PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (109) (2014), 119–132 DOI: 10.2298/PIM1409119K

SYSTEMS OF ABSTRACT TIME-FRACTIONAL EQUATIONS

Marko Kostić

Communicated by Stevan Pilipović

Abstract. We analyze systems of abstract time-fractional equations in certain classes of sequentially complete locally convex spaces. We also consider arbitrary matrices of operators as generators of fractional regularized resolvent families, improving in such a way the results known for semigroups of operators.

1. Introduction and preliminaries

The fractional calculus is one of the active research fields in mathematical analysis, primarily from its importance in modeling of various problems in engineering, physics, chemistry and other sciences. Presumably the first systematic exposition on abstract time-fractional equations with Caputo fractional derivatives is that of Bazhlekova [2]. In this fundamental work, the abstract time-fractional equations with Caputo fractional derivatives have been studied by converting them into equivalent abstract Volterra equations [17].

The reading of paper [7] by Kisyński served as a starting point for the genesis of this paper. We shall prove a generalization of the assertion [7, Theorem 1, (a) ⇒ (b)] for abstract time-fractional equations (Theorem 2.1, Remark 2.1). The second aim of the paper is to generalize [3, Theorem 14.1] to abstract time-fractional equations (Theorem 2.2), and to clarify some classes of sequentially complete locally convex spaces in which the above-mentioned result admits a reformulation (Theorem 2.3).

Throughout the paper, we assume that E is a Hausdorff sequentially complete locally convex space, SCLCS for short, and that the abbreviation ⊛ stands for the fundamental system of seminorms which defines the topology of E. By L(E) is denoted the space of all continuous linear mappings from E into E. The domain, resolvent set, spectrum and range of a closed linear operator A acting on E are denoted by D(A), ρ(A), σ(A) and R(A), respectively. Put D_{∞} (A) := ⋂ _{n∈ℕ} D(A^{n}). Suppose F is a linear subspace of E. Then the part of A in F, denoted by A_{|F} , is a linear operator defined by D(A_{|F} ) := {x ∈ D(A) ∩ F : Ax ∈ F } and A_{|F} x := Ax, x ∈ D(A_{|F} ). If B ∈ L(E) and m ∈ ℕ, then we define the operator B_{m} ∈ L(E^{m}) (sometimes also denoted by B) by B_{m}(x_{1}, …, x_{m}) := (Bx_{1}, …, Bx_{m}), (x_{1}, …, x_{m}) ∈ E^{m}. Given s ∈ ℝ in advance, set ⌊s⌋ := sup{l ∈ ℤ : l ⩽ s} and ⌈s⌉ := inf{l ∈ ℤ : s ⩽ l}. The Gamma function is denoted by Γ(·) and the principal branch is always used to take the powers. Set 0^{α} := 0 and g_{α} (t) := t^{α−1} /Γ(α) (α > 0, t > 0). We refer the reader to [15, pp. 99–102] for the basic material concerning integration in SCLCSs, and to [9] for the definition and elementary properties of analytic functions with values in SCLCSs. The reader may consult [16] for the basic properties of distribution spaces used henceforward.

2010 Mathematics Subject Classification: Primary 47D03, 47D06; Secondary 47D60, 47D62, 47D99.

Partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.

The following definition has been recently introduced in [8, 9].

Definition 1.1. Let α > 0 and let A be a closed linear operator on E. A strongly continuous family (R_{α} (t))_{t⩾0} in L(E) is said to be a (global) (g_{α} , C)- regularized resolvent family having A as a subgenerator iff the following holds:

(a) R_{α} (t)A ⊆ AR_{α} (t), t ⩾ 0, R_{α} (0) = C and CA ⊆ AC,
(b) R_{α} (t)C = CR_{α} (t), t ⩾ 0, and
(c) R_{α} (t)x = Cx + ∫_{0}^{t} g_{α} (t − s)AR_{α} (s)x ds, t ⩾ 0, x ∈ D(A);

(R_{α} (t))_{t⩾0} is said to be exponentially equicontinuous if there exists ω ∈ ℝ such that the family {e^{−ωt} R_{α} (t) : t ⩾ 0} is equicontinuous.

The integral generator Â of (R_{α} (t))_{t⩾0} is defined by

Â := { (x, y) ∈ E × E : R_{α} (t)x − Cx = ∫_{0}^{t} g_{α} (t − s)R_{α} (s)y ds, t ⩾ 0} .

Suppose that (R_{α} (t))_{t⩾0} is exponentially equicontinuous and that the following equality holds:

R_{α} (t)x − Cx = A ∫_{0}^{t} g_{α} (t − s)R_{α} (s)x ds, t ⩾ 0, x ∈ E.

Then Â is the maximal subgenerator of (R_{α} (t))_{t⩾0} with respect to the set inclusion and Â = C^{−1} AC. Notice also that the above equality holds provided that A is densely defined and that (R_{α} (t))_{t⩾0} is locally equicontinuous [9].

Let α > 0, β ∈ ℝ and the Mittag-Leffler function E_{α,β} (z) be defined by E_{α,β} (z) := ∑ _{n=0}^{∞} z^{n}/Γ(αn+β), z ∈ ℂ. In this place, we assume that 1/Γ(αn+β)=0 if αn + β ∈ −ℕ_{0}. Set, for short, E_{α} (z) := E_{α,1} (z), z ∈ ℂ. Then it is well known that, for every α > 0, there exists c_{α} > 0 such that:

(1.1)  E_{α} (t) ⩽ c_{α} exp(t^{1/α} ), t ⩾ 0.

The asymptotic expansion of the entire function E_{α,β} (z) is given in the following lemma (cf. [19, Theorem 1.1]).

Lemma 1.1. Let 0 < σ < \frac{1}{2}π. Then, for every z ∈ ℂ ∖ \{0\} and m ∈ ℕ ∖ \{1\}:

E_{α,β} (z) = \frac{1}{α} ∑_{s} Z_{s}^{1−β} e^{Z_{s}} − ∑_{j=1}^{m−1} \frac{z^{−j}}{Γ(β − αj)} + O(|z|^{−m} ), |z| → ∞,

where Z_{s} is defined by Z_{s} := z^{1/α} e^{2πis/α} and the first summation is taken over all those integers s satisfying | argz + 2πs| < α(\frac{π}{2} + σ).

Let E be one of the spaces L^{p}(ℝ^{n}) (1 ⩽ p ⩽ ∞), C_{0}(ℝ^{n}), C_{b}(ℝ^{n}), BUC(ℝ^{n}), C^{σ}(ℝ^{n}) (0 < σ < 1) and let 0 ⩽ l ⩽ n. Put ℕ_{0}^{l} := {η ∈ ℕ_{0}^{n} : η_{l+1} = ⋯= η_{n} = 0} and recall that the space E_{l} is defined by E_{l} := {f ∈ E : f^{(η)} ∈ E for all η ∈ ℕ_{0}^{l}}. The totality of seminorms (q_{η}(f) := ||f^{(η)}||_{E}, f ∈ E_{l}; η ∈ ℕ_{0}^{l}) induces a Fréchet topology on E_{l} (cf. the proof of [20, Lemma 5.6, p. 25]). Put D^{η}· := (−i)^{|η|}·^{(η)} (η ∈ ℕ_{0}^{n}).

