δn(j)=0ifn=jand δn(n)=1forn,j∈Z. Given a Banach space X,Xis the dual of X,andB(X) is the set of linear bounded operators on X;givenA∈B(X), we A∈B(X) is the adjoint of the operator A.WedenotebyχIthe indicator function of a set I(i.e., χI(n)=1ifn∈Iand χI(n)=0ifn/∈I). Furthermore, Inand Jnare the Bessel functions. The usual set numbers N,N0=N∪{0},Z,R,andCare used. Furthermore, is the gamma function, βis the Wright function (Sect. A.1), Eα,βis the Mittag-Leffler function, Inand Jnare the Bessel functions (Sect. A.2), and the stable Lévy distribution is denoted by ft,α (Sect. A.3). 2ABanachalgebraframework Given 1 ≤p≤∞, we recall that the Banach spaces (p(Z), ·p) are formed by biinfinite sequences f=(f(n))n∈Z⊂Csuch that fp:= ∞  n=–∞f(n) p1 p <∞,1≤p<∞, González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 4 of 32 f∞:= sup n∈Zf(n)<∞. We recall the natural embeddings 1(Z)→p(Z)→∞(Z)for1≤p≤∞and that the dual of p(Z) is identified with p(Z), where 1 p+1 p=1for1<p<∞and p=1ifp=∞. In the case of f∈1(Z)andg∈p(Z), we define (f∗g)(n):= ∞  j=–∞ f(n–j)g(j), n∈Z. From Young’s inequality it follows that f∗g∈p(Z). Note that (1(Z), ∗)isacommutative Banach algebra with identity δ0:= χ{0}.Weobservethatδ1∗δ1=δ2and, in general, δn∗δm= δn+mfor n,m∈Z. The Gelfand transform associated with (1(Z),∗) is the discrete Fourier transform F: 1(Z)→C(T) (or Fourier series), where ˆ f(θ):=F(f)eiθ:=  n∈Z f(n)einθ,θ∈T. We recall that the spectrum of f,denotedσ1(Z)(f), is defined by σ1(Z)(f):=λ∈C:(λδ0–f)–1 ∈1(Z). In what follows, we consider the general theory of commutative Banach algebras as a framework. We collect the results that will be of our interest in the following theorem. Theorem 2.1 The following properties hold: (i) The spectrum Spec(1(Z)) is compact and,consequently,homeomorphic to the unit complex circle T:= {z∈C:|z|=1}. (ii) σ1(Z)(f)⊂{z∈C;|z|<f1},and (λδ0–f)–1 = n≥0 λ–n–1fn,f1<|λ|. (2.1) (iii) The algebra 1(Z)is a semisimple regular Banach algebra,and the discrete Fourier transform Fis injective. (iv) F(f∗g)=F(f)F(g),and σ1(Z)(f)=F(f)(T), f∈1(Z). (2.2) Proof Thefirstclaimfollowsfromthefactthatthealgebra1(Z) has an identity; see, for example, [35], and the second one can be found in [35, p. 116]. The proof of (ii) is straightforward. From [35, Theorem 4.7.4] it follows that 1(Z)issemisimpleandFis injective. By [35, Corolary 7.2.3] 1(Z) is a regular Banach algebra. Statement (iv) is taken from [35, Theorem 3.4.1.].  We observe that the range of the Gelfand transform is the Wiener algebra A(T), the pointwise algebra of absolutely convergent Fourier series, that is, F(eiθ)=n∈Zf(n)eiθn, (θ∈T)withf∈1(Z). For F∈A(T), we also write F(z)=n∈Zf(n)znfor |z|≤1. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 5 of 32 The inverse discrete Fourier transform is given by the expressions F–1(F)(n)= 1 2ππ –π Feiθe–inθdθ=1 2πi|z|=1 F(z)dz zn+1 ,n∈Z, for F∈A(T) (and for other functions in larger sets). The classical formulation of Wiener’s lemma characterizes the functions F∈A(T)that are invertible in A(T) as follows. For F∈A(T)whereF(eiθ)=n∈Zf(n)eiθnfor θ∈T, F(eiθ)=0forallθ∈Tif and only if 1/F∈A(T), that is, (1/F)(eiθ)=n∈Zg(n)eiθnwith (g(n))n∈Z∈1(Z); in this case, f∗g=δ0[32, Theorem 5.5]. Recall the definition of the classical Mittag-Leffler function (see (A.3)). We now introduce the following definition. Definition 2.2 For α,β> 0, we define the vector-valued Mittag-Leffler function Eα,β: 1(Z)→1(Z), by Eα,β(a):= ∞  j=0 aj (αj+β),a∈1(Z). Note that E1,1(a)= ∞  j=0 aj j!