PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 49–75 DOI: 10.2298/PIM1614049K

DEGENERATE MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS IN LOCALLY CONVEX SPACES

Marko Kostić

Abstract. We investigate, in the setting of sequentially complete locally convex spaces, degenerate multi-term fractional differential equations with Caputo derivatives. The obtained theoretical results are illustrated with some examples.

1. Introduction and preliminaries

Let n ∈ ℕ∖{1}, 0 ⩽ α_{1} < ⋯< α_{n}, let A_{1}, …, A_{n−1} be closed linear operators on a Hausdorff sequentially complete locally convex space E, and let f: [0, ∞) → E be a continuous function. The well-posedness of the following multi-term fractional differential equation has been analyzed in a series of recent papers (cf. [20, Section 2.10] for an extensive survey of results on abstract multi-term fractional differential equations with Caputo fractional derivatives)

D_{t}^{α_{n}} u(t) + ∑_{i=1}^{n−1} A_{i}D_{t}^{α_{i}} u(t) = f(t), t ⩾ 0; u^{(j)} (0) = u_{j} , j = 0, …, ⌈α_{n}⌉ − 1.

Define m_{i} := ⌈α_{i}⌉, i ∈ ℕ_{n−1} , T_{i,L} u(t) := A_{i}D_{t}^{α_{i}} u(t), if t ⩾ 0, i ∈ ℕ_{n−1} and α_{i} > 0, and T_{i,R} u(t) := D_{t}^{α_{i}} A_{i}u(t), if t ⩾ 0 and i ∈ ℕ_{n−1} . Henceforth it will be assumed that, for every t ⩾ 0 and i ∈ ℕ_{n−1} , T_{i}u(t) denotes either T_{i,L} u(t) or T_{i,R} u(t). In this paper, we will consider the following degenerate abstract multi-term problem:

(1.1)  ∑_{i=1}^{n−1} T_{i}u(t) = f(t), t ⩾ 0.

2010 Mathematics Subject Classification: Primary 35R11, 45D05; Secondary 45N05, 47D99.

Key words and phrases: abstract multi-term fractional differential equations, Caputo fractional derivatives, degenerate (a, k)-regularized C-resolvent families, well-posedness, locally convex spaces.

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

Communicated by Stevan Pilipović.

In such a way, we will continue our previous research studies [22, 27], where we have looked at generation of degenerate fractional resolvent operator families and hypercyclic properties of degenerate multi-term fractional differential equations.

In order to subject initial conditions to equation (1.1), we shall follow the approach from [27]. First of all, assume that α > 0 and m = ⌈α⌉. Let us recall that the Caputo fractional derivative D_{t}^{α} u [6, 20] is defined for those functions u ∈ C^{m−1} ([0, ∞) : E) for which g_{m−α} ∗(u−∑ _{j=0}^{m−1} u^{(j)} (0)g_{j+1} ) ∈ C^{m}([0, ∞) : E), by

D_{t}^{α} u(·) = \frac{d^{m}}{dt^{m}} [g_{m−α} ∗ ( u − ∑_{j=0}^{m−1} u^{(j)} (0)g_{j+1} ) ].

If α ∈ ℕ, then the Caputo fractional derivative D_{t}^{α} u(·) is defined iff u ∈ C^{m}([0, ∞) : E); in this case, D_{t}^{α} u(·) = (d^{α} /dt^{α})u(·) ∈ C([0, ∞) : E). The following facts can be proved by using the equality [6, (1.21)], induction and closedness of A:

(a) Suppose that l ∈ ℕ and u, Au ∈ C^{l}([0, ∞) : E). Then u^{(j)} (t) ∈ D(A), t ⩾ 0 and Au^{(j)} (t) = (Au)^{(j)} (t), t ⩾ 0 (0 ⩽ j ⩽ l).

(b) Suppose that the Caputo fractional derivatives D_{t}^{α} u and D_{t}^{α} Au are defined. Then u^{(j)} (t) ∈ D(A), t ⩾ 0, Au^{(j)} (t) = (Au)^{(j)} (t), t ⩾ 0 (0 ⩽ j ⩽ m − 1), D_{t}^{α} u(t) ∈ D(A), t ⩾ 0, AD_{t}^{α} u(t) = D_{t}^{α} Au(t), t ⩾ 0, and

J_{t}^{α} AD_{t}^{α} u(t) = J_{t}^{α} D_{t}^{α} Au(t) = Au(t) − ∑_{j=0}^{m−1} Au^{(j)} (0)g_{j+1} (t), t ⩾ 0,

where J_{t}^{α} u(t) := (g_{α} ∗ u)(t), t ⩾ 0. Set P_{λ} := ∑ _{i=1}^{n−1} λ^{α_{i} −α_{n−1}} A_{i}, λ ∈ ℂ ∖ {0}, I := {i ∈ ℕ_{n−1} : α_{i} > 0 and T_{i,L} u(t) appears on the left hand side of (1.1)}, Q := maxI, if I ≠ ∅ and Q := m_{Q} := 0, if I = ∅. We will subject the following initial conditions to the equation (1.1), cf. (a)–(b):

(1.2) u^{(j)} (0) = u_{j} , 0 ⩽ j ⩽ m_{Q} −1 and (A_{i}u)^{(j)} (0) = u_{i,j} if m_{i} −1 ⩾ j ⩾ m_{Q}.

If T_{n−1} u(t) = T_{n−1,L} u(t), then (1.2) reads u^{(j)} (0) = u_{j}, 0 ⩽ j ⩽ m_{n−1} − 1. If this is not the case, then choice (1.2) may be nonoptimal, the index i ∈ ℕ_{n−1} has to satisfy the inequality m_{i} − 1 ⩾ m_{Q} in the second equality appearing in (1.2), and we cannot expect the existence of solutions of problem (1.1)–(1.2), in general (consider, for example, the case n = 3, A_{2} = A_{1} and u_{2,0} ≠ u_{1,0}); furthermore, for any index i ∈ ℕ_{n−1} satisfying the inequality m_{i}−1 ⩾ m_{Q} and for every nonnegative integer k ∈ [m_{Q}, m_{i} − 1], we need to introduce exactly one initial value u_{i,k} .

The most important subcases of problem (1.1))–(1.2) are the following fractional Sobolev degenerate equations:

(DFP)_{R} : \{ \begin{matrix}D_{t}^{α} Bu(t) = Au(t) + f(t), & t ⩾ 0, \\ Bu(0) = Bx; (Bu)^{(j)} (0) = 0, & 1 ⩽ j ⩽ ⌈α⌉ − 1, \end{matrix}

and

(DFP)_{L} : \{ \begin{matrix}BD_{t}^{α} u(t) = Au(t) + f(t), & t ⩾ 0, \\ u(0) = x; u^{(j)} (0) = 0, & 1 ⩽ j ⩽ ⌈α⌉ − 1, \end{matrix}

where α > 0. For further information concerning the wellposedness of Sobolev first order degenerate equations, the reader may consult the monographs by Favini, Yagi [9], Krein [31], Carroll, Showalter [7], Melnikova, Filinkov [34] and Sviridyuk, Fedorov [45], as well as the papers [1,10–14,35,50,51,55]. The well-posedness of various types of degenerate Sobolev equations of second order have been analyzed in [2,4,7,9,15,22,36,44,52,56]. The corresponding results on degenerate Sobolev equations with integer higher-order derivatives can be found in [3], [9, Section 5.7], [45–49, 56].

For the purpose of study of abstract multi-term problem (1.1)–(1.2), we introduce the classes of exponentially equicontinuous k-regularized C-resolvent (i, j)- propagation families (Subsection 2.1) and exponentially equicontinuous (k; C)-reg-ularized resolvent (i, j)-propagation families (Subsection 2.2). We investigate subordination principles, regularity properties, existence and uniqueness of solutions of problem (1.1)–(1.2) and its integral analogons; besides this, we clarify some results on the C-wellposedness of the equation (DFP)_{L} in Subsection 2.2 (cf. Abdelaziz, Neubrander [1] for the case α = 1). In a follow-up research [25], we are going to analyze various types of degenerate Volterra integro-differential equations by using results from the theory of multivalued linear operators [9, 34].

Before we explain the notation used throughout the paper, it should be noticed that we take under consideration some equations that are valuable only from the mathematical point of view and do not have any physical significance, for now at least.

Unless specifed otherwise, we shall always assume that E is a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short; the abbreviation ⊛ stands for the fundamental system of seminorms which defines the topology of E. If X is also an SCLCS over the field of complex numbers, then we denote by L(E, X) the space of all continuous linear mappings from E into X; L(E) ≡ L(E, E). By E^{∗} we denote the dual space of E; if E is a Banach space, then we denote by ∥x∥ the norm of an element x ∈ E. If A is a linear operator acting on E, then the domain and range of A will be denoted by D(A) and R(A), respectively. Since no confusion seems likely, we will identify A with its graph. Given s ∈ ℝ in advance, set ⌈s⌉ := inf{l ∈ ℤ : s ⩽ l}. Define Σ_{α} := {z ∈ ℂ ∖ {0} : | arg(z)| < α} (α ∈ (0, π]). By AC_{loc}([0, ∞)) we denote the space consisting of all complex valued functions that are absolutely continuous on any finite interval [0, T] (T > 0). The Gamma function is denoted by Γ(·) and the principal branch is always used to take the powers; the convolution like mapping ∗ is given by f ∗ g(t) := ∫_{0}^{t} f(t − s)g(s) ds. Set g_{ζ}(t) := t^{ζ−1} /Γ(ζ), 0^{ζ} := 0 (ζ > 0, t > 0), ℕ_{l} := {1, …, l}, ℕ_{l}^{0} := {0, 1, …, l} (l ∈ ℕ) and g_{0}(t) := the Dirac δ-distribution.

Fairly complete information about fractional calculus and fractional differential equations can be obtained by consulting [6, 8, 16–22, 27, 28, 32, 39, 40]. In the sequel, we shall use the following fact about Caputo fractional derivatives: Assume that α > 0, m = ⌈α⌉, β ∈ (0, α) and the Caputo fractional derivative D_{t}^{α} u(·) is defined. Then we know that the Caputo fractional derivative D_{t}^{β}u(·) is also defined and the following equality holds:

(1.3)  D_{t}^{β}u(t) = (g_{α−β} ∗ D_{t}^{α} u(·))(t) + ∑_{j=⌈β⌉}^{m−1} u^{(j)} (0)g_{j+1−β} (t), t ⩾ 0.

The Mittag-Leffler function E_{α,β} (z) is defined by

E_{α,β} (z) := ∑_{n=0}^{∞} \frac{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. The asymptotic behaviour of the entire function E_{α,β} (z) is given in the following auxiliary lemma; cf. also [32, 42] and the formulae [6, (1.27)–(1.28)] for the case 0 < α < 2, which is the most important in our analysis.

Lemma 1.1. [53] 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 | arg(z) + 2πs| < α(\frac{π}{2} + σ).

Throughout the paper, we shall always assume that the function k(t) is a scalarvalued continuous kernel on [0, ∞). The following conditions on function k(t) will be used occasionally:

(P1): k(t) is Laplace transformable, i.e., it is locally integrable on [0, ∞) and there exists β ∈ ℝ such that k̃(λ) := L(k)(λ) := lim_{b→∞} ∫_{0}^{b}e^{−λt} k(t)dt :=

∫_{0}^{∞} e^{−λt} k(t)dt exists for all λ ∈ ℂ with Reλ > β. Put abs(k) :=inf{Reλ : k̃(λ) exists}, δ̃(λ) := 1 and denote by L^{−1} the inverse Laplace transform.

(P2): k(t) satisfies (P1) and k̃(λ) ≠ 0, Reλ > β for some β ⩾ abs(k). Let γ ∈ (0, 1) and ω ∈ ℝ. Recall that the Wright function Φ_{γ}(·) is defined by Φ_{γ}(t) := L^{−1} (E_{γ}(−λ))(t), t ⩾ 0. Following [54, Definition 1.1.3], we say that a function h : (ω, ∞) → E belongs to the class LT − E iff there exists a function f ∈ C([0, ∞) : E) such that for each p ∈ ⊛ there exists M_{p} > 0 satisfying p(f(t)) ⩽ M_{p}e^{ωt}, t ⩾ 0 and h(λ) = ∫_{0}^{∞} e^{−λt} f(t) dt, λ > ω; if this is the case, then we know that the function λ ↦ h(λ), λ > ω can be analytically extended to the right half plane {λ ∈ ℂ : Reλ > ω}.