In the proofs of our main results, we will make use of the functional calculus for commuting generators of bounded C_{0}-groups ([3]). Denote by F and F^{−1} the n-dimensional Fourier transform and its inverse transform, respectively. That is

(Ff)(ξ) := ∫_{ℝ^{n}} e^{i(x,ξ)} f(x) dx, ξ ∈ ℝ^{n} and F^{−1} := (2π)^{−n} F̂,

where ˆ denotes the reflection in 0. Let (E, ∥ · ∥) be a complex Banach space, n ∈ ℕ and iA_{j} , 1 ⩽ j ⩽ n be commuting generators of bounded C_{0}-groups on E. Set A := (A_{1}, …, A_{n}) and A^{η} := A_{1}^{η_{1}} ⋯ A_{n}^{η_{n}} for any η = (η_{1}, …, η_{n}) ∈ ℕ_{0}^{n}. If ξ = (ξ_{1}, …, ξ_{n}) ∈ ℝ^{n} and u ∈ A := {f ∈ C_{0}(ℝ^{n}) : Ff ∈ L^{1}(ℝ^{n})}, put |ξ| := (∑ _{j=1}^{n} ξ_{j}^{2})^{1/2} , (ξ, A) := ∑ _{j=1}^{n} ξ_{j} A_{j} and

u(A)x := (2π)^{−n} ∫_{ℝ^{n}} Fu(ξ)e^{−i(ξ,A)} x dξ, x ∈ E.

Then u(A) ∈ L(E), u ∈ A and there exists a constant M < ∞ such that ∥u(A)∥ ⩽ M∥Fu∥_{L^{1} (ℝ^{n} )} , u ∈ A.

Put Z_{n} := FD(ℝ^{n}) and assume that Z_{n} is equipped with the topology transported by F from D(ℝ^{n}). By Z′_{n} we denote the strong dual of Z_{n}. It is clear that Z_{n} = F^{−1} D(ℝ^{n}) and that the dual mapping of F_{|Z_{n}} : Z_{n} → D(ℝ^{n}) is an isomorphism of D′(ℝ^{n}) onto Z′_{n}. We have the following equality:

⟨FT, Fφ⟩ = (2π)^{n}⟨T, φ̂⟩, T ∈ D′(ℝ^{n}), φ ∈ D(ℝ^{n}).

The operator ∂/∂x_{j} : Z′_{n} → Z′_{n} is defined as the dual operator of −∂/∂x_{j} : Z_{n} → Z_{n}, so that ∂/∂x_{j} FT = F(iξ_{j} T), T ∈ D′(ℝ^{n}), 1 ⩽ j ⩽ n the actions of F on (D′(ℝ^{n}))^{m} and of F^{−1} on (Z′_{n})^{m} are coordinatewise.

Let m, n, d ∈ ℕ and let M_{m} denote the ring of all complex matrices of format m × m. Define P (x) := ∑ _{|η|⩽d} A_{η}x^{η}, x ∈ ℝ^{n} (A_{η} ∈ M_{m}), P (∂/∂x) := ∑ _{|η|⩽d} A_{η}(∂/∂x)^{η}, and P̃(ξ) := ∑ _{|η|⩽d} i^{|η|}A_{η}ξ^{η}. Denote by λ_{1}(ξ), …, λ_{m}(ξ) the eigenvalues of P̃(ξ) (ξ ∈ ℝ^{n}).

In the sequel, we will always consider the case in which the space E is barreled, so that every (g_{α} , C)-regularized resolvent family (R_{α} (t))_{t⩾0} in E is locally equicontinuous (cf. [9, Remark 2.2]). The results obtained in this paper can be used in the analysis of the following system of abstract time-fractional equations with α > 0:

(1.2) D_{t}^{α} u⃗(t) = P (∂/∂x)_{|E} u⃗(t), t > 0; u^{(k)} (0) = x⃗_{k}, k = 0, 1, …, ⌈α⌉ − 1,

where D_{t}^{α} denotes the Caputo fractional derivative of order α ([2, 10]). If x⃗_{k} ∈ C(D(P (∂/∂x)_{|E})), k = 0, 1, …, ⌈α⌉ − 1, then the unique solution of (1.2) is given by:

u⃗(t) = R_{α} (t)C^{−1} x⃗_{0} + ∑_{j=1}^{⌈α⌉} ∫_{0}^{t} \frac{(t − s)^{j−1}}{(j − 1)!} R_{α} (s)C^{−1} x⃗_{j−1} ds, t ⩾ 0.

Let α > 0. Then it is not difficult to prove that the formula G_{α} (t)f⃗ := E_{α} (t^{α} P̃(ξ))f⃗, t ⩾ 0, f⃗ ∈ (D′(ℝ^{n}))^{m}, determines a global (g_{α} , I)-regularized resolvent family on (D′(ℝ^{n}))^{m}, and that the integral generator of (G_{α} (t))_{t⩾0} is the multiplication operator P̃(ξ)_{|(D′(ℝ^{n} ))^{m}} ∈ L((D′(ℝ^{n}))^{m}). Furthermore, the formula

R_{α} (t)f⃗ := FE_{α} (t^{α} P̃(ξ))F^{−1} f⃗, t ⩾ 0, f⃗ ∈ (Z′_{n})^{m},

determines a global (g_{α} , I)-regularized resolvent family (R_{α} (t))_{t⩾0} on (Z′_{n})^{m}. The operator P (∂/∂x)_{|(Z′_{n} )^{m}} ∈ L((Z′_{n})^{m}) is the integral generator of (R_{α} (t))_{t⩾0}, and (G_{α} (t))_{t⩾0} as well as (R_{α} (t))_{t⩾0} can be extended to the whole complex plane. The following holds:

(i) Let α ∈ (0, ∞) ∖ ℕ and f⃗ ∈ (Z′_{n})^{m}.

(i.1) The mapping z ↦ R_{α} (z)f⃗, z ∈ ℂ ∖ (−∞, 0] is analytic.

(i.2) The mapping t ↦ R_{α} (t)f⃗, t ⩾ 0 belongs to the space C^{⌊α⌋} ([0, ∞) :

(Z′_{n})^{m}).

(i.3) For every compact set K ⊆ ℂ ∖ (−∞, 0], the family {R_{α} (z) : z ∈

K} ⊆ L((Z′_{n})^{m}) is equicontinuous.
(ii) Let α ∈ ℕ and f⃗ ∈ (Z′_{n})^{m}.
(ii.1) The mapping z ↦ R_{α} (z)f⃗, z ∈ ℂ is entire.

(ii.2) For every compact set K ⊆ ℂ, the family {R_{α} (z) : z ∈ K} ⊆

L((Z′_{n})^{m}) is equicontinuous.

