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MARKO KOSTI ´
C
Abstract degenerate Volterra inclusions in locally convex spaces
Marko Kosti´c
Faculty of Technical Sciences
University of Novi Sad
Trg D. Obradovi´ca 6
21125 Novi Sad, Serbia
E-mail: marco.s@verat.net
Contents
1. Abstract degenerate Volterra inclusions in locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1. Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2. Multivalued linear operators in lo cally convex spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3. Laplace transform of functions with values in sequentially complete locally convex
spaces ............................................................................ 15
1.4. Abstract degenerate Volterra integro-differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5. Multivalued linear operators as subgenerators of (a, k)-regularized C-resolvent so-
lution operator families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.1. Differential and analytical properties of (a, k)-regularized C-resolvent families 42
References ............................................................................... 53
[3]
Abstract
In this paper, we analyze the abstract degenerate Volterra integro-differential equations in se-
quentially complete locally convex spaces by using multivalued linear operators and vector-valued
Laplace transform. We follow the method which is based on the use of (a, k)-regularized C-
resolvent families generated by multivalued linear operators and which suggests a very general
way of approaching abstract Volterra equations. Among many other themes, we consider the
Hille-Yosida type theorems for (a, k)-regularized C-resolvent families, differential and analytical
properties of (a, k)-regularized C-resolvent families, the generalized variation of parameters for-
mula, and subordination principles. We also introduce and analyze the class of (a, k)-regularized
(C1, C2)-existence and uniqueness families. The main purpose of third section, which can be
viewed of some independent interest, is to introduce a relatively simple and new theoretical
concept useful in the analysis of operational properties of Laplace transform of non-continuous
functions with values in sequentially complete locally convex spaces. This concept coincides with
the classical concept of vector-valued Laplace transform in the case that Xis a Banach space.
Acknowledgements. The author is partially supported by grant 174024 of Ministry of Science
and Technological Development, Republic of Serbia.
2010 Mathematics Subject Classification: Primary 34G25, 45D05, 47D06; Secondary 46G12,
47D60, 47D62.
Key words and phrases: abstract degenerate Volterra inclusions, abstract degenerate fractional
differential equations, (a, k)-regularized C-resolvent families, multivalued linear operators,
locally convex spaces.
[4]
1. Abstract degenerate Volterra inclusions in locally convex
spaces
1.1. Introduction and preliminaries. As mentioned in the abstract, the main aim
of this paper is to analyze the abstract degenerate Volterra integro-differential equations
in sequentially complete locally convex spaces by using multivalued linear operators (cf.
[JP] and [MK2] 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 only mention that we have not been able to find any
appropriate reference in the existing literature which treats abstract degenerate fractional
inclusions associated with the use of Caputo fractional derivatives; furthermore, we have
not been able to find any reference where such elementary notion, like C-regularized
semigroup, or fractionally integrated semigroup, generated by a multivalued linear op-
erator has been defined. It is also worth noting that there exists, by now, only a few
relevant references which treat the abstract degenerate Volterra equations ([FT]-[FLT],
[K], [MK7]-[MK8]), for which we can freely say that contain a set of partial results that
are not so easily linked and not enough for creating a stable and consistent theory. 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. 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 [C] and [FY]. We introduce the notion of a C-resolvent of a multivalued
linear operator, reconsider the assertions from [FY, Chapter I] and state a generalization
of [MK2, Proposition 2.1.14] for C-resolvents of multivalued linear operators. Following
the approach of C. Knuckles and F. Neubrander [KN], 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 [MK2, 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
[5]
6 Marko Kosti´c
extends the corresponding one introduced by T.-J. Xiao and J. Liang ([XL1], 1997), and
coincides with the classical concept of vector-valued Laplace transform in the case that
the state space Xis one of Banach’s [ABHN]. Concerning the integration of functions
with values in sequentially complete locally convex spaces, we follow the approach of
C. Martinez and M. Sanz (cf. [MS, pp. 99-102] for more details). Once we have proved
the formula for partial integration in Theorem 1.10, 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 1.12(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 A. Yagi [Y],
written over the last twenty five years, have concerned applications of multivalued linear
operators to abstract degenerate differential equations (cf. [CS], [DU], [FY] and [MF]-
[OZ] for the primary source of information on this subject). In Section 4, we analyze the
following abstract degenerate Volterra inclusion:
Bu(t)⊆ A
t
Z
0
a(t−s)u(s)ds +F(t), t ∈[0, τ),(1.1)
where a∈L1
loc([0, τ )), a 6= 0,A:X→P(Y) and B:X→P(Y) are given multivalued
linear operators acting between sequentially complete locally convex spaces Xand Y, and
F:X→P(Y) is a given mutivalued mapping, as well as the following fractional Sobolev
inclusions:
(DFP)R:Dα
tBu(t)∈ Au(t) + F(t), t ≥0,
(Bu)(j)(0) = Bxj,0≤j≤ dαe − 1,
where we assume that B=Bis single-valued, and
(DFP)L:BDα
tu(t)⊆ Au(t) + F(t), t ≥0,
u(j)(0) = xj,0≤j≤ dαe − 1.
Here, Dα
tu(t) denotes the Caputo fractional derivative of function u(t).We define various
types of solutions of problems (1.1), (DFP)Rand (DFP)L.In Theorem 1.18 and Theorem
1.19, we reconsider the main results of research of M. Kim [K], while in Theorem 1.20 we
prove an extension of [KN, Theorem 3.5] for abstract degenerate fractional differential
inclusions. Subordination principles are clarified in Theorem 1.21 and Theorem 1.22 fol-
lowing the methods proposed by J. Pr¨uss [JP, Section 4] and E. Bazhlekova [EB, Section
3] (cf. [FD] and [MK7]-[MK11] for similar results known in degenerate case).
Following the old ideas of R. deLaubenfels [DL], in Section 5 we introduce and analyze
the class of (a, k)-regularized (C1, C2)-existence and uniqueness families (cf. [MK2, Sec-
tion 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
Abstract degenerate Volterra inclusions... 7
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). It is well known (see e.g.
[FY, Theorem 2.4], [KN, Theorem 3.6] and [K, p. 169]) that Hille-Yosida’s type estimates
for the resolvent of a multivalued operator Aimmediately implies that Ais single-valued
in a certain sense. In part (ii) of Theorem 1.33, we will prove a similar assertion pro-
vided that the Hille-Yosida condition (1.39) below holds. For the validity of Theorem
1.33(ii), we have found the condition k(0) 6= 0 very important to be satisfied; in other
words, the existence of above-mentioned single-valued branch of Acan be proved exactly
in non-convoluted or non-integrated case, so that we have arrived to a diametrically op-
posite conclusion to that stated on l. 7-13, p. 169 of [K]. Nevertheless, the existence or
non-existence of such a single-valued branch of Ais not sufficient for obtaining a fairly
complete information on the well-posednesss of inclusion (1.1) with B=I(the reading
of papers [K]-[KN] has strongly influenced us to write this paper, and compared with
the results of [K], here we do not need the assumption that a(t) is a normalized func-
tion of local bounded variation). In the remaining part of Section 5, we enquire into the
possibility to extend the most important results from [MK2, 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 equicon-
tunuous (a, k)-regularized C-resolvent families, the generalized variation of parameters
formula, and subordination principles; in a separate subsection, we analyze differential
and analytical properties of (a, k)-regularized C-resolvent families.
For the sake of brevity and better exposition, and because of some similarity with our
previous researches of non-degenerate case, we have been forced to write this paper in a
half-expository manner, including only the most relevant details of proofs of our struc-
tural results. It is beyond the scope of this paper to further examine various subclasses
of degenerate convoluted C-semigroups and degenerate convoluted C-cosine functions
in locally convex spaces, as well as perturbations and approximation properties of ab-
stract degenerate Volterra integro-differential equations; these themes will be considered
somewhere else.
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 stimu-
lating and enlightening discussions during the research.
We use the standard terminology throughout the paper. By Xwe denote a Hausdorff
sequentially complete locally convex space over the field of complex numbers, SCLCS
for short. If Yis 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 Xinto 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 Zis denoted by ~Z.The symbol
IX(IY) denotes the identity operator on X(Y); if there is no risk for confusion, we
will also write Iin place of IX.By X∗we denote the dual space of X. Let 0 < τ ≤
8 Marko Kosti´c
∞.A strongly continuous operator family (W(t))t∈[0,τ)⊆L(X, Y ) is said to be locally
equicontinuous iff, for every T∈(0, τ ) and for every p∈~Y,there exist qp∈~Xand
cp>0 such that p(W(t)x)≤cpqp(x), x ∈X, t ∈[0, T ]; the notions of equicontinuity
of (W(t))t∈[0,τ)and the exponential equicontinuity of (W(t))t≥0are defined similarly.
Notice that (W(t))t∈[0,τ)is automatically locally equicontinuous in case that the space
Xis barreled ([MV]).
By Bwe denote the family consisting of all bounded subsets of X. Define pB(T) :=
supx∈Bp(T x), p ∈~Y,B∈ B, T ∈L(X, Y ).Then pB(·) is a seminorm on L(X, Y ) and
the system (pB)(p,B)∈~Y×B induces the Hausdorff locally convex topology on L(X, Y ).If
Yis 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 Aby D(A), N(A) and R(A),respectively. Since no confusion seems likely, we
will identify Awith its graph. Set pA(x) := p(x) + p(Ax), x ∈D(A), p ∈~. Then the
calibration (pA)p∈~induces the Hausdorff sequentially complete locally convex topology
on D(A); we denote this space simply by [D(A)].
Suppose that Vis a general topological vector space. 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 [ABHN], [MK2, 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 →Xis equivalent with its weak
analyticity.
A function f: [0, T ]→X, where 0 < T < ∞,is said to be H¨older continuous with
the exponent r∈(0,1] iff for each p∈~Xthere exists M≥1 such that p(f(t)−f(s)) ≤
M|t−s|r,provided 0 ≤t, s ≤T, while a function f: [0,∞)→Xis said to be locally
H¨older continuous with the exponent riff its restriction on any finite interval [0, T ]
is H¨older continuous with the same exponent. By ACloc([0,∞)) we denote the space
consisting of all functions f: [0,∞)→Xwhose restriction on any finite interval [0, T ]
(T > 0) is absolutely continuous.
Let 0 < τ ≤ ∞ and a∈L1
loc([0, τ )).Then we say that the function a(t) is a kernel
on [0, τ ) iff for each f∈C([0, τ )) the assumption Rt
0a(t−s)f(s)ds = 0, t ∈[0, τ )
implies f(t)=0, t ∈[0, τ ).Given s∈Rin advance, set bsc:= sup{l∈Z:l≤s}
and dse:= inf{l∈Z:s≤l}.The Gamma function is denoted by Γ(·) and the principal
branch is always used to take the powers. Set gζ(t) := tζ−1/Γ(ζ) (ζ > 0, t > 0), g0(t) := δ-
distribution and, by common consent, 0ζ:= 0.For any angle α∈(0, π],we define
Σα:= {z∈C:z6= 0,|arg(z)|< α}.Set C+:= {λ∈C:<λ > 0}.
Now we will 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 µwe denote a locally finite Borel measure defined on Ω.A function
f: Ω →Xis said to be µ-measurable iff there exists a sequence (fn) in XΩof simple
functions (cf. [MK2, Definition 1.1.1(i)] for the notion) such that limn→∞ fn(t) = f(t)
for a.e. t∈Ω.
Definition 1.1.Let K⊆Ω be a compact set, and let a function f:K→Xbe strongly
Abstract degenerate Volterra inclusions... 9
measurable. Then it is said that f(·) is (µ-)integrable if there is a sequence (fn)n∈Nof
simple functions such that limn→∞ fn(t) = f(t) a.e. t∈Kand for all > 0 and each
p∈~there is a number n0=n0(, p) such that
Z
K
pfn−fmdµ ≤(m, n ≥n0).(1.2)
In this case we define Z
K
f dµ := lim
n→∞ Z
K
fndµ.
Due to (1.2), we have that (p(fn))n∈Nis a Cauchy sequence in the space L1(K, µ),
so that the limit p(f) = limn→∞ p(fn) is µ-integrable. Similarly we can prove that each
function p(fn−f) is µ-integrable and the sequence of its corresponding integrals converges
to zero. Recall that every continuous function f:K→Xis µ-integrable.
Definition 1.2.(i) A function f: Ω →Xis said to be locally µ-integrable if, for
every compact set K⊆Ω,the restriction f|K:K→Xis µ-integrable.
(ii) A function f: Ω →Xis said to be µ-integrable if it is locally integrable and if
additionally Z
Ω
p(f)dµ < ∞, p ∈~.(1.3)
If this is the case, we define Z
Ω
f dµ := lim
n→∞ Z
Kn
f dµ,
with (Kn)n∈Nbeing an expansive sequence of compact subsets of Ω with the prop-
erty that Sn∈NKn= Ω.
The above definition does not depend on the choice of sequence (Kn)n∈N.Moreover,
pZΩ
f dµ≤ZΩ
p(f)dµ, p ∈~.(1.4)
It is not difficult to verify that the µ-integrablity of a function f:K→X, resp. f: Ω →
X, implies that for each x∗∈X∗,one has:
*x∗,Z
K
f dµ+=Z
Kx∗, f dµ, resp. *x∗,Z
Ω
f dµ+=Z
Ωx∗, f dµ. (1.5)
Definition 1.2 is equivalent with the definition of Bochner integral, provided that Xis
a Banach space. Furthermore, every continuous function f: Ω →Xsatisfying (1.3) is
µ-integrable and the following holds:
Theorem 1.3.(i) (The Dominated Convergence Theorem) Suppose that (fn)is a se-
quence of µ-integrable functions from XΩand (fn)converges pointwisely to a func-
tion f: Ω →X. Assume that, for every p∈~,there exists a µ-integrable function
Fp: Ω →[0,∞)such that p(fn)≤Fp, n ∈N.Then f(·)is a µ-integrable function
and limn→∞ RΩfndµ =RΩf dµ.
10 Marko Kosti´c
(ii) Let Ybe a SCLCS, and let T:X→Ybe a continuous linear mapping. If f: Ω →X
is µ-integrable, then T f : Ω →Yis likewise µ-integrable and
TZ
Ω
f dµ =Z
Ω
T f dµ. (1.6)
(iii) Let Ybe a SCLCS, and let T:D(T)⊆X→Ybe a closed linear mapping.
If f: Ω →D(T)is µ-integrable and T f : Ω →Yis likewise µ-integrable, then
RΩf dµ ∈D(T)and (1.6) holds.
Recent decades have witnessed a fast growing applications of fractional calculus and
fractional differential equations to diverse scientific and engineering fields (cf. [EB], [KD],
[KST], [IP], [SKM] and references cited therein for further information). In this paper,
we mainly use the Caputo fractional derivaties. Let ζ > 0.Then the Caputo fractional
derivative Dζ
tu([EB], [MK2]) is defined for those functions u∈Cdζe−1([0,∞) : X) for
which gdζe−ζ∗(u−Pdζe−1
j=0 u(j)(0)gj+1)∈Cdζe([0,∞) : X),by
Dζ
tu(t) := ddζe
dtdζe"gdζe−ζ∗ u−dζe−1
X
j=0
u(j)(0)gj+1!#.
Define Cr([0, T ] : X) to be the vector space consisting of H¨older continuous functions
f: [0, T ]→Xwith the exponent r; if r0∈(0,∞)\N,then we define Cr0([0, T ] : X) as the
vector space consisting of those functions f: [0, T ]→Xfor which f∈Cbr0c([0, T ] : X)
and f(br0c)∈Cr0−br0c([0, T ] : X).We need this notion because we would like to observe
that the existence of Caputo fractional derivative Dζ
tuimplies u∈Cdζe((0,∞) : X)∩
Cζ([0, T ] : E),for each finite number T > 0. A proof is left to the interested reader.
We refer the reader to [EB] for the notion of a Riemann-Liouville fractional derivative
Dα
tu(t) of order α > 0.The Mittag-Leffler function Eβ,γ (z) (β > 0, γ ∈R) is defined by
Eβ,γ (z) := ∞
X
k=0
zk
Γ(βk +γ), z ∈C.
Set, for short, Eβ(z) := Eβ,1(z), z ∈C.If β∈(0,1),then we define the Wright function
Φβ(·) by
Φβ(t) := L−1Eβ(−λ)(t), t ≥0,
where L−1denotes the inverse Laplace transform. For further information about the
Mittag-Leffler and Wright functions, we refer the reader to [EB], [MK2] and references
cited there.
1.2. Multivalued linear operators in locally convex spaces. A multivalued map
(multimap) A:X→P(Y) is said to be a multivalued linear operator (MLO) iff the
following holds:
(i) D(A) := {x∈X:Ax6=∅} is a linear subspace of X;
(ii) Ax+Ay⊆ A(x+y), x, y ∈D(A) and λAx⊆ A(λx), λ ∈C, x ∈D(A).
If X=Y, then we say that Ais 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
Abstract degenerate Volterra inclusions... 11
x∈D(A), λ 6= 0.Furthermore, for any x, y ∈D(A) and λ, η ∈Cwith |λ|+|η| 6= 0,
we have λAx+ηAy=A(λx +ηy).If Ais an MLO, then A0 is a linear manifold in Y
and Ax=f+A0 for any x∈D(A) and f∈ Ax. Set R(A) := {Ax:x∈D(A)}.The
set A−10 = {x∈D(A) : 0 ∈ Ax}is called the kernel of Aand it is denoted henceforth
by N(A) or Kern(A).The inverse A−1of an MLO is defined by D(A−1) := R(A) and
A−1y:= {x∈D(A) : y∈ Ax}. It is checked at once that A−1is an MLO in X, as well
as that N(A−1) = A0 and (A−1)−1=A.If N(A) = {0},i.e., if A−1is single-valued,
then Ais said to be injective. It is worth noting that Ax=Ayfor some two elements x
and y∈D(A),iff Ax∩ Ay6=∅; moreover, if Ais injective, then the equality Ax=Ay
holds iff x=y.
For any mapping A:X→P(Y) we define ˇ
A:= {(x, y) : x∈D(A), y ∈ Ax}.Then
Ais an MLO iff ˇ
Ais a linear relation in X×Y, i.e., iff ˇ
Ais a linear subspace of X×Y.
If A,B:X→P(Y) are two MLOs, then we define its sum A+Bby D(A+B) :=
D(A)∩D(B) and (A+B)x:= Ax+Bx, x ∈D(A+B).It can be simply verified that
A+Bis likewise an MLO.
Let A:X→P(Y) and B:Y→P(Z) be two MLOs, where Zis an SCLCS.
The product of Aand Bis defined by D(BA) := {x∈D(A) : D(B)∩ Ax6=∅} and
BAx:= B(D(B)∩ Ax).Then BA :X→P(Z) is an MLO and (BA)−1=A−1B−1.
The scalar multiplication of an MLO A:X→P(Y) with the number z∈C, zAfor
short, is defined by D(zA) := D(A) and (zA)(x) := zAx, x ∈D(A).It is clear that
zA:X→P(Y) is an MLO and (ωz)A=ω(zA) = z(ωA), z, ω ∈C.
Suppose that X0is a linear subspace of X, and A:X→P(Y) is an MLO. Then we
define the restriction of operator Ato the subspace X0,A|X0for short, by D(A|X0) :=
D(A)∩X0and A|X0x:= Ax, x ∈D(A|X0).Clearly, A|X0:X0→P(Y) is an MLO.
It is well known that an MLO A:X→P(Y) is injective (resp., single-valued) iff
A−1A=I|D(A)(resp., AA−1=IY
|R(A)).
The integer powers of an MLO A:X→P(X) are defined recursively as follows:
A0=: I; if An−1is defined, set
D(An) := x∈D(An−1) : D(A)∩ An−1x6=∅,
and
Anx:= AAn−1x=[
y∈D(A)∩An−1xAy, x ∈D(An).
We can prove inductively that (An)−1= (An−1)−1A−1= (A−1)n=: A−n, n ∈Nand
D((λ−A)n) = D(An), n ∈N0.Moreover, if Ais single-valued, then the above definitions
are consistent with the usual definition of powers of A.
If A:X→P(Y) and B:X→P(Y) are two MLOs, then we write A⊆Biff
D(A)⊆D(B) and Ax⊆ Bxfor all x∈D(A).Assume now that a linear single-valued
operator S:D(S)⊆X→Yhas domain D(S) = D(A) and S⊆ A,where A:X→P(Y)
is an MLO. Then Sis called a section of A; if this is the case, we have Ax=Sx +A0,
x∈D(A) and R(A) = R(S) + A0.
We say that an MLO operator A:X→P(Y) is closed if for any nets (xτ) in D(A) and
(yτ) in Ysuch that yτ∈ Axτfor all τ∈Iwe have that the suppositions limτ→∞ xτ=x
12 Marko Kosti´c
and limτ→∞ yτ=yimply x∈D(A) and y∈ Ax.
We introduce the notion of a relatively closed MLO as follows ([KN]). We say that
an MLO A:X→P(Y) is relatively closed iff there exist auxiliary SCLCSs XAand
YAsuch that D(A)⊆XA→X, R(A)⊆YA→Yand Ais closed in XA×YA; i.e.,
the assumptions D(A)3xτ→xas τ→ ∞ in XAand Axτ3yτ→yas τ→ ∞
in YAimplies that x∈D(A) and y∈ Ax. A relatively closed operator will also be
called XA×YA-closed. For example, let A, B :D⊆X→Ybe closed linear operators
with the same domain D. Then the operator A+Bis not necessarily closed but it is
always [D(A)] ×Y-closed (cf. [K, p. 170]). Examples presented in [KN] can be simply
reformulated for operators acting on locally convex spaces, as well:
Example 1.4.(i) If A:X→P(Y) is an MLO, then A:X→P(Y) is likewise an
MLO. This shows that any MLO has a closed linear extension, in contrast to the
usually considered single-valued linear operators.
(ii) Let A:D(A)⊆X→Ybe a single-valued linear operator that is XA×YA-closed,
let B:X→P(Y) be an MLO that is XB×YB-closed, and let YA→YB.Then the
MLO S=A+Bis XS×YB-closed, where XS:= D(A)∩XBand the topology on
XSis induced by the system (sp,q,r) of fundamental seminorms, defined as follows:
