Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 20, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS OF ORDER 2 < α < 3 ON UMD SPACES JICHAO ZHANG, SHANGQUAN BU Abstract. In this article, we study the ℓp-maximal regularity for the frac- tional difference equation ∆αu(n) = Tu(n) + f(n), (n ∈ N0). We introduce the notion of α-resolvent sequence of bounded linear operators defined by the parameters T and α, which gives an explicit representation of the solution. Using Blunck’s operator-valued Fourier multipliers theorems on ℓp(Z;X), we give a characterization of the ℓp-maximal regularity for 1 < p < ∞ and X is a UMD space. 1. Introduction In this article, we study the unique representation of solutions and the ℓp- maximal regularity for the fractional difference equation ∆αu(n) = Tu(n) + f(n), (n ∈ N0) (1.1) with the initial conditions u(0) = u(1) = u(2) = 0, where T is a bounded linear operator defined on a Banach space X, f : N0 → X is an X-valued sequence, 2 < α < 3 and 1 < p < ∞. Here we denote by N0 the set of non-negative integers, the discrete fractional operator ∆α corresponds to sampling, by means of the Poisson distribution, of the Riemann-Liouville fractional derivative [17] (see the precise definition in the second section). The fractional difference equation (1.1) is the counterparts of fractional differ- ential equations in discrete time which arises as models for several biological and physical applications. Much literature have been devoted to such problems [2, 6]. For instance, Blunck studied the ℓp-maximal regularity for (1.1) with the initial con- dition u(0) = 0 when α = 1, and T substitutes for T−I. He established the following result: when the underlying Banach space X is a UMD space and 1 < p < ∞, equa- tion (1.1) has the ℓp-maximal regularity if and only if {z : |z| = 1, z ̸= 1} ⊂ ρ(T ) and the set { (z − 1)(z − T )−1 : |z| = 1, z ̸= 1 } is Rademacher bounded (R-bounded) [7]. 2020 Mathematics Subject Classification. 47A10, 35R11, 35R20, 43A22. Key words and phrases. Fractional difference equation; maximal regularity; UMD space; R-bounded. ©2024. This work is licensed under a CC BY 4.0 license. Submitted March 12, 2023. Published February 26, 2024. 1 2 J. ZHANG, S. BU EJDE-2024/20 Later, Lizama further considered the ℓp-maximal regularity for (1.1) with the initial condition u(0) = 0 when 0 < α ≤ 1, and proved that when the underlying Banach space X is a UMD space, 1 < p < ∞ and {z1−α(z− 1)α : |z| = 1, z ̸= 1} ⊂ ρ(T ), then (1.1) has the ℓp-maximal regularity if and only if the set{ z1−α(z − 1)α[z1−α(z − 1)α − T ]−1 : |z| = 1, z ̸= 1 } is R-bounded [18]. In the case 1 < α ≤ 2, 1 < p < ∞ and X is a UMD space, Lizama and Arcila showed that (1.1) with the initial conditions u(0) = u(1) = 0, has the ℓp-maximal regularity if and only if {z2−α(z − 1)α : |z| = 1, z ̸= 1} ⊂ ρ(T ), and the set { z2−α(z − 1)α[z2−α(z − 1)α − T ]−1 : |z| = 1, z ̸= 1 } is R-bounded [19]. See [6, 15], [17]-[21] for further results on the corresponding quasi-linear equations. The fractional difference equation (1.1) when 0 < α < 2 was studied in [18, 19]. Meanwhile, when 2 < α < 3, the existence and uniqueness of solution of (1.1), as well as the ℓp-maximal regularity for (1.1), are open topics that deserve to be investigated, the objective of this paper is to solve this problem. When 2 < α < 3, we first introduce a sequence of bounded linear operators Pα(n) defined by the parameters T and α, that we call α-resolvent sequence, which will give an explicit representation of solution for (1.1). Precisely, the sequence (Pα(n))n∈N0 is defined by Pα(0) = Pα(1) = Pα(2) = I, and Pα(n+ 3)− 2Pα(n+ 2) + Pα(n+ 1) = T (kα−2 ∗ Pα)(n) + kα−2(n+ 3)I + (1− α)kα−2(n+ 2)I + (α− 1)(α− 2) 2 kα−2(n+ 1)I for n ∈ N0, where kα−2 is defined by (2.4). We show that when f : N0 → X is given, the fractional differential equation (1.1) has a unique solution u : N0 → X given by u(n) = (hα ∗ Pα ∗ f)(n− 3) for n ≥ 3, where the function hα : N0 → R is defined by hα(n+ 3) = −(1− α)hα(n+ 2)− (α− 1)(α− 2)hα(n+ 1)/2 for n ∈ N0, and hα(0) = 1, hα(1) = α − 1, hα(2) = α(α − 1)/2. We notice that similar α-resolvent sequences have been used in the representation of solutions of (1.1) when 0 < α ≤ 2 in [18, 19]. Concerning the ℓp-maximal regularity for (1.1), we show that when X is a UMD space, 1 < p < ∞ and {z3−α(z − 1)α : |z| = 1, z ̸= ±1} ⊂ ρ(T ), then (1.1) has the ℓp-maximal regularity if and only if the set{ z3−α(z − 1)α[z3−α(z − 1)α − T ]−1 : |z| = 1, z ̸= ±1 } (1.2) is R-bounded. Our main tool is the operator-valued Fourier multipliers theorems on ℓp(Z;X) by Blunck [7], we will transform the ℓp-maximal regularity for (1.1) to an operator-valued Fourier multiplier problem on ℓp(Z;X). It is clear that the R-boundedness of the set (1.2) does not depend on the space parameter p. Thus when X is a UMD space and{ z3−α(z − 1)α : |z| = 1, z ̸= ±1 } ⊂ ρ(T ), EJDE-2024/20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS 3 if (1.1) has the ℓp-maximal regularity for some 1 < p < ∞, then (1.1) has the ℓp- maximal regularity for all 1 < p < ∞. Since every norm bounded subset of B(X) is actually R-bounded when X is a Hilbert space, we deduce that if X is a Hilbert space, 1 < p < ∞, and {z3−α(z − 1)α : |z| = 1, z ̸= ±1} ⊂ ρ(T ), then (1.1) has the ℓp-maximal regularity if and only if the set (1.2) is norm bounded. Concerning applications of the fractional difference equation (1.1), it has been considered as a time-stepping scheme in many fields of sciences such as linear vis- coelasticity theory for describing the behavior of polymeric materials, see [14] [22]. Our results also reveal a close relation between time-stepping scheme and linear vis- coelasticity theory, which are supported and coincident with very recent research in mechanical engineering [9]. In terms of time-stepping scheme, the connection stated in our paper can be interpreted as a methodology to identify the desired (and probably best) time stepping scheme in terms not only of the mathematical model but also of the characteristics of the real material that it models. This article is organized as follows: In section 2, we recall some basic concepts on UMD spaces, R-boundedness, fractional difference operators, the discrete time Fourier transform and Blunck’s Fourier multipliers theorems for operator-valued symbols on UMD spaces. Section 3 is devoted to the study of the α-resolvent sequence defined by T and α, which gives a representation of solution for (1.1). In the last section, we give a characterization of the ℓp-maximal regularity of (1.1) when 1 < p < ∞ and X is a UMD space. 2. Preliminaries In this section, we briefly recall some basic notions about UMD spaces, R- boundedness, fractional difference operators, the discrete time Fourier transform and Blunck’s Fourier multipliers theorems, which will be fundamental in our inves- tigation. Let X be a Banach space. We denote by S(N0;X) the of all X-valued sequences u : N0 → X. Similarly we denote by S(Z;X) the set consisting of all X-valued sequences u : Z → X. The forward Euler operator ∆ : S(N0;X) → S(N0;X) is defined as ∆u(n) := u(n+ 1)− u(n) where n ∈ N0. For every m ∈ N, we define recursively the m-th order forward difference operator ∆m : S(N0;X) → S(N0;X) by ∆m = ∆m−1∆. Let f ∈ S(N0;C) and g ∈ S(N0;X). The finite convolution f ∗ g ∈ S(N0;X) is defined by (f ∗ g)(n) := n∑ j=0 f(n− j)g(j) (2.1) for n ∈ N0. It is easy to verify that if h ∈ S(N0;C), then (f ∗ h ∗ g)(n) = ((f ∗ h) ∗ g)(n) = (f ∗ (h ∗ g))(n) = ∑ i+j+k=n f(i)h(j)g(k) (2.2) for n ∈ N0. The following definition of fractional sum, used the previous works (see [1, 4, 5, 12]), was formally presented by Lizama in [17]. Let 0 < β ≤ 1 and u ∈ S(N0;X) 4 J. ZHANG, S. BU EJDE-2024/20 be given. The fractional sum of u of order β is