Observe also that the above assertions continue to hold for (G_{α} (t))_{t⩾0} and f⃗ ∈ (D′(ℝ^{n}))^{m}, and that, for every z ∈ ℂ, R_{α} (z)Z_{n}^{m} ⊆ Z_{n}^{m} and R_{α} (z)(FE′(ℝ^{n}))^{m} ⊆ (FE′(ℝ^{n}))^{m}. This implies that (R_{α} (t)_{|Z_{n}^{m}} )_{t⩾0}, resp. (R_{α} (t)_{|(FE′(ℝ^{n} ))^{m}} )_{t⩾0}, is a locally equicontinuous (g_{α} , I)-regularized resolvent family generated by P (∂/∂x)_{|Z_{n}^{m}} , resp. P (∂/∂x)_{|(FE′(ℝ^{n} ))^{m}} .

2. Formulation and proof of main results

In the following theorem, we will transfer the assertion of [7, Theorem 1,

(a) ⇒ (b)] to abstract time-fractional equations.

Theorem 2.1. Suppose ω ⩾ 0, 0 < α ⩽ 2 and

(2.1)  sup_{z∈σ(P̃(ξ))} ℜ(z^{1/α} ) ⩽ ω.

Let E be one of the spaces listed below:

(i) E = (S(ℝ^{n}))^{m} or E = (S′(ℝ^{n}))^{m}.

(ii) E = X_{n}, where X is L^{p}(ℝ^{n}) (1 ⩽ p ⩽ ∞), C_{0}(ℝ^{n}), C_{b}(ℝ^{n}), BUC(ℝ^{n})

or C^{σ}(ℝ^{n}) (0 < σ < 1). (iii) E = {f⃗ ∈ (L^{2}(ℝ^{n}))^{m} : (P (∂/∂x))^{l} f⃗ ∈ (L^{2}(ℝ^{n}))^{m} for all l ∈ ℕ}, with the

topology induced by the following family of seminorms:

∥f⃗∥_{l} := ∥(P (∂/∂x))^{l} f⃗∥_{(L^{2} (ℝ^{n} ))^{m}} (f⃗ ∈ E, l ∈ ℕ_{0}).

Then the operator P (∂/∂x)_{|E} is the integral generator of a global (g_{α} , I)-regularized resolvent family (S_{α} (t))_{t⩾0} on E satisfying that, for every p ∈ ⊛ and ϵ > 0, there exist M ⩾ 1 and q ∈ ⊛ such that:

(2.2)  p(e^{−(ω+ϵ)t} S_{α} (t)) ⩽ Mq(x), t ⩾ 0, x ∈ E.

Proof. We will prove the theorem provided that α ∈ (0, 2) ∖ {1}. Let ϵ > 0 be fixed and let Γ_{ϵ} denote the boundary of the region {z ∈ ℂ : ℜ(z^{1/α} ) ⩽ ω + ϵ}.

I. The case 1 < α < 2. Suppose a positively oriented curve C_{ξ} encircles the spectrum of P̃(ξ) and is a subset of {z ∈ ℂ : ℜ(z^{1/α} ) ⩽ ω + ϵ/2} (ξ ∈ ℝ^{n}). Notice that, for every ξ_{0} ∈ ℝ^{n}, there exists an open neighborhood U_{ξ_{0}} of ξ_{0} such that C_{ξ_{0}} encircles the spectrum of P̃(ξ) for all ξ ∈ U_{ξ_{0}} . Let z_{0} > (ω + 2ϵ)^{α}. Using [7, Theorem II, (iii)] and the Cauchy integral formula, we obtain that there exists v ∈ ℕ such that:

E_{α} (t^{α} P̃(ξ)) = (P̃(ξ) − z_{0}I)^{2} \frac{1}{2πi} ∮_{C_{ξ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{2}} (zI − P̃(ξ))^{−1} dz

= (P̃(ξ) − z_{0}I)^{2}[a_{0}(t, ξ)I + ⋯+ a_{m−1} (t, ξ)P̃(ξ)^{m−1} ], t ⩾ 0, ξ ∈ ℝ^{n}, (2.3)

where a_{j} (t, ξ) can be written as a finite sum, with coefficients independent of t and ξ, of terms like

S_{i_{1} ,...,i_{m} ;l} (t, ξ) = λ_{1}^{i_{1}} (ξ) ⋯ λ_{m}^{i_{m}} (ξ)\frac{1}{2πi} ∮_{C_{ξ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{2}} \frac{dz}{(z − λ_{1}(ξ)) ⋯ (z − λ_{l}(ξ))} ,

where t ⩾ 0, ξ ∈ ℝ^{n}, i_{j} ∈ ℕ_{0} for 1 ⩽ j ⩽ m and i_{1} + ⋯+ i_{m} ⩽ v. By Lemma 1.1 and (2.1) (cf. also the proof of [11, Theorem 2.1]), we get that there exists M_{ϵ} > 0 such that |E_{α} (t^{α} z)| ⩽ M_{ϵ}e^{(ω+ϵ)t} for all t ⩾ 0 and z ∈ ℂ with ℜ(z^{1/α} ) ⩽ ω + ϵ/2. Since dist(Γ_{ϵ/2}, Γ_{ϵ}) := κ(ω, α, ϵ) > 0, the residue theorem implies that, for every t ⩾ 0 and ξ ∈ ℝ^{n}:

S_{i_{1} ,...,i_{m} ;l} (t, ξ) = λ_{1}^{i_{1}} (ξ) ⋯ λ_{m}^{i_{m}} (ξ)\frac{1}{2πi} ∫_{Γ_{ϵ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{2}} \frac{dz}{(z − λ_{1}(ξ)) ⋯ (z − λ_{l}(ξ))} ,

which yields the existence of a number σ > 0 such that:

|S_{i_{1} ,...,i_{m} ;l} (t, ξ)| ⩽ \frac{M_{ϵ}}{2π} (1 + |ξ|)^{σ}e^{(ω+ϵ)t} ∫_{Γ_{ϵ}} \frac{d|z|}{|z − z_{0} |^{2}} ,

provided i_{j} ∈ ℕ_{0} for 1 ⩽ j ⩽ m and i_{1} + ⋯+ i_{m} ⩽ v. In combination with (2.3), the above implies that there exist N_{ϵ} > 0 and σ_{1} > 0 such that:

(2.4)  ∥E_{α} (t^{α} P̃(ξ))∥ ⩽ N_{ϵ}(1 + |ξ|)^{σ_{1}} e^{(ω+ϵ)t} , t ⩾ 0, ξ ∈ ℝ^{n}.

Now we will prove that, for every multi-index η ∈ ℕ_{0}^{n} with |η| > 0, there exist N_{ϵ,η} > 0 and σ_{η} > 0 such that:

(2.5)  ∥D^{η}(E_{α} (t^{α} P̃(ξ)))∥ ⩽ N_{ϵ,η}(1 + |ξ|)^{σ_{η}} e^{(ω+ϵ)t} ,; t ⩾ 0, ξ ∈ ℝ^{n}.