The reader may consult [54] and [20] for further information concerning the Laplace transform of functions with values in sequentially complete locally convex spaces (cf. [5] for the Banach space case). In the sequel, we shall use the following uniqueness type theorem for the Laplace transform.

Lemma 1.2. Let ω ⩾ 0, and let f_{1}, f_{2} ∈ C([0, ∞) : E) satisfy that for each p ∈ ⊛ there exists M′_{p} > 0 such that p(f_{1}(t)) + p(f_{2}(t)) ⩽ M′_{p}e^{ωt}, t ⩾ 0. Suppose that a continuous function g: [0, ∞) → ℂ satisfies (P1) and that A is a closed linear operator on E satisfying that for λ > a,

∫_{0}^{∞} e^{−λt} f_{1}(t)dt ∈ D(A),
A ∫_{0}^{∞} e^{−λt} f_{1}(t)dt = ∫_{0}^{∞} e^{−λt} f_{2}(t)dt + ∫_{0}^{∞} e^{−λt} g(t)x dt, λ > a,

for some x ∈ E. Then, for every t ⩾ 0, one has f_{1}(t) ∈ D(A) and Af_{1}(t) = f_{2}(t) + g(t)x.

In [22], we have recently considered the C-wellposedness of the following degenerate abstract Volterra equation:

(1.4)  Bu(t) = f(t) + ∫_{0}^{t} a(t − s)Au(s)ds, t ⩾ 0,

where t ↦ f(t), t ⩾ 0 is a continuous E-valued mapping, a ∈ L_{loc}^{1}([0, ∞)) and A, B are closed linear operators with domain and range contained in E (cf. also [11–13] and [18]). Following Xiao and Liang [55, 56], we have introduced in [22] the class of exponentially equicontinuous (a, k)-regularized C-resolvent families for (1.4) as follows.

Definition 1.1. [22] Suppose that the functions a(t) and k(t) satisfy (P1), as well as that R(t): D(B) → E is a linear mapping (t ⩾ 0). Let C ∈ L(E) be injective. Then the operator family (R(t))_{t⩾0} is said to be an exponentially equicontinuous (a, k)-regularized C-resolvent family for (1.4) iff there exists ω ⩾ max(0, abs(a), abs(k)) such that the following holds:

(i) The mapping t ↦ R(t)x, t ⩾ 0 is continuous for every fixed element x ∈ D(B). (ii) The family {e^{−ωt} R(t) : t ⩾ 0} is equicontinuous, i.e., for every p ∈ ⊛, there

exist c > 0 and q ∈ ⊛ such that p(e^{−ωt} R(t)x) ⩽ cq(x), x ∈ D(B), t ⩾ 0. (iii) For every λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0, the operator B − ã(λ)A is

injective, C(R(B)) ⊆ R(B − ã(λ)A) and

k̃(λ)(B − ã(λ)A)^{−1} CBx = ∫_{0}^{∞} e^{−λt} R(t)x dt, x ∈ D(B).

If k(t) = g_{r+1} (t) for some r ⩾ 0, then it is also said that (R(t))_{t⩾0} is an exponentially equicontinuous r-times integrated (a, C)-regularized resolvent family for (1.4); an exponentially equicontinuous 0-times integrated (a, C)-regularized resolvent family for (1.4) is also said to be an exponentially equicontinuous (a, C)- regularized resolvent family for (1.4).

For further information concerning the applications of (a, k)-regularized Cresolvent families to non-degenerate Volterra integro-differential equations (cf. Definition 1.1 with B = I), the reader may consult the monograph [20]. For abstract non-scalar Volterra equations, we refer the reader to [25, 29, 30, 40].

2. Degenerate k-regularized C-resolvent propagation families for problem (1.1)–(1.2)

Let us recall that n ∈ ℕ ∖ {1}, 0 ⩽ α_{1} < ⋯< α_{n−1} , as well as that A_{1}, …, A_{n−1} are closed linear operators on E and that f: [0, ∞) → E is a continuous function. Let the set I and number Q be defined as above.

Suppose, for the time being, that the initial values u_{j} ∈ E (0 ⩽ j ⩽ m_{Q} − 1) satisfy u_{j} ∈ D(A_{i}), provided i ∈ ℕ_{n−1} , T_{i}u(t) = T_{i,R} u(t) and 0 ⩽ j ⩽ m_{i} − 1 (put u_{i,j} := A_{i}u_{j} in this case), and u_{i,j} ∈ E, provided i ∈ ℕ_{n−1} , T_{i}u(t) = T_{i,R} u(t) and m_{i} − 1 ⩾ j ⩾ m_{Q}. We start this section by introducing the notion of a strong solution of problem (1.1)–(1.2).

Definition 2.1. A function u ∈ C([0, ∞) : E) is said to be a strong solution of problem (1.1)–(1.2) iff the term T_{i}u(t) is well defined and continuous for any t ⩾ 0, i ∈ ℕ_{n−1} , and (1.1)–(1.2) holds identically on [0, ∞).

Now we would like to observe the following fact. If Q > 0, then we can consider the problem obtained from problem (1.1) by replacing some of the terms T_{i,R} (t), for 1 ⩽ i ⩽ Q, with the corresponding terms of form T_{i,L} (t). By (1.3) and (b), it readily follows that a strong solution of problem (1.1)–(1.2)) is also a strong solution of the problem described above, when endowed with initial conditions (1.2). It is also worth noting that we have considered in [27, Remark 11(i)] problem (1.1) endowed with slightly different initial conditions, and that the existence of strong solutions of certain classes of multi-term problems with hypercyclic behaviour has been investigated in [27, Theorem 10] (cf. [27, Example 13] for an interesting application of the above-mentioned theorem involving the Ornstein-Uhlenbeck operators on L^{2}-type spaces).

Define now, for every i ∈ ℕ_{n−1} and t ⩾ 0,

T_{i,L} u(t) := g_{α_{n−1} −α_{i}} ∗ A_{i}[u(·) − ∑_{j=0}^{m_{i} −1} u_{j} g_{j+1} (·)](t), if T_{i}u(t) = T_{i,L} u(t), (2.1)

T_{i,R} u(t) := g_{α_{n−1} −α_{i}} ∗ [A_{i}u(·) − ∑_{j=0}^{m_{i} −1} u_{i,j} g_{j+1} (·)](t), if T_{i}u(t) = T_{i,R} u(t). (2.2)

Let T_{i}u(t) denote, as before, exactly one of the terms T_{i,L} u(t) or T_{i,R} u(t). Integrating equation (1.1) α_{n−1} times, the foregoing arguments imply that any strong solution t ↦ u(t), t ⩾ 0 of problem 1.1–(1.2) satisfies the following integral equation

(2.3)  ∑_{i=1}^{n−1} T_{i}u(t) = (g_{α_{n−1}} ∗ f)(t), t ⩾ 0.

This motivates us to introduce the following definition.

Definition 2.2. Let u_{j} ∈ E (0 ⩽ j ⩽ m_{Q} −1), let u_{i,j} ∈ E, provided i ∈ ℕ_{n−1} ,

T_{i}u(t) = T_{i,R} u(t) and 0 ⩽ j ⩽ m_{i} − 1, and let V ⊆ ℕ_{n−1} . Then a continuous Evalued function t ↦ u(t), t ⩾ 0 is said to be a V-mild solution of (2.3) iff (i)–(v) hold where (i) g_{α_{n−1} −α_{i}} ∗ [u(·) − ∑ _{j=0}^{m_{i} −1} u_{j} g_{j+1} (·)](t) ∈ D(A_{i}) for all t ⩾ 0 and i ∈ I ∩ V, the

mapping t ↦ A_{i}(g_{α_{n−1} −α_{i}} ∗ [u − ∑ _{j=0}^{m_{i} −1} u_{j} g_{j+1} ])(t), t ⩾ 0 is well-defined and continuous for all i ∈ I ∩ V,

(ii) the mapping t ↦ (g_{α_{n−1} −α_{i}} ∗ A_{i}[u − ∑ _{j=0}^{m_{i} −1} u_{j} g_{j+1} ])(t), t ⩾ 0 is continuous

for all i ∈ I ∖ V, (iii) (g_{α_{n−1} −α_{i}} ∗ u)(t) ∈ D(A_{i}) for all t ⩾ 0 and i ∈ (ℕ_{n−1} ∖ I) ∩ V, the mapping

t ↦ A_{i}(g_{α_{n−1} −α_{i}} ∗ u)(t), t ⩾ 0 is continuous for all i ∈ (ℕ_{n−1} ∖ I) ∩ V, (iv) the mapping t ↦ (g_{α_{n−1} −α_{i}} ∗ A_{i}u)(t), t ⩾ 0 is well-defined and continuous for

all i ∈ (ℕ_{n−1} ∖ I) ∖ V,

(v) for every t ⩾ 0, the following holds:

∑_{i∈I∩V} A_{i}( g_{α_{n−1} −α_{i}} ∗ [u(·) − ∑_{j=0}^{m_{i} −1} u_{j} g_{j+1} (·)]) (t)

+ ∑_{i∈I∖V} ( g_{α_{n−1} −α_{i}} ∗ A_{i}[u(·) − ∑_{j=0}^{m_{i} −1} u_{j} g_{j+1} (·)]) (t)

+ ∑_{i∈(ℕ_{n−1} ∖I)∖V} (g_{α_{n−1} −α_{i}} ∗ A_{i}u)(t) + ∑_{i∈(ℕ_{n−1} ∖I)∩V} A_{i}(g_{α_{n−1} −α_{i}} ∗ u)(t)

= ∑_{i∈ℕ_{n−1} ∖I} ∑_{j∈ℕ_{m_{i} −1}^{0}} g_{α_{n−1} −α_{i} +1+j} (t)u_{i,j} + (g_{α_{n−1}} ∗ f)(t), t ⩾ 0.

If V = ∅ (V = N_{n−1} ), then we also say that u(t) is a strong (mild) solution of (2.3).

Any strong solution of problem (1.1)–(1.2) is also a strong solution of problem (2.3), and any V-mild solution of problem (2.3) is also a V′-mild solution of (2.3) provided that V, V′ ⊆ ℕ_{n−1} and V ⊆ V′. As already observed in [27] for the problem (DFP)_{L}, a sufficiently smooth strong solution of the problem (2.3) need not be a strong solution of problem (1.1)–(1.2) in the case that I ≠ ∅. The situation is quite complicated even in the case that I = ∅ because then we can only prove that a strong solution of problem (2.3) satisfies the equation

∑_{i∈ℕ_{n−1}} ( g_{m_{n−1} −m_{i}} ∗g_{m_{i} −α_{i}} ∗[A_{i}u(·)− ∑_{j=0}^{m_{i} −1} u_{i,j} g_{1+j} (·)]) (t) = (g_{α_{n−1}} ∗f)(t), t ⩾ 0,

which does not imply, in general, that the function

t ↦ g_{m_{i} −α_{i}} ∗ [A_{i}u − ∑_{j=0}^{m_{i} −1} u_{i,j} g_{1+j} ](t), t ⩾ 0

is m_{i}-times continuously differentiable for i ∈ ℕ_{n−1} (the problem (DFP)_{R} is an exception, cf. [27]). Because of that, we shall primarily consider degenerate integral equation (2.3) in the sequel.

Remark 2.1. Before dividing our further research into two separate subsections, it should be observed that we can further generalize the abstract form of problem (1.1) by assuming that some of the terms T_{i}u(t) can be expressed as sums of terms like A′_{i}D_{t}^{α_{i}} (B′_{i}D_{t}^{β_{i}} u(t)) and D_{t}^{α_{i}} A′′_{i}(D_{t}^{β_{i}} B′′_{i}u(t)), with A′_{i}, B′_{i}, A′′_{i}, B′′_{i} being closed linear operators on E and β_{i} ⩾ 0 (cf. [44, Chapter VI] for corresponding examples). It would take too long to go into further details concerning this topic here.