=ea;E2,1(a)= ∞  j=0 aj (2j)!. The set exp(1(Z)) := {ea;a∈1(Z)}is the connected component of δ0in the set of regular elements in 1(Z)[35, Theorem 6.4.1]. We follow the usual terminology in semigroup theory: the element ais called the generator of the entire group (eza)z∈C; the cosine and sine functions are defined as Cos(z,a):= E2,1(z2a)andSin(z,a):=zE2,2(z2a). We have Sin(z,a)=[0,z] Cos(s,a)ds,z∈C, for a∈1(Z); see [10, Sects. 3.1 and 3.14]. Moreover, the Laplace transform of an entire group or a cosine function is connected with the resolvent of its generator as follows: (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, for example, [10, p. 213]. Example 2.3 For α,β>0,wehave Eα,β(zδ0)=Eα,β(z)δ0;Eα,β(zδ1)= ∞  j=0 zjδj (αj+β). In particular, ezδ1=∞ j=0 zjδj j!and Cos(z,δ1)=∞ j=0 z2jδj (2j)! are generated by δ1. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 6 of 32 Considering generalized versions of the Mittag-Leffler function, as well as of other hypergeometric series, as presented, for example, in [3–5], more examples can be easily derived. In the next proposition, we collect some basic properties of these vector-valued MittagLeffler functions. As usual, we consider Bochner vector-valued integration in the Banach space 1(Z); see, for example, [41, Sect. 1.2]. For the definition of the Wright function γ, see the Appendix, formula (A.1). Proposition 2.4 For α,β>0and a ∈1(Z), we have: (i) Eα,β(a)1≤Eα,β(a1). (ii) F(Eα,β(a)) = Eα,β(F(a));in particular,F(eaz)=ezF(a)and F(Cos(z,a)) = Cos(F(z),a)for z∈C. (iii) σ1(Z)(Eα,β(a)) = Eα,β(σ1(Z)(a)). (iv) The following Laplace transform formula holds: ∞ 0 e–λttαk+β–1E(k) α,βtαadt =k!λα–βλα–a–1(k+1),(λ)>a1/α 1, (2.4) for k∈N∪{0}. (v) For 0<γ<1,Eγ,1(a)=∞ 0γ(t)eta dt. Proof Proofs of parts (i) and (ii) are straightforward. Part (iii) is the spectral mapping theoremshownin[35, Theorem 6.2.1]. Since the algebra 1(Z)issemisimple(seeTheorem2.1), formulae in (iv) and (v) are direct consequences of the scalar identities [39, formula (180), p. 21].  Given a∈1(Z), the modified Mittag-Leffler function Sα,β:(0,∞)→1(Z), which we define by Sα,β(t,a):=tβ–1Eα,βtαa,t> 0, (2.5) is a (gα,gβ)-regularized resolvent family generated by ain the algebra 1(Z). For the definition of (gα,gβ)-regularized resolvent families and more detail in the general case of linear and bounded operators in a Banach space, we refer the reader to [2,Sect.4]andthesurvey [36]. We introduce the functions ψα,β(t,s):=tβ–1 ∞  n=0 (–st–α)n n!(–αn+β),s,t>0, for 0 < α<1andβ>0.Notethatψα,1–α(s,t)=t–αα(stα)for0<α<1. A direct consequence of [2, Theorem 12] is the following subordination theorem. Theorem 2.5 Let 0<η1,0<η2,and a ∈1(Z), and let Sη1,η2be defined in (2.5). Then Sαη1,αη2+β(t,a)=∞ 0 ψα,β(t,s)Sη1,η2(s,a)ds,t>0, for 0<α<1and β≥0. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 7 of 32 Note that in the case of η1=2,η2=1,andα=β=1 2in Theorem 2.5,weobtainthe well-known relation between cosine and semigroup operators generated by a,knownas the Weierstrass formula: eat =1 √πt∞ 0 e–s2 4tCos(s,a)ds,t> 0, (2.6) for a∈1(Z); see, for example, [10, Theorem 3.14.17]. A nice application of the classical Wiener lemma is the invariance of spectrum for convolution operators defined on p(Z)for1≤p≤∞. This issue is contained in the following theoremthatisthekeyabstractresultinthispaper. Theorem 2.6 For a ∈1(Z), we define A(b)(n):=(a∗b)(n), n∈Z,b∈p(Z). (2.7) Then A ∈B(p(Z)) for all 1≤p≤∞.Moreover,A=a1,and for all 1≤p≤∞,we have the following identities: σB(p(Z))(A)=σ1(Z)(a)=F(a)(T). (2.8) For all a ∈1(Z), we have that eza is an entire group in p(Z)with generator a,and for all 1≤p≤∞,we have the following identities: σB(p(Z))eza=σ1(Z)eza=ezF(a)(T),z∈C. (2.9) Proof From Young’s inequality it follows that A∈B(p(Z)). Since the algebra p(Z)hasthe identity δ0, the property of the norm follows. For identities (2.8),wereferto[32, Corollary 5.20]. Finally, for the spectral mapping theorem (2.9), we use (2.8)and[35,Theorem 6.2.1].  