Noticing that D^{−1} = adj(D)/ det(D) for every regular matrix D ∈ M_{m}, we obtain that there exist l_{η} ∈ ℕ and polynomials q_{ij}^{η} (ξ, z) in (n + 1) variables such that, for every ξ ∈ ℝ^{n} and z ∈ ρ(P̃(ξ)):

D^{η}(zI − P̃(ξ))^{−1} = \frac{[q_{ij}^{η} (ξ, z)]_{1⩽i,j⩽m}}{(z − λ_{1}(ξ))^{l_{η}} ⋯ (z − λ_{m}(ξ))^{l_{η}}} .

Set N_{η} := max{dg(q_{ij}^{η} (ξ, z)) : 1 ⩽ i, j ⩽ m} + 2. By the Cauchy integral formula, one has:

E_{α} (t^{α} P̃(ξ)) = (P̃(ξ)−z_{0}I)^{N_{η}} \frac{1}{2πi} ∮_{C_{ξ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{N_{η}}} (zI −P̃(ξ))^{−1} dz, t ⩾ 0, ξ ∈ ℝ^{n}.

Further on,

D^{η} \frac{1}{2πi} ∮_{C_{ξ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{N_{η}}} (zI − P̃(ξ))^{−1} dz

= \frac{1}{2πi} ∮_{C_{ξ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{N_{η}}} D^{η}(zI − P̃(ξ))^{−1} dz

= \frac{1}{2πi} ∮_{C_{ξ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{N_{η}}} \frac{[q_{ij}^{η} (ξ, z)]_{1⩽i,j⩽m} dz}{(z − λ_{1}(ξ))^{l_{η}} ⋯ (z − λ_{m}(ξ))^{l_{η}}}

= \frac{1}{2πi} ∫_{Γ_{ϵ}} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{N_{η}}} \frac{[q_{ij}^{η} (ξ, z)]_{1⩽i,j⩽m} dz}{(z − λ_{1}(ξ))^{l_{η}} ⋯ (z − λ_{m}(ξ))^{l_{η}}} (2.6)

⩽ Mκ(ω, α, ϵ)^{−ml_{η}} \frac{M_{ϵ}}{2π} e^{(ω+ϵ)t} (1 + |ξ|)^{N_{η} −2} ∫_{Γ_{ϵ}} \frac{(1 + |z|)^{N_{η} −2}}{|z − z_{0} |^{N_{η}}} d|z|, (2.7)

where (2.6) follows from the residue theorem. Using the matrix differentiation rules and (2.7), we immediately obtain (2.5).

(i): Let E = (S(ℝ^{n}))^{m}. By the invariance of E under the Fourier transform, it follows that S_{α} (t) := R_{α} (t)_{|E} ∈ L(E) for all t ⩾ 0. Having in mind [7, Theorem B] and (2.4)–(2.5), we get that, for every p ∈ ⊛, there exist M ⩾ 1 and q ∈ ⊛ such that (2.2) holds. Let x⃗ ∈ E, let t ⩾ 0 and let x⃗_{n} be a sequence in Z_{n}^{m} such that lim_{n→∞} x⃗_{n} = x⃗ in E. Suppose that p is a continuous seminorm on E. Let M ⩾ 1 and q ∈ ⊛ be such that (2.2) holds. Then p_{|Z_{n}^{m}} is a continuous seminorm on Z_{n}^{m}, and the strong continuity of (S_{α} (t))_{t⩾0} simply follows from the following estimate:

p(S_{α} (t)x⃗ − S_{α} (s)x⃗) ⩽ M(e^{(ω+ϵ)t} + e^{(ω+ϵ)s} ) + p_{|Z_{n}^{m}} (R_{α} (t)x⃗_{n} − R_{α} (s)x⃗_{n}).

Therefore, (S_{α} (t))_{t⩾0} is an exponentially equicontinuous (g_{α} , I)-regularized resolvent family generated by P (∂/∂x)_{|E}. The proof is quite similar in the case E = (S′(ℝ^{n}))^{m}.

(ii): Suppose first X ≠ C^{σ}(ℝ^{n}). Then estimates (2.4)–(2.5), taken together with the product rule and the Bernstein’s lemma [1, Lemma 8.2.1], imply that there exists a sufficiently large v ∈ ℕ such that, for given t ⩾ 0 in advance, every entry of the matrix f_{t}(ξ) ≡ [E_{α} (t^{α} P̃(ξ))(1 + |ξ|^{2})^{−v} ] belongs to A. Then it is not difficult to prove that the expression (W_{α} (t) ≡ f_{t}(−i∂/∂x_{1}, …, −i∂/∂x_{n}))_{t⩾0} (cf. the previous section for the definition of functional calculus) determines an exponentially bounded (g_{α} , (1 − Δ)^{−v} )-regularized resolvent family generated by P (∂/∂x)_{|X} . Furthermore, ∥W_{α} (t)∥_{X} = O(e^{(ω+ϵ)t} ), t ⩾ 0. By the definition of topology of E, it follows that (R_{α} (t) ≡ W_{α} (t)W_{α} (0)^{−1} )_{t⩾0} is an exponentially equicontinuous (g_{α} , I)-regularized resolvent family generated by P (∂/∂x)_{|E}, and that, for every p ∈ ⊛, there exist M ⩾ 1 and q ∈ ⊛ such that (2.2) holds. Keeping in mind the assertion [6, b), p. 374], a similar proof works in the case X = C^{σ}(ℝ^{n}) (0 < σ < 1).

(iii): Let Q be the totality of indexes q = (j_{1}, …, j_{s}) where 1 ⩽ s ⩽ m and 1 ⩽ j_{1} < ⋯< j_{s} ⩽ m. By [4, Lemma 3] (cf. also [18, Lemma 10.1]), we obtain that there exist absolute constants α_{p,q}^{k} (0 ⩽ p ⩽ m − 1, q ∈ Q, 0 ⩽ k ⩽ m − 1) such that, for every t ⩾ 0 and ξ ∈ ℝ^{n},

E_{α} (t^{α} P̃(ξ)) = (P̃(ξ) − z_{0}I)^{m+1}

× ∑_{k=0}^{m−1} ( ∑_{0⩽p⩽m−1,q∈Q} α_{p,q}^{k} \frac{1}{2πi} ∮_{C_{ξ}} \frac{z^{p}E_{α} (t^{α} z)}{(z − z_{0})^{m+1} ∏ _{j∈q} (z − λ_{j} (ξ))} dz) P̃(ξ)^{k}

= ∑_{k=0}^{m−1} ( ∑_{0⩽p⩽m−1,q∈Q} α_{p,q}^{k} \frac{1}{2πi} ∮_{C_{ξ}} \frac{z^{p}E_{α} (t^{α} z)}{(z − z_{0})^{m+1} ∏ _{j∈q} (z − λ_{j} (ξ))} dz)

× (P̃(ξ) − z_{0}I)^{m+1} P̃(ξ)^{k}.