defined by ∆−βu(n) := (kβ ∗ u)(n) = n∑ j=0 kβ(n− j)u(j) (2.3) for n ∈ N0, where kβ(j) := Γ(β + j) Γ(β)Γ(j + 1) (2.4) for j ∈ N0, where Γ is the Gamma function. It is clear that kβ(0) = 1, kβ(1) = β, kβ(n) = β(β + 1), . . . , (β + n− 1)/n! (2.5) for n ≥ 2. Definition 2.1. Let α > 0, α /∈ N and u ∈ S(N0;X) be given. The fractional difference operator of order α of u is defined by ∆αu(n) := ∆m∆−(m−α)u(n) (2.6) for n ∈ N0, where m ∈ N is the unique integer m satisfying m − 1 < α < m. For more information about fractional difference operators, we refer the readers to [11] Remark 2.2. We notice that when 0 < β ≤ 1 and u ∈ S(N0;X), the value of ∆−βu(n) defined by (2.3) depends on all u(k) when 0 ≤ k ≤ n. If v ∈ S(N0;X) and k ≥ 1 are given, we let u ∈ S(N0;X) satisfy u(n) = v(n− k) for n ≥ k. Then ∆−βu(n) = n∑ j=0 kβ(j)u(n− j) = ∆−βv(n− k) + n∑ j=n−k+1 kβ(j)u(n− j) for n ≥ k. Thus the equality ∆−βu(n) = ∆−βv(n− k) (2.7) is not necessarily true when n ≥ k. Meanwhile if u(j) = 0 where 0 ≤ j ≤ k − 1, then (2.7) is true. Similarly, let α > 0, α /∈ N, if v ∈ S(N0;X) and k ≥ 1 are given, we let u ∈ S(N0;X) satisfy u(n) = v(n− k) for n ≥ k, then the equality ∆αu(n) = ∆αv(n− k) (2.8) is not necessarily true when n ≥ k. But if u(j) = 0 for 0 ≤ j ≤ k − 1, then (2.8) is true. This is the main reason that we only consider the equation (1.1) with the simpler initial conditions u(0) = u(1) = u(2) = 0, instead of the general initial conditions u(0) = x0, u(1) = x1 and u(2) = x2. Let u ∈ S(Z;X). The discrete time Fourier transform of u is û(z) := ∞∑ j=−∞ z−ju(j) for all |z| = 1, whenever it exists. We notice that the Fourier transform of û is sometimes also denoted by F(u). It is clear that if f ∈ S(N0;C) and g ∈ S(N0;X), then (̂f ∗ g)(z) = f̂(z)ĝ(z) (2.9) when both sides are well defined for all |z| = 1. EJDE-2024/20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS 5 Let 0 < α ≤ 1 and let kα be defined by (2.4). It follows from (2.5) that the Fourier transform of kα is given by k̂α(z) = zα (z − 1)α (2.10) for |z| = 1 and z ̸= 1. This implies that if 0 < α, β < 1 and α+ β = 1, then kα ∗ kβ = k1 (2.11) by (2.9). We say that a Banach space X is a UMD space if for all 1 < p < ∞, there exists a constant C > 0 (depending only on p and X) such that for any martingale (gn)n≥0 ⊂ Lp(Ω,Σ, µ;X) and all scalars |εn| = 1, n = 1, 2 . . . , N , the following inequality holds:∥∥g0 + N∑ n=1 εn(gn − gn−1) ∥∥ Lp(Ω,Σ,µ;X) ≤ C ∥∥gN∥∥ Lp(Ω,Σ,µ;X) . It is well known that Lp-spaces, Schatten class Sp, and Sobolev spaces Wm,p are UMD spaces when 1 < p < ∞. UMD spaces have played very important part in vector-valued harmonic analysis and probability theory, see [8]. Let X be a Banach space, we denote by B(X) the space of all bounded linear operators on X. Let rj be the j-th Rademacher function defined on [0, 1] by rj(t) = sgn(sin(2jt)) whenever j ≥ 1. Definition 2.3. Let X be a Banach spaces. A set W ⊂ B(X) is said to be Rademacher bounded [10, 13], if there exists C ≥ 0 such that∥∥ n∑ j=1 rjTjxj ∥∥ L1([0,1];X) ≤ C ∥∥ n∑ j=1 rjxj ∥∥ L1([0,1];X) for all T1, T2, . . . , Tn ∈ W , x1, x2, . . . , xn ∈ X and n ∈ N. Remark 2.4. It is clear that if W1,W2 ⊂ B(X) are R-bounded sets, then W1W2 := {ST : S ∈ W1,T ∈ W2} and W1 +W2 := {S + T : S ∈ W1, T ∈ W2} are still R-bounded. It is easy to see that if W is a bounded subset of the complex plane, then the set {µI : µ ∈ W} is also R-bounded, where I stands for the identity operator. This follows easily from the Kahane’s contraction principle [16]. Let X be a Banach space and let G : (−π, 0) ∪ (0, π) → B(X) be bounded and measurable. Let f ∈ S(Z;X) with finite support, i.e., the set {n ∈ Z : f(n) ̸= 0} is finite. Then the function t → G(t)(Ff)(eit) defined on (−π, 0) ∪ (0, π) is bounded and measurable. Thus its inverse Fourier transform F−1 [ G(·)(Ff)(ei·) ] (n) = 1 2π ∫ 2π 0 G(t)(Ff)(eit)eintdt makes sense for all n ∈ Z. Let 1 ≤ p < ∞ be given. We say that G is an ℓp-Fourier multiplier if there exists a constant C > 0 independent from f such that(∑ n∈Z ∣∣F−1 [ G(·)(Ff)(ei·) ] (n) ∣∣p)1/p ≤ C (∑ n∈Z |f(n)|p )1/p for all f ∈ S(Z;X) with finite support. In this case, there exists a unique bounded linear operator TG ∈ B(ℓp(Z;X)) such that G(t)(Ff)(eit) = F(TGf)(e it) (2.12) 6 J. ZHANG, S. BU EJDE-2024/20 for t ∈ (−π, 0) ∪ (0, π) and f ∈ S(Z;X) with finite support. Here we used the fact that the set of all elements f ∈ S(Z;X) with finite support is dense in ℓp(Z;X). We recall two results obtained by Blunck [7] concerning ℓp-Fourier multipliers which will be fundamental in our investigation. Theorem 2.5. Let X be a UMD space and let 1 < p < ∞. Assume that G : (−π, 0) ∪ (0, π) → B(X) is differentiable and the sets{ G(t) : t ∈ (−π, 0) ∪ (0, π) } , { (eit − 1)(eit + 1)G′(t) : t ∈ (−π, 0) ∪ (0, π) } are R-bounded. Then G is an ℓp-Fourier multiplier. Theorem 2.6. Let X be a Banach space and 1 ≤ p < ∞. Let G : (−π, 0)∪(0, π) → B(X) be continuous and bounded. Assume that G is an ℓp-Fourier multiplier. Then the set {G(t) : t ∈ (−π, 0) ∪ (0, π)} is R-bounded. 3. Representation of solutions In this section, we introduce a special sequence of bounded operators, called α-resolvent sequence, which will give an explicit representation of the solution for the fractional difference equation (1.1). 3.1. α-Resolvent Sequence (Pα(n))n∈N0 . We notice that the fractional difference equation (1.1) in the case 0 < α ≤ 1 with the initial condition u(0) = 0 was previously studied by Lizama [18]. Later, Lizama and Murillo-Arcila [19] further studied the fractional difference equation (1.1) in the case 1 < α < 2 with initial conditions u(0) = u(1) = 0. We will introduce a similar α-resolvent sequence used in [18, 19]. Definition 3.1. Let X be a Banach space, T ∈ B(X) and let 2 < α < 3. We let (Pα(n))n∈N0 ⊂ B(X) determined by: (i) Pα(0) = Pα(1) = Pα(2) = I; (ii) Pα(n+ 3)− 2Pα(n+ 2) + Pα(n+ 1) = T (Pα ∗ kα−2)(n) + kα−2(n+ 3)I + (1− α)kα−2(n+ 2)I + (α−1)(α−2) 2 kα−2(n+ 1)I for n ∈ N0. (Pα(n))n∈N0 is called the α-resolvent sequence generated by T . Remark 3.2. Let 2 < α < 3 and assume that z3−α(z − 1)α ∈ ρ(T ) for all |z| = 1, z ̸= ±1. Then the Fourier transform of Pα is P̂α(z) = [ z3 + (1− α)z2 + (α− 1)(α− 2) 2 z ][ z3−α(z − 1)α − T ]−1 (3.1) for |z| = 1, z ̸= ±1. Indeed, taking the Fourier transform on both sides of (ii) in Definition 3.1 and using (2.5) and (2.9), we obtain z3P̂α(z)− z3 − z2 − z − 2 [ z2P̂α(z)− z2 − z ] + zP̂α(z)− z = T P̂α(z)k̂ α−2(z) + z3k̂α−2(z)− z3 − z2(α− 2)− (α− 1)(α− 2) 2 z + (1− α) [ z2k̂α−2(z)− z2 − z(α− 2) ] + (α− 1)(α− 2) 2 [ zk̂α−2(z)− z ] for |z| = 1, z ̸= ±1. This implies that[ z3 − 2z2 + z − T k̂α−2(z) ] P̂α(z) = [ z3 + (1− α)z2 + (α− 1)(α− 2) 2 z ] k̂α−2(z) EJDE-2024/20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS 7 for |z| = 1, z ̸= ±1. It follows from (2.10) that P̂α(z) = [ z3 + (1− α)z2 + (α− 1)(α− 2) 2 z ][ z3−α(z − 1)α − T ]−1 (3.2) for |z| = 1, z ̸= ±1. For the proof of the main result of this section, we need the following results. Lemma 3.3. Let 2 < α < 3, b : N0 → C, and P : N0 → X, where X is a Banach space. Then ∆α(b ∗ P )(n) = (b ∗∆αP )(n) + b(n+ 3)P (0) + b(n+ 2) [ P (1)− αP (0) ] + b(n+ 1) [ P (2)− αP (1) + α(α− 1) 2 P (0) ] . Proof. By (2.1)-(2.3) and (2.6), we have ∆α(b ∗ P )(n) = ∆3∆−(3−α)(b ∗ P )(n) = ∆−(3−α)(b ∗ P )(n+ 3)− 3∆−(3−α)(b ∗ P )(n+ 2) + 3∆−(3−α)(b ∗ P )(n+ 1)−∆−(3−α)(b ∗ P )(n) = (k3−α ∗ b ∗ P )(n+ 3)− 3(k3−α ∗ b ∗ P )(n+ 2) + 3(k3−α ∗ b ∗ P )(n+ 1)− (k3−α ∗ b ∗ P )(n) = n+3∑ j=0 b(j)(k3−α ∗ P )(n+ 3− j)− 3 n+2∑ j=0 b(j)(k3−α ∗ P )(n+ 2− j) + 3 n+1∑ j=0 b(j)(k3−α ∗ P )(n+ 1− j)− n∑ j=0 b(j)(k3−α ∗ P )(n− j) for n ∈ N0. It follows that ∆α(b ∗ P )(n) = n∑ j=0 b(j) [ (k3−α ∗ P )(n+ 3− j)− 3(k3−α ∗ P )(n+ 2− j) + 3(k3−α ∗ P )(n+ 1− j)− (k3−α ∗ P )(n− j) ] + b(n+ 1)(k3−α ∗ P )(2) + b(n+ 2)(k3−α ∗ P )(1) + b(n+ 3)(k3−α ∗ P )(0)− 