2.1. Exponentially equicontinuous k-regularized C-resolvent (i, j) - propagation families for (1.1)–(1.2). Following the method employed in the papers [55,56] and [22], we introduce the notion of an exponentially equicontinuous k-regularized C-resolvent propagation family for problem (1.1)–(1.2) as follows (cf. the problem (2.3) with I = ∅, x = u_{i,j} , the other initial values being zeroes, and then apply the formula [6, (1.23)] for the Laplace transform of Caputo derivatives of the α^{th} order).

Definition 2.3. Suppose that the function k(t) satisfies (P1), as well as that 1 ⩽ i ⩽ n − 1, 0 ⩽ j ⩽ m_{i} − 1 and R_{i,j} (t): D(A_{i}) → E is a linear mapping (t ⩾ 0). Let the operator C ∈ L(E) be injective. Then the operator family (R_{i,j} (t))_{t⩾0} is said to be an exponentially equicontinuous k-regularized C-resolvent (i, j)-propagation family for problem (1.1)–(1.2) iff there exists ω ⩾ max(0, abs(k)) such that the following holds:

(i) The mapping t ↦ R_{i,j} (t)x, t ⩾ 0 is continuous for every fixed element x ∈ D(A_{i}).

(ii) The family {e^{−ωt} R_{i,j} (t) : t ⩾ 0} is equicontinuous, i.e., for every p ∈ ⊛, there

exist c > 0 and q ∈ ⊛ such that p(e^{−ωt} R_{i,j} (t)x) ⩽ cq(x), x ∈ D(A_{i}), t ⩾ 0. (iii) For every λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0, the operator P_{λ} is injective,

C(R(A_{i})) ⊆ R(P_{λ}) and

(2.4)  λ^{α_{i} −α_{n−1} −j} k̃(λ)P_{λ}^{−1} CA_{i}x = ∫_{0}^{∞} e^{−λt} R_{i,j} (t)x dt, x ∈ D(A_{i}).

If k(t) = g_{r+1} (t) for some r ⩾ 0, then it is also said that (R_{i,j} (t))_{t⩾0} is an exponentially equicontinuous r-times integrated C-regularized resolvent (i, j)- propagation family for (1.1)–(1.2); an exponentially equicontinuous 0-times integrated C-regularized resolvent (i, j)-propagation family for (1.1)–(1.2) is also said to be an exponentially equicontinuous C-regularized resolvent (i, j)-propagation family for (1.1)–(1.2).

Before we state the following important extension of [56, Theorem 3.1], it is worth noting that we do not use here the condition CA_{i} ⊆ A_{i}C, in contrast to the corresponding definitions from [55, 56] and [22], and that the existence of an exponentially equicontinuous k-regularized C-resolvent (i, 0)-propagation family for problem (1.1)–(1.2) implies the existence of an exponentially equicontinuous k-regularized C-resolvent (i, j)-propagation family for problem (1.1)–(1.2) (j ∈ ℕ_{m_{i} −1}^{0} ); if this is the case, we have R_{i,j} (t)x = (g_{j} ∗ R_{i,0} (·)x)(t), t ⩾ 0, j ∈ ℕ_{m_{i} −1}^{0} , x ∈ D(A_{i}). Observe also that the uniqueness theorem for Laplace transform implies that there exists at most one exponentially equicontinuous kregularized C-resolvent (i, j)-propagation family for problem (1.1)–(1.2) and that the assertions of [22, Remark 2.3(iv), Proposition 2.4, Theorem 2.5] can be reformulated in our context.

Theorem 2.1. Suppose that 1 ⩽ i ⩽ n − 1, 0 ⩽ j ⩽ m_{i} − 1 and there exists an exponentially equicontinuous k-regularized C-resolvent (i, j)-propagation family (R_{i,j} (t))_{t⩾0} for problem (1.1)–(1.2).

(i) Assume that there exists l ∈ ℕ_{n−1} such that the following condition:

(C.1) For every v ∈ ℕ_{n−1} ∖ {l} and x ∈ D(A_{i}), there exist a number ω_{0} > ω and a continuous E-valued function t ↦ f_{i,j,v} (t; x), t ⩾ 0 such that, for every p ∈ ⊛, there exists M_{p} > 0 with p(f_{i,j,v} (t; x)) ⩽ M_{p}e^{ωt}, t ⩾ 0 (v ∈ ℕ_{n−1} ∖{l}) and that, for every λ ∈ ℂ with Reλ > ω_{0} and k̃(λ) ≠ 0,

∫_{0}^{∞} e^{−λt} f_{i,j,v} (t; x) dt = λ^{α_{i} −α_{n−1} −j+α_{v} −α_{n−1}}  k̃(λ)A_{v}P_{λ}^{−1} CA_{i}x

holds. Then for each v_{0} ∈ D(A_{i}) the function u(t) := R_{i,j} (t)v_{0}, t ⩾ 0 is a mild solution of the integral equation

∑_{v=1}^{n−1} A_{v}(g_{α_{n−1} −α_{v}} ∗ u)(t) = (g_{α_{n−1} −α_{i} +j} ∗ k)(t)CA_{i}v_{0}, t ⩾ 0, (2.5)

defined in the same way as in Definition 2.2(ii). (ii) Let ∅ ≠ K ⊆ ℕ_{n−1} . If the following conditions hold,

(C.2) For every l ∈ K and x ∈ D(A_{i}), and for every v ∈ ℕ_{n−1} ∖ {l}, there exist a number ω_{l,v} > ω and a continuous E-valued function t ↦ g_{i,j,l,v} (t; x), t ⩾ 0 such that, for every p ∈ ⊛, there exists M_{p,l,v} > 0 with p(g_{i,j,l,v} (t; x)) ⩽ M_{p,l,v} e^{ω_{l,v} t}, t ⩾ 0 (l ∈ K, v ∈ ℕ_{n−1} ∖ {l}) and that, for every λ ∈ ℂ with Reλ > ω_{l,v} and k̃(λ) ≠ 0,

∫_{0}^{∞} e^{−λt} g_{i,j,l,v} (t; x)dt = λ^{α_{i} −α_{n−1} −j+α_{v} −α_{l}}  k̃(λ)A_{v}P_{λ}^{−1} CA_{i}x

(C.3) For every l ∈ K, there exist a number ω_{l} > ω and a continuous function h_{l}: [0, ∞) → ℂ satisfying (P1) and

h̃_{l}(λ) = k̃(λ)λ^{α_{i} −α_{l} −j} , Reλ > ω_{l},

then for each v_{0} ∈ D(A_{i}) the function u(t) = R_{i,j} (t)v_{0}, t ⩾ 0 satisfies that the mappings t ↦ A_{l}u(t), t ⩾ 0 are well-defined, continuous and that for each p ∈ ⊛ there exist M_{p} > 0 and ω_{0} > ω with p(A_{l}u(t) − h_{l}(t)CA_{i}v_{0}) ⩽ M_{p}e^{ω_{0} t}, t ⩾ 0 (l ∈ K). Furthermore, for every t ⩾ 0,

∑_{l∈K} (g_{α_{n−1} −α_{l}} ∗ A_{l}u)(t) + ∑_{l∈ℕ_{n−1} ∖K} A_{l}(g_{α_{n−1} −α_{l}} ∗ u)(t) (2.6)

= (g_{α_{n−1} −α_{i} +j} ∗ k)(t)CA_{i}v_{0}.

(iii) Suppose that (C.1) holds. Let v_{0} ∈ ⋂ _{i=1}^{n−1} D(A_{i}), let ∅ ≠ K ⊆ ℕ_{n−1} , and let CA_{p} ⊆ A_{p}C for all p ∈ ℕ_{n−1} . If the following condition holds, (C.4) For every l ∈ K and for every v ∈ ℕ_{n−1} ∖ {i}, there exist a number

ω_{l,v} > ω and a continuous function h_{l,v} : [0, ∞) → E satisfying that, for every p ∈ ⊛, there exists M_{p,l,v} > 0 with p(h_{l,v} (t)) ⩽ M_{p,l,v} e^{ω_{l,v} t}, t ⩾ 0 and h̃_{l,v} (λ) = k̃(λ)λ^{α_{v} −α_{n−1} −j} A_{l}P_{λ}^{−1} CA_{v}v_{0}, Reλ > ω_{l,v} , k̃(λ) ≠ 0, then the function u(t) = R_{i,j} (t)v_{0}, t ⩾ 0 satisfies that the mappings t ↦ A_{l}u(t), t ⩾ 0 are well-defined, continuous and that for each p ∈ ⊛ there exist M_{p} > 0 and ω_{0} > ω with p(A_{l}u(t) − (g_{j} ∗ k)(t)CA_{l}v_{0}) ⩽ M_{p}e^{ω_{0} t}, t ⩾ 0 (l ∈ K). Furthermore, for every t ⩾ 0, (2.6) holds.

(iv) Suppose that CA_{p} ⊆ A_{p}C, p ∈ ℕ_{n−1} and k(t) satisfies (P2), as well as that n = 3 or that n ⩾ 4 and the following condition holds: (C.5) For every p ∈ ⊛ and l ∈ ℕ_{n−1} ∖ {i}, there exist numbers λ_{p,l}, σ_{p,l} > 0,

a seminorm q_{p,l} ∈ ⊛ and a function h_{p,l} : (λ_{p,l}, ∞) → (0, ∞) such that

p(P_{λ}^{−1} CA_{l}x) ⩽ [q_{p,l}(x) + q_{p,l}(A_{l}x)]h_{p,l}(λ), λ > λ_{p,l}, x ∈ D(A_{l}), and

lim_{λ→+∞} e^{−λσ_{p,l}} h_{p,l}(λ) = 0.

Then the function u(t) = R_{i,j} (t)v_{0}, t ⩾ 0 is a unique mild solution of integral equation (2.5), provided that v_{0} ∈ D(A_{i}) and the assumptions of (i) hold. Furthermore, the function u(t) = R_{i,j} (t)v_{0}, t ⩾ 0 is a unique function satisfying that the mapping t ↦ A_{l}u(t), t ⩾ 0 is well-defined, continuous (l ∈ K) and that (2.6) holds, provided that v_{0} ∈ D(A_{i}) and the assumptions of (ii) hold, resp. v_{0} ∈ ⋂ _{i=1}^{n−1} D(A_{i}) and the assumptions of (iii) hold.