Then one gets the existence of a number K_{ϵ} > 0 such that, for every t ⩾ 0, ξ ∈ ℝ^{n}, 0 ⩽ p ⩽ m − 1 and q ∈ Q,

∥ \frac{1}{2πi} ∮_{C_{ξ}} \frac{z^{p}E_{α} (t^{α} z)}{(z − z_{0})^{m+1} ∏ _{j∈q} (z − λ_{j} (ξ))} dz∥

= ∥ \frac{1}{2πi} ∫_{Γ_{ϵ}} \frac{z^{p}E_{α} (t^{α} z)}{(z − z_{0})^{m+1} ∏ _{j∈q} (z − λ_{j} (ξ))} dz∥

⩽ K_{ϵ}e^{(ω+ϵ)t} ∫_{Γ_{ϵ}} \frac{(1 + |z|)^{m−1}}{|z − z_{0} |^{m+1}} d|z|,

and the proof of [7, Theorem 1(v)] can be repeated verbatim.

II. The case 0 < α < 1. Although technically complicated, the proof of the theorem in this case is almost the same as the proof of the theorem in the case I. The essential change is only the passing from the integration along the curve C_{ξ}, by using the residue theorem, to the integration along Γ_{ϵ}. Put k_{0} := ⌈1/2α⌉ and suppose first that ω = 0. Then ℜ(re^{iθ}) ⩽ 0 (r > 0, θ ∈ (−π, π]) iff

θ ∈ ( ⋃_{k′=0}^{k_{0} −1} ±[α\frac{(4k′+ 1)π}{2} , α\frac{(4k′+ 3)π}{2} ]) ∩ (−π, π] =: S_{α} .

Furthermore, the set Φ_{α} := {z ∈ ℂ ∖ {0} : arg(z) ∈ (−π, π] ∖ S_{α} } has a finite number of connected components, which implies that Γ_{ϵ} = {(ϵ/ cos(θ/α))^{α} e^{iθ} : θ ∈ (−π, π] ∖ S_{α} } can be represented as a finite union of smooth curves. Set Γ_{ϵ,R} := Γ_{ϵ} ∩ {z ∈ ℂ : |z| = R} (R > 0). Then there exists M_{ϵ} > 0 such that |E_{α} (t^{α} z)| ⩽ M_{ϵ}e^{ϵt}, t ⩾ 0, z ∈ ⋃ _{R>0} (Γ_{ϵ,R})^{∘}. This implies that, for every ξ ∈ ℝ^{n}, there exists a sufficiently large R_{ξ} > 0 such that, for every R ⩾ R_{ξ}, the path of integration C_{ξ}, in any of the integrals considered in the case I, can be deformed into the curve Γ_{ϵ,R}. Now the claimed assertion follows by observing that the distance between ∂Φ_{α} and Γ_{ϵ} is positive, and that

lim_{R→∞} ∮_{z∈Γ_{ϵ,R} ,|z|=R} \frac{E_{α} (t^{α} z)}{(z − z_{0})^{2}} dz = 0.

If ω > 0, then ℜ(re^{iθ}) ⩽ ω (r > 0, θ ∈ (−π, π]) is equivalent to

θ ∈ S_{α} or (θ ∈ (−π, π] ∖ S_{α} and r ⩽ (ω/(cos(θ/α)))^{α} ),

so that the proof follows similarly as in the case ω = 0. □

Remark 2.1. (i) Let (E, ∥·∥) be a complex Banach space and let iA_{j}, 1 ⩽ j ⩽ n be commuting generators of bounded C_{0}-groups on E. For a polynomial matrix P (x) = ∑ _{|η|⩽d} P_{η}x^{η} (P_{η} ∈ M_{m}), we define P (A) ≡ ∑ _{|η|⩽d} P_{η}A^{η} with a maximal domain. Then it is well known that P (A) is closable. Suppose ω ⩾ 0 and

sup_{x∈ℝ^{n}} {ℜ(λ(x)^{1/α} ) : λ(x) ∈ σ(P (x))} ⩽ ω.

Then the proof of Theorem 2.1(I) implies that there exists a sufficiently large σ > 0 such that \overline{P (A)} is the integral generator of a global (g_{α} , (1 + |A|^{2})^{−σ} )-regularized resolvent family (S_{α} (t))_{t⩾0} on E^{m} satisfying that, for every ϵ > 0, there exists M_{ϵ} ⩾ 1 such that ||S_{α} (t)|| ⩽ M_{ϵ}e^{(ω+ϵ)t} , t ⩾ 0. Disappointingly, our method produces a completely imprecise estimate for the lower bound of σ; the additional difficulty is that the equality

D^{η}(E_{α} (t^{α} P (x))) = ∑_{j=1}^{|η|} t^{αj} E_{α}^{(j)} (t^{α} P (x))Q_{j} (x), t ⩾ 0, x ∈ ℝ^{n}, m = 1,

where Q_{j} (x) are complex polynomials of degree ⩽ Nj −|η| (1 ⩽ j ⩽ |η|), cannot be so easily interpreted in the matricial case m > 1. Using distributional techniques, the above generation result remains true, with suitable modifications, in the case that E is L^{∞} (ℝ^{n}), C_{b}(ℝ^{n}) or C^{σ}(ℝ^{n}) (0 < σ < 1).

(ii) In contrast to [7, Theorem 1], Theorem 2.1(ii) covers the case E = X_{n}, where X is L^{p}(ℝ^{n}) (p ∈ [1, ∞) ∖ {2}), C_{0}(ℝ^{n}) or C^{σ}(ℝ^{n}) (0 < σ < 1). Notice also that it is not clear how one can transfer the implication [7, Theorem 1, (b) ⇒ (a)] to abstract time-fractional equations.

Now we will prove the following extension of [3, Theorem 14.1].

Theorem 2.2. Let (E, ∥ · ∥) be a complex Banach space and let iA_{j}, 1 ⩽ j ⩽ n be commuting generators of bounded C_{0}-groups on E. Suppose α > 0 and P (x) = ∑ _{|η|⩽d} P_{η}x^{η} (P_{η} ∈ M_{m}, x ∈ ℝ^{n}) is a polynomial matrix. Then there exists an injective operator L(E) ∋ C with a dense range such that the operator \overline{P (A)} is the integral generator of a global (g_{α} , C_{m})-regularized resolvent family (W_{α} (t))_{t⩾0} on E^{m}. Furthermore, the mapping t ↦ W_{α} (t), t ⩾ 0 can be extended to the entire complex plane and the following holds:

(i) R(W_{α} (z)) ⊆ D_{∞} (P (A)), z ∈ ℂ and

(2.8)  \overline{P (A)}∫_{0}^{z} g_{α} (z − s)W_{α} (s)x⃗ds = W_{α} (z)x⃗ − C_{m} x⃗, z ∈ ℂ, x⃗ ∈ E^{m}.

