ABSTRACT DEGENERATE VOLTERRA INCLUSIONS IN LOCALLY CONVEX SPACES

DEGENERATE VOLTERRA INCLUSIONS IN LOCALLY CONVEX SPACES


Introduction and preliminaries
The main aim of this paper is to analyze the abstract degenerate Volterra integrodifferential equations in sequentially complete locally convex spaces by using multivalued linear operators (cf. [68] and [36] for a comprehensive survey of results on abstract non-degenerate Volterra equations), as well as to introduce a new theoretical approach to the Laplace transform of functions with values in sequentially complete locally convex spaces. To outline the motivation of our research, let us mention that there exists only a few published papers in the existing literature treating the abstract degenerate Volterra equations ( [16], [18]- [21], [31]) and the abstract degenerate fractional inclusions associated with the use of Caputo fractional derivatives ( [45]- [49], [51]). In this paper, we make an attempt to perform the first systematic exploration of abstract degenerate Volterra equations and abstract degenerate fractional differential equations in locally convex spaces, contributing also to the theories of abstract degenerate differential equations of first and second order (for pioneering results about semigroups of operators in locally convex spaces, we refer the reader to the papers [33,34,82]). A great number of our results seems to be new even in the Bahach space setting.
The organization and main ideas of this paper can be briefly described as follows. In the second section of paper, we will take a preliminary and incomplete look at the multivalued linear operators in locally convex spaces; for more details, we refer the reader to the monographs [9,17]. We introduce the notion of a C-resolvent of a multivalued linear operator, reconsider the assertions from [17, Chapter I] and state a generalization of [36, Proposition 2.1.14] for C-resolvents of multivalued linear operators. Following the approach of Knuckles and Neubrander [32], we introduce the notion of a relatively closed multivalued linear operator in locally convex space. The generalized resolvent equations continue to hold in our framework.
As mentioned in [36,Section 1.2], only a few noteworthy facts has been said about the Laplace transform of functions with values in sequentially complete locally convex spaces. In Section 3, we propose a new theoretical approach to the Laplace transform of functions with values in sequentially complete locally convex spaces. This concept extends the corresponding one introduced by Xiao and Liang ( [80], 1997), and coincides with the classical concept of vector-valued Laplace transform in the case that the state space X is one of Banach's [1]. Concerning the integration of functions with values in sequentially complete locally convex spaces, we follow the approach of Martinez and Sanz (cf. [61, pp. 99-102] for more details); for Pettis integration in locally convex spaces and some applications to abstract differential inclusions of first order, we refer the reader to [28,29,56]. Once we have proved the formula for partial integration in Theorem 3.1, we have an open door to consider various operational properties of Laplace transform by using the methods already known in the Banach space case. The non-possibility of establishing Fubini-Tonelli theorem in this concept of integration additionally hinders our research and does not able us to fully transfer some assertions from the Banach space case to the general locally convex space case; for example, in Theorem 3.3(vi) we consider the Laplace transform of finite convolution product and there it is almost inevitable to impose the condition that the function f (t) is continuous.
A large number of research papers, starting presumably with that of Yagi [81], written over the last twenty five years, have concerned applications of multivalued linear operators to abstract degenerate differential equations (cf. [8], [13], [17] and [63]- [65] for the primary source of information on this subject). In Section 4, we analyze the abstract degenerate Volterra inclusion where a ∈ L 1 loc ([0, τ )), a = 0, A : X → P (Y ) and B : X → P (Y ) are given multivalued linear operators acting between sequentially complete locally convex spaces X and Y , and F : X → P (Y ) is a given mutivalued mapping, as well as the fractional Sobolev inclusions D α t Bu(t) ∈ Au(t) + F(t), t ≥ 0, (Bu) (j) (0) = Bx j , 0 ≤ j ≤ α − 1, (1.2) where we assume that B = B is single-valued, and Here, D α t u(t) denotes the Caputo fractional derivative of function u(t). We define various types of solutions of problems (1.1), (1.2) and (1.3). In Theorems 4.3 and 4.5, we reconsider the main results of research of Kim [31], while in Theorem 4.6 we prove an extension of [32,Theorem 3.5] for abstract degenerate fractional differential inclusions. Subordination principles are clarified in Theorem 4.8 and Theorem 4.9 following the methods proposed by Prüss [68,Section 4] and Bazhlekova [5, Section 3] (cf. [22] and [42]- [46] for similar results known in degenerate case).
Following the old ideas of deLaubenfels [11], in Section 5 we introduce and analyze the class of (a, k)-regularized (C 1 , C 2 )-existence and uniqueness families (cf. [36,Section 2.8] for non-degenerate case). Later on, we single out the class of (a, k)regularized C-resolvent families for special considerations. We focus our attention on the analysis of Hille-Yosida's type theorems for (a, k)-regularized C-resolvent families generated by multivalued linear operators (as in all previous researches of non-degenerate case, we introduce the notion of a subgenerator of an (a, k)regularized C-resolvent family and investigate the most important properties of subgenerators; our analysis is based on the use of vector-valued Laplace transform). It is well known (see e.g. [17,Theorem 2.4], [32,Theorem 3.6] and [31, p. 169]) that Hille-Yosida's type estimates for the resolvent of a multivalued operator A immediately implies that A is single-valued in a certain sense. In part (ii) of Theorem 5.12, we will prove a similar assertion provided that the Hille-Yosida condition (5.17) below holds. For the validity of Theorem 5.12(ii), we have found the condition k(0) = 0 very important to be satisfied; in other words, the existence of above-mentioned single-valued branch of A can be proved exactly in non-convoluted or non-integrated case, so that we have arrived to a diametrically opposite conclusion to that stated on l. 7-13, p. 169 of [31]. Nevertheless, the existence or non-existence of such a single-valued branch of A is not sufficient for obtaining a fairly complete information on the well-posedness of inclusion (1.1) with B = I (the reading of papers [31,32] has strongly influenced us to write this paper, and compared with the results of [31], here we do not need the assumption that a(t) is a normalized function of local bounded variation). In the remainder of Section 5, we enquire into the possibility to extend the most important results from [36, Section 2.1, Section 2.2] to (a, k)-regularized C-resolvent families generated by multivalued linear operators, and present several examples and possible applications of our abstract theoretical results. We clarify the complex characterization theorem for the generation of exponentially equicontinuous (a, k)-regularized C-resolvent families, the generalized variation of parameters formula, and subordination principles; in two separate subsections, we analyze differential and analytical properties of (a, k)-regularized C-resolvent families as well as the case in which some of the regularizing operators C and C 2 is not injective. We provide several illustrative examples, including applications to fractional Maxwell's equations, fractional linearized Benney-Luke equation and backward Poisson heat equation.
Because of some similarity with our previous researches of non-degenerate case, we have decided to write this paper in a half-expository manner, including only the most relevant details of proofs of our structural results. The author would like to express his appreciation and sincere thanks to Prof. Vladimir Fedorov (Chelyabinsk, Russia) and Prof. Rodrigo Ponce (Talca, Chile) for many stimulating and enlightening discussions during the research.
We use the standard terminology throughout the paper. By X we denote a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short. If Y is also an SCLCS over the same field of scalars as X, then we denote by L(X, Y ) the space consisting of all continuous linear mappings from X into Y ; L(X) ≡ L(X, X). By X ( , if there is no risk for confusion), we denote the fundamental system of seminorms which defines the topology of X; the fundamental system of seminorms which defines the topology on an arbitrary SCLCS Z is denoted by Z . The symbol I X (I Y ) denotes the identity operator on X (Y ); if there is no risk for confusion, then we also write I in place of I X . By X * we denote the dual space of X. Let 0 < τ ≤ ∞. A strongly continuous operator family (W (t)) t∈[0,τ ) ⊆ L(X, Y ) is said to be locally equicontinuous if and only if, for every T ∈ (0, τ ) and for every p ∈ Y , there exist q p ∈ X and c p > 0 such that p(W (t)x) ≤ c p q p (x), x ∈ X, t ∈ [0, T ]; the notions of equicontinuity of (W (t)) t∈[0,τ ) and the exponential equicontinuity of (W (t)) t≥0 are defined similarly. Notice that (W (t)) t∈[0,τ ) is automatically locally equicontinuous in case that the space X is barreled ( [62]).