The element ain the theorem is also called the symbol of the operator A. Remark 2.7 It is also straightforward to check that the adjoint operator of Ais again a convolution operator given by A(g)(n):=(˜ a∗g)(n), where ˜ a(n)=a(–n), n∈Z. 3 Some finite difference operators in 1(Z) An important case of finite difference operators is given by sequences in the set cc(Z):=a∈1(Z):∃m∈Z+:a(n)=0,∀|n|>m). In such a case, the discrete Fourier transform of a∈cc(Z) is the trigonometric polynomial F(a)eiθ= m  j=–m a(j)eijθ. (3.1) González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 8 of 32 It is interesting to observe that if m j=–ma(j)=0,then0∈σ1(Z)(a). This immediately follows from (2.8). In this paper, we concentrate our study on the operators that appear in the seminal paper of Bateman [16]. Definition 3.1 For f∈p(Z)with1≤p≤∞, we define the following operators: (1) –f(n):=f(n)–f(n+1)=((δ0–δ–1)∗f)(n); (2) ∇f(n):=f(n)–f(n–1)=((δ0–δ1)∗f)(n); (3) df(n):=f(n+1)–2f(n)+f(n–1)=((δ–1 –2δ0+δ1)∗f)(n);and (4) ddf(n):=f(n+2)–2f(n)+f(n–2)=((δ–2 –2δ0+δ2)∗f)(n) for n∈Z. We remark that when considering the above-defined operators in the context of numerical analysis, the operators –and ∇are related to the Euler scheme of approximation, and the operator dcorresponds to the second-order central difference approximation for the second-order derivative. The operator dd appears in Bateman’s paper [16, p. 506] in connection with the equations of Born and Karman on crystal lattices in vibration. 3.1 The operator – The forward difference operator f(n):=f(n+1)–f(n)isaclassicaloperatorusedin approximation theory and in the theory of difference equations. Considering it as an operator from p(Z)top(Z), our main result is as follows. Theorem 3.2 The operator –f=a∗f,where a := δ0–δ–1,possesses the following properties: (1) The norm is given by =2; (2) The Fourier transform is F(a)(z)=1–z,|z|=1; (3) For all 1≤p≤∞,the spectrum is given by σB(p(Z))(–)={z∈T:|z–1|=1}; (4) For |λ+1|>1, (λδ0+a)–1 = j≥0 δ–j (1 + λ)j+1 . (5) The associated group is e–za(n)=e–zz–n (–n)! χ–N0(n),z∈C,n∈Z,and its generator is –a. (6) The norm of the group is given by e–ta1=1,t>0; (7) The associated cosine function is Cos(z,–a)(n)= √π (–n)! (z 2)–n+1 2J–n–1 2(z)χ–N0(n)for z∈Cand n∈Z. Proof (1) The Minkowski inequality shows that ≤2. Then observe that δ0∈p(Z) with δ0p=1satisfiesδ0p= 2, proving the claim. (2) Follows immediately from the definition of the discrete Fourier transform. (3) Follows from formula (2.8)inTheorem2.6 and (2). To prove (4), we apply (2.1)toget (λδ0+a)–1 =(λ+1)δ0–δ–1–1 = j≥0 δ–j (1 + λ)j+1 González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 9 of 32 for |λ+1|> 1. We show (5) directly: e–za(n)=ezδ–1 ∗e–zδ0(n)=ezδ–1 ∗e–zδ0(n)=e–zz–n (–n)!χ–N0(n) for z∈Cand n∈Z.Thenorme–ta1=1fort>0isstraightforwardfrom(5).Finally,to show (7), we apply the Laplace transform and formula (A.10)toget √π (–n)! ∞ 0 e–λtt 2–n+1 2 J–n–1 2(t)dt =λ (λ2+1) –n+1 ,λ>1, for n≤0. By (4) we have that λ (λ2+1) –n+1 =λλ2+a–1(n), n≤0, and we apply (2.3) to conclude the claimed equality and identify the generator of the cosine function with –a. We remark that groups generated by are treated in [1, Sect. 2] and cosine functions in [16, Introduction]. 