(ii) The mapping z ↦ W_{α} (z), z ∈ ℂ ∖ (−∞, 0] is analytic.

(iii) The mapping z ↦ W_{α} (z), z ∈ ℂ is entire, provided that α ∈ ℕ.

Proof. Let 2|k, k > 1/α, a > 0 and C := (e^{−a|x|^{kd}} )(A). Then C ∈ L(E), C is injective and D_{∞} (A_{1}^{2} + ⋯ + A_{n}^{2}) ⊇ R(C) is dense in E (cf. [12, p. 152]). Assume that P (x)^{l} = [p_{ij;l} (x)]_{1⩽i,j⩽m} for l ∈ ℕ_{0} and x ∈ ℝ^{n}. Define

W_{α} (z)x⃗ := ∑_{l=0}^{∞} \frac{z^{αl}}{Γ(αl + 1)} P (A)^{l}C_{m} x⃗, z ∈ ℂ, x⃗ ∈ E^{m}.

Then it is checked at once that

W_{α} (z) := [ ∑_{l=0}^{∞} \frac{z^{αl}}{Γ(αl + 1)} ( p_{ij;l} (x)e^{−a|x|^{kd}} ) (A)]_{1⩽i,j⩽m} , z ∈ ℂ.

Let ϵ ∈ (0, 1) be fixed. Then there exists a constant M_{1} < ∞ such that, for every multi-index η ∈ ℕ_{0}^{n} with |η| ⩽ k_{1} ≡ 1 + ⌊n/2⌋,

|p_{ij;l}^{(η)} (x)| ⩽ M_{1}^{l}(1 + |x|)^{ld}, x ∈ ℝ^{n}, 1 ⩽ i, j ⩽ m, l ∈ ℕ_{0} and

|(e^{−a|x|^{kd}} )^{(η)}(x)| ⩽ M_{1}e^{−ϵa|x|^{kd}} , x ∈ ℝ^{n}. (2.9)

The asymptotic formula for the Gamma function combined with the choice of k implies that lim_{l→∞} Γ(\frac{2ld+n}{kd} )^{1/2l} Γ(αl + 1)^{(−1)/l} = 0 and that the mapping z ↦ ∑ _{l=0}^{∞} \frac{z^{l}}{Γ(αl+1)} Γ(\frac{2ld+n}{kd} )^{1/2} , z ∈ ℂ is entire. Furthermore, a direct computation shows that there exists a constant M_{3} < ∞ such that

(2.10) ( ∫_{ℝ^{n}} (1 + |x|)^{2ld}e^{−2ϵa|x|^{kd}} dx) ^{1/2} ⩽ M_{3}^{l}[1 + Γ( \frac{2ld + n}{kd} )^{1/2} ], l ∈ ℕ_{0}.

Taking into account (1.1) and (2.9)–(2.10), an elementary calculus involving Bernstein’s lemma, the dominated convergence theorem and the product rule, implies that there exist constants M_{2}, M_{4} < ∞ such that, for 1 ⩽ i, j ⩽ m and z ∈ ℂ,

∥∑_{l=0}^{∞} \frac{z^{αl}}{Γ(αl + 1)} ( p_{ij;l} (x)e^{−a|x|^{kd}} ) (A)∥

⩽ M_{2} ∑_{l=0}^{∞} \frac{|z|^{αl}}{Γ(αl + 1)} ∥F( p_{ij;l} (x)e^{−a|x|^{kd}} ) ∥_{L^{1} (ℝ^{n} )}

⩽ M_{2} ∑_{|η|⩽k_{1}} ∑_{l=0}^{∞} \frac{|z|^{αl}}{Γ(αl + 1)} ∥D^{η}(p_{ij;l} (x)e^{−a|x|^{kd}} )∥_{L^{2} (ℝ^{n} )}

⩽ M_{1}M_{2}M_{4} ∑_{l=0}^{∞} \frac{|z|^{αl}}{Γ(αl + 1)} M_{1}^{l}( ∫_{ℝ^{n}} (1 + |x|)^{2ld}e^{−2ϵa|x|^{kd}} dx) ^{1/2}

⩽ M_{1}M_{2}M_{4} ∑_{l=0}^{∞} \frac{|z|^{αl}}{Γ(αl + 1)} M_{1}^{l}M_{3}^{l}[1 + Γ( \frac{2ld + n}{kd} )^{1/2} ]

⩽ M_{1}M_{2}M_{4}c_{α} e^{|z|M_{1}^{1/α} M_{3}^{1/α}}

+ M_{1}M_{2}M_{4} ∑_{l=0}^{∞} \frac{(|z|^{α} M_{1}M_{3})^{l}}{Γ(αl + 1)} Γ( \frac{2ld + n}{kd} )^{1/2} < ∞.

Hence, W_{α} (z) ∈ L(E), z ∈ ℂ. It is clear that W_{α} (0) = C_{m} and that the mapping t ↦ W_{α} (t), t ⩾ 0 is strongly continuous. It is straightforward to prove that C_{m} \overline{P (A)} ⊆ \overline{P (A)}C_{m}, W_{α} (z)\overline{P (A)} ⊆ \overline{P (A)}W_{α} (z), z ∈ ℂ and W_{α} (z)C_{m} = C_{m}W_{α} (z), z ∈ ℂ. Using the dominated convergence theorem and the closedness of \overline{P (A)}, we get that:

\overline{P (A)}∫_{0}^{z} g_{α} (z − s) W_{α} (s)x⃗ ds

= \overline{P (A)}∑_{l=0}^{∞} ∫_{0}^{z} g_{α} (z − s)g_{αl+1} (s)\overline{P (A)}^{l}C_{m} x⃗ ds

= \overline{P (A)}∑_{l=0}^{∞} g_{α(1+l)+1} (z)\overline{P (A)}^{l}C_{m} x⃗

= ∑_{l=0}^{∞} g_{α(1+l)+1} (z)\overline{P (A)}^{l+1} C_{m} x⃗

= W_{α} (z)x⃗ − C_{m} x⃗, z ∈ ℂ, x⃗ ∈ E^{m}.