By B we denote the family consisting of all bounded subsets of X. Define p B (T ) := sup x∈B p(T x), p ∈ Y , B ∈ B, T ∈ L(X, Y ). Then p B (·) is a seminorm on L(X, Y ) and the system (p B ) (p,B)∈ Y ×B induces the Hausdorff locally convex topology on L(X, Y ). If Y is continuously embedded in X, we will use the notation Y → X. Suppose that A is a closed linear operator acting on X. Then we denote the domain, kernel space and range of A by D(A), N (A) and R(A), respectively. Since no confusion seems likely, we will identify A with its graph. Set p A (x) := p(x) + p(Ax), x ∈ D(A), p ∈ . Then the calibration (p A ) p∈ induces the Hausdorff sequentially complete locally convex topology on D(A); we denote this space simply by [D(A)].
Suppose that V is a general topological vector space (the consistent and stable theory of abstract degenerate Volterra integro-differential equations in non-locally convex spaces has not been yet created; see [26] for some results established in non-degenerate case). As it is well-known, a function f : Ω → V , where Ω is an open non-empty subset of C, is said to be analytic if it is locally expressible in a neighborhood of any point z ∈ Ω by a uniformly convergent power series with coefficients in V . The reader may consult [1], [36, Section 1.1] and references cited there for the basic information about vector-valued analytic functions. In our framework, the analyticity of a mapping f : Ω → X is equivalent with its weak analyticity.
A function f : [0, T ] → X, where 0 < T < ∞, is said to be Hölder continuous with the exponent r ∈ (0, 1] if for each p ∈ X there exists M ≥ 1 such that p(f (t) − f (s)) ≤ M |t − s| r , provided 0 ≤ t, s ≤ T , while a function f : [0, ∞) → X is said to be locally Hölder continuous with the exponent r if its restriction on any finite interval [0, T ] is Hölder continuous with the same exponent. By AC loc ([0, ∞)) we denote the space consisting of all functions f : [0, ∞) → X whose restriction on any finite interval [0, T ] (T > 0) is absolutely continuous.
Let 0 < τ ≤ ∞ and a ∈ L 1 loc ([0, τ )). Then we say that the function a(t) is a kernel on [0, τ ) if for each f ∈ C([0, τ )) the assumption t 0 a(t − s)f (s) ds = 0, t ∈ [0, τ ) implies f (t) = 0, t ∈ [0, τ ). 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 g ζ (t) := t ζ−1 /Γ(ζ) (ζ > 0, t > 0), g 0 (t) := δ-distribution and, by common consent, 0 ζ := 0. For any angle α ∈ (0, π], we define Σ α := {z ∈ C : z = 0, | arg(z)| < α}. Set C + := {λ ∈ C : λ > 0}. Now we repeat some basic facts and definitions about integration of functions with values in SCLCSs. Unless stated otherwise, by Ω we denote a locally compact, separable metric space and by µ we denote a locally finite Borel measure defined on Ω. A function f : Ω → X is said to be µ-measurable if and only if there exists a sequence (f n ) in X Ω of simple functions (cf. [36, Definition 1.1.1(i)] for the notion) such that lim n→∞ f n (t) = f (t) for a.e. t ∈ Ω. Definition 1.1. Let K ⊆ Ω be a compact set, and let a function f : K → X be strongly measurable. Then it is said that f (·) is (µ-)integrable if there is a sequence (f n ) n∈N of simple functions such that lim n→∞ f n (t) = f (t) a.e. t ∈ K and for all > 0 and each p ∈ there is a number n 0 = n 0 ( , p) such that In this case we define From (1.4), we have that (p(f n )) n∈N is a Cauchy sequence in the space L 1 (K, µ), so that the limit p(f ) = lim n→∞ p(f n ) is µ-integrable. Similarly we can prove that each function p(f n − f ) is µ-integrable and the sequence of its corresponding integrals converges to zero. Recall that every continuous function f : K → X is µ-integrable. Definition 1.2. (i) A function f : Ω → X is said to be locally µ-integrable if, for every compact set K ⊆ Ω, the restriction f |K : K → X is µ-integrable.
(ii) A function f : Ω → X is said to be µ-integrable if it is locally integrable and if additionally (1.5) If this is the case, we define with (K n ) n∈N being an expansive sequence of compact subsets of Ω with the property that n∈N K n = Ω.
The above definition does not depend on the choice of sequence (K n ) n∈N . Moreover, (1.6) Definition 1.2 is equivalent with the definition of Bochner integral, provided that X is a Banach space. Furthermore, every continuous function f : Ω → X satisfying (1.5) is µ-integrable and the following holds.
is a sequence of µ-integrable functions from X Ω and (f n ) converges pointwise to a function f : Ω → X. Assume that, for every p ∈ , there exists a µ-integrable (ii) Let Y be a SCLCS, and let T : X → Y be a continuous linear mapping.
Recent decades have witnessed a fast growing applications of fractional calculus and fractional differential equations to diverse scientific and engineering fields (cf. [5,14,30,67,70] and references cited therein for further information). In this paper, we mainly use the Caputo fractional derivatives. Let ζ > 0. Then the Caputo fractional derivative D ζ t u [5,36] is defined for those functions u ∈ C ζ −1 ([0, ∞) : Define C r ([0, T ] : X) to be the vector space consisting of Hölder continuous functions f : [0, T ] → X with the exponent r; if r ∈ (0, ∞) \ N, then we define C r ([0, T ] : X) as the vector space consisting of those functions f : [0, T ] → X for which f ∈ C r ([0, T ] : X) and f ( r ) ∈ C r − r ([0, T ] : X). Without going into further details, we will only observe here that the existence of Caputo fractional derivative D ζ t u implies u ∈ C ζ ((0, ∞) : X) ∩ C ζ ([0, T ] : E), for each finite number T > 0. A proof is left to the interested reader.
We refer the reader to [5] for the notion of a Riemann-Liouville fractional deriv- , z ∈ C.

Multivalued linear operators in locally convex spaces
A multivalued map (multimap) A : X → P (Y ) is said to be a multivalued linear operator (MLO) if and only if the following holds: If X = Y , then we say that A is an MLO in X. An almost immediate consequence of definition is that Ax + Ay = A(x + y) for all x, y ∈ D(A) and λAx = A(λx) for all x ∈ D(A), λ = 0. Furthermore, for any x, y ∈ D(A) and λ, η ∈ C with |λ| + |η| = 0, we have λAx + ηAy = A(λx + ηy). If A is an MLO, then A0 is a linear manifold in Y and Ax = f + A0 for any x ∈ D(A) and f ∈ Ax. Set Suppose that X is a linear subspace of X, and A : X → P (Y ) is an MLO. Then we define the restriction of operator A to the subspace X , A |X for short, by D(A |X ) := D(A) ∩ X and A |X x := Ax, x ∈ D(A |X ). Clearly, A |X : X → P (Y ) is an MLO. It is well known that an MLO A : X → P (Y ) is injective (resp., single-valued) if and only if A −1 A = I |D(A) (resp., AA −1 = I Y |R(A) ). The integer powers of an MLO A : X → P (X) are defined recursively as follows: We can prove inductively that ( We say that an MLO operator A : X → P (Y ) is closed if for any nets (x τ ) in D(A) and (y τ ) in Y such that y τ ∈ Ax τ for all τ ∈ I we have that the suppositions lim τ →∞ x τ = x and lim τ →∞ y τ = y imply x ∈ D(A) and y ∈ Ax.
We introduce the notion of a relatively closed MLO as follows [32]. We say that an MLO A : X → P (Y ) is relatively closed if and only if there exist auxiliary and y ∈ Ax. A relatively closed operator will also be called X A × Y A -closed. For example, let A, B : D ⊆ X → Y be closed linear operators with the same domain D. Then the operator A + B is not necessarily closed but it is always [D(A)] × Y -closed (cf. [31, p. 170]). Examples presented in [32] can be simply reformulated for operators acting on locally convex spaces, as well: This shows that any MLO has a closed linear extension, in contrast to the usually considered single-valued linear operators. ( and the topology on X S is induced by the system (s p,q,r ) of fundamental seminorms, defined as follows: s p,q,r (x) =: p(x)+p(Ax)+q(x)+r(Ax), x ∈ X S (p ∈ X , q ∈ X B , r ∈ and the topology on X C is induced by the system (s p,q ) of fundamental seminorms, defined as follows: s p,q (x) =: p(x)+p(Ax)+q(Ax), x ∈ X C (p ∈ X , q ∈ Y B ).