3.2 The operator ∇ This operator corresponds to the classical backward difference operator. Theorem 3.3 The operator ∇f=a∗f,where a := δ0–δ1,possesses the following properties: (1) ∇=2; (2) F(a)(z)=1–1 z; (3) For all 1≤p≤∞,we have σB(p(Z))(∇)={z∈T:|z–1|=1}; (4) For |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–ta1=1,t>0; (7) Cos(z,–a)=√π n!(z 2)n+1 2Jn–1 2(z)χN0(n),z∈C,n∈Z. Proof The proofs of statements (1), (2), (3), and (4) follow the lines of Theorem 3.2.For statement (5), we have e–za(n)=ezδ1∗e–zδ0(n)=e–zzn n!χN0(n) for z∈Cand n∈Z. Claim (6) follows from (5). Finally, we check (7) as follows. We apply Laplace transform and formula (A.10)toget √π n!∞ 0 e–λtt 2n+1 2 Jn–1 2(t)dt =λ (λ2+1) n+1 ,λ>1, González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 16 of 32 Now we apply the Lévy subordination principle (4.2)toa=δ–1 –δ0or a=δ1–δ0and by Theorem 4.4 obtain the following result. Note that this formula is also obtained from (A.12). Corollary 4.5 Let 0<α<1,and let fs,αbe the Lévy stable process defined by (A.11). Then ∞  j=1 k–αj(n)(–t)j j!=∞ 0 ft,α(s)e–ssn n!ds,t>0,n≥1. In particular,when α=1 2,we have ∞  j=1 k–j 2(n)(–t)j j!=∞ 0 t √4πs3e–t2 4se–ssnds,n≥1. 4.2 The operator (–d)α The operator (–d)αfor 0 < α≤1, called the fractional discrete Laplacian, has been deeply treated in [22,23,26,37]. In [37, Sect. 3] the sequence (–δ–1 +2δ0–δ1)αis denoted by Kα d. To follow the notation in that paper, we write Kα d(n):= 1 2ππ –π4sin2(θ/2)αe–inθdθ=(–1)n(2α+1) (1 + α+n)(1 + α–n) for n∈Zand α>0[37, Formula (22)]. In the case 1 + α+n∈–N0,Kα d(n)=0.Then |Kα d(n)|∼(2α+1) π|n|–2α–1 as n→±∞. We summarize the main properties of the kernel Kα din the following result. Theorem 4.6 For 0<α<1,we have 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 particular, Kα d=∞  j=0 k–α –∗k–α(j)δj, (4.9) where k–α –(n):=k–α(–n). Moreover,σ(Kα d)=[0,4 α], and  Kα d 1=2(1 + 2α) (1 + α)2. Proof Identity (4.7) follows from (4.1). To show (4.8), we apply the discrete Fourier transform to obtain that FKα +∗Kα –(z)=1–1 zα (1 – z)α=2–z+1 zα =FKα d(z) González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 17 of 32 for z∈T. Since the discrete Fourier transform is one-to-one, we obtain the equality. To prove (4.9), we note that the right-hand side evaluated at n∈Zis equal to (k–α –∗kα)(n), and we have 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 the result follows from (4.8). The spectrum is given in [37, Theorem 1.3 (iii)], and the norm of Kαis calculated in [37, Lemma 3.2].  An interesting consequence is the following corollary, which seems to be a new formula for binomials of noninteger entries. Corollary 4.7 Let α∈(0, 1) and n ∈N0.We have the following equality: 2α α+n=∞  j=0 α j+nα j. Proof The combinatorial equality is a straightforward consequence of the explicit expression of the kernel convolutions Kα +,Kα –,andKα d. The following result collects the main results on the fractional discrete semigroup. For other results, see also [37]. Theorem 4.8 For any 0<α<1,we have that the fractional discrete semigroup generated by –Kα dis given by e–zKα d(n)=(–1) n∞  k=1 (–1)kzk k! (2kα+1) (1 + kα+n)(1 + kα–n)+δ0(n) for n ∈Zand z ∈C.Moreover: (i) The discrete Fourier transform of e–zKα dis given by Fe–zKα deiθ=e–z(4sin2(θ 2))α,θ∈[–π,π),z∈C. (ii) e–tKα d(n)≥0,and e–tKα d1=1for n∈Zand t≥0,that is,it is a Markovian semigroup. (iii) σ(e–zKα d)={e–z(4sin2(θ 2))α:θ∈[–π,π)}. Proof The fractional discrete semigroup generated by –Kα dis given in [37, Theorem 1.3]. There the entire group (e–zKα d)z∈Cis written as Lα z. Statement (i) follows from Proposition 2.4(ii) combined with (4.7)inTheorem4.6. The proof of (ii) is contained in [37,Theorem 1.3(v)]. Finally, to prove (iii), we use (2.9)inTheorem2.6 and (4.7)inTheorem4.6. We also apply the Lévy subordination principle (4.2) to the semigroup generated by a= δ–1 –2δ0+δ1to obtain the following result. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 18 of 32 Corollary 4.9 Let 0<α<1,and let fs,αbe the Lévy stable process defined by (A.11). For n∈Zand 0<t<1,we have ∞  j=0 Kαj d(n)(–t)j j!=∞ 0 ft,α(s)e–2sIn(2s)ds; in particular,for α=1 2, ∞  j=0 K j 2 d(n)(–t)j j!=∞ 0 t √4πs3e–t2 4se–2sIn(2s)ds. 4.3 The operator (–dd)α Since the element (δ2–2δ0+δ–2) generates a uniformly bounded C0-semigroup, we consider the fractional power (δ2–2δ0+δ–2)αfor 0 < α<1.Forsimplicity,wewriteKα dd instead of (δ2–2δ0+δ–2)α. Theorem 4.10 Let 0<α<1. (i) We have 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)=0for n∈Z. (iii) Kα dd1=2(1+2α) (1+α)2. (iv) F(Kα dd)(eiθ)=(4sin2(θ))αfor θ∈[–π,π),and σ(Kα dd)=[0,4 α]. Proof (i) For n∈Z,wehavethat 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, wherewehaveapplied[31, Formula 3.631 (8)]. Parts (ii), (iii), and (iv) are straightforward from part (i).  Now we consider the entire group (e–zKα dd )z∈Cgenerated by –Kα dd. Theorem 4.11 For 0<α<1,we have: (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)=0for n∈Z. (iii) e–tKα dd (n)≥0and e–tKα dd 1=1for n∈Zand t≥0,that is,it is a Markovian semigroup. (iv) F(e–zKα dd )(eiθ)=e–z(4sin2(θ))αfor θ∈[–π,π),and σe–zKα dd =σe–zKα dd =e–zuα|u∈[0,4]. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 19 of 32 Proof By [47, Theorem 1, p. 263] we have that e–tKα dd (n)=∞ 0 ft,α(s)e–2sIn 2(2s)dsχ2Z(n)=e–tKα dn 2χ2Z(n), and we conclude equalities (i) and (ii). Parts (iii) and (iv) are proved from similar properties of e–zKα d. Remark 4.12 The spectrum of the discrete fractional Laplacian –(–d)αis determined by σ(–(–d)α)=[–4 α,0], recovering the result of Lizama and Roncal [37], Theorem 1.3(iii). A similar result is obtained for the operator –(–dd)αgiven by σ(–(–dd)α)=[–4 α,0].For the discrete fractional difference operators –and ∇,wehaveσ((–)α) = [–(1+eiT)α]and σ(∇α) = [–(1 + eiT)α], respectively. 5 Fundamental solutions for semidiscrete evolution equations In this section, we consider the operator Bf (n):=(b∗f)(n)withb∈1(Z), f∈p(Z), p∈ [1,∞], and n∈Z. Our objective is obtaining a fundamental representation of solutions for the following semidiscrete fractional evolution equation: ⎧ ⎨ ⎩ Dβ tu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), ut(n,0)=φ(n), n∈Z, where β∈(0,2]. For a sufficiently regular function v,wedenotebyDβ tthe Caputo derivative of order βgiven by Dβ tv(t)= 1 (1 – β)t 0 (t–s)–βv(s)ds =g1–β∗v(t), t>0, for 0 < β<1and Dβ tv(t)= 1 (2 – β)t 0 (t–s)1–βv(s)ds =g2–β∗v(t), t>0, for 1 < β<2.Forβ=1andβ= 2, we consider the usual firstand second-order derivatives. Note that lim β→1–Dβ tv(t)=v(t), lim β→2–Dβ tv(t)=v(t), t>0; however, lim β→0+Dβ tv(t)=v(t)–v(0), lim β→1+Dβ tv(t)=v(t)–v(0), t> 0; (5.1) see, for example, [17,30]. To begin with, we consider the semidiscrete Cauchy problem (1.1) given in the introduction, ⎧ ⎨ ⎩ ∂tu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z,(5.2) González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 20 of 32 and its fundamental solution, which is obviously given by Duhamel’s formula u(n,t)=eBtϕ(n)+t 0 