Therefore, (W_{α} (t))_{t⩾0} is a global (g_{α} , C_{m})-regularized resolvent family which does have \overline{P (A)} as a subgenerator. Furthermore, (2.8) holds and C_{m}^{−1} \overline{P (A)}C_{m} = \overline{P (A)} [21], which implies that \overline{P (A)} is, in fact, the integral generator of (W_{α} (t))_{t⩾0}. This completes the proof of (i). The proofs of (ii) and (iii) are left to the reader. □

Remark 2.2. (i) If m = 1, p_{11}(x) = ∑ _{|α|⩽d} a_{α} x^{α}, x ∈ ℝ^{n} (a_{α} ∈ ℂ) and

E is a function space on which translations are uniformly bounded and strongly continuous (for example, L^{p}(ℝ^{n}) with p ∈ [1, ∞), C_{0}(ℝ^{n}) or BUC(ℝ^{n}); notice also that E can be composed of functions defined on some bounded domain [3, 12, 21]), then the natural choice for A_{j} is i∂/∂x_{j} (1 ⩽ j ⩽ n). In this case, \overline{P (A)} is just the operator ∑ _{|α|⩽d} a_{α} i^{|α|} (∂/∂x)^{α} with its maximal distributional domain. By Theorem 2.2, we infer that for each α > 0 there exists a dense subset E_{0,α} of L^{p}(ℝ^{n}) such that the abstract Cauchy problem:

D_{t}^{α} u(t, x) = ∑_{|α|⩽d} a_{α} i^{|α|} (∂/∂x)^{α} u(t, x), t > 0, x ∈ ℝ^{n};

\frac{∂^{l}}{∂t^{l}} u(t, x)_{|t=0} = f_{l}(x), x ∈ ℝ^{n}, l = 0, 1, …, ⌈α⌉ − 1,

has a unique solution provided f_{l}(·) ∈ E_{0,α} , l = 0, 1, …, ⌈α⌉−1. A similar assertion can be proved in the case that E is L^{∞} (ℝ^{n}), C_{b}(ℝ^{n}) or C^{σ}(ℝ^{n}) (0 < σ < 1).

(ii) The results stated in Remark 2.1(i), Theorem 2.2, and the first part of this remark, can be reformulated for (systems of) abstract time-fractional equations considered in E_{l}-type spaces.

We close the paper with the following theorem.

Theorem 2.3. (i) Suppose α > 0 and X is S(ℝ^{n}) or S′(ℝ^{n}). Then there exists an injective operator C ∈ L(X) with dense range such that the operator P (∂/∂x)_{|E} is the integral generator of a global (g_{α} , C_{m})-regularized resolvent family (W_{α} (t))_{t⩾0} on E ≡ X^{m}. Furthermore, the mapping t ↦ W_{α} (t), t ⩾ 0 can be extended to the entire complex plane and the properties (i)–(ii) stated in the first section remain true with R_{α} (·) and (Z′_{n})^{m} replaced by W_{α} (·) and E, respectively.

(ii) Suppose α > 0, X is L^{2}(ℝ^{n}) and E = {f⃗ ∈ (L^{2}(ℝ^{n}))^{m} : (P (∂/∂x))^{l} f⃗ ∈ (L^{2}(ℝ^{n}))^{m} for all l ∈ ℕ}. Then there exists an injective operator C ∈ L(X) such that the operator P (∂/∂x)_{|E} is the integral generator of a global (g_{α} , C_{m}_{|E})- regularized resolvent family (W_{α} (t))_{t⩾0} on E. Furthermore, R(C_{m}_{|E}) is dense in E, the mapping t ↦ W_{α} (t), t ⩾ 0 can be extended to the entire complex plane and the properties (i)–(ii) stated in the first section remain true with R_{α} (·) and (Z′_{n})^{m} replaced by W_{α} (·) and E, respectively.

Proof. Suppose first that E = (S(ℝ^{n}))^{m}. Let a > 0, 2|k and k > 1/α. Define

W_{α} (z) := FE_{α} (z^{α} P̃(ξ))e^{−a|ξ|^{kd}} F^{−1} , z ∈ ℂ

and Cf := Fe^{−a|ξ|^{kd}} F^{−1} f, f ∈ S(ℝ^{n}). Let P̃(ξ)^{l} = [p_{ij;l} (ξ)]_{1⩽i,j⩽m} (l ∈ ℕ_{0}, ξ ∈ ℝ^{n}). Then it is obvious that:

W_{α} (z)f⃗ = [ ∑_{l=0}^{∞} \frac{z^{αl}}{Γ(αl + 1)} F( p_{ij;l} (ξ)e^{−a|ξ|^{kd}} ) F^{−1} ]_{1⩽i,j⩽m} f⃗, z ∈ ℂ, f⃗ ∈ E.

Since F and F^{−1} are topological isomorphisms of S(ℝ^{n}), we immediately obtain that L(X) ∋ C is injective. Clearly, R(C) is dense in X and L(E) ∋ W_{α} (0) = C_{m} is injective. In order to prove that W_{α} (z) ∈ L(E) for every z ∈ ℂ, it suffices to show (cf. [16, Theorem 8.19-8.21]) that, for every multi-index η ∈ ℕ_{0}^{n}, there exist M_{η} ⩾ 1 and N_{η} ∈ ℕ such that:

(2.11)  ∥D^{η}[E_{α} (z^{α} P̃(ξ))e^{−a|ξ|^{kd}} ]∥ ⩽ M_{η}(1 + |ξ|)^{N_{η}} , ξ ∈ ℝ^{n}.

It can be easily seen that there exists M_{1} ⩾ 1 such that, for every η ∈ ℕ_{0}^{n}, one has |p_{ij;l}^{(η)} (ξ)| ⩽ M_{1}^{l}(1 + |ξ|)^{ld}, ξ ∈ ℝ^{n}, 1 ⩽ i, j ⩽ m, l ∈ ℕ_{0}, which implies by (1.1) that, for every z ∈ ℂ, ξ ∈ ℝ^{n} and l ∈ ℕ_{0}:

∥E_{α} (z^{α} P̃(ξ))∥ ⩽ E_{α} (M_{1}|z|^{α} (1 + |ξ|)^{d}) ⩽ c_{α} e^{M_{1}^{1/α} |z|(1+|ξ|)^{d/α}} .

Further on,

∥ \frac{∂}{∂ξ_{j}} E_{α} (z^{α} P̃(ξ))∥

= ∥∑_{l=1}^{∞} \frac{z^{αl}}{Γ(αl + 1)} [P̃(ξ)^{l−1} ( \frac{∂}{∂ξ_{j}} P̃(ξ)) + ⋯+ ( \frac{∂}{∂ξ_{j}} P̃(ξ)) P̃(ξ)^{l−1} ]∥

⩽ ∑_{l=1}^{∞} \frac{|z|^{αl}}{Γ(αl + 1)} lM_{1}^{l−1} (1 + |ξ|)^{d(l−1)}

⩽ ∑_{l=1}^{∞} \frac{|z|^{αl}}{Γ(αl + 1)} (2M_{1})^{l}(1 + |ξ|)^{dl}

⩽ c_{α} e^{|z|(2M_{1} )^{1/α} (1+|ξ|)^{d/α}} , z ∈ ℂ, ξ ∈ ℝ^{n}, 1 ⩽ j ⩽ n.

Continuing in this way, we obtain that, for every η ∈ ℕ_{0}^{n}, there exists b_{η} ⩾ 1 such that:

∥D^{η}E_{α} (z^{α} P̃(ξ))∥ ⩽ ∑_{l=1}^{∞} \frac{|z|^{αl}}{Γ(αl + 1)} (b_{η}M_{1})^{l}(1 + |ξ|)^{dl}
⩽ c_{α} e^{|z|b_{η}^{1/α} M_{1}^{1/α} (1+|ξ|)^{d/α}} , z ∈ ℂ, ξ ∈ ℝ^{n}.