(iv) Let A : D(A) ⊆ X → Y and B : D(B) ⊆ X → Y be two single-valued linear operators. Set Then A is an MLO in X, and the following holds: (a) If one of the operators A, B is bounded and the other closed, then A is closed.
If A : X → P (Y ) is an MLO, then we define the adjoint A * : Y * → P (X * ) of A by its graph A * := y * , x * ∈ Y * × X * : y * , y = x * , x for all pairs (x, y) ∈ A .
It is simply verified that A * is a closed MLO, and that y * , y = 0 whenever y * ∈ D(A * ) and y ∈ A0. Furthermore, A * is single-valued provided that A is In the remaining part of this section, we will analyze the C-resolvent sets of multivalued linear operators in locally convex spaces. Our standing assumptions will be that A is an MLO in X, as well as that C ∈ L(X) is injective (the only exception will be Subsection 5.2, where C can be possibly non-injective) and CA ⊆ AC (this is equivalent to say that, for any (x, y) ∈ X × X, we have the implication (x, y) ∈ A ⇒ (Cx, Cy) ∈ A; by induction, we immediately get that CA k ⊆ A k C for all k ∈ N). Then the C-resolvent set of A, ρ C (A) for short, is defined as the union of those complex numbers λ ∈ C for which We can almost trivially construct examples of MLOs for which ρ(A) = ∅ and ρ C (A) = ∅; for example, let Y be a proper closed linear subspace of X, let A be an MLO in Y , and let λ ∈ C so that (λ − A) −1 ∈ L(Y ). Taking any injective operator C ∈ L(X) with R(C) ⊆ Y , and looking A = A X as an MLO in X, it is clear that λ ∈ ρ C (A X ) and ρ(A X ) = ∅. In general case, if ρ C (A) = ∅, then for any λ ∈ ρ C (A) we have A0 = N ((λI − A) −1 C), as well as λ ∈ ρ C (A), A ⊆ C −1 AC and ((λ − A) −1 C) k (D(A l )) ⊆ D(A k+l ), k, l ∈ N 0 ; here it is worth noting that the equality A = C −1 AC holds provided, in addition, that ρ(A) = ∅ (see the proofs of [12, Proposition 2.1, Lemma 2.3]). The basic properties of C-resolvent sets of single-valued linear operators [35,36] continue to hold in our framework (observe, however, that there exist certain differences that we will not discuss here). For example, if ρ(A) = ∅, then A is closed; it is well known that this statement does not hold if ρ C (A) = ∅ for some C = I (cf. [12,Example 2.2]). Arguing as in the proofs of [17, Theorem 1.7-Theorem 1.9], we can deduce the validity of the following important theorem, which will be frequently used in the sequel.
It is well known that ρ C (A) need not be an open subset of C if C = I and A is a single-valued linear operator (cf. [12,Example 2.5]) and that ρ(A) is an open subset of C, provided that X is a Banach space and A is an MLO in X (cf. [17,Theorem 1.6]). The regular C-resolvent set of A, ρ r C (A) for short, is defined as the union of those complex numbers λ ∈ ρ C (A) for which (λ − A) −1 C ∈ R(X), where R(X) denotes the set of all regular bounded linear operators A ∈ L(X), i.e., the operators A ∈ L(X) for which there exists a positive constant c > 0 such that for each seminorm p ∈ there exists another seminorm q ∈ such that p(A n x) ≤ c n q(x), x ∈ X, n ∈ N; the regular resolvent set of A, ρ r (A) for short, is then defined by ρ r (A) := ρ r I (A). By the argumentation contained in the proof of [17, Theorem 1.6], it readily follows that ρ r (A) is always an open subset of C.
The generalized resolvent equations hold for C-resolvents of multivalued linear operators; more precisely, we have the following theorem which can be proved by induction.
(i) Let x ∈ X, k ∈ N 0 and λ, z ∈ ρ C (A) with z = λ. Then the following holds: Then the following holds: We close this subsection with the observation that the notion of C-resolvent set of a given MLO can be also introduced in the case that C is not injective. If this is the case, Theorem 2.4, Theorem 2.6 and an analogue of Proposition 2.5 continue to hold ( [38]).

Laplace transform of functions with values in sequentially complete locally convex spaces
In this section, we assume that µ = dt is the Lebesgue measure on Ω = [0, ∞) and f : [0, ∞) → X is a locally Lebesgue integrable function (in the sense of Definition 1.2(i)). As in the Banach space case, we will denote the space consisting of such functions by L 1 loc ([0, ∞) : X); similarly we define the space L 1 ([0, τ ] : X) for 0 < τ < ∞. It is clear that (1.7) implies x * , f (·) ∈ L 1 loc ([0, ∞)) for x * ∈ X * . The first normalized antiderivative t → f [1]   A few auxiliary results on integration in sequentially complete locally convex spaces is collected in the following theorem, which seems to be new and not considered elsewhere in the existing literature: loc ([0, ∞)) and f ∈ C([0, ∞) : X), then gf ∈ L 1 loc ([0, ∞) : X). (iii) (The partial integration) Suppose that g ∈ AC loc ([0, ∞)). Then, for every Proof. Fix a number τ ∈ (0, ∞). Let (f n ) n∈N be a sequence of simple functions in X [0,τ ] such that lim n→∞ f n (t) = f (t) a.e. t ∈ K = [0, τ ] and for all > 0 and each p ∈ there is a number n 0 = n 0 ( , p) such that (1.4) holds. Then and that for all > 0 and p = | · | there is a number n 0 = n 0 ( , p) such that (1.4) holds with the functions f n (·) and f m (·) replaced respectively with s n (·) and s m (·). Clearly, (s n f n ) n∈N is a sequence of simple functions in X [0,τ ] such that lim n→∞ s n (t)f n (t) = g(t)f (t) a.e. t ∈ [0, τ ]. Furthermore, it can be easily seen that This proves (i). To prove (ii), observe first that using Definition 1.1 we can directly prove that a function g 1 f 1 (·) belongs to the space L 1 ([0, τ ] : X), provided that f 1 : and that for all > 0 there is a number n 0 = n 0 ( , p) such that (1.4) holds. Therefore, (gf n ) n∈N is a sequence in L 1 ([0, τ ] : X) and lim n→∞ g(t)f n (t) = g(t)f (t) a.e. t ∈ [0, τ ]. Making use of the dominated convergence theorem (Theorem 1.3(i)), we obtain that gf ∈ L 1 loc ([0, ∞) : X), as claimed. By (i) and (ii), the both integral in (3.1) are well-defined. Let x * ∈ X * . Using the partial integration in the Lebesgue integral and (1.7), we obtain that Since x * was arbitrary, it readily follows from (1.7) that (3.1) holds. The proof of the theorem is thereby complete.
In the remaining part of this section, we are concerned with the existence of Laplace integral for λ ∈ C. Iff (λ 0 ) exists for some λ 0 ∈ C, then we define the abscissa of convergence off (·) by abs X (f ) := inf λ :f (λ) exists ; otherwise, abs X (f ) := +∞. It is said that f (·) is Laplace transformable, or equivalently, that f (·) belongs to the class (P1)-X, if and only if abs X (f ) < ∞. Assuming that there exists a number ω ∈ R such that for each seminorm p ∈ there exists a number M p > 0 satisfying that as the infimum of all numbers ω ∈ R with the above property; if there is no such a number ω ∈ R, then we define ω X (f ) := +∞. Further on, we abbreviate ω X (f ) (abs X (f )) to ω(f ) (abs(f )), if there is no risk for confusion. Define (iii) Suppose that λ ∈ C and the limit lim τ →∞ t 0 e −λs p(f (s)) ds exists for any p ∈ . Thenf (λ) exists, as well.
Recall [79], a function h(·) belongs to the class LT − X if and only if there exist a function g ∈ C([0, ∞) : X) and a number ω ∈ R such that ω(g) ≤ ω < ∞ and h(λ) = (Lg)(λ) for λ > ω; as observed in [36, Section 1.2], the assumption h ∈ LT − X immediately implies that the function λ → h(λ), λ > ω can be analytically extended to the right half plane {λ ∈ C : λ > ω}. In the sequel, the set of all originals g(·) whose Laplace transform belongs to the class LT − X will be abbreviated to LT or − X. Keeping this observation and the equations (3.2)-(3.3) in mind, we can simply prove that the mapping λ →f (λ), λ > abs(f ) is analytic, provided that f ∈ (P1)−X. If this is the case, the following formula holds: In the following theorem, we will collect various operational properties of vectorvalued Laplace transform.