eB(t–s)g(n,s)ds,n∈Z,t≥0. Analogously, in the case of the second-order semidiscrete Cauchy problem ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ttu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z, u(n,0)=ψ(n), n∈Z, (5.3) we have that the fundamental solution is given by D’Alembert formula u(n,t)=Cos(t,B)ϕ(n)+Sin(t,B)ψ(n)+t 0 Sin(t–s,B)f(s)ds, where Cos(t,B)andSin(t,B)aregeneratedbyB. We now consider fractional in time generalizations. Given 0 < β≤1, we first consider the equation ⎧ ⎨ ⎩ Dβ tu(n,t)=Bu(n,t)+g(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z.(5.4) We recall that Eα,β(b)(withb∈1(Z)) is the vector-valued Mittag-Leffler function given in Definition 2.2. The main result is the following theorem. Theorem 5.1 Let ϕ,φ∈p(Z), and let g :Z×R+→Cbe such that for each t ∈R+,g(·,t)∈ p(Z), and sups∈[0,t]g(·,s)p<∞with 1≤p≤∞. (i) For 0<β<1,the function u(n,t)=Eβ,1tβb∗ϕ(n) +t 0 (t–s)β–1Eβ,β(t–s)βb∗g(·,s)(n)ds,n∈Z, is the unique solution of the initial value problem (5.4). Moreover,u(·,t)belong to p(Z)for t>0. (ii) For 1<β<2,the function u(n,t)=Eβ,1tβb∗ϕ(n)+tEβ,2tβb∗φ(n) +t 0 (t–s)β–1Eβ,β(t–s)βb∗g(·,s)(n)ds,n∈Z, is the unique solution of the initial value problem (1.5). Moreover,u(·,t)belong to p(Z)for t>0. Proof Since the algebra 1(Z) is semisimple (see Theorem 2.1), the formulae in (i) and (ii) are direct consequences of the scalar identities, which in case 0 < α<1canbefoundin González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 21 of 32 [36, Sect. 3.3, formula (8)] combined with [36, Sect. 1.2]. The case 1 < α< 2 follows from [36, Sect. 3.3, formula (11)]. See also the references therein.  Remark 5.2 Now we consider the behavior of the solution as βtends to the integer parameter, that is, β= 1, 2. For simplicity, we consider the homogeneous case g=0.Asβ→1–, the solution of equation (1.4) converges to semigroup family operators E1,1(tb), and as β→2–, the solution of equation (1.5), u(·,t)=Eβ,1tβb∗ϕ+tEβ,2tβb∗φ,t>0, converges to unique mild solution of second-order Cauchy problem, that is, the sum of a cosine function and a sine function generated by b;see[10, Corollary 3.14.8]. However, as in the scalar case, as β→1+, the solution of equation (1.5)convergesto u(·,t)=E1,1(bt)+tE1,2(tb), t>0. Note that this function is a solution of the following first-order modified Cauchy problem: ⎧ ⎨ ⎩ v(n,t)=Bv(n,t)+φ(n), n∈Z,t>0, v(n,0)=ϕ(n), n∈Z, for φ,ϕ∈p(Z). This fact is in accordance with the interpolation property of the Caputo fractional derivative; see (5.1). The fundamental solutions uβ,1 for systems (1.4)and(1.5) are obtained by requiring that the initial value ψand the initial velocity φbe the sequences ψ=δ0and φ=0.In the case 1 < β≤2 (including the wave equation), a second fundamental solution uβ,2 is given by ψ=0andφ=δ0;see[26, Remark 3.2]. A consequence of Theorems 5.1 and 2.5 is the following subordination theorem for fundamental solutions, which extends [26, Corollary 3.5]. Corollary 5.3 Let uβ,1 and uβ,2 be the fundamental solutions of problems (1.4)and (1.5), and let αbe the Wright function defined by (A.1). (i) Let 0<β<1.Then uβ,1(n,t)=∞ 0 β(τ)u1,1n,τtβdτ,n∈Z,t>0. (ii) Let 1<β<2.Then uβ,1(n,t)=∞ 0 β 2(τ)u2,1n,τtβ 2dτ, uβ,2(n,t)=t 0 (t–u)–β 2 (1 – β 2)∞ 0 β 2(τ)u2,2n,τuβ 2dτdu for n∈Zand t>0. González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 22 of 32 Remark 5.4 The Wright function 1 3can be expressed in terms of the Airy function Ai(z), that is, 1 3(z)=32 3Aiz 31 3,z∈C; see, for example, [28]. A integral representation of the Airy function is