Proof. Let v_{0} ∈ D(A_{i}). Due to the condition (C.1) and Lemma 1.2, we have that the function t ↦ A_{v}(g_{α_{n−1} −α_{v}} ∗ R_{i,j} (·)v_{0})(t), t ⩾ 0 is well-defined, continuous and that for each p ∈ ⊛ there exist M′_{p} > 0 and ω′ > ω with p(A_{v}(g_{α_{n−1} −α_{v}} ∗ R_{i,j} (·)v_{0})(t)) ⩽ M′_{p}e^{ω′t}, t ⩾ 0 (v ∈ ℕ_{n−1} ∖ {l}); furthermore,

∫_{0}^{∞} e^{−λt} A_{v}(g_{α_{n−1} −α_{v}} ∗ R_{i,j} (·)v_{0})(t)dt = k̃(λ)λ^{α_{i} −α_{n−1} −j+α_{v} −α_{n−1}} A_{v}P_{λ}^{−1} CA_{i}v_{0},

for any v ∈ ℕ_{n−1} ∖ {l} and for any λ ∈ ℂ with Reλ > ω′ and k̃(λ) ≠ 0. Using the identity

(2.7) k̃(λ)λ^{α_{i} −α_{n−1} −j+α_{l} −α_{n−1}} A_{l}P_{λ}^{−1} CA_{i}v_{0}

= k̃(λ)λ^{α_{i} −α_{n−1} −j} [CA_{i}v_{0} − ∑_{v∈ℕ_{n−1} ∖{l}} λ^{α_{v} −α_{n−1}} A_{v}P_{λ}^{−1} CA_{i}v_{0}],

for any λ ∈ ℂ with Reλ > ω′ and k̃(λ) ≠ 0, and Lemma 1.2, it readily follows that the function t ↦ A_{l}(g_{α_{n−1} −α_{l}} ∗ R_{i,j} (·)v_{0})(t), t ⩾ 0 is well-defined, continuous and that

A_{l}(g_{α_{n−1} −α_{l}} ∗ R_{i,j} (·)v_{0})(t)= (g_{α_{n−1} −α_{i} +j} ∗ k)(t)CA_{i}v_{0}

− ∑_{v∈ℕ_{n−1} ∖{l}} A_{l}(g_{α_{n−1} −α_{v}} ∗ R_{i,j} (·)v_{0})(t), t ⩾ 0,

proving that the function u(t) = R_{i,j} (t)v_{0}, t ⩾ 0 is a mild solution of integral equation (2.5). Suppose now that conditions (C.2)–(C.3) hold, as well as that v_{0} ∈ D(A_{i}) and ∅ ≠ K ⊆ ℕ_{n−1} . Clearly, (C.2) implies (C.1) with any l ∈ K. Similarly as in the proof of (i), the conditions (C.2)–(C.3) in combination with equation (2.7), multiplied by λ^{α_{n−1} −α_{l}} , imply that there exists a sufficiently large number ω′_{l} > ω such that

A_{l} ∫_{0}^{∞} e^{−λt} R_{i,j} (t)v_{0} dt = h̃_{l}(λ)CA_{i}v_{0} − ∫_{0}^{∞} e^{−λt} ∑_{v∈ℕ_{n−1} ∖K} g_{i,j,l,v} (t; v_{0})dt,

for any l ∈ K and for any λ ∈ ℂ with Reλ > ω′_{l}. Then we can use the assertion (i) and Lemma 1.2 to complete the proof of (ii). In order to prove (iii), observe first that the assumptions v_{0} ∈ ⋂ _{i=1}^{n−1} D(A_{i}) and CA_{p} ⊆ A_{p}C, p ∈ ℕ_{n−1} imply that

P_{λ}^{−1} C(λ^{α_{i} −α_{n−1}} A_{i}v_{0})+ ∑_{v∈ℕ_{n−1} ∖{i}}P_{λ}^{−1} C(λ^{α_{v} −α_{n−1}} A_{v}v_{0}) = P_{λ}^{−1} CP_{λ}v_{0} = Cv_{0}, (2.8)

provided Reλ > ω and k̃(λ) ≠ 0. Making use of (2.4) and (2.8), we obtain that, for any such value of complex parameter λ, the following holds:

A_{l} ∫_{0}^{∞} e^{−λt} R_{i,j} (t)v_{0} dt= λ^{−j} k̃(λ)A_{l}P_{λ}^{−1} C(λ^{α_{i} −α_{n−1}} A_{i}v_{0})

= λ^{−j} k̃(λ)A_{l}[Cv_{0} − ∑_{v∈ℕ_{n−1} ∖{i}} λ^{α_{v} −α_{n−1}} P_{λ}^{−1} CA_{v}v_{0}].

Keeping in mind the last equation, as well as condition (C.4) and Lemma 1.2, the proof of (iii) follows instantly. We will prove the uniqueness of solutions in (iv) only in the case that v_{0} ∈ D(A_{i}) and the assumptions of (i) hold. Let t ↦ u(t), t ⩾ 0 be a mild solution of integral equation (2.5) with v_{0} = 0. Convoluting the function u(·) with g_{ξ}(·), for a sufficiently large number ξ > 0, we may assume without of generality that, for every v ∈ ℕ_{n−1} , the mapping t ↦ A_{v}u(t), t ⩾ 0 is well-defined and continuous. Set, for every t ⩾ 0 and ζ > 0, v_{t,ζ}(λ) := (g_{ζ} ∗ e^{λ·})(t) − λ^{−ζ} e^{tλ}, λ > 0; v_{t,0}(λ) := 0 (t ⩾ 0, λ > 0). Then the mapping t ↦ v_{t,ζ}(λ) is continuous in t ⩾ 0, for any fixed numbers ζ ⩾ 0 and λ > 0, and by [54, Lemma 1.5.5, p. 23], there exists M ⩾ 1 such that the mapping λ ↦ v_{t,ζ}(λ), λ > 0 satisfies

(2.9) |v_{t,ζ}(λ)| ⩽ M[(1 + t)^{ζ−1} λ^{−1} (1 + λ^{1−ζ} ) + t^{ζ−1} λ^{−1} ], λ > 0, t > 0, ζ > 0.

Keeping in mind that CA_{p} ⊆ A_{p}C, p ∈ ℕ_{n−1} , we have that, for every t ⩾ 0 and λ > 0,

λ^{α_{i} −α_{n−1}} ∫_{0}^{t} e^{λ(t−s)} A_{i}Cu(s)ds + ∫_{0}^{t} v_{t−s,α_{n−1} −α_{i}} (λ)A_{i}Cu(s)ds

= C ∫_{0}^{t} e^{λ(t−s)} (g_{α_{n−1} −α_{i}} ∗ A_{i}u)(s)ds

= (−C) ∑_{v∈ℕ_{n−1} ∖{i}} ∫_{0}^{t} e^{λ(t−s)} (g_{α_{n−1} −α_{v}} ∗ A_{v}u)(s)ds

= − ∑_{v∈ℕ_{n−1} ∖{i}} [λ^{α_{v} −α_{n−1}} ∫_{0}^{t} e^{λ(t−s)} A_{v}Cu(s)ds +∫_{0}^{t}v_{t−s,α_{n−1} −α_{v}} (λ)A_{v}Cu(s)ds],

which clearly implies that, for every λ > ω, σ > 0 and t ⩾ 0, the following holds:

λ^{α_{i} −α_{n−1} −j} k̃(λ)e^{−λσ} ∫_{0}^{t} e^{λ(t−s)} Cu(s)ds (2.10)

= −λ^{α_{i} −α_{n−1} −j} k̃(λ)e^{−λσ} P_{λ}^{−1} C ∫_{0}^{t} v_{t−s,α_{n−1} −α_{i}} (λ)A_{i}u(s)ds

− λ^{α_{i} −α_{n−1} −j} k̃(λ)e^{−λσ} ∑_{v∈ℕ_{n−1} ∖{i}} P_{λ}^{−1} C ∫_{0}^{t} v_{t−s,α_{n−1} −α_{v}} (λ)A_{v}u(s)ds.

By (2.4) and (2.10), we obtain that, for every λ > ω, σ > 0 and t ⩾ 0, the following holds:

e^{−λσ} ∫_{0}^{t} e^{λ(t−s)} Cu(s)ds

= −\frac{λ^{α_{n−1} +j−α_{i}} e^{−λσ}}{k̃(λ)} ∫_{0}^{∞} e^{−λs} R_{i,j} (s)( ∫_{0}^{t} v_{t−r,α_{n−1} −α_{i}} (λ)u(r)dr) ds

− e^{−λσ} ∑_{v∈ℕ_{n−1} ∖{i}} P_{λ}^{−1} CA_{v} ∫_{0}^{t} v_{t−s,α_{n−1} −α_{v}} (λ)u(s)ds.

For the estimation of the first addend on the right-hand side of the above equality, we can use the fact that there exist numbers σ_{0} > 0 and M′ ⩾ 1 such that

(2.11)  \frac{e^{−λσ_{0}}}{|k̃(λ)|} ⩽ M′, λ > ω + 1;

cf. the proof of [22, Theorem 2.8]. Keeping in mind (2.9) and (2.11), it can be simply proved that, for every σ > σ_{0} and for every p ∈ ⊛, we have (2.12)

lim_{λ→+∞} p( \frac{λ^{α_{n−1} +j−α_{i}} e^{−λσ}}{k̃(λ)} ∫_{0}^{∞} e^{−λs} R_{i,j} (s)( ∫_{0}^{t} v_{t−r,α_{n−1} −α_{i}} (λ)u(r)dr) ds) = 0.

If n ⩾ 4, then condition (C.5) in combination with the previous equality and (2.9) shows that, for every p ∈ ⊛, there exists a sufficiently large number σ_{p} > 0 such that lim_{λ→+∞} e^{−λσ_{p}} p((e^{λ·} ∗ Cu)(t)) = 0, t ⩾ 0; the same holds in the case that n = 3 because then we can use, instead of condition (C.5), equation (2.8) and the arguments already seen in proving equation (2.12), to conclude that

lim_{λ→+∞} p( e^{−λσ} ∑_{v∈ℕ_{n−1} ∖{i}} P_{λ}^{−1} CA_{v} ∫_{0}^{t} v_{t−s,α_{n−1} −α_{v}} (λ)v(s)ds) = 0,

for any σ > σ_{0} and t ⩾ 0. In such a way, we obtain that for each p ∈ ⊛ the following holds: lim_{λ→+∞} ∫_{0}^{t} e^{λ(t−s−σ)} Cu(s)ds = 0, t ⩾ 0, σ > σ_{p}. By the dominated convergence theorem, it readily follows that for each p ∈ ⊛ we have: lim_{λ→+∞} p(∫_{0}^{t−σ} e^{λ(t−s−σ)} Cu(s)ds) = 0, t ⩾ σ > σ_{p}. Therefore,

lim_{λ→+∞} ∫_{0}^{t} e^{λ(t−s)} Cu(s)ds = 0, t ⩾ 0.

Since C is injective, we can apply [20, Lemma 2.1.33(iii)] (cf. [38, Lemma 1.4.4, p. 100] for the Banach space case) to complete the proof. □

The uniqueness of solutions of integral equation (2.5), resp. (2.6), can be proved even in the case of non-existence of an exponentially equicontinuous k-regularized C-resolvent (i, j)-propagation family for problem (1.1)–(1.2). Strictly speaking, the proof of Theorem 2.1 implies the following uniqueness type theorem for degenerate multi-term problems (cf. [10, Theorem 3.1] for a pioneering result on the uniqueness of degenerate first order equations):

Theorem 2.2. Suppose that CA_{p} ⊆ A_{p}C for all p ∈ ℕ_{n−1} , V ⊆ ℕ_{n−1} and the requirements in (C.5) hold for every seminorm p ∈ ⊛ and for every number l ∈ ℕ_{n−1} . Then there exists at most one mild solution t ↦ u(t), t ⩾ 0 of integral equation (2.5) with v_{0} = 0, resp. there exists at most one continuous E-valued function t ↦ u(t), t ⩾ 0 satisfying that the mapping t ↦ A_{l}u(t), t ⩾ 0 is welldefined, continuous (l ∈ K) and that (2.6) holds with v_{0} = 0. In particular, there exists at most one V-mild solution of problem (2.3) and there exists at most one strong solution of problem (1.1)–(1.2).

Remark 2.2. Suppose again that the general assumptions of Theorem 2.1 hold, i.e., that 1 ⩽ i ⩽ n − 1, 0 ⩽ j ⩽ m_{i} − 1 and there exists an exponentially equicontinuous k-regularized C-resolvent (i, j)-propagation family (R_{i,j} (t))_{t⩾0} for problem (1.1)–(1.2).

(i) Suppose that k(t) satisfies (P2) and that, for every l ∈ ℕ_{n−1} ∖ {i}, there exists j_{l} ∈ ℕ_{m_{l} −1}^{0} such that there exists an exponentially equicontinuous kregularized C-resovent (l, j_{l})-propagation family for problem (1.1)–1.2. By the proof of Theorem 2.1(iv), we have that condition (C.5) automatically holds.

(ii) The uniqueness of solutions of non-degenerate integral equations has been recently considered in [30]. It ought to be observed that we must impose the additional condition CA_{p} ⊆ A_{p}C, p ∈ ℕ_{n−1}^{0} in the formulation of Theorem 3.2 in [30] in order for its proof to work.

(iii) Let ∅ ≠ K ⊆ ℕ_{n−1} . Suppose that v_{0} ∈ D(A_{i}) and conditions (C.2)–(C.3) hold, or that v_{0} ∈ ⋂ _{i=1}^{n−1} D(A_{i}), CA_{p} ⊆ A_{p}C for all p ∈ ℕ_{n−1} , and the conditions (C.1) and (C.4) hold. Let u(t) be the solution of (2.6), satisfying the properties stated above. Consider now equation (2.3) and the notion introduced in Definition 2.2 with indexes i, j replaced by i′, j′. Then the following holds:

(a) If i ∈ ℕ_{n−1} ∖ I, k(t) = 1, u_{j′} = 0 (0 ⩽ j′ ⩽ m_{i′} − 1), u_{i′,j′} = CA_{i}v_{0}, provided i′ = i and j′ = j, and u_{i′,j′} = 0, otherwise, then u(t) is a (ℕ_{n−1} ∖ K)-mild solution of (2.3) with f(t) = 0.