3b(n+ 1)(k3−α ∗ P )(1) − 3b(n+ 2)(k3−α ∗ P )(0) + 3b(n+ 1)(k3−α ∗ P )(0) = n∑ j=0 b(j)∆3(k3−α ∗ P )(n− j) + b(n+ 1) [ k3−α(0)P (2) + k3−α(1)P (1) + k3−α(2)P (0) ] + b(n+ 2) [ k3−α(0)P (1) + k3−α(1)P (0) ] + b(n+ 3)k3−α(0)P (0)− 3b(n+ 1) [ k3−α(0)P (1) + k3−α(1)P (0) ] − 3b(n+ 2)k3−α(0)P (0) + 3b(n+ 1)k3−α(0)P (0) = (∆αP ∗ b)(n) + b(n+ 3)P (0) + b(n+ 2) [ P (1)− αP (0) ] 8 J. ZHANG, S. BU EJDE-2024/20 + b(n+ 1) [ P (2)− αP (1) + α(α− 1) 2 P (0) ] where n ∈ N0. This completes the proof. □ Lemma 3.4. Let X be a Banach space, T ∈ B(X), 2 < α < 3 and let (Pα(n))n∈N0 be the α-resolvent sequence given by Definition 3.1. Then ∆αPα(n) = TPα(n) = T where 0 ≤ n ≤ 2. Proof. By (2.1), (2.3) and (2.6), we have ∆αPα(0) = ∆3∆−(3−α)Pα(0) = ∆−(3−α)Pα(3)− 3∆−(3−α)Pα(2) + 3∆−(3−α)Pα(1)−∆−(3−α)Pα(0) = (k3−α ∗ Pα)(3)− 3(k3−α ∗ Pα)(2) + 3(k3−α ∗ Pα)(1)− (k3−α ∗ Pα)(0) = 3∑ j=0 k3−α(3− j)Pα(j)− 3 2∑ j=0 k3−α(2− j)Pα(j) + 3 1∑ j=0 k3−α(1− j)Pα(j)− k3−α(0)Pα(0) = Pα(3) + k3−α(1)I − 2k3−α(2)I + k3−α(3)I − I. (3.3) It follows from Definition 3.1 that Pα(3) = I + T (kα−2 ∗ Pα)(0) + kα−2(3)I + (1− α)kα−2(2)I + (α− 1)(α− 2) 2 kα−2(1)I. (3.4) Therefore, (3.3) and (3.4) imply that ∆αPα(0) = I + T (kα−2 ∗ Pα)(0) + kα−2(3)I + (1− α)kα−2(2)I + (α− 1)(α− 2) 2 kα−2(1)I + k3−α(1)I − 2k3−α(2)I + k3−α(3)I − I = T (kα−2 ∗ Pα)(0) = TPα(0) = T. We have Pα(4) = 2Pα(3)− I + T (kα−2 ∗ Pα)(1) + kα−2(4)I + (1− α)kα−2(3)I + (α− 1)(α− 2) 2 kα−2(2)I. (3.5) EJDE-2024/20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS 9 Applying (2.1), (2.3) and (2.6) again, we have ∆αPα(1) = ∆3∆−(3−α)Pα(1) = ∆−(3−α)Pα(4)− 3∆−(3−α)Pα(3) + 3∆−(3−α)Pα(2)−∆−(3−α)Pα(1) = (k3−α ∗ Pα)(4)− 3(k3−α ∗ Pα)(3) + 3(k3−α ∗ Pα)(2)− (k3−α ∗ Pα)(1) = 4∑ j=0 k3−α(4− j)Pα(j)− 3 3∑ j=0 k3−α(3− j)Pα(j) + 3 2∑ j=0 k3−α(2− j)Pα(j)− 1∑ j=0 k3−α(1− j)Pα(j) = Pα(4)− αPα(3) + k3−α(2)I − 2k3−α(3)I + k3−α(4)I + (α− 1)I. (3.6) By using (3.4), (3.5) and (3.6), we deduce that ∆αPα(1) = T + kα−2(4)I + (3− 2α)kα−2(3)I + 3(α− 1)(α− 2) 2 kα−2(2)I − (α− 1)(α− 2)2 2 kα−2(1)I + k3−α(2)I − 2k3−α(3)I + k3−α(4)I = T. We have Pα(5) = 2Pα(4)− Pα(3) + T (kα−2 ∗ Pα)(2) + kα−2(5)I + (1− α)kα−2(4)I + (α− 1)(α− 2) 2 kα−2(3)I. (3.7) A similar argument used in (3.6) shows that ∆αPα(2) = Pα(5)− αPα(4) + α(α− 1) 2 Pα(3) + k3−α(5)I − 2k3−α(4)I + k3−α(3)I − k3−α(2)I + 2k3−α(1)I − k3−α(0)I. (3.8) By using (3.4), (3.5), (3.7) ,and (3.8), we deduce that ∆αPα(2) = T + α(α− 3) 2 I + kα−2(5)I + (3− 2α)kα−2(4)I + (α− 2)(2α− 3)kα−2(3)I + (2− α)(1− α)(5− 2α) 2 kα−2(2)I − (α− 1)(3− α)(α− 2)2 4 kα−2(1)I + k3−α(5)I − 2k3−α(4)I + k3−α(3)I − k3−α(2)I + 2k3−α(1)I = T. This completes the proof. □ Lemma 3.5. Let X be a Banach space, T ∈ B(X), 2 < α < 3 and let (Pα(n))n∈N0 be the α-resolvent sequence given by Definition 3.1. Then ∆αPα(n) = TPα(n) for all n ∈ N0. Proof. First note that ∆αkα−2(n) = ∆3∆−(3−α)kα−2(n) = ∆3(k3−α ∗ kα−2)(n) = ∆3k1(n) = 0 (3.9) 10 J. ZHANG, S. BU EJDE-2024/20 where n ∈ N0. By (ii) of Definition 3.1, we obtain n∑ j=0 k3−α(j)Pα(n+ 3− j)− 2 n∑ j=0 k3−α(j)Pα(n+ 2− j) + n∑ j=0 k3−α(j)Pα(n+ 1− j) = T n∑ j=0 k3−α(j)(kα−2 ∗ Pα)(n− j) + n∑ j=0 k3−α(j)kα−2(n+ 3− j)I + (1− α) n∑ j=0 kα−2(n+ 2− j)k3−α(j)I + (α− 1)(α− 2) 2 n∑ j=0 kα−2(n+ 1− j)k3−α(j)I where n ∈ N0. This implies that ∆−(3−α)Pα(n+ 3)− 2∆−(3−α)Pα(n+ 2) + ∆−(3−α)Pα(n+ 1) − Pα(2)k 3−α(n+ 1)− Pα(1)k 3−α(n+ 2) − k3−α(n+ 3)Pα(0) + 2k3−α(n+ 1)Pα(1) + 2k3−α(n+ 2)Pα(0)− k3−α(n+ 1)Pα(0) = T∆−(3−α)(kα−2 ∗ Pα)(n) + ∆−(3−α)kα−2(n+ 3)I + (1− α)∆−(3−α)kα−2(n+ 2)I + (α− 1)(α− 2) 2 ∆−(3−α)kα−2(n+ 1)I − kα−2(0)k3−α(n+ 3)I − kα−2(1)k3−α(n+ 2)I − kα−2(2)k3−α(n+ 1)I − (1− α)kα−2(1)k3−α(n+ 1)I − (1− α)kα−2(0)k3−α(n+ 2)I − (α− 1)(α− 2) 2 kα−2(0)k3−α(n+ 1)I for n ∈ N0. Thus ∆−(3−α)Pα(n+ 3)− 2∆−(3−α)Pα(n+ 2) + ∆−(3−α)Pα(n+ 1) = T∆−(3−α)(kα−2 ∗ Pα)(n) + ∆−(3−α)kα−2(n+ 3) + (1− α)∆−(3−α)kα−2(n+ 2) + (α− 1)(α− 2) 2 ∆−(3−α)kα−2(n+ 1) for n ∈ N0. Consequently ∆αPα(n+ 3)− 2∆αPα(n+ 2) + ∆αPα(n+ 1) = T∆α(Pα ∗ kα−2)(n) by (3.9) for n ∈ N0. It follows from Lemma 3.3 that ∆α(kα−2 ∗ Pα)(n) = (∆αkα−2 ∗ Pα)(n) + kα−2(0)Pα(n+ 3) + [ kα−2(1)− αkα−2(0) ] Pα(n+ 2) + [ kα−2(2)− αkα−2(1) + α(α− 1) 2 kα−2(0) ] Pα(n+ 1) = Pα(n+ 3)− 2Pα(n+ 2) + Pα(n+ 1) EJDE-2024/20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS 11 for n ∈ N0. We conclude that ∆2∆αPα(n+ 1) = ∆2TPα(n+ 1) for n ∈ N0. The conclusion follows easily from Lemma 3.4. The proof is complete. □ 3.2. Formula of Solution. With the help of the α-resolvent sequence (Pα(n))n∈N0 , we are able to give the exact expression of the solution of (1.1). Definition 3.6. For 2 < α < 3, we define the function hα : N0 → R by hα(0) = 1, hα(1) = α− 1, hα(2) = α(α−1) 2 , and hα(n+ 3) + (1− α)hα(n+ 2) + (α− 1)(α− 2) 2 hα(n+ 1) = 0 (3.10) for n ≥ 0. Remark 3.7. Let (hα(n))n∈N0 be defined by (3.10). Then the Fourier transform of hα is ĥα(z) = z3 z3 + (1− α)z2 + (α−1)(α−2) 2 z . (3.11) Indeed, taking the Fourier transform on both sides of (3.10), we have z3ĥα(z)− z3 − (α− 1)z2 − α(α− 1) 2 z + (1− α) [ z2ĥα(z)− z2 − (α− 1)z ] + (α− 1)(α− 2) 2 [ zĥα(z)− z ] = 0 when |z| = 1, which implies that[ z3 + (1− α)z2 + (α− 1)(α− 2) 2 z ] ĥα(z) = z3 (3.12) for |z| = 1. Thus (3.11) holds for all |z| = 1. Now we are ready to state the main result of this section. Theorem 3.8. Let X be a Banach space, T ∈ B(X), 2 < α < 3 and let f ∈ S(N0;X) be given. Then (1.1) has a unique solution u ∈ S(N0;X) defined by u(0) = u(1) = u(2) = 0, and u(n) = (hα ∗ Pα ∗ f)(n− 3) (3.13) for n ≥ 3. Proof. Let u ∈ S(N0;X). By Lemma 3.3, Lemma 3.5 and Definition 3.6, we have ∆α ( hα ∗ Pα ) (n) = (hα ∗∆αPα)(n) + hα(n+ 3)Pα(0) + hα(n+ 2) [ Pα(1)− αPα(0) ] + hα(n+ 1) [ Pα(2)− αPα(1) + α(α− 1) 2 Pα(0) ] = (Thα ∗ Pα)(n) + hα(n+ 3) + (1− α)hα(n+ 2) + (α− 1)(α− 2) 2 hα(n+ 1) = T (hα ∗ Pα)(n) (3.14) 12 J. ZHANG, S. BU EJDE-2024/20 for n ∈ N0. We deduce from Lemma 3.3, Lemma 3.5 and (3.14) that ∆α ( hα ∗ Pα ∗ f ) (n) = (∆α(hα ∗ Pα) ∗ f)(n) + (hα ∗ Pα)(0)f(n+ 3) + [ (hα ∗ Pα)(1)− α(hα ∗ Pα)(0) ] f(n+ 2) + [ (hα ∗ Pα)(2)− α(hα ∗ Pα)(1) + α(α− 1) 2 (hα ∗ Pα)(0) ] f(n+ 1) = ( ∆α(hα ∗ Pα) ∗ f ) (n) + f(n+ 3) = T (Pα ∗ hα ∗ f)(n) + f(n+ 3) (3.15) where n ∈ N0. We claim that ∆αu(n) = ∆α(Pα ∗ hα ∗ f)(n− 3) (3.16) where n ≥ 3. Indeed, since u(0) = u(1) = u(2) = 0, we have ∆−(3−α)u(n) = (k3−α ∗ u)(n) = n∑ j=0 k3−α(j)u(n− j) = n−3∑ j=0 k3−α(j)(hα ∗ Pα ∗ f)(n− 3− j) = ∆−(3−α)(hα ∗ Pα ∗ f)(n− 3) for n ≥ 3, which clearly implies that (3.16) is true as ∆α = ∆3∆−(3−α) by (2.6). It follows from (3.15) and (3.16) that ∆αu(n) = Tu(n) + f(n) (3.17) for n ≥ 3. Now we are going to show that equality (3.17) remains valid when 0 ≤ n ≤ 2. Indeed, it follows from (3.13) that u(3) = (hα ∗ Pα ∗ f)(0) = (hα ∗ Pα)(0)f(0) = hα(0)Pα(0)f(0) = f(0); (3.18) u(4) = (hα ∗ Pα ∗ f)(1) = (hα ∗ Pα)(1)f(0) + (hα ∗ Pα)(0)f(1) = [ hα(1)Pα(0) + hα(0)Pα(1) ] f(0) + hα(0)Pα(0)f(1) = αf(0) + f(1); (3.19) u(5) = (hα ∗ Pα ∗ f)(2) = (hα ∗ Pα)(2)f(0) + (hα ∗ Pα)(1)f(1) + (hα ∗ Pα)(0)f(2) = [ hα(2)Pα(0) + hα(1)Pα(1) + hα(0)Pα(2) ] f(0) + [ hα(1)Pα(0) + hα(0)Pα(1) ] f(1) + hα(0)Pα(0)f(2) = α(α+ 1) 2 f(0) + αf(1) + f(2). (3.20) On the other hand, using the conditions u(0) = u(1) = u(2) = 0 and (3.18)-(3.20), we have ∆αu(0) = ∆3∆−(3−α)u(0) = (k3−α ∗ u)(3) = k3−α(0)u(3) = u(3) = f(0); EJDE-2024/20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS 13 ∆αu(1) = ∆3∆−(3−α)u(1) = (k3−α ∗ u)(4)− 3(k3−α ∗ u)(3) = k3−α(0)u(4) + k3−α(1)u(3)− 3k3−α(0)u(3) = u(4)− αu(3) = f(1); ∆αu(2) = ∆3∆−(3−α)u(2) = (k3−α ∗ u)(5)− 3(k3−α ∗ u)(4) + 3(k3−α ∗ u)(3) = k3−α(0)u(5) + k3−α(1)u(4) + k3−α(2)u(3) − 3k3−α(0)u(4)− 3k3−α(1)u(3) + 3k3−α(0)u(3) = u(5)− αu(4) + α(α− 1) 2 u(3) = f(2). Thus ∆αu(n) = Tu(n) + f(n) where 0 ≤ n ≤ 2 as u(0) = u(1) = u(2) = 0. We have shown that u ∈ S(N0;X) given by (3.13) is a solution of (1.1). It remains to show that the solution is unique. It is clear that we only need to show that 0 ∈ S(N0;X) is the unique solution