given by the improper Riemann integral: Ai(x)= 1 π∞ 0 cost3 3+xtdt,x∈R. This function appears in several applied problems, in particular, in the Schrödinger equation of quantum physics and in optics (study of caustics); see more details in [44]. By Corollary 5.3(i) we conclude that E1 3,1t1 3b(n)=32 3∞ 0 Aiτ 31 3eτt1 3b(n)dτ,n∈Z,t>0, for b∈1(Z). The particular case of Theorem 5.1 with B=–(–A)α,whereAis the infinitesimal generator of an uniformly bounded C0-semigroup in B(p(Z)), has received a special attention. In [34, Theorem 3.3] and [26, Theorem 3.1] the time/space fractional evolution equations (1.4)and(1.5)oforders0<β≤1and1<β≤2, respectively, are solved, where B=–(–d)α,anddis the discrete Laplacian operator. Both proofs are based on the explicit expressions of vector-valued Mittag-Leffler functions Eβ,1(–tβKα d), Eβ,2(–tβKα d), and Eβ,β(–tβKα d). As a consequence of the results in Sect. 4, we can easily give a general version, which extends both results. Corollary 5.5 Let ϕ,φ∈p(Z), and let g :Z×R+→Cbe such that for each t ∈R+, g(·,t)∈p(Z)and sups∈[0,t]g(·,s)p<∞with 1≤p≤∞.For a ∈1(Z)generating a uniformly continuous semigroup in 1(Z), we write (–a)αfor the fractional powers given in Definition 4.1 and B(f):=–(–a)α∗fforf∈p(Z)and 0<α<1.Thenthesamerepresentation of the fundamental solutions given in Theorem 5.1 with b =–(–a)αholds. 6 Applications We study some concrete examples that appear in various applied fields. 6.1 The discrete Nagumo equation Let us consider the linear part of the discrete Nagumo equation, which can be written as follows: ⎧ ⎨ ⎩ ∂tu(n,t)=du(n,t)–ku(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z,(6.1) where 0 < k< 1/2. The discrete Nagumo equation is used as a model for the spread of genetic traits and for the propagation of nerve pulses in a nerve axon, neglecting recovery; González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 23 of 32 see [48] and references therein. Using Theorem 3.4(3), we obtain σet(d–kI)=etσ(d–kI)=ets :t≥0,–4 – k≤s≤–k. This implies that the unique solution of equation (6.1) is uniformly asymptotically stable, that is, u(n,t)=et(d–kI)ϕ(n)→0ast→∞. Moreover, using Theorem 3.4(4) and the semigroup property, we can obtain the following representation of the fundamental solution: u(n,t)=e–tkIetdϕ(n):=e–tkI ∗etd∗ϕ(n)= n  j=0 e–tkI ∗etd(n–j)ϕ(j) =e–2t n  j=0 n–j  l=0 (–kt)l l!In–j–l(2t)ϕ(j). Since σ(–(–d)α)=[–4 α,0](seeRemark4.12), we have that the same asymptotic behavior also holds for the fundamental solution of the fractional Laplacian version for the discrete Nagumo equation [37,Sect.7]: ⎧ ⎨ ⎩ ∂tu(n,t)=–(–d)αu(n,t)–ku(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z. 6.2 The semidiscrete transport equation associated with the r-difference operator Let us consider the semidiscrete transport equation ⎧ ⎨ ⎩ ∂tu(n,t)=ru(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z,(6.2) where r>0,andris the r-forward difference operator defined by rf(n):=f(n+1)– rf (n); see [7, Sect. 5.5]. Observe that r=+(1–r)I,whereIis the identity operator. Then by perturbation semigroup theory the unique solution of (6.2)hastheformu(n,t)= et(+(1–r)I)ϕ(n), n∈Z. By the spectral mapping (2.9)inTheorem2.6 we obtain that σet(+(1–r)I)=etσ(+(1–r)I)). Hence by Theorem 3.2(3)wededucethatσ(+(1–r)I)={z∈T:|z+r|=1}. Therefore for any r> 1, we have that the upper bound of the spectrum of B=+(1–r)Iis negative, that is, ωσ+(1–r)I:= supz:z∈σ+(1–r)I<0, González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 24 of 32 and, consequently, we obtain that