Taken together with the product rule and (2.9), the last estimate immediately implies (2.11). The strong continuity of (W_{α} (t))_{t⩾0} follows form the estimate |t^{αl} − s^{αl} | ⩽ l|t^{α} −s^{α} | max(t, s)^{l−1} , t, s ⩾ 0 and the previously given arguments. Applying twice the Darboux inequality, one yields that, for every z ∈ ℂ∖(−∞, 0], there exists a constant κ(z) ⩾ 1 such that:

(2.12)  | \frac{(z + h)^{αl} − z^{αl}}{h} − αlz^{αl−1} | ⩽ |h|κ(z)^{αl} ,

for any h ∈ ℂ with |h| < dist(z, (−∞, 0]). Using (2.12), it readily follows that the mapping z ↦ W_{α} (z)f⃗, z ∈ ℂ ∖ (−∞, 0] is analytic for every fixed f⃗ ∈ E and that, for every z ∈ ℂ ∖ (−∞, 0] and f⃗ ∈ E:

\frac{d}{dz}W_{α} (z)f⃗ = [ ∑_{l=1}^{∞} \frac{αlz^{αl−1}}{Γ(αl + 1)} F( p_{ij;l} (ξ)e^{−a|ξ|^{kd}} ) F^{−1} ]_{1⩽i,j⩽m} f⃗.

One obtains similarly that, for every f⃗ ∈ E, the mapping z ↦ W_{α} (z)f⃗, z ∈ ℂ is entire, provided α ∈ ℕ, and that the mapping t ↦ W_{α} (t)f⃗, t ⩾ 0 is in C^{⌊α⌋} ([0, ∞) : E) . The rest of the proof in the case E = (S(ℝ^{n}))^{m} is simple. The proof of the theorem in the case E = (S′(ℝ^{n}))^{m} is similar and therefore omitted.

To prove (ii), set W_{α} (z)f⃗ := FE_{α} (z^{α} P̃(ξ))e^{−a|ξ|^{kd}} F^{−1} f⃗, z ∈ ℂ, f⃗ ∈ (L^{2}(ℝ^{n}))^{m} and Cf := Fe^{−a|ξ|^{kd}} F^{−1} f, f ∈ L^{2}(ℝ^{n}), where a > 0, 2|k and k > 1/α. Certainly, R(C_{m}_{|E}) is dense in E and there exists c > 0 such that:

max_{1⩽j⩽m} |λ_{j} (ξ)| ⩽ c(1 + |ξ|)^{d}, ξ ∈ ℝ^{n}.

Keeping in mind the proofs of [7, Theorem 1(v)] and Theorem 2.1(iii), it suffices to show that there exists a non-negative function t ↦ f(t), t ⩾ 0 such that, for every p ∈ {0, …, m − 1}, q ∈ Q, t ⩾ 0 and ξ ∈ ℝ^{n}:

(2.13)  ∥ \frac{1}{2πi} ∮_{C_{ξ}} \frac{z^{p}E_{α} (t^{α} z)}{(z − 3c(1 + |ξ|)^{d})^{m+1} ∏ _{j∈q} (z − λ_{j} (ξ))} e^{−a|ξ|^{kd}} dz∥ ⩽ f(t).

Using the Cauchy theorem and (1.1), we easily infer that there exists μ > 0 such that:

∥ \frac{1}{2πi} ∮_{C_{ξ}} \frac{z^{p}E_{α} (t^{α} z)}{(z − 3c(1 + |ξ|)^{d})^{m+1} ∏ _{j∈q} (z − λ_{j} (ξ))} e^{−a|ξ|^{kd}} dz∥

= ∥ \frac{1}{2πi} ∮_{|z|=2c(1+|ξ|)^{d}} \frac{z^{p}E_{α} (t^{α} z)}{(z − 3c(1 + |ξ|)^{d})^{m+1} ∏ _{j∈q} (z − λ_{j} (ξ))} e^{−a|ξ|^{kd}} dz∥

⩽ μe^{μt} + e^{μt|ξ|^{d/α} −a|ξ|^{kd}} , t ⩾ 0, ξ ∈ ℝ^{n}.

Now the existence of function t ↦ f(t), t ⩾ 0 satisfying (2.13) follows from the choice of k. The proof of the theorem is completed. □

References

1. W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, 2001.

2. E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, PhD Thesis, Eindhoven University of Technology, Eindhoven, 2001.

3. R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lect. Notes Math. 1570, Springer, New York, 1994.

4. E. A. Gorin, On quadratic integrability of solution of partial differential equations with constant coefficients, Siberian Math. J. 2 (1961), 221–232 [in Russian].

5. M. Hieber, Integrated semigroups and differential operators on L^{p} spaces, Math. Ann. 291 (1991), 1–16.

6. M. Hieber, A. Holderrieth, F. Neubrander, Regularized semigroups and systems of linear partial differential equations, Ann. Scuola Norm. Sup. Pisa 19 (1992), 363–379.

7. J. Kisyński, The Petrovskiĭ correctness and semigroups of operators, preprint, 2009.

8. M. Kostić, (a, k)-regularized C-resolvent families: regularity and local properties, Abstr. Appl. Anal. vol. 2009, Art. ID 858242, 27 pages, 2009.

9. , Abstract Volterra equations in locally convex spaces, Science China Math. 55 (2012), 1797–1825.

10. , Generalized Semigroups and Cosine Functions, Mathematical Institute Belgrade, 2011.

11. , Abstract differential operators generating fractional resolvent families, Acta Math. Sinica, to appear.

12. Y. Lei, W. Yi, Q. Zheng, Semigroups of operators and polynomials of generators of bounded strongly continuous groups, Proc. London Math. Soc. 69 (1994), 144–170.

13. F.-B. Li, M. Li, Q. Zheng, Fractional evolution equations governed by coercive differential operators, Abstr. Appl. Anal. 2009, Art. ID 438690, 14 pp. 34G10.

14. M. Li, Q. Zheng, J. Zhang, Regularized resolvent families, Taiwanese J. Math. 11 (2007), 117–133.

15. C. Martinez, M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Stud., Amsterdam, 2001.

16. S. Pilipović, B. Stanković, Prostori Distribucija, SANU, Novi Sad, 2000.

17. J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993.

18. T. Ushijima, On the generation and smoothness of semi-groups of linear operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 (1972), 65–127; Corrections 20 (1973), 187–189.

19. R. Wong, Y.-Q. Zhao, Exponential asymptotics of the Mittag-Leffler function, Constr. Approx. 18 (2002), 355–385.

20. T.-J. Xiao, J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, Springer–Verlag, Berlin, 1998.

21. Q. Zheng, Y. Li, Abstract parabolic systems and regularized semigroups, Pacific J. Math. 182 (1998), 183–199.

Faculty of Technical Sciences (Received 12 07 2011) University of Novi Sad 21125 Novi Sad Serbia

    marco.s@verat.net