Using the proof of [1, Theorem 1.7.7], Theorem 3.1(iii), as well as the equations (1.6) and (3.5), we can simply prove that the Post-Widder inversion formula holds in our framework: The situation is much more complicated if we consider the Phragmén-Doetsch inversion formula for the Laplace transform of functions with values in SCLCSs. The following result of this type will be sufficiently general for our purposes: Theorem 3.5. Let f ∈ (P1)-X and t ≥ 0. Then the following holds:  .
The following converse of Theorem 3.3(iv) simply follows from an application of Theorem 3.5.
The method proposed by Xiao and Liang in [80] provides a sufficiently enough framework for the theoretical study of real and complex inversion methods for the Laplace transform of functions with values in SCLSCs, as well as for the studies of analytical properties and approximation of Laplace transform (see e.g. [79, Section 1.1.1] and [36, Section 1.2] for more details); this method can be successfully applied in the analysis of subordination principles for abstract time-fractional inclusions, as well (cf. Theorem 4.8 below). It is also worth noting that there exists a great number of theoretical results from the monograph [1], not mentioned so far, which can be reconsidered for the Laplace transformable functions with values in SCLSCs; for example, all structural results from [1, Section 4.1] continue to hold in our framework. Due primarily to the space limitations, in this paper we will not be able to consider many other important questions concerning the vector-valued Laplace transform of functions with values in SCLCSs.
At the end of this section, we would like to briefly explain how we can extend the definition of Laplace transformable functions to the multivalued ones. Let We denote the set of all sections, resp., all continuous sections, of F by sec(F), resp., sec c (F). Suppose now that τ = ∞ and any function f ∈ sec(F) belongs to the class (P1)-X. Then we define abs X (F) := sup{abs X (f ) : f ∈ sec(v)}; F(·) is said to be Laplace transformable if and only if abs X (F) < ∞.

Abstract degenerate Volterra integro-differential inclusions
In the following general definition, we introduce various types of solutions to the abstract degenerate inclusions (1.1), (1.2) and (1.3).
, and let A : X → P (Y ), B : X → P (Y ) be two given mappings (possibly non-linear).
(i) A function u ∈ C([0, τ ) : X) is said to be a pre-solution of (1.1) if and only if (a * u)(t) ∈ D(A) and u(t) ∈ D(B) for t ∈ [0, τ ), as well as (1.1) holds. By a solution of (1.1), we mean any pre-solution u(·) of (1.1) satisfying additionally that there exist functions for t ≥ 0, and the requirements of (1.2) hold; a pre-solution of (1.2) is any p-solution of (1.2) that is continuous for t ≥ 0. Finally, a solution of (1.2) is any pre-solution u(·) of (1.2) satisfying additionally that there exists a function (iii) By a pre-solution of (1.3), we mean any continuous X-valued function t → u(t), t ≥ 0 such that the term t → D α t u(t), t ≥ 0 is well defined and continuous, as well as that D α t u(t) ∈ D(B) and u(t) ∈ D(A) for t ≥ 0, and that the requirements of (1.3) hold; a solution of (1.3) is any pre-solution u(·) of (1.3) satisfying additionally that there exist functions u α, Before proceeding further, we want to observe that the existence of solutions to (1.1), (1.2) or (1.3) immediately implies that sec c (F) = ∅, as well as that any strong solution of (1.1) is already a solution of (1.1), provided that A and B are MLOs with A being closed; this can be simply verified with the help of Theorem 2.3. The notion of a (pre-)solution of problems (1.2) and (1.3) can be similarly defined on any finite interval [0, τ ) or [0, τ ], where 0 < τ < ∞, and extends so the notion of a strict solution of [17, problem (E) pp. [33][34] is continuous single-valued). We refer the reader to [23]- [24] and [45] for related results about the wellposedness of problem (1.2), as well as to the monograph [15] by Dragoni, Macki, Nistri and Zecca for some other concepts of solutions to the abstract differential inclusions in abstract spaces.
In our further work, it will be assumed that A and B are multivalued linear operators. Observe that we cannot consider the qualitative properties of solutions of problems (1.1), (1.2) or (1.3) in full generality by a simple passing to the multivalued linear operators B −1 A or AB −1 (see the definition of a solution of (1.1)). Concerning this question, we have the following remark. (i) Suppose that u(·) is a pre-solution (or, equivalently, solution) of problem If, in addition to this, B ∈ L(X, Y ) and u(·) is a strong solution of problem (1.1) with the above requirements being satisfied, then the mappings t → (v) Suppose that C Y ∈ L(Y ) is injective and the closed graph theorem holds for the mappings from Y into Y . Then we define the set , as well as that This is an extension of [17, Theorem 1.14] and holds even in the case that the operator C Y does not commute with AB −1 , when we define the C Yresolvent set of the operator λ − AB −1 in the same way as before. Observe also that the assumption D(A) ⊆ D(B), which has been used in [17, Section 1.6], is not necessary for the validity of (4.1).
Furthermore, if C = I, X = Y and B ∈ L(X, Y ), then ρ B (A) ⊆ ρ(B −1 A) and the previous equality holds.
Consider now the case in which the operator A is closed, the operator B = B is single-valued and the function F(t) = f (t) is Y -continuous at each point t ≥ 0. Then any pre-solution u(·) of problem (1.2) is already a solution of this problem, and Theorem 2.3 in combination with the identity [5, (1.21)] implies that Then it can be simply verified that u(·) is a solution of problem (1.2); it is noteworthy that we do not need the assumption on closedness of A in this direction. Even in the case that A = A is a closed single-valued linear operator, a corresponding statement for the problem (1.3) cannot be proved. Suppose, finally, that the operators A and B are closed, u(·) is a solution of problem (1.2), the function The proof of following important theorem can be deduced by using Theorem 2.3, Theorem 3.3[(iv),(vi)], Theorem 3.5 and the argumentation already seen in the proof of [31, Theorem 3.1] (cf. also [32, Fundamental Lemma 3.1]); observe that we do not use the assumption on the exponential boundedness of function u(t) here. After formulation, we will only include the most relevant details needed for the proof of implication (iii) ⇒ (iv).
and ω > max(0, ω X (u), abs Y A (Bu), abs Y A (F), abs X A (a * u)). Consider the following assertions: is single-valued, then the above is equivalent.
Sketch of proof for (iv) ⇒ (v). Suppose that for any section u B ∈ sec(Bu) there is a section f ∈ sec(F) such that (4.2) holds. Let a number λ ∈ N with λ > ω and a(λ) = 0 be temorarily fixed. Then there exists a sequence (λ n ) n∈N in (λ, ∞) such thatã(λ n ) = 0 and lim n→+∞ λ n = λ. [2] (t) dt (λ ∈ N, λ > ω) and now we can apply Theorem 3.5, along with the X A ×Y A -closedness of A, in order to see that u [2] B (t)−f [2] (t) ∈ A(a * u) [2] (t), t ≥ 0. This simply implies (4.3).  If Ω is a non-empty open subset of C and G : Ω → X is an analytic mapping that it is not identically equal to the zero function, then we can simply prove that for each zero λ 0 of G(·) there exists a uniquely determined natural number n ∈ N such that G (j) (λ 0 ) = 0 for 0 ≤ j ≤ n − 1 and G (n) (λ 0 ) = 0. Owing to this fact, we can repeat almost literally the arguments given in the proof of [31,Theorem 3.2] to verify the validity of the following Ljubich uniqueness type theorem: and there exist a sequence (λ k ) k∈N of complex numbers and a number ω > abs(|a|) such that lim k→∞ λ k = +∞,ã(λ k ) = 0, k ∈ N, and 1 Then there exists a unique pre-solution of (1.1), with τ = ∞, satisfying that u ∈ In the following extension of [36, Theorem 2.1.34], we will prove one more Ljubich's uniqueness criterium for abstract Cauchy problems with multivalued linear operators (cf. also [32,Theorem 3.5] and [36,Theorem 2.10.44]).