of the equation ∆αu(n) = Tu(n), (n ∈ N0); u(0) = u(1) = u(2) = 0. (3.21) Let u ∈ S(N0;X) be a solution of (3.21). We first show that u(3) = 0. The identity ∆αu(0) = Tu(0) implies that ∆αu(0) = 0, or, equivalently, ∆3∆−(3−α)u(0) = ∆−(3−α)u(3)− 3∆−(3−α)u(2) + 3∆−(3−α)u(1) + ∆α−3u(0) = (k3−α ∗ u)(3)− 3(k3−α ∗ u)(2) + 3(k3−α ∗ u)(1) + (k3−α ∗ u)(0) = k3−α(0)u(3) = u(3) = 0 as u(0) = u(1) = u(2) = 0 by assumption. Assume that u(n) = 0 for all 3 ≤ n ≤ k for some k ≥ 3. We are going to show that u(k + 1) = 0. Since k − 2 < k, we have u(k − 2) = 0 by assumption. Thus ∆αu(k − 2) = Tu(k − 2) = 0, or, equivalently, ∆3∆−(3−α)u(k − 2) = ∆−(3−α)u(k + 1)− 3∆−(3−α)u(k) + 3∆−(3−α)u(k − 1) + ∆−(3−α)u(k − 2) = (k3−α ∗ u)(k + 1)− 3(k3−α ∗ u)(k) + 3(k3−α ∗ u)(k − 1) + (k3−α ∗ u)(k − 2) = k3−α(0)u(k + 1) = u(k + 1) = 0 as u(k − 2) = u(k − 1) = u(k) = 0 by assumption. Consequently u(n) = 0 for all n ∈ N0. This concludes the proof. □ 4. Characterization of the ℓp-maximal regularity Let T ∈ B(X), where X is a Banach space. Let f : N0 → X be an X-valued sequence and 1 < p < ∞. In this section, we will study the ℓp-maximal regularity for the discrete time evolution equation of fractional order ∆αu(n) = Tu(n) + f(n), (n ∈ N0); u(0) = u(1) = u(2) = 0, (4.1) where 2 < α < 3 is given. For all f ∈ S(N0;X), the unique solution u ∈ S(N0;X) of (4.1) is given by u(0) = u(1) = u(2) = 0, and u(n) = (Pα ∗ hα ∗ f ) (n− 3), (4.2) 14 J. ZHANG, S. BU EJDE-2024/20 for n ≥ 3 by Theorem 3.8. This means that ∆αu(0) = ∆αu(1) = ∆αu(2) = 0, and ∆αu(n) = T (Pα ∗ hα ∗ f ) (n− 3) + f(n) (4.3) for n ≥ 3. Now we introduce the following definition concerning maximal regularity, which is motivated by the case α = 1 and α = 2, see for instance [7]. Definition 4.1. Let 1 < p < ∞. We say that (4.1) has the ℓp-maximal regularity if (Rαf)(n) := T (Pα ∗ hα ∗ f)(n) = T n∑ j=0 (Pα ∗ hα)(n− j)f(j), (n ∈ N0) (4.4) defines a bounded linear operator Rα ∈ B(ℓp(N0;X)). It is clear that (4.1) has the ℓp-maximal regularity if and only if for all f ∈ ℓp(N0;X), the unique solution u of (4.1) given by (4.2) satisfies ∆αu ∈ ℓp(N0;X) by (4.3). We say that T ∈ B(X) satisfies the assumption (Cα) if z3−α(z − 1)α ∈ ρ(T ) for all |z| = 1, z ̸= ±1. Now we are ready to state the main result of this section. Theorem 4.2. Let X be a UMD space, 2 < α < 3 and let 1 < p < ∞. Assume that T ∈ B(X) satisfies the assumption (Cα). Then the following statements are equivalent: (i) Equation(4.1) has the ℓp-maximal regularity; (ii) the set{ e(3−α)it(eit − 1)α [ e(3−α)it(eit − 1)α − T ]−1 : t ∈ (−π, 0) ∪ (0, π) } (4.5) is R-bounded. Proof. (ii) implies (i). Assume that the set (4.5) is R-bounded. We are going to show that (4.1) has the ℓp-maximal regularity. Let gα(t) = e3it(1− e−it)α, G(t) = gα(t) [ gα(t)− T ]−1 for t ∈ (−π, 0) ∪ (0, π). Then g′α(t) = 3igα(t) + αigα(t) eit − 1 = (3i+ αi eit − 1 )gα(t) G′(t) = g′α(t) gα(t) G(t)− g′α(t) gα(t) G2(t) = (3i+ αi eit − 1 )(G(t)−G2(t)) for t ∈ (−π, 0) ∪ (0, π). Thus the sets{ G(t) : t ∈ (−π, 0) ∪ (0, π) } , { (eit − 1)(eit + 1)G′(t) : t ∈ (−π, 0) ∪ (0, π) } are R-bounded by assumption and Remark 2.4. Using Theorem 2.5, there exists an operator Tα ∈ B(ℓp(Z;X)) such that T̂αf(e it) = G(t)f̂(eit) (4.6) when t ∈ (−π, 0)∪(0, π) for all f ∈ ℓp(Z;X) with finite support. The trivial identity T [ gα(t)− T ]−1 = gα(t) [ gα(t)− T ]−1 − I EJDE-2024/20 MAXIMAL REGULARITY FOR FRACTIONAL DIFFERENCE EQUATIONS 15 for t ∈ (−π, 0) ∪ (0, π) together with (4.6) implies that T [ gα(t)− T ]−1 f̂(eit) = gα(t) [ gα(t)− T ]−1 f̂(eit)− f̂(eit) = G(t)f̂(eit)− f̂(eit) = T̂αf(e it)− f̂(eit) (4.7) when t ∈ (−π, 0) ∪ (0, π) for all f ∈ ℓp(Z;X) with finite support. It follows from Remark 3.2 and Remark 3.7 that P̂α ∗ hα(e it) = e3it [ gα(t)− T ]−1 (4.8) for t ∈ (−π, 0) ∪ (0, π). This together with (4.8) implies that R̂αf(e it) = e3itT [ gα(t)− T ]−1 f̂(eit) when t ∈ (−π, 0) ∪ (0, π) for all f ∈ ℓp(N0;X) with finite