(b) If i ∈ I, k(t) = 1, u_{i′,j′} = 0 (i′ ∈ ℕ_{n−1} , j′ ∈ ℕ_{m_{i′} −1}^{0} ), u_{j′} = Cv_{0}, provided j′ = j, u_{j′} = 0, otherwise, and CA_{i} ⊆ A_{i}C, then u(t) is a (ℕ_{n−1} ∖ K)-mild solution of (2.3) with f(t) = 0, provided that for each i′ ∈ {s ∈ I ∖ {i} : m_{s} − 1 ⩾ j} one has A_{i′}Cv_{0} = 0.

(iv) Making use of [54, Theorem 1.1.9], Lemma 1.2 and the formula [6, (1.23)], we can clarify some sufficent conditions for the existence of terms A_{p}D_{t}^{α_{p}} u(t) and D_{t}^{α_{p}} A_{p}u(t) (p ∈ ℕ_{n−1} ). Unfortunately, it is very hard to verify these conditions in practical situations because we do not know the precise values of elements R_{i,j} (0)x, R′_{i,j} (0)x, … (x ∈ D(A_{i})).

The notion of an exponentially equicontinuous (equicontinuous), analytic kregularized C-resolvent (i, j)-propagation family (R_{i,j} (t))_{t⩾0} for problem (1.1)– (1.2) is introduced in the following definition.

Definition 2.4. Suppose that 1 ⩽ i ⩽ n − 1, 0 ⩽ j ⩽ m_{i} − 1, 0 < α ⩽ π and there exists an exponentially equicontinuous k-regularized C-resolvent (i, j)- propagation family (R_{i,j} (t))_{t⩾0} for problem (1.1)–(1.2). Suppose, further, that the function k(t) satisfies (P1), as well as that C ∈ L(E) is an injective mapping. Then it is said that (R_{i,j} (t))_{t⩾0} is an exponentially equicontinuous (equicontinuous), analytic k-regularized C-resolvent (i, j)-propagation family (R_{i,j} (t))_{t⩾0} for problem (1.1)–(1.2), of angle α, iff the following holds:

(i) For every x ∈ D(A_{i}), the mapping t ↦ R_{i,j} (t)x, t > 0 can be analytically extended to the sector Σ_{α}; since no confusion seems likely, we shall denote the extension by the same symbol.

(ii) For every x ∈ D(A_{i}) and β ∈ (0, α), one has lim_{z→0,z∈Σ_{β}} R_{i,j} (z)x = R_{i,j} (0)x. (iii) For every β ∈ (0, α), there exists ω_{β} ⩾ max(0, abs(k)) (ω_{β} = 0) such that the

family {e^{−ω_{β} z} R_{i,j} (z) : z ∈ Σ_{β}} is equicontinuous, i.e., for every p ∈ ⊛, there exist c > 0 and q ∈ ⊛ such that p(e^{−ω_{β} z} R_{i,j} (z)x) ⩽ cq(x), x ∈ D(A_{i}), z ∈ Σ_{β}.

The proof of following theorem can be given by using the arguments given in that of [24, Theorem 3.7].

Theorem 2.3. Assume that the function k(t) satisfies (P1), 1 ⩽ i ⩽ n − 1, 0 ⩽ j ⩽ m_{i} − 1, ω ⩾ max(0, abs(k)), α ∈ (0, π/2] and the operator C ∈ L(E) is injective. Assume, further, that for every λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0, we have that the operator P_{λ} is injective and C(R(A_{i})) ⊆ R(P_{λ}). Let for each x ∈ D(A_{i}) there is an analytic function q_{x}: ω + Σ_{\frac{π}{2} +α} → E such that

q_{x}(λ) = λ^{α_{i} −α_{n−1} −j} k̃(λ)P_{λ}^{−1} CA_{i}x, Reλ > ω, k̃(λ) ≠ 0.

Suppose that, for every β ∈ (0, α) and p ∈ ⊛, there exist c_{p,β} > 0 and r_{p,β} ∈ ⊛ such that p((λ − ω)q_{x}(λ)) ⩽ c_{p,β} r_{p,β} (x), x ∈ D(A_{i}), λ ∈ ω + Σ_{β+(π/2)} and that, for every x ∈ D(A_{i}), there exists the limit lim_{λ→+∞} λq_{x}(λ) in E. Then there exists an exponentially equicontinuous k-regularized C-resolvent (i, j)-propagation family (R_{i,j} (t))_{t⩾0} for problem (1.1)–(1.2), of angle α, and for each β ∈ (0, α) the family {e^{−ωz} R_{i,j} (z) : z ∈ Σ_{β}} is equicontinuous.

Differential properties of (a, k)-regularized C-resolvent families have been investigated in [22, Theorems 3.4 and 3.5]; these assertions can be simply reformulated for exponentially equicontinuous (analytic) k-regularized C-resolvent (i, j)- propagation families in locally convex spaces. As the following theorem shows, this is also the case with the assertion of [22, Theorem 3.3].

Theorem 2.4. Suppose that 1 ⩽ i ⩽ n − 1, 0 ⩽ j ⩽ m_{i} − 1, 0 < γ < 1,

0 ⩽ j′ ⩽ m_{i} − 1, and there exists an exponentially equicontinuous k-regularized C-resolvent (i, j)-propagation family (R_{i,j} (t))_{t⩾0} for problem (1.1)–(1.2) satisfying that the family {e^{−ωt} R_{i,j} (t) : t ⩾ 0} is equicontinuous for some ω ⩾ max(0, abs(k)). Assume that k(t) satisfies (P1) and there exists a scalar-valued continuous kernel k_{γ}(t) on [0, ∞), satisfying (P1), and a positive real number η > 0 such that

k̃_{γ}(λ) = λ^{γ−1+j′−γj} k̃(λ^{γ}), λ > η.

Then there exists an exponentially equicontinuous k-regularized C-resolvent (i, j′)- propagation family (R_{i,j′,γ} (t))_{t⩾0} for problem (1.1)–(1.2), with α_{i} replaced by γα_{i} (i ∈ ℕ_{n−1} ) , and (R_{i,j′,γ} (t))_{t⩾0} is given by R_{i,j′,γ} (0) := R_{i,j} (0),

R_{i,j′,γ} (t)x := ∫_{0}^{∞} t^{−γ} Φ_{γ}(st^{−γ} )R_{i,j} (s)x ds, x ∈ D(A_{i}), t > 0.

Furthermore, the family {e^{−ω^{1/γ} t} R_{i,j′,γ} (t) : t ⩾ 0} is equicontinuous and, for every ζ ⩾ 0, the equicontinuity of the family {e^{−ωt} (1 + t^{ζ})^{−1} R_{i,j} (t) : t ⩾ 0}, resp. {e^{−ωt} t^{−ζ} R_{i,j} (t) : t ⩾ 0}, implies the equicontinuity of the family

{e^{−ω^{1/γ} t} (1 + t^{γζ})^{−1} (1 + ωt^{ζ(1−γ)} )^{−1} R_{i,j′,γ} (t) : t ⩾ 0},
resp. {e^{−ω^{1/γ} t} t^{−γζ} (1 + ωt^{ζ(1−γ)} )^{−1} R_{i,j′,γ} (t) : t ⩾ 0},

and the following holds:

(i) The mapping t ↦ R_{i,j′,γ} (t)x, t > 0 admits an analytic extension to the sector Σ_{min (( \frac{1}{γ} −1) \frac{π}{2} ,π)} for all x ∈ D(A_{i}).

(ii) If ω = 0 and ε ∈ (0, min(( \frac{1}{γ} − 1) \frac{π}{2} , π)), then the family \{R_{i,j′,γ} (z) : z ∈ Σ_{min (( \frac{2}{γ} −1) \frac{π}{2} ,π)−ε} \} is equicontinuous and lim_{z→0,z∈Σ_{min (( \frac{1}{γ} −1) \frac{π}{2} ,π)−ε}} R_{i,j′,γ} (z)x

= R_{i,j′,γ} (0)x for all x ∈ D(A_{i}). (iii) If ω > 0 and ε ∈ (0, min((\frac{1}{γ} − 1)\frac{π}{2} , \frac{π}{2} )), then there exists δ_{γ,ε} > 0 such that the

family \{e^{−δ_{γ,ε} Re z} R_{i,j′,γ} (z) : z ∈ Σ_{min (( \frac{1}{γ} −1) \frac{π}{2} , \frac{π}{2} )−ε} \} is equicontinuous. Moreover, lim_{z→0,z∈Σ_{min (( \frac{1}{γ} −1) \frac{π}{2} , \frac{π}{2} )−ε}} R_{i,j′,γ} (z)x = R_{i,j′,γ} x for all x ∈ D(A_{i}).

Remark 2.3. Using the proof of [6, Theorem 3.1] and an elementary argumentation, it can be simply verified that any of conditions (C.1)–(C.5) is invariant under the action of subordination principle described in Theorem 2.4.

Before illustrating our abstract results from this subsection by some examples, we would like to observe that the analysis carried out in [40, Theorem 4.1, p. 101], [22, Theorem 2.6(i)] and Theorem 2.1 can be used in the study of the following degenerate integral equation:

∑_{j∈K} (a_{j} ∗ A_{j} u)(t) + ∑_{j∈ℕ_{n−1} ∖K} A_{j} (a_{j} ∗ u)(t) = f(t), t ⩾ 0,

where ∅ ≠ K ⊆ ℕ_{n−1} , f ∈ C([0, ∞) : E), and the functions a_{1}(t), …, a_{n−1} (t) satisfy certain properties. For the sake of brevity and better exposition, we shall only refer the reader to [29, Theorem 4.4] for the corresponding result in the case of non-degenerate equations.

Example 2.1. (cf. also [29, Example 5.1(i)]) Suppose that c_{l} ∈ ℂ ∖ 0 (1 ⩽ l ⩽ n − 1), as well as that A and B are closed linear operators on E, and A_{l} = c_{l}B for 1 ⩽ l ⩽ n − 1. We consider the following degenerate multi-term problem:

(2.13)  D_{t}^{α_{n}} Bu(t) + ∑_{l=1}^{n−1} c_{l}D_{t}^{α_{l}} Bu(t) = D_{t}^{α} Au(t), t ⩾ 0,

equipped with the initial conditions of the form (1.2). Here 0 ⩽ α_{1} < ⋯< α_{n}, 0 ⩽ α < α_{n}, and

P_{λ} = ∑_{l=1}^{n−1} c_{l}λ^{α_{l} −α_{n}} B − λ^{α−α_{n}} A + B, λ ∈ ℂ ∖ {0}.

(i) (a) Suppose 0 < δ ⩽ 2, σ ⩾ 1, \frac{πδ}{2(α_{n} −α)} − \frac{π}{2} > 0, 0 ⩽ j ⩽ ⌈α_{n}⌉ − 1, and there exists an exponentially equicontinuous (σ − 1)-times integrated C-resolvent propagation family (R(t))_{t⩾0} for problem (1.4), with a(t) = g_{δ}(t). Put σ′ := max(1, 1 − j + (α_{n} − α)(σ − 1)δ^{−1} ) and θ := min( \frac{π}{2} , \frac{πδ}{2(α_{n} −α)} − \frac{π}{2} ). Then, for every sufficiently small number ε > 0, there exists ω_{ε} > 0 such that C(R(B)) ⊆ R(P_{λ}) for all λ ∈ ω_{ε} + Σ_{\frac{π}{2} δ−ε} and that the family \{|λ|^{\frac{δ−σ}{δ}} (1 + |λ|^{\frac{1}{δ}} )(λB − A)^{−1} CBx : λ ∈ ω_{ε} + Σ_{\frac{π}{2} δ−ε} , x ∈ D(B)\} is equicontinuous. Noting also that

arg( λ^{α_{n} −α} + ∑_{l=1}^{n−1} c_{l}λ^{α_{l} −α} ) ≈ (α_{n} − α) arg(λ), λ → ∞, arg(λ) < \frac{π}{α_{n} − α} ,

our choice of θ implies that, for every sufficiently small number ε > 0, there exists ω′_{ε} > 0 such that, for every λ ∈ ω′_{ε} + Σ_{\frac{π}{2} +θ−ε} , one has

λ^{α_{n} −α} + ∑_{l=1}^{n−1} c_{l}λ^{α_{l} −α} ∈ ω_{ε} + Σ_{\frac{π}{2} δ−ε} .