Then, for every x 0 , · · ·, x α −1 ∈ X, there exists at most one pre-solution of the initial value problem (1.2) with B = I.
Proof. It suffices to show that the zero function is the only pre-solution of problem (1.2) with B = I and the initial values x 0 , · · ·, x α −1 chosen to be zeroes. Let u(·) be a pre-solution of such a problem. Set z n (t) : Then it can be easily checked with the help of Theorem 2.4(i) that z n (·) is a solution of the initial value problem: Assume, further, that f β ∈ LT or − Y A is single-valued and there exists a presolution (or, equivalently, solution) u(t) We define Let p ∈ X B be fixed. Then the condition p u β (t) = O 1 + t ξp e ωpt for some ξ p ≥ 0, (4.5) resp., resp., Furthermore, the following holds: ). If, for every p ∈ , one has ω p = 0, then for each θ ∈ (0, min(( 1 γ − 1) π 2 , π)) the following holds: Proof. The proofs of (i)-(iii) follows similarly as in that of [5,Theorem 3.3], while the proof that the condition (4.5), resp. (4.6), implies (4.7), resp. (4.8), follows similarly as in that of [36,Theorem 2.4.2]. Furthermore, it can be easily seen that the estimate (4.4) holds for solution u α (·). By Theorem 4.3, we should only show Since u β ∈ LT or − X B , the proof of [5, Theorem 3.1] immediately implies that u α ∈ LT or − X B , as well as that by performing the Laplace transform (the convergence of last integral is taken for the topology of X A ). This simply implies that (the convergence of this integral is taken for the topology of Y A ) and Bu α (λ) = B u α (λ) for all sufficiently large values of λ > ω. The proof of (4.9) now follows from a simple computation.
We can similarly prove the following subordination principles for abstract degenerate Volterra inclusions in locally convex spaces (cf. [68,Section 4] and [36, Theorem 2.1.8, Theorem 2.8.7] for more details concerning non-degenerate case and, especially, the case in which b(t) = g 1 (t) or b(t) = g 2 (t)).
Assume, further, that f β ∈ LT or − Y A is single-valued and there exists a presolution (or, equivalently, solution) u(t) Let c(t) be completely positive and let there exist a function f a ∈ LT or − Y A such that Let Furthermore, the function t → u a,r (t) ∈ X B , t ≥ 0 is locally Hölder continuous with the exponent r ∈ (0, 1]. (ii) In Theorem 4.9, we have faced ourselves with a loss of regularity for solutions of the subordinated problem. Even in the case that X = Y and B = I, it is not so simple to prove the existence of a solution of problem (1.1), with τ = ∞, a(t) and F = f a , without imposing some additional unpleasant conditions. In the next section, we will introduce various types of solution operator families for the abstract Volterra inclusion (1.1) and there we will reconsider the problem of loss of regularity for solutions of the subordinated problem once more (cf. Theorem 5.7).

Multivalued linear operators as subgenerators of (a, k)-regularized C-resolvent solution operator families
In [36, Section 2.8], the class of (a, k)-regularized (C 1 , C 2 )-existence and uniqueness families has been introduced and analyzed within the theory of abstract nondegenerate Volterra equations. The main aim of this section is to consider multivalued linear operators in locally convex spaces as subgenerators of (a, k)-regularized (C 1 , C 2 )-existence and uniqueness families, as well as to consider in more detail the class of (a, k)-regularized C-resolvent families. Unless specified otherwise, we assume that 0 < τ ≤ ∞, k ∈ C([0, τ )), k = 0, a ∈ L 1 loc ([0, τ )), a = 0, A : X → P (X) is an MLO, C 1 ∈ L(Y, X), C 2 ∈ L(X) is injective, C ∈ L(X) is injective and CA ⊆ AC.
As an immediate consequence of definition, we have that R(R 1 (0)−k(0)C 1 ) ⊆ A0 as well as that R 2 (t)A is single-valued for any t ≥ 0, and R 2 (t)y = 0 for any y ∈ A0 and t ≥ 0. Now we will extend the definition of an (a, k)-regularized C-resolvent family subgenerated by a single-valued linear operator (cf. [36, Definition 2.1.1]).
It would take too long to consider the notion of q-exponential equicontinuity for the classes of mild (a, k)-regularized C 1 -existence families and mild (a, k)regularized C 2 -uniqueness families (cf. [36,Section 2.4] for more details about non-degenerate case). If k(t) = g α+1 (t), where α ≥ 0, then it is also said that (R(t)) t∈[0,τ ) is an α-times integrated (a, C)-resolvent family; 0-times integrated (a, C)-resolvent family is further abbreviated to (a, C)-resolvent family. We will accept a similar terminology for the classes of mild (a, k)-regularized C 1 -existence families and mild (a, k)-regularized C 2 -uniqueness families; in the case of consideration of convoluted C-semigroups, it will be always assumed that the condition (5.1) holds with a(t) = 1 and the operator C 1 replaced by C. Let us mention in passing that the operator semigroups generated by multivalued linear operators have been analyzed by A. G. Baskakov in [4].
The following proposition can be proved with the help of Theorems 2.3 and 3.1(ii).
is a mild (a, k)-regularized (C 1 , C 2 )-existence and uniqueness family with a subgenerator A and (R(t)) t∈[0,τ ) ⊆ L(X) is an (a, k)-regularized C-resolvent family with a subgen- is a mild (a, k)-regularized C 2 -uniqueness family with a subgenerator A. Furthermore, the following holds: Suppose that, for every x ∈ D(A) and y ∈ R(A), the mapping t → R(t)x, t ∈ [0, τ ) is continuous in X 1 A and the mapping t → R(t)y, t ∈ [0, τ ) is continuous in X 2 A . Then ((b * R)(t)) t≥0 is a (a, k)regularized C-regularized family with a subgenerator A.
Although the parts (i) and (ii) of the above proposition have been stated for X 1 A × X 2 A -closed subgenerators, the most important case in our further study will be that in which X 1 A = X 2 A = X. This is primarily caused by the following fact: Let A be a subgenerator of a mild (a, k)-regularized C 1 -existence family (mild (a, k)-regularized C 2 -uniqueness family; mild (a, k)-regularized C-resolvent family) Suppose that (R 1 (t), R 2 (t)) t∈[0,τ ) is a mild (a, k)-regularized (C 1 , C 2 )-existence and uniqueness family with a subgenerator A. Arguing as in non-degenerate case (cf. the paragraph directly preceding [36, Definition 2.8.3]), we may conclude that The integral generator of mild (a, k)-regularized C 2 -uniqueness family (R 2 (t)) t∈[0,τ ) (mild (a, k)-regularized (C 1 , C 2 )-existence and uniqueness family (R 1 (t), R 2 (t)) t∈[0,τ ) ) is defined by we define the integral generator of an (a, k)-regularized C-regularized family (R(t)) t∈[0,τ ) in the same way as above. The integral generator A int is an MLO in X which is, in fact, the maximal subgenerator of (R 2 (t)) t∈[0,τ ) ((R(t)) t∈[0,τ ) ) with respect to the set inclusion; furthermore, the assumption R 2 (t) The local equicontinuity of (R 2 (t)) t∈[0,τ ) ((R(t)) t∈[0,τ ) ) immediately implies that A int is closed. Observe that, in the above definition of integral generator, we do not require that the function a(t) is a kernel on [0, τ ), as in non-degenerate case. In the case of resolvent families, the following holds: (i) Suppose that (R(t)) t∈[0,τ ) is locally equicontinuous and A is a closed subgenerator of (R(t)) t∈[0,τ ) . Then for each y ∈ Ax and z ∈ Bx. (iv) Let A be a subgenerator of (R(t)) t∈[0,τ ) , and let λ ∈ ρ C (A) (λ ∈ ρ(A)).
(ii) Suppose that n ∈ N, X and Y are Banach spaces, A : D(A) ⊆ X → Y is closed, B ∈ L(X, Y ) and (V (t)) t≥0 ⊆ L(X) is a degenerate exponentially bounded n-times integrated semigroup generated by linear operators A, B, in the sense of [63, Definition 1.5.3]. Then the arguments mentioned above show that (V (t)) t≥0 is an exponentially bounded n-times integrated (g 1 , I)-regularized family (semigroup) with a closed subgenerator B −1 A.