support. Hence Rα is a bounded and linear operator on ℓp(N0;X) by (4.7). Here we have used the fact that the set of all f ∈ ℓp(N0;X) with finite support is dense in ℓp(N0;X). We have shown that (4.1) has the ℓp-maximal regularity. Hence (ii) implies (i). (i)implies (ii). Assume that (i) holds. Then (Rαf)(n) = { T (Pα ∗ hα ∗ f)(n), n ≥ 3, 0, otherwise. (4.9) defines a bounded linear operator Rα ∈ B(ℓp(N0;X)). It follows from (4.8) that R̂αf(e it) = e3itT [ gα(t)− T ]−1 f̂(eit) (4.10) when t ∈ (−π, 0) ∪ (0, π) for all f ∈ ℓp(N0;X) with finite support. Since it is clear that Rα is a convolution operator, Rα is translation invariant on ℓp(Z;X). Thus Rα extends to a bounded linear operator on ℓp(Z;X). In other words, the function t → e3itT [ gα(t) − T ]−1 defined on (−π, 0) ∪ (0, π) is an ℓp-Fourier multiplier by (2.12). Hence the set{ e3itT [ gα(t)− T ]−1 : t ∈ (−π, 0) ∪ (0, π) } is R-bounded by Theorem 2.6. The trivial equality T [ gα(t)− T ]−1 + I = gα(t) [ gα(t)− T ]−1 for t ∈ (−π, 0) ∪ (0, π) implies that the set{ gα(t) [ gα(t)− T ]−1 : t ∈ (−π, 0) ∪ (0, π) } is R-bounded. This completes the proof. □ Since the second condition in Theorem 4.2 does not depend on the parameter 1 < p < ∞, we have the following immediate consequence. Corollary 4.3. Let X be a UMD space, 2 < α < 3 and let T ∈ B(X) satisfy the assumption (Cα). If (4.1) has the ℓp-maximal regularity for some 1 < p < ∞, then it has the ℓp-maximal regularity for all 1 < p < ∞. Let H be a Hilbert space, then each bounded subset W ⊂ B(H) is actually R-bounded [3]. This together with Theorem 4.2 gives the following result. 16 J. ZHANG, S. BU EJDE-2024/20 Corollary 4.4. Let H be a Hilbert space, 2 < α < 3, 1 < p < ∞ and let T ∈ B(H) satisfy the assumption (Cα). Then (4.1) has the ℓp-maximal regularity if and only if there exists C > 0 such that∣∣e(3−α)it(eit − 1)α [ e(3−α)it(eit − 1)α − T ]−1∣∣ ≤ C (4.11) for all t ∈ (−π, 0) ∪ (0, π). Now we give a concrete example that Corollary 4.4 is applicable. Let H be a Hilbert space, 1 < p < ∞, 2 < α < 3 and let T ∈ B(H)) satisfy σ(T ) ⊂ {z ∈ C : |z| > 2α}. Then for every f ∈ ℓp(N0;H), there is a unique u ∈ ℓp(N0;H) such that (4.1) holds. Indeed, an easy calculation gives sup t∈[−π,π] ∣∣(eit − 1)α ∣∣ ≤ 2α. This together with the assumption σ(T ) ⊂ {z ∈ C : |z| > 2α} implies that e(3−α)it(eit−1)α ∈ ρ(T ) for all t ∈ [−π, π], and since the function t → |(e(3−α)it(eit− 1)α−T )−1| is continuous on the closed interval [−π, π], the condition (4.11) is valid. Acknowledgments. This work was supported by the NSF of China (Grant No. 12171266). We are grateful to the anonymous referees for the careful reading the manuscript and for providing valuable comments. References [1] T. Abdeljawad, F. M. 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Murillo-Arcila; ℓp-maximal regularity for a class of fractional difference equa- tions on UMD spaces: the case 1 < α ≤ 2, Banach J. Math. Anal., 11 (2017), 188-206. [20] C. Lizama, M. Murillo-Arcila; Maximal regularity in ℓp spaces for discrete time fractional shifted equations, J. Differential Equations., 263 (2017), 3175-3196. [21] C. Lizama, M. Murillo-Arcila; Well posedness for semidiscrete abstract fractional Cauchy problems with finite delay, J. Comput. Appl. Math., 339 (2018), 356-366. [22] C. Lizama, M. Murillo-Arcila; Discrete maximal regularity for Volterra equations and nonlocal time-stepping schemes, Discrete Contin. Dyn. Syst., 40 (2020), 509-528. Jichao Zhang School of Science, Hubei University of Technology, Wuhan 430068, China Email address: 156880717@qq.com Shangquan Bu Department of Mathematical Science, Tsinghua University, Beijing 100084, China Email address: bushangquan@tsinghua.edu.cn 1. Introduction 2. Preliminaries 3. Representation of solutions 3.1. -Resolvent Sequence (P(n))nN0 3.2. Formula of Solution 4. Characterization of the p-maximal regularity Acknowledgments References