for any r> 1, the unique solution of equation (6.2)is uniformly asymptotically stable, that is,  ertϕ →0ast→∞, uniformly with respect to ϕ≤1. Of course, this result can be also directly deduced from Theorem 3.2(5). Analogously, using the fact that r–eit> 0 implies r–eitα>0 for any 0 < α<1andr> 1, we can deduce from Theorem 4.4 that the same property of asymptotic stability remains true for the unique solution of the fractional semidiscrete transport equation ⎧ ⎨ ⎩ ∂tu(n,t)=–(– r)αu(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z.(6.3) 6.3 The De Juhasz equation We consider the following semidiscrete equation: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ttu(n,t)=du(n,t)–2ku(n,t), n∈Z,t>0, u(n,0)=ϕ(n), n∈Z, ut(n,0)=ψ(n), n∈Z, (6.4) where k> 0. This equation can be found in the seminal paper of Bateman [16]inconnection with surges in springs and connected systems of springs. We call it the De Juhasz equation because, according Bateman’s paper, De Juhasz deduced for the first time the modeling of such equation in mechanical theory. Following Bateman’s paper, this semidiscrete equation is obtained when the concentrated masses on a light string are mounted on springs arranged either along a straight line or on the circumference of a circle or helix [16, Sect. 5, formula (5.1)]. Applying Theorem 3.4 and considering the operator B=d–2kI, we obtain σ(d–2kI)=[–4–2k,–2k], and therefore σCos(t,B)=cos(t√s):s∈[2k,4+2k]. In particular, this implies that on the Hilbert space 2(Z), we have Cos(t)≤1, and, consequently, the unique solution of (6.4)whenψ≡0mustbebounded.Thisextendsthe previous result of Bateman [16,Sect.5],whostudied(6.4) with the initial conditions ψ≡0 and ϕ(n)=δ0(n). González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 25 of 32 6.4 The Caputo–Fabrizio derivative We recall that given a sufficiently regular function uand 0 < α<1,theCaputo–Fabrizio derivative of order αis defined as [19] CFDαu(t)= 1 1–αt 0 e–α 1–α(t–s)u(s)ds. Note that the Caputo–Fabrizio derivative has been very recently used to propose a new mathematical modeling of human liver [12], HIV [13], parallel RCL circuits [8], the Rubella disease model [14], epidemic childhood diseases [11], and COVID-19 [15]. However, with the exception of the implicit solution for the linear model in the scalar case, proposed by Losada and Nieto in [38], so far no explicit formulas have been proposed for the solution of the fractional Cauchy problem in the context of Banach algebras. We consider the equation ⎧ ⎨ ⎩ CFDα tu(n,t)=Bu(n,t)+g(n,t), n∈Z,t≥0, u(n,0)=ϕ(n), n∈Z,(6.5) where we recall that Bf (n):=(b∗f)(n)withb∈1(Z), f∈p(Z), p∈[1,∞], and n∈Z. Since Bis bounded, assuming that 1 1–α∈ρ(B), we obtain the following representation for the solution of (6.5): u(n,t)=T(t)ϕ(n)+t 0 T(t–s)h(n,s)ds,n∈Z,t> 0, (6.6) where T(t):=eα 1–α(1–(1–α)B)–1te–α 1–αt,t≥0, (6.7) and h(n,t):=I–(1–α)B–1(1 – α)gt(n,t)+αg(n,t). (6.8) Indeed, since Bis bounded, from [38, Proposition 2] and taking into account [38,formula (8) and the explicit formula for M(α) given in Remark p. 89] we know that the unique solution of problem (6.5)isgivenbytheuniquesolutionoftheproblem ut(n,t)=αBI–(1–α)B–1u(n,t) +I–(1–α)B–1(1 – α)gt(n,t)+αg(n,t) (note that there is a small but important misprint in [38, p. 90, l. 16], where we must read ˜σ(t) instead of ˜σ(t)). Using the identity (1 – α)B1–(1–α)B–1 =I–(1–α)B–1 –I, González-Camus et al. Advances in Difference Equations ( 2021) 2021:35 Page 32 of 32 11. 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