(iii) Let n ∈ N 0 . Due to Theorem 5.5(ii), the notion of an exponentially bounded (a, k)-regularized C-resolvent family extends the notion of a degenerate exponentially bounded n-times integrated semigroup generated by an MLO, introduced in [63, Definition 1.6.6, Definition 1.6.8].
The proof of following extension of [36, Theorems 2.1.8(i) and 2.8.7(i)] is standard and therefore omitted; we can similarly reformulate Theorem 4.8 and [36, Proposition 2.1.16] for the class of mild (a, k)-regularized (C 1 , C 2 )-existence and uniqueness families ((a, k)-regularized C-resolvent families). Here it is only worth noting that the existence of a mild (a, k 1 )-regularized C 1 -existence family (R 0,1 (t)) t≥0 in the second part of theorem is not automatically guaranteed by the denseness of A (even in the case that the operator A = A is single-valued, it seems that the condition C 1 A ⊆ AC 1 is necessary for such a mild existence family to exist).
(ii) Suppose that (R 2 (t)) t∈[0,τ ) is a locally equicontinuous mild (a, k)-regularized C 2 -uniqueness family with a subgenerator A. Then every strong solution u(t) of (1.1) with B = I and F = f ∈ C([0, τ ) : X) satisfies Furthermore, the problem (1.1) has at most one pre-solution provided, in addition, that the functions a(t) and k(t) are kernels on on [0, τ ) and the function F(t) is single-valued. Theorem 5.9. (i) Suppose that (R(t)) t∈[0,τ ) is a locally equicontinuous (a, k)-regularized C-resolvent family generated by A, the equation (5.1) holds for each y = x ∈ X, with R 1 (·) and C 1 replaced therein with R(·) and C, respectively, k(t) is a kernel on [0, τ ), u, f ∈ C([0, τ ) : X), and (5.10) holds with R 2 (·) and C 2 replaced therein with R(·) and C, respectively. Then u(t) is a solution of the abstract Volterra inclusion (1.1) with B = I and F = f .
(ii) Suppose that the functions a(t) and k(t) are kernels on [0, τ ), and A is a closed MLO in X. Consider the following assertions: (a) A is a subgenerator of a locally equicontinuous (a, k)-regularized C-resolvent family (R(t)) t∈[0,τ ) satisfying the equation (5.1) for each y = x ∈ X, with R 1 (·) and C 1 replaced therein by R(·) and C, respectively. (b) For every x ∈ X, there exists a unique solution of (1.1) with B = I and F(t) = f (t) = k(t)Cx, t ∈ [0, τ ). Then (a) ⇒ (b). If, in addition, X is a Fréchet space, then the above are equivalent.
Before proceeding further, it should be noticed that some additional conditions ensure the validity of implication (b) ⇒ (a) in complete locally convex spaces. We will explain this fact for the problem (1.3), where after integration we have a(t) = g α (t). Assume that there exists a unique solution of problem (1.3) with B = I, F(t) ≡ 0, x 0 ∈ C(D(A)) and x j = 0, 1 ≤ j ≤ α − 1. If, in addition to this, X is complete, A is closed, CA ⊆ AC and for each seminorm p ∈ and T > 0 there exist q ∈ and c > 0 such that p(u(t; Cx)) ≤ cq(x), x ∈ D(A), t ∈ [0, T ], then the arguments used in non-degenerate case (see e.g. [40, p. 304]) show that A is a subgenerator of a locally equicontinuous (g α , C)-resolvent family (R α (t)) t≥0 .
The proof of following complex characterization theorem for (a, k)-regularized C-resolvent families is left to the reader as an easy exercise.
In the first part of following example, we will briefly explain how one can use multiplication operators for construction of local integrated semigroups generated by multivalued operators; in the second part of example, we will apply the complex characterization theorem for proving the existence of a very specific exponentially equicontinuous, convoluted fractional resolvent family (cf. [42, Example 2.5] for an example of a locally defined solution of an abstract degenerate multi-term fractional problem).
. Then it is very simple to prove that, for every α ∈ (0, 1), the resolvent set of the multivalued linear operator A := B −1 A contains the exponential region E(α, 1) := {x + iy : x ≥ 1, |y| ≤ e αx }, as well as Furthermore, the operator A generates a local once integrated semigroup (S 1 (t)) t∈[0,1] , given by (ii) Put X := {f ∈ C ∞ ([0, ∞)) : lim x→+∞ f (k) (x) = 0 for all k ∈ N 0 } and ||f || k := k j=0 sup x≥0 |f (j) (x)|, f ∈ X, k ∈ N 0 . Then the topology induced by these norms turns X into a Fréchet space (cf. also [ . In a recent research study with Pilipović and Velinov [53], we have shown that A cannot be the generator of any local integrated semigroup in X, as well as that A generates an ultradistribution semigroup of Beurling class. Set A := B −1 A. We will prove that there exists a sufficiently large number ω > 0 such that for each s > 1 and d > 0 the operator family {e −d|λ| 1/s (λ − A) −1 : λ > ω, λ ∈ Σ απ/2 } ⊆ L(X) is equicontinuous, which immediately implies by Theorem 5.10 that A generates an exponentially equicontinuous (g α , L −1 (e −d|λ| α/s ))-regularized resolvent family. It is clear that the resolvent of A will be given by ∈ J, our first task will be to estimate the derivatives of function 1/λ − (· + ie · ) outside the interval J. In order to do that, observe first that any complex number λ ∈ C \ S, where S := {x + ie x : x ≥ 0}, belongs to the resolvent set of A and Set Ω := {λ ∈ C : λ ≥ a| λ| 1/s + b} and denote by Γ the upwards oriented boundary of the region Ω. Inductively, we can prove that for each number n ∈ N there exist complex polynomials P j (z) = j l=0 a j,l z l (1 ≤ j ≤ n) such that deg(P j ) = j, |a j,l | ≤ (n + 1)! (1 ≤ j ≤ n, 0 ≤ l ≤ j) and Suppose λ ∈ Ω and x ≥ 0. If | λ − e x | ≥ 1, then we have the estimate λ − e x 2k ≤ 2 2j 1 + | λ| 2j , k ∈ N 0 , 0 ≤ j < k.

(5.12)
If | λ − e x | < 1, then λ > 0, 0 ≤ x < ln( λ + 1), and Let ω > 0 be such that {λ ∈ Σ απ/2 : λ > ω } ⊆ Ω. Combining (5.11)-(5.13), it can be simply proved that for each number n ∈ N there exists a finite constant c n > 0 such that for λ ∈ Σ απ/2 and λ > ω . We can similarly prove an estimate of type (5.14) for the derivatives of function (λm b (x)−(x+ie x )) −1 on the interval J, which is well-defined for λ ∈ Σ απ/2 because of assumption 0 ≤ m b (x) ≤ 1, x ≥ 0 and the condition Σ απ/2 ∩{x+ie x : x ∈ J} = ∅. In actual fact, an induction argument shows that for each number n ∈ N there exist numbers a j,l1,···,ls such that |a j,l1,···,ls | ≤ (n + 1)! (1 ≤ j ≤ n, 0 ≤ l ≤ j) and that, for every x ∈ J and λ ∈ Σ απ/2 , is a positive real number and |(λm a (x)) mj | ≤ c mj |λ| mj for all λ ∈ Σ απ/2 with λ > ω, where the number ω > ω is sufficiently large, (5.15) shows that for each number n ∈ N there exists a finite number c n > 0 such that for λ ∈ Σ απ/2 and λ > ω. By (5.14) and (5.16), we have that the operator family {e −d|λ| 1/s (λ − A) −1 : λ ∈ Σ απ/2 , λ > ω} ⊆ L(X) is equicontinuous, as claimed. Now we would like to tell something more about the importance of condition k(0) = 0 in the part (ii) of subsequent theorem. If all the necessary requirements hold, the arguments contained in the proof of [32,Theorem 3.6] imply the existence of a global (a, k * g 1 )-regularized C-resolvent family (R 1 (t)) t≥0 subgenerated by A, which additionally satisfies that for each t ≥ 0 the operator R 1 (t)A is single-valued on D(A). Then it is necessary to differentiate the equality R 1 (t)x − (k * g 1 )(t)Cx = t 0 a(t−s)R 1 (s)Ax ds, t ≥ 0, x ∈ D(A) and to employ the fact that (  Cx, x ∈ X, λ ∈ D(H), satisfies that the mapping λ → H(λ)x, λ ∈ D(H) is infinitely differentiable for every fixed x ∈ X and, for every p ∈ , there exist c p > 0 and r p ∈ such that Then, for every r ∈ (0, 1], the operator A is a subgenerator of a global (a, k * g r )regularized C-resolvent family (R r (t)) t≥0 satisfying that, for every p ∈ , and that, for every p ∈ and B ∈ B, the mapping t → p B (R r (t)), t ≥ 0 is locally Hölder continuous with exponent r; furthermore, (R r (t)) t≥0 is a mild (a, k * g r )regularized C-existence family having A as subgenerator, and the following holds: is a mild (a, k)-regularized C-existence family having A as subgenerator. Then A is a well-defined single-valued closed linear operator in D(A), and moreover, A is the integral generator of a global (a, k)-regularized C -resolvent family (S(t)) t≥0 ⊆ L(D(A)) satisfying that the family In the following proposition, which extends the assertions of [59, Proposition 2.5] and [36, Proposition 2.1.4(ii)], we will reconsider the condition k(0) = 0 from Theorem 5.12 once more. A straightforward proof is omitted.