Put now, for every x ∈ D(B) and λ ∈ ω′_{ε} + Σ_{\frac{π}{2} +θ−ε} ,

q_{x}(λ) := λ^{−j−σ′} P_{λ}^{−1} CBx.

Then q_{x} : ω′_{ε} + Σ_{\frac{π}{2} +θ−ε} → E is an analytic function and, for every β ∈ (0, θ) and p ∈ ⊛, there exist c_{p,β} > 0 and r_{p,β} ∈ ⊛ such that p((λ − ω′_{ε})q_{x}(λ)) ⩽ c_{p,β} r_{p,β} (x), x ∈ D(B), λ ∈ ω′_{ε}+Σ_{\frac{π}{2} +θ−ε} . By the proof of [5, Proposition 4.1.3, p. 248], we have that lim_{Re λ→+∞} λ^{δ−σ+1} (λ^{δ}B − A)^{−1} CBx = R_{i,j} (0)x, x ∈ D(B), which simply implies that, for every x ∈ D(B), there exists the limit lim_{λ→+∞} λq_{x}(λ) in E. Therefore, Theorem 2.3 implies that there exists an exponentially equicontinuous, analytic (σ′−1)-times integrated C-resolvent (n, j)-propagation family (R_{n,j} (t))_{t⩾0} for problem (2.13), of angle θ (with the clear meaning).

(b) Suppose 0 < δ ⩽ 2, σ ⩾ 1, 0 ⩽ j ⩽ ⌈α_{n}⌉−1, γ ∈ (0, \frac{π}{2} ] and \frac{δ( \frac{π}{2} +γ)}{(α_{n} −α)} − \frac{π}{2} > 0.

Put σ_{1} := σ′ and θ_{1} := min(\frac{π}{2} , \frac{δ( \frac{π}{2} +γ)}{(α_{n} −α)} − \frac{π}{2} ). Arguing similarly as in (a), one can prove the following: Suppose that for each ε ∈ (0, \frac{π}{2} + γ) there exists ω_{ε} > 0 such that for each x ∈ D(B) there exists an analytic function q_{x} : ω_{ε} + Σ_{\frac{π}{2} +γ−ε} → E satisfying that

q_{x}(λ) = λ^{δ−σ} (λ^{δ}B − A)^{−1} CBx, λ ∈ ω_{ε} + Σ_{\frac{π}{2} +γ−ε} , x ∈ D(B),

and that for each p ∈ ⊛ there exist c_{p} > 0 and q_{p} ∈ ⊛ so that

p(q_{x}(λ)) ⩽ c_{p} \frac{q_{p}(x)}{1 + |λ|} , λ ∈ ω_{ε} + Σ_{\frac{π}{2} +γ−ε} , x ∈ D(B).

Then the existence of limit lim_{Re λ→+∞} λ^{δ−σ+1} (λ^{δ}B − A)^{−1} CBx in E, for all x ∈ D(B), implies that there exists an exponentially equicontinuous, analytic (σ_{1} − 1)-times integrated C-resolvent (n, j)-propagation family (R_{n,j} (t))_{t⩾0} for problem (2.13), of angle θ_{1}; if there is an element x ∈ D(B) such that the limit lim_{Re λ→+∞} λ^{δ−σ+1} (λ^{δ}B − A)^{−1} CBx does not exist in E, then the above holds with any number σ_{2} > σ_{1}. For the purpose of illustration of obtained results, assume now that n ∈ ℕ and iA′_{l}, 1 ⩽ l ⩽ n are commuting generators of bounded C_{0}-groups on a Banach space E. Put A′ := (A′_{1}, …, A′_{n}); we refer the reader to [57] and [22, Section 4] for the definition of a closable operator P (A′), where P (x) is a complex polynomial in n variables, and for more details about functional calculus for commuting generators of bounded C_{0}-groups (if E = L^{p}(ℝ^{n}) for some p ∈ [1, ∞), then the obvious choice is A′_{l} := −i∂/∂x_{l}, with maximal distibutional domain). Suppose 0 < δ < 2, ω ⩾ 0, P_{1}(x) and P_{2}(x) are non-zero complex polynomials, N_{1} = dg(P_{1}(x)), N_{2} = dg(P_{2}(x)), β > \frac{n}{2} \frac{(N_{1} +N_{2} )}{min (1,δ)} (resp. β ⩾ n|\frac{1}{p}− \frac{1}{2}|\frac{(N_{1} +N_{2} )}{min (1,δ)} , if E = L^{p}(ℝ^{n}) for some 1 < p < ∞), P_{2}(x) ≠ 0, x ∈ ℝ^{n} and

(2.14)  sup_{x∈ℝ^{n}} Re( ( \frac{P_{1}(x)}{P_{2}(x)})^{1/δ}) ⩽ ω.

Set

(2.15)  R_{δ}(t) := ( E_{δ}( t^{δ} \frac{P_{1}(x)}{P_{2}(x)}) (1 + |x|^{2})^{−β/2} ) (A′), t ⩾ 0.

By [22, Theorem 4.3], we know that (R_{δ}(t))_{t⩾0} ⊆ L(E) is a global exponentially bounded (g_{δ}, R_{δ}(0))-regularized resolvent family for problem (1.4) with B = \overline{P_{2}(A′)}, A = \overline{P_{1}(A′)} and a(t) = g_{δ}(t). By the conclusion in (a), it readily follows that there exists an exponentially equicontinuous, analytic C-resolvent (n, j)-propagation family (R_{n,j} (t))_{t⩾0} for problem (2.13), of angle θ = min(\frac{π}{2} , \frac{πδ}{2(α_{n} −α)} − \frac{π}{2} ). Since condition (ii.1) given in the formulation of [22, Theorem 2.8] holds, with a(t) = g_{δ}(t) and k(t) = 1, it is not diificult to prove, with the help of our previous consideration and the results concerning the Laplace transform of analytic vector-valued functions (see e.g. [24, Section 3]) that conditions (C.1) and (C.5) hold for (2.13), as well as that condition (C.4) holds for (2.13) provided that α_{n−1} ⩽ α; we need the last condition because the inclusion λ^{α_{n−1} −α_{n} −1} AP_{λ}^{−1} Cx ∈ LT − E has to be satisfied (x ∈ E), it is also worth noting that we do not need the condition α_{n−1} ⩽ α for the existence of solutions of the integral equation

Bu(t) + ∑_{l=1}^{n−1} c_{l}(g_{α_{n} −α_{l}} ∗ Bu)(t) = A(g_{α_{n} −α} ∗ u)(t) + CBv_{0}, t ⩾ 0,

cf. (2.6). It is also worth noting that we can refine the results on C-wellposednes of equation (2.13) by using the estimates quoted in [22, Remark 4.4(ii)] and that we can similarly consider equation (2.13) in E_{l}-type spaces (cf. [22, Remark 4.5]).

(ii) (cf. [29, Example 5.1(i)-(b)] and [22, Example 3.8] for more details) Let s > 1, 0 ⩽ j ⩽ ⌈α_{n}⌉ − 1, k_{a,b}(t) := L^{−1} (exp(−aλ^{b}))(t), t ⩾ 0 (a > 0, b ∈ (0, 1)),

E := \{f ∈ C^{∞} [0, 1]; ∥f∥ := sup_{p⩾0} \frac{∥f^{(p)}∥_{∞}}{p!^{s}} < ∞\}

A′:= −d/ds, D(A′) := {f ∈ E; f′ ∈ E, f(0) = 0}.

Let P_{1}(z) = ∑ _{l=0}^{N_{1}} a_{l,1}z^{l}, z ∈ ℂ, a_{N_{1} ,1} ≠ 0 be a complex non-zero polynomial, and let P_{2}(z) = ∑ _{l=0}^{N_{2}} a_{l,2}z^{l}, z ∈ ℂ, a_{N_{2} ,2} ≠ 0 be a complex non-zero polynomial so that N_{1} = dg(P_{1}) ⩾ 1 + dg(P_{2}) = 1 + N_{2} (we leave to the interested reader the analysis of the case N_{1} ⩽ N_{2}, in which we always have the existence of integrated solution resolvent (n, j)-propagation families for problem (2.13)). Set A := P_{1}(A′) and B := P_{2}(A′). Using the consideration given in [22, Example 3.8], we can prove that there exist numbers b > 0 and c > 0 such that

(2.16)  ∥(λB − A)^{−1} ∥ = O(e^{b|λ|^{1/(N_{1} −N_{2} )s} +c|λ|^{1/(N_{1} −N_{2} )}} ), λ ∈ ℂ,

and that, for every complex non-zero polynomial P (z) with dg(P) ⩽ N_{1}, there exists ζ > 0 such that

(2.17)  ∥(λB − A)^{−1} P (A′)f∥ ⩽ ζ∥f∥e^{b|λ|^{1/(N_{1} −N_{2} )s} +c|λ|^{1/(N_{1} −N_{2} )}} ,

for all λ ∈ ℂ and f ∈ D(P (A′)) (observe that the above estimates can be used in proving the existence of convoluted solutions of fractional analogs of equation (λ − Δ)u_{t} = αΔ − βΔ^{2} (α, β > 0, λ ∈ ℝ), in contrast with the assertions of [22, Theorems 4.2 and 4.3] which can be applied only in the case that λ > 0; as observed by G. A. Sviridyuk, this equation is important in evolution modeling of some problems appearing in the theory of liquid filtration, see e.g. [9, p. 6]). Let θ ∈ (0, π/2], b′ = (α_{n} − α)/(N_{1} − N_{2}) and let b′ ⩽ π/(π + 2θ). Owing to (2.16), (2.17) and Theorem 2.3, we obtain that there is a sufficiently large number a′ > 0 such that there exists an exponentially equicontinuous, analytic k_{a′,b′}-regularized I-resolvent (n, j)-propagation family (R_{n,j} (t))_{t⩾0} for problem (2.13), of angle θ, satisfying conditions (C.1)–(C.5). Denote, as before, T_{l,L} u(t) = BD_{t}^{α_{l}} u(t), t ⩾ 0 if l ∈ ℕ_{n} and α_{l} > 0, T_{l,R} u(t) = D_{t}^{α_{l}} Bu(t), t ⩾ 0 if l ∈ ℕ_{n}, T_{0,L} u(t) = AD_{t}^{α} u(t), t ⩾ 0 if α > 0, and T_{0,R} u(t) = D_{t}^{α} Au(t), t ⩾ 0. Let T_{l}u(t) be either T_{l,L} u(t) or T_{l,R} u(t) (l ∈ ℕ_{n}^{0}). Then it can be easily seen that for each x ∈ D(B) the function u(t) = R_{n,j} (t)x, t ⩾ 0 is a unique strong solution of the problem

T_{n}u(t) + ∑_{l=1}^{n−1} c_{l}T_{l}u(t) = T_{0}u(t) + (k_{a′,b′}^{(m_{n} )} ∗ g_{j+m_{n} −α_{n}} )(t), t ⩾ 0,

with all initial values chosen to be zeroes. Observe, finally, that the analysis contained in [29, Example 5.4] can be used for the construction of hypoanalytic exponentially equicontinuous k-regularized I-resolvent (n, j)-propagation families for the problem

∑_{l∈K} P_{l}(A′)D_{t}^{α_{l}} u(t) + ∑_{l∈ℕ_{n−1} ∖K} D_{t}^{α_{l}} P_{l}(A′)u(t) = 0, t ⩾ 0,

where P_{1}(z), …, P_{n−1} (z) are complex non-zero polynomials satysfying certain properties, and K ⊆ ℕ_{n−1} .

2.2. Exponentially equicontinuous (a, k) -regularized C-resolvent families generated by A, B ; exponentially equicontinuous (k; C) -regular-ized resolvent (i, j) -propagation families for (1.1)–(1.2). In this subsection, we shall mainly consider the C-wellposedness of problem (DFP)_{L} with A and B being closed linear operators on E. Set p_{B}(x) := p(x) + p(Bx), x ∈ D(B), p ∈ ⊛. Then the calibration (p_{B})_{p∈⊛} induces the Hausdorff sequentially complete locally convex topology on D(B). We shall denote this space simply by [D(B)].

Following the consideration given in [1, Section 2], we introduce the following definition.

Definition 2.5. Suppose that the functions a(t) and k(t) satisfy (P1), as well as that R(t) ∈ L(E, [D(B)]) for all t ⩾ 0. Let C ∈ L(E) be injective, and let CA ⊆ AC and CB ⊆ BC. Then the operator family (R(t))_{t⩾0} is said to be an exponentially equicontinuous (a, k)-regularized C-resolvent family generated by A, B iff there exists ω ⩾ max(0, abs(a), abs(k)) such that the following holds:

(i) The mappings t ↦ R(t)x, t ⩾ 0 and t ↦ BR(t)x, t ⩾ 0 are continuous for every fixed element x ∈ E.

(ii) The family {e^{−ωt} R(t) : t ⩾ 0} ⊆ L(E, [D(B)]) is equicontinuous, i.e., for every p ∈ ⊛, there exist c > 0 and q ∈ ⊛ such that

p(e^{−ωt} R(t)x) + p(e^{−ωt} BR(t)x) ⩽ cq(x), x ∈ E, t ⩾ 0.

(iii) For every λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0, the operator B − ã(λ)A is injective, R(C) ⊆ R(B − ã(λ)A) and

k̃(λ)(B − ã(λ)A)^{−1} Cx = ∫_{0}^{∞} e^{−λt} R(t)x dt, x ∈ E.

If k(t) = g_{r+1} (t) for some r ⩾ 0, then it is also said that (R(t))_{t⩾0} is an exponentially equicontinuous r-times integrated (a, C)-regularized resolvent family generated by A, B; an exponentially equicontinuous 0-times integrated (a, C)- regularized resolvent family generated by A, B is also said to be an exponentially equicontinuous (a, C)-regularized resolvent family generated by A, B.

Before going any further, it should be noticed that we have already constructed some examples of (g_{α} , k)-regularized C-resolvent families generated by A, B in Example 2.1(ii).

Remark 2.4. Suppose that the functions a(t) and k(t) satisfy (P1), as well as that CA ⊆ AC and CB ⊆ BC.

(i) It is clear that an exponentially equicontinuous (a, k)-regularized C-resolvent family generated by A, B, if exists, must be unique.

(ii) If for each λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0 the operator B commutes with (B − ã(λ)A)^{−1} C, then the operator family (BR(t))_{t⩾0} is an exponentially equicontinuous (a, k)-regularized C-resolvent family for (1.4), and the condition (ii.1) stated in the formulation of [22, Theorem 2.8] holds. Furthermore, for each t ⩾ 0 the operator BR(t) can be continuously extended from D(B) to the whole space E.

(iii) Assume that (R(t))_{t⩾0} is an exponentially equicontinuous (a, k)-regularized Cresolvent family for (1.4) and that there exists a strongly continuous operator family (R̂(t))_{t⩾0} ⊆ L(E) such that R̂(t)x = R(t)x, t ⩾ 0, x ∈ D(B) (the last condition automatically follows from the previous one if E is complete and B is densely defined; cf. [22, Remark 2.3(iv)]). If B^{−1} ∈ L(E) and BR(t) ⊆ R(t)B, t ⩾ 0, then (R(t)B^{−1} )_{t⩾0} is an exponentially equicontinuous (a, k)-regularized C-resolvent family generated by A, B.

The proof of the following theorem can be deduced by using slight modifications of the proofs of [1, Proposition 2.1, Lemma 2.2] and the fact that the assertion of [21, Lemma 2.4] continues to hold in SCLCSs.

Theorem 2.5. Let (R(t))_{t⩾0} be an exponentially equicontinuous (a, k)-regular-ized C-resolvent family generated by A, B, and let abs(|a|) < ∞. Then the following holds:

(i) For every x ∈ E and for every λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0, we have

k̃(λ)B(B − ã(λ)A)^{−1} Cx = ∫_{0}^{∞} e^{−λt} BR(t)x dt.

(ii) R(t)Bx = k(t)Cx + ∫_{0}^{t} a(t − s)R(s)Ax ds, t ⩾ 0, x ∈ D(A) ∩ D(B). (iii) ∫_{0}^{t} a(t − s)R(s)x ds ∈ D(A) ∩ D(B), t ⩾ 0, x ∈ E. (iv) BR(t)x = k(t)Cx + A ∫_{0}^{t} a(t − s)R(s)x ds, t ⩾ 0, x ∈ E.

(v) R(t)B(D(A) ∩ D(B)) ⊆ D(A) ∩ D(B), t ⩾ 0. (vi) B(B −ã(λ)A)^{−1} CAx = A(B −ã(λ)A)^{−1} CBx for every x ∈ D(A)∩D(B) and

for every λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0; AR(t)Bx = BR(t)Ax, t ⩾ 0, x ∈ D(A) ∩ D(B).

(vii) Suppose that the function k(t) is differentiable at a point t_{0} ⩾ 0 and that a ∈ AC_{loc}([0, ∞)). If λ ∈ ℂ satisfies Reλ > ω and k̃(λ) ≠ 0, then for every j ∈ ℕ_{0}, z ∈ ℂ and for every complex polynomial P (·), we have

( \frac{d}{dt}[(z(B − ã(λ)A)^{−1} C − P (C))^{j} R(t)Bx])_{t=t_{0}}

= (z(B − ã(λ)A)^{−1} C − P (C))^{j} ( \frac{d}{dt}R(t)Bx)_{t=t_{0}} .

(viii) Let x ∈ D(A) ∩ D(B). Then the function t ↦ u(t), t ⩾ 0, defined by u(t) := R(t)Bx, t ⩾ 0 satisfies u ∈ C([0, ∞) : [D(A)]) ∩ C([0, ∞) : [D(B)]) and

Bu(t) = k(t)CBx + ∫_{0}^{t} a(t − s)Au(s)ds, t ⩾ 0.

Remark 2.5. (i) Suppose that x ∈ D(A)∩D(B), α > 0 and there exists an exponentially equicontinuous (g_{α} , C)-regularized resolvent family (R(t))_{t⩾0} generated by A, B. Using the identity R(t)Bx = Cx+∫_{0}^{t} g_{α} (t−s)R(s)Ax ds, t ⩾ 0, it readily follows that the mapping t ↦ R(t)Bx, t ⩾ 0 is (m − 1)-times continuously differentiable on [0, ∞), where m = ⌈α⌉. Furthermore, it can be easily verified that the Caputo derivative D_{t}^{α} R(t)Bx is well defined as well as that D_{t}^{α} R(t)Bx = R(t)Ax, t ⩾ 0. Keeping in mind Remark 2.2(ii) and Proposition 2.5(vi), we get that the function u(t) := R(t)Bx, t ⩾ 0 is a unique solution of the following Cauchy problem:

u ∈ C([0, ∞) : [D(A)]) ∩ C([0, ∞) : [D(B)]) ∩ C^{m−1} ([0, ∞) : E),
BD_{t}^{α} u(t) = Au(t), t ⩾ 0,
u(0) = Cx; u^{(j)} (0) = 0, 1 ⩽ j ⩽ m − 1.

In Theorem 2.6 below, we extend this result to the class of exponentially equicontinuous (g_{α} , g_{αl+1} )-regularized C-resolvent families generated by A, B (l ∈ℕ).

(ii) Now we would like to illustrate the conclusion deduced in the first part of this remark to degenerate fractional equations associated with the abstract differential operators [20,57]. For the sake of simplicity, we shall only consider the equations of order α ∈ (0, 2); cf. [22, Subsection 4.1] for further information concerning the case α = 2. Assume that n ∈ ℕ and iA_{j} , 1 ⩽ j ⩽ n are commuting generators of bounded C_{0}-groups on a Banach space E. Suppose again that 0 < α < 2, ω ⩾ 0, P_{1}(x) and P_{2}(x) are non-zero complex polynomials, N_{1} = dg(P_{1}(x)), N_{2} = dg(P_{2}(x)), β > \frac{n}{2} \frac{(N_{1} +N_{2} )}{min (1,α)} (resp. β ⩾ n|\frac{1}{p} − \frac{1}{2}|\frac{(N_{1} +N_{2} )}{min (1,α)} , if E = L^{p}(ℝ^{n}) for some 1 < p < ∞), P_{2}(x) ≠ 0, x ∈ ℝ^{n} and that (2.14) holds with δ replaced by α. Define (R_{α} (t))_{t⩾0} as in (2.15), with δ replaced by α; C ≡ R_{α} (0). Then we know that (R_{α} (t))_{t⩾0} ⊆ L(E) is a global exponentially bounded (g_{α} , R_{α} (0))-regularized resolvent family for the problem

(P)_{R} : \{ \begin{matrix}D_{t}^{α} \overline{P_{2}(A)}u(t) = \overline{P_{1}(A)}u(t), t ⩾ 0, \\ u(0) = Cx; u^{(j)} (0) = 0, 1 ⩽ j ⩽ ⌈α⌉ − 1; \end{matrix}

cf. Definition 1.1 with a(t) = g_{α} (t) and k(t) = 1. Furthermore, the analysis contained in [22, Remark 4.4(i)] implies that there exists an exponentially bounded, strongly continuous operator family (G_{α} (t))_{t⩾0} such that G_{α} (t)x = \overline{P_{2}(A)}^{−1} R_{α} (t)x, t ⩾ 0, x ∈ E and λ^{α−1} (λ^{α} B − A)^{−1} Cx = ∫_{0}^{∞} e^{−λt} G_{α} (t)x dt for any x ∈ E and λ > 0 sufficiently large. Hence, (G_{α} (t))_{t⩾0} is an exponentially equicontinuous (g_{α} , C)-regularized resolvent family generated by \overline{P_{1}(A)}, \overline{P_{2}(A)}, so that for each f ∈ D(\overline{P_{1}(A)}) ∩ D(\overline{P_{2}(A)}), the function u(t) := R_{α} (t)x, t ⩾ 0 is a unique solution of the following Cauchy problem:

(P)_{L} : \{ \begin{matrix}u ∈ C([0, ∞) : [D(\overline{P_{1}(A)})]) ∩ C([0, ∞) : [D(\overline{P_{2}(A)})]), \\ \overline{P_{2}(A)}D_{t}^{α} u(t) = \overline{P_{1}(A)}u(t), t ⩾ 0, \\ u(0) = Cx; u^{(j)} (0) = 0, 1 ⩽ j ⩽ ⌈α⌉ − 1. \end{matrix}

The consideration is quite similar in the case that the requirements of [22, Theorem 4.3] hold (cf. also [22, Remark 4.4(ii)] and [22, Remark 4.5], which enables us to consider the wellposedness of problem (P)_{L} in E_{l}-type spaces [54]).

We shall employ the following auxiliary lemma in the proof of Theorem 2.6 mentioned above.

Lemma 2.1. (cf. [20, Corollary 2.1.20]) Suppose α > 0, l ∈ ℕ, z ∈ ℂ, A is a subgenerator of an exponentially equicontinuous (g_{α} , g_{lα+1} )-regularized C-resolvent family (S_{l,α} (t))_{t⩾0} on E, z − A is injective, R(C) ⊆ R((z − A)^{l}) and (z − A)^{−1} C ∈ L(E), …, (z − A)^{−l} C ∈ L(E). Set, for every x ∈ E and t ⩾ 0,

S_{α} (t)x := (−1)^{l}S_{l,α} (t)x + ∑_{j=0}^{l−1} (−1)^{j+1} \binom{l}{j} z^{l−j} [L^{−1} ( \frac{r^{αj}}{(r^{α} − z)^{l}} ) ∗ S_{l,α} (·)x](t)

+ ∑_{j=1}^{l} (−1)^{l−j} L^{−1} ( \frac{r^{α−1}}{(r^{α} − z)^{l+1−j}} ) (t)(z − A)^{−j} Cx.

Then (S_{α} (t))_{t⩾0} is an exponentially equicontinuous (g_{α} , (z − A)^{−l} C)-regularized resolvent family with a subgenerator A.

Now we state the following important extension of [1, Theorem 2.2] (cf. the forthcoming monograph [26] for more details about applications of Theorem 2.6 in the study of analytical solutions to fractional Barenblatt–Zheltov–Kochina equation in finite domains).

Theorem 2.6. Suppose that α > 0, l ∈ ℕ, z ∈ ℂ, there exists an exponentially equicontinuous (g_{α} , g_{lα+1} )-regularized C-resolvent family (S_{l,α} (t))_{t⩾0} generated by A, B, the operator zB−A is injective and x ∈ D(A)∩D(B)∩D(((zB−A)^{−1} B)^{l}C). Define

u(t) := (−1)^{l}S_{l,α} (t)Bx + ∑_{j=0}^{l−1} (−1)^{j+1} \binom{l}{j} z^{l−j} [L^{−1} ( \frac{r^{αj}}{(r^{α} − z)^{l}} ) ∗ S_{l,α} (·)Bx](t)

+ ∑_{j=1}^{l} (−1)^{l−j} L^{−1} ( \frac{r^{α−1}}{(r^{α} − z)^{l+1−j}} ) (t)((zB − A)^{−1} B)^{j} Cx, t ⩾ 0 .

Then the function u(t) is a unique solution of the problem (DFP)_{L} with f(t) ≡ 0 and the initial value x replaced by ((zB−A)^{−1} B)^{l}Cx (we will designate this problem by (DFP)_{L,l} in the sequel).

Proof. The uniqueness of solutions follows similarly as in Remark 2.5(i) and we shall only prove that the function u(t) is a solution of the problem (DFP)_{L,l} .

Denote x_{j} := ((zB − A)^{−1} B)^{j} Cx (j ∈ ℕ_{l}^{0}), F_{j,l} (t) := L^{−1} (\frac{r^{αj}}{(r^{α} −z)^{l}} )(t), t > 0

(0 ⩽ j ⩽ l − 1) and G_{j,l} (t) := L^{−1} (\frac{r^{α−1}}{(r^{α} −z)^{l+1−j}} )(t), t ⩾ 0 (1 ⩽ j ⩽ l). Then the function F_{j,l} (t) is continuous on (0, ∞), locally integrable on [0, ∞) and exponentially bounded on [1, ∞) (0 ⩽ j ⩽ l − 1), while the function G_{j,l} (t) is continuous and exponentially bounded on [0, ∞) (1 ⩽ j ⩽ l); cf. [20]. Set m := ⌈α⌉. By Theorem 2.5(ii), we have that the mapping t ↦ S_{l,α} (t)Bx, t ⩾ 0 is (m − 1)-times continuously differentiable and

\frac{d^{m−1}}{dt^{m−1}} S_{l,α} (t)Bx = g_{αl+2−m} (t)Cx + ∫_{0}^{t} g_{α+1−m} (t − s)S_{l,α} (s)Ax ds, t ⩾ 0;

hence (\frac{d^{j}}{dt^{j}} S_{l,α} (t)Bx)_{t=0} = 0, 0 ⩽ j ⩽ m − 1. This simply implies that the mapping t ↦ [F_{j,l} ∗ S_{l,α} (·)Bx](t), t ⩾ 0 is (m − 1)-times continuously differentiable as well as that

\frac{d^{m−1}}{dt^{m−1}} [F_{j,l} ∗ S_{l,α} (·)Bx](t) = [F_{j,l} ∗ \frac{d^{m−1}}{dt^{m−1}} S_{l,α} (·)Bx](t), t ⩾ 0,

provided 0 ⩽ j ⩽ l − 1; hence, (\frac{d^{p}}{dt^{p}} [F_{j,l} ∗ S_{l,α} (·)Bx](t))_{t=0} = 0, 0 ⩽ p ⩽ m − 1 (0 ⩽ j ⩽ l − 1). Now it is not difficult to prove that

D_{t}^{α} S_{l,α} (t)Bx = g_{lα+1−α} (t)Cx + S_{l,α} (t)Ax, t ⩾ 0,

D_{t}^{α} [F_{j,l} ∗ S_{l,α} (·)Bx](t) = g_{lα+1−α} (t)Cx + S_{l,α} (t)Ax, t ⩾ 0 (0 ⩽ j ⩽ l − 1).

Suppose, for the time being, that the assumptions of Lemma 2.1 hold. Since for each x ∈ D(A) the function v(t) := S_{α} (t)x, t ⩾ 0 is a unique solution of the problem

\{ \begin{matrix}D_{t}^{α} v(t) = Av(t), t ⩾ 0, \\ v(0) = Cx; v^{(j)} (0) = 0, 1 ⩽ j ⩽ m − 1, \end{matrix}

we may conclude from the above (by plugging l = 1, 2, … successively in Lemma 2.1) that for each j ∈ ℕ_{l} the function G_{j,l} (t) is (m − 1)-times continuously differentiable on [0, ∞) as well as that (\frac{d^{p}}{dt^{p}} G_{j,l} (t))_{t=0} = 0, 1 ⩽ p ⩽ m − 1 (1 ⩽ j ⩽ l) and that the Caputo derivative D_{t}^{α} G_{j,l} (t) is well defined (1 ⩽ j ⩽ l). Since G_{l,l} (t) = E_{α} (zt^{α}), t ⩾ 0, it readily follows that the function u(t) satisfies u(0) = x_{l} and u^{(j)} (0) = 0, 1 ⩽ j ⩽ m − 1. It remains to be proved that BD_{t}^{α} = Au(t), t ⩾ 0. Carrying out a straightforward computation, it can be easily seen that this equality holds iff

(−1)^{l}g_{αl+1−α} (t)BCx + ∑_{j=0}^{l−1} (−1)^{j+1} \binom{l}{j} z^{l−j} [F_{j,l} ∗ g_{αl+1−α} ](t)BCx

+ ∑_{j=1}^{l} (−1)^{l−j} D_{t}^{α} G_{j,l} (t)Bx_{j} = ∑_{j=1}^{l} (−1)^{l−j} G_{j,l} (t)Ax_{j} , t ⩾ 0

iff

(−1)^{l}g_{αl+1−α} (t)BCx + ∑_{j=0}^{l−1} (−1)^{j+1} \binom{l}{j} z^{l−j} [F_{j,l} ∗ g_{αl+1−α} ](t)BCx

+ ∑_{j=1}^{l} (−1)^{l−j} D_{t}^{α} G_{j,l} (t)Bx_{j} = ∑_{j=1}^{l} (−1)^{l−j} G_{j,l} (t)[zBx_{j} − Bx_{j−1} ], t ⩾ 0.

This is true because the coefficients of Bx_{j}, for every fixed number j ∈ ℕ_{l}^{0}, on both sides of previous equality are equal (cf. also the proof of [20, Theorem 2.1.19]). □

Suppose that the operator B is injective, x ∈ D(AB^{−1} ), α > 0 and there exists an exponentially equicontinuous (g_{α} , C)-regularized resolvent family (R(t))_{t⩾0} generated by A, B. Then it can be easily checked that the function u(t) := R(t)x, t ⩾ 0 is a unique solution of problem (DFP)_{L} with f(t) ≡ 0 and the initial value x replaced by CB^{−1} x. We leave to the interested reader the problem of transferring this conclusion, as well as the other ones from [1, Remark 2.4], to degenerate fractional equations whose solutions are goverened by (g_{α} , g_{αl+1} )-regularized Cresolvent families generated by A, B (l ∈ ℕ).

Assume now that n ∈ ℕ ∖ {1}, 0 ⩽ α_{1} < ⋯< α_{n−1} , and A_{1}, …, A_{n−1} are closed linear operators on E. In the analysis of existence and uniqueness of integral equations associated with the problem (1.1)–(1.2), we can also use the notion of an exponentially equicontinuous (analytic) (k; C)-regularized resolvent (i, j)-propagation family.

Definition 2.6. (cf. Definition 2.3 and Definition 2.4) Suppose that the function k(t) satisfies (P1), as well as that 1 ⩽ i ⩽ n − 1, 0 ⩽ j ⩽ m_{i} − 1 and R_{i,j} (t) ∈ L(E, [D(A_{i})]) for all t ⩾ 0. Let the operator C ∈ L(E) be injective.

(i) Then the operator family (R_{i,j} (t))_{t⩾0} is said to be an exponentially equicontinuous (k; C)-regularized resolvent (i, j)-propagation family for problem (1.1)– (1.2) iff there exists ω ⩾ max(0, abs(k)) such that the following holds:

(a) The mappings t ↦ R_{i,j} (t)x, t ⩾ 0 and t ↦ A_{i}R_{i,j} (t)x, t ⩾ 0 are continuous for every fixed element x ∈ E.

(b) The family {e^{−ωt} R_{i,j} (t) : t ⩾ 0} ⊆ L(E, [D(A_{i})]) is equicontinuous, i.e., for every p ∈ ⊛, there exist c > 0 and q ∈ ⊛ such that

p(e^{−ωt} R_{i,j} (t)x) + p(e^{−ωt} A_{i}R_{i,j} (t)x) ⩽ cq(x), x ∈ E, t ⩾ 0.

(c) For every λ ∈ ℂ with Reλ > ω and k̃(λ) ≠ 0, the operator P_{λ} is injective, R(C) ⊆ R(P_{λ}) and

λ^{α_{i} −α_{n−1} −j} k̃(λ)P_{λ}^{−1} Cx = ∫_{0}^{∞} e^{−λt} R_{i,j} (t)x dt, x ∈ E.

(ii) Let (R_{i,j} (t))_{t⩾0} be an exponentially equicontinuous (k; C)-regularized resolvent (i, j)-propagation family for problem (1.1)–(1.2). Then it is said that (R_{i,j} (t))_{t⩾0} is an exponentially equicontinuous (equicontinuous), analytic (k; C)- regularized resolvent (i, j)-propagation family for problem (1.1)–(1.2), of angle α, iff the following holds:

(a) For every x ∈ E, the mappings t ↦ R_{i,j} (t)x, t > 0 and t ↦ A_{i}R_{i,j} (t)x, t > 0 can be analytically extended to the sector Σ_{α}; since no confusion seems likely, we shall denote these extensions by the same symbols.

(b) For every x ∈ E and β ∈ (0, α), one has lim_{z→0,z∈Σ_{β}} R_{i,j} (z)x = R_{i,j} (0)x and lim_{z→0,z∈Σ_{β}} A_{i}R_{i,j} (z)x = A_{i}R_{i,j} (0)x.

(c) For every β ∈ (0, α), there exists ω_{β} ⩾ max(0, abs(k)) (ω_{β} = 0) such that the family {e^{−ω_{β} z} R_{i,j} (z) : z ∈ Σ_{β}} ⊆ L(E, [D(A_{i})]) is equicontinuous, i.e., for every p ∈ ⊛, there exist c > 0 and q ∈ ⊛ such that

p(e^{−ω_{β} z} R_{i,j} (z)x) + p(e^{−ω_{β} z} A_{i}R_{i,j} (z)x) ⩽ cq(x), x ∈ E, z ∈ Σ_{β}.

Exponentially equicontinuous (analytic) (k; C)-regularized resolvent (i, j)-prop-agation families yield results very similar to those obtained by k-regularized Cresolvent (i, j)-propagation families. Without going into a deeper analysis, we shall only observe that the assertions of Theorem 2.1(i)–(iii), Remark 2.2(i), (iii), Theorem 2.3 and Theorem 2.4 can be restated for exponentially equicontinuous (k; C)-regularized resolvent (i, j)-propagation families. Details can be left to the interested reader.

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Faculty of Technical Sciences (Received 11 10 2014) University of Novi Sad, Novi Sad, Serbia

    marco.s@verat.net