(ii) ([31]- [32]) Here we would like to observe, without going into full details, that we can similarly prove some results on the existence and uniqueness of analytical solutions of the abstract Volterra equation where c(·) is a completely positive function. Fractional Maxwell's equations have gained much attention in recent years (see e.g. [10], [27], [60], [74], [83] and references cited therein for more details on the subject). Here we want to briefly explain how we can use the analysis of Consider the following abstract time-fractional Maxwell's equations: in R 3 , where E (resp. H) denotes the electric (resp. magnetic) field intensity, B (resp. D) denotes the electric (resp. magnetic) flux density, and where J is the current density. It is assumed that the medium which fills the space R 3 is linear but possibly anisotropic and nonhomogeneous, which means that D = E, B = µH and J = σE + J with some 3 × 3 real matrices (x), µ(x), σ(x) (x ∈ R 3 ) and J being a given forced current density. Let any component of (x), µ(x), σ(x) be a bounded, measurable function in R 3 , let the conditions [17, (2.23)-(2.25)] hold, and let f (t) = −(J (·, t) 0) T . Then we can formulate the problem (5.21) in the abstract form (5.22) in the space X := {L 2 (R 3 )} 6 , using the bounded self-adjoint operator B of multiplication by c(x) acting in X, and A being the closed linear operator in X given by [17, (2.27)].
We end this example with the observation that Theorem 4.8 and Theorem 4.9 can be successfully applied in the analysis of a large class of abstract degenerate Volterra integro-differential equations that are subordinated, in a certain sense, to degenerate differential equations of first and second order for which we know that are well posed [17,22,25,71,73,75,76].
Concerning the adjoint type theorems, it should be noticed that the assertions of [36, Theorem 2.1.12(i)/(ii); Theorem 2.1.13] continue to hold for (a, k)-regularized C-regularized families subgenerated by closed multivalued linear operators. Furthermore, it is not necessary to assume that the operator A is densely defined in the case of consideration of [36, Theorem 2.1.12(i)].
is a unique solution of the following abstract time-fractional inclusion: Furthermore, v ∈ C 1 ([0, τ ) : X) provided that r ≥ 1 or x = 0 and r ≥ 0.
In the following example, we consider a time-fractional analogue of the linearized Benney-Luke equation in L 2 -spaces and there we will meet some interesting examples of exponentially bounded, analytic fractional resolvent families of bounded operators whose angle of analyticity can be strictly greater than π/2; in our approach, we do not use neither multivalued linear operators nor relatively p-radial operators ( [17], [73]). The method employed by G. A. Sviridyuk and V. E. Fedorov [73] for the usually considered Benney-Luke equation of first order can be very hepful for achieving the final conclusions stated in (i)-(ii), as well as for the concrete choice of the state space X 0 below (cf. also [ [42]).
Since D((−∆) 1/2 ) ∩ X 0 is dense in X 0 and (T 2 (t)) t≥0 is bounded, the usual arguments shows that (T 2 (t)) t≥0 is strongly continuous. Now we can proceed as in the proof of Theorem 4.8 in order to see that, for every η ∈ (0, 2), (T η (t)) t≥0 is an exponentially bounded, analytic (g η , I)-regularized resolvent family of angle θ. A straightforward computation shows that, for every η ∈ (0, 2], the integral generator A of (T η (t)) t≥0 is a closed single-valued operator in X 0 , given by in particular, A is an extension of the operator B −1 A |X0 . It is also clear that (T η (t)) t≥0 is a mild (g η , I)-existence family generated by A. Keeping in mind the identity [5, (1.25)], we can carry out a direct computation showing that the homogeneous counterpart of problem (5.25) ≡ (5.25) with x j = 0 for 1 ≤ j ≤ ζ − 1, has an exponentially bounded pre-solution u h,0 (t) = T η (t)x 0 , t ≥ 0 for any x k ∈ D(A)∩X 0 (0 ≤ k ≤ η − 1), which seems to be an optimal result in the case that η ≤ 1. Concerning the homogeneous counterpart of problem (5.25) with x 0 = 0, its solution u h,1 (t) has to be find in the form u h,1 (t) = t 0 T η (s)x 1 ds, t ≥ 0. Consider first the case η ∈ (1, 2). Then for each k ∈ N with λ k = λ, we have On the other hand, expanding the function E η ( αλ k −βλ 2 k λ−λ k · η ) − 1 in a power series we obtain that The previous two equalities together imply that d Using again the asymptotic expansion formula [5, (1.28)], we obtain that the above series converges for any x 1 ∈ X 0 and belongs to D(B) provided, in addition, that x 1 ∈ D(B) ∩ X 0 . In this case, the equality BD η t u h,1 (t) = Au h,1 (t), t ≥ 0 readily follows, so that the function u h (t) := u h,0 (t) + u h,1 (t), t ≥ 0 is a pre-solution of problem (1.3) provided that x 0 ∈ D(A) ∩ X 0 and x 1 ∈ D(B) ∩ X 0 (with X = Y = L 2 (Ω) in Definition 4.1(iii)); furthermore, the mappings t → u h (t) ∈ L 2 (Ω), t > 0 and t → Bu h (t) ∈ L 2 (Ω), t > 0 can be analytically extended to the sector Σ θ . The situation is slightly different in the case that η = 2 since we cannot use the formula [5, (1.28)]; then a simple computation shows that, formally, for every t ≥ 0, Hence, the function u h (t) := u h,0 (t)+u h,1 (t), t ≥ 0 is a pre-solution of problem (1.3) with x 0 ∈ D(A) ∩ X 0 and x 1 ∈ D((−∆) 3/2 ) ∩ X 0 . The range of any pre-solution of problem (5.25) with f = 0 must be contained in X 0 , so that the uniqueness of solutions of problem (5.25) follows from its linearity and Proposition 5.8(ii). Before considering the inhomogeneous problem (5.25), we would like to observe that the assumptions (x, y) ∈ A and x ∈ D(A) imply (x, y) ∈ B −1 A |X0 . Keeping in mind this remark, Theorem 2.3, as well as the fact that the assertion of [68, Proposition 2.1(iii)] admits a reformulation in our framework, we can simply prove that for any function h ∈ W 1,1 On the other hand, the operator B annihilates any function from span{φ k : k|λ = λ k } so that the function t → k|λ k =λ f (t),φ k βλ 2 k −αλ k φ k , t ≥ 0 is a pre-solution of problem (5.25) with f = k|λ k =λ f (·), φ k φ k , provided that the following condition holds (A1) : D η t f (t), φ k exists in L 2 (Ω) for k|λ = λ k , x 0 , φ k = 0 for k|λ = λ k , Summa summarum, we have the following: (5.26), and the condition (A1) holds. Then there exists a unique pre-solution of problem (5.25). (ii) η = 2 : Suppose x 1 ∈ D((−∆) 3/2 ) ∩ X 0 and the remaining assumptions from (i) hold. Then there exists a unique pre-solution of problem (5.25). Observe also that our results on the well-posedness of fractional analogue of the Benney-Luke equation, based on a very simple approach, are completely new provided that η > 1, as well as that we have obtained some new results on the wellposedness of the inhomogeneous Cauchy problem P η,f in the case that η < 1 (cf. [22,Theorem 4.2] for the first result in this direction).
The classes of exponentially equicontinuous, analytic (a, k)-regularized C 1 -existence families and (a, k)-regularized C 2 -uniqueness families can be introduced and analyzed, as well. For the sequel, we need the following notion.
Definition 5.21. Let X = Y , and let A be a subgenerator of a C 1 -existence family (R 1 (t)) t≥0 (cf. Definition 5.1(i) with a(t) ≡ 1 and k(t) ≡ 1). Then (R 1 (t)) t≥0 is said to be entire if, for every x ∈ X, the mapping t → R 1 (t)x, t ≥ 0 can be analytically extended to the whole complex plane.
Using the arguments in the proof of [41,Theorem 3.15], we can deduce the following result.
Theorem 5.22. Suppose r ≥ 0, θ ∈ (0, π/2), A is a closed MLO and −A is a subgenerator of an exponentially equicontinuous, analytic r-times integrated Csemigroup (S r (t)) t≥0 of angle θ. Then there exists an operator C 1 ∈ L(X) such that A is a subgenerator of an entire C 1 -existence family in X.
Remark 5.23. (i) It ought to be observed that we do not require the injectivity of operator C 1 here. The operators T α (z) and S α,z0 (z), appearing in the proof of [41,Theorem 3.15], annulate on the subspace A0.
(ii) Theorem 5.22 is closely linked with the assertions of [40, Theorem 2.1, Theorem 2.2]. These results can be extended to abstract degenerate fractional differential inclusions, as well.
Example 5.24. In a great number of research papers, many authors have considered infinitely differentiable semigroups generated by multivalued linear operators of form AB −1 or B −1 A, where the operators A and B satisfy the condition [17, (3.14)], or its slight modification, with certain real constants 0 < β ≤ α ≤ 1, γ ∈ R and c, C > 0 (in our notation, we have A = L and B = M ). The validity of this condition with α = 1 (see e.g. [17,Example 3.3,3.6]) immediately implies by Theorem 5.19 and Remark 4.2(v) that the operator AB −1 generates an exponentially bounded, analytic σ-times integrated semigroup of angle Σ arcctan(1/c) , provided that σ > 1 − β; in the concrete situation of [17,Example 3.4,3.5], the above holds with the operator AB −1 replaced by B −1 A.
Unfortunately, this fact is not sufficiently enough for taking up a fairly complete study of the abstract degenerate Cauchy problems that are subordinated to those appearing in the above-mentioned examples and, concerning this question, we will only want to mention that the subordination fractional operator families can be constructed since the semigroups considered in [17, Chapter III] have a removable singularity at zero (cf. the proof of [5, Theorem 3.1], [47] and the forthcoming monograph [38] for more details). On the other hand, from the point of view of possible applications of Theorem 5.22, it is very important to know that the operators AB −1 or B −1 A generate exponentially bounded, analytic integrated semigroups. This enables us to consider the abstract degenerate Cauchy problems that are backward to those appearing in [17,. For example, we can consider the following modification of the backward Poisson heat equation in the space L p (Ω): where Ω is a bounded domain in R n , b > 0, m(x) ≥ 0 a.e. x ∈ Ω, m ∈ L ∞ (Ω) and 1 < p < ∞. Let B be the multiplication in L p (Ω) with m(x), and let A = ∆ − b act with the Dirichlet boundary conditions. Then Theorem 5.22 implies that there exists an operator C 1 ∈ L(L p (Ω)) such that A = −AB −1 is a subgenerator of an entire C 1 -existence family; hence, for every u 0 ∈ R(C 1 ), the problem (5.31) has a unique solution t → u(t), t ≥ 0 which can be extended entirely to the whole complex plane. Furthermore, it can be proved that the set of all initial values u 0 for which there exists a unique solution of problem (5.31) is dense in L p (Ω) provided that there exists a constant d > 0 such that |m(x)| ≥ d a.e. x ∈ Ω.
Observe, finally, that the Riemann-Liouville fractional derivative D ζ t u(t) need not be defined here.
Now we are ready to formulate the following extension of [36, Theorem 2.1.24].

5.2.
Non-injectivity of regularizing operators C 2 and C. In this subsection, we consider multivalued linear operators as subgenerators of mild (a, k)regularized (C 1 , C 2 )-resolvent operator families and (a, k)-regularized C-resolvent operator families. We use the same notion and notation as before but now we allow that the operators C 2 and C are possibly non-injective (see Definition 5.1-Definition 5.2). Without any doubt, this choice has some obvious displeasing consequences on the uniqueness of corresponding abstract Volterra integro-differential inclusions (see Proposition 5.8(ii) and Theorem 5.9(ii)).
In the following definition, we introduce the notion of an (a, k, C)-subgenerator of any strongly continuous operator family (Z(t)) t∈[0,τ ) ⊆ L(X). This definition extends the corresponding ones introduced by Kuo [54, 55, Definition 2.4] in the setting of Banach spaces, where it has also been assumed that the operator A = A is linear and single-valued.
The real representation theorem for generation of degenerate (a, k)-regularized C-resolvent families can be also formulated but the assertion of Theorem 5.12(ii) is not attainable in the case that the operator C is not injective. The assertion of Theorem 5.7 continues to hold with minimal terminological changes. Since the identity (5.24) holds for degenerate (a, k)-regularized C-resolvent families, with C being not injective, Proposition 5.15 can be reformulated without substantial difficulties, as well, but we cannot prove the uniqueness of solutions of corresponding abstract time-fractional inclusions.
As already mentioned, the adjoint type theorems [36, Theorem 2.1.12(i)/(ii); Theorem 2.1.13] continue to hold for (a, k)-regularized C-regularized families subgenerated by closed multivalued linear operators and it is not necessary to assume that the operator A is densely defined in the case of consideration of [36, Theorem 2.1.12(i)]. All this remains true if the operator C is not injective, when we also do not need to assume that R(C) is dense in X.
If C is not injective, then we introduce the notion of (exponential equicontinuous) analyticity of degenerate (a, k)-regularized C-resolvent families in the same way as in Definition 5.16. Then Theorem 5.18 does not admit a satisfactory reformulation in our new frame. On the other hand, the assertion of Theorem 5.19 can be rephrased by taking into consideration the conditions (d)-(e) from Theorem 5.33. Differential properties of degenerate (a, k)-regularized C-resolvent families clarified in Theorem 5.26-Theorem 5.27 continue to hold after a reformulation of the same type.
During the peer-review process, the author has published several research papers about degenerate (a, k)-regularized C-resolvent families and their applications. Various subclasses of degenerate convoluted C-semigroups and degenerate convoluted C-cosine functions in locally convex spaces have been investigated in [50]. Perturbation results for abstract degenerate Volterra integro-differential equations have been examined in [51], while the approximation and convergence of degenerate (a, k)-regularized C-resolvent families have been examined in [52].

Conclusions and final remarks
In this research article, we have analyzed the abstract degenerate Volterra integrodifferential equations in sequentially complete locally convex spaces. We have systematically investigated the class of degenerate (a, k)-regularized C-resolvent families subgenerated by multivalued linear operators and examined many interesting topics including the generation of (a, k)-regularized C-resolvent families, smoothing properties of (a, k)-regularized C-resolvent families and subordination principles. We have also examined the class of mild (a, k)-regularized C 1 -existence families, the class of mild (a, k)-regularized C 2 -uniqueness families and provided a new theoretical concept of vector-valued Laplace transform. In addition to the above, we have presented many useful comments, open problems, examples and illustrative applications of our theoretical results.
The material of this paper has recently been published as a part of the research monograph [38]; the almost periodic type solutions of the abstract degenerate Volterra integro-differential equations have recently been analyzed in the research monograph [37]. We close the paper with the observation that we have obeyed the multivalued linear operators approach here; this approach, although very dominant when compared with the other existing methods and theoretical strategies in this theory, is not sufficiently adequate to cover all related problems regarding the abstract degenerate Volterra integro-differential equations. For some other concepts of solution operator families, we may refer to [42,43,45,46].
Finally, we would like to emphasize that almost anything relevant has been said about the existence and uniqueness of the positive solutions to the abstract degenerate Volterra integro-differential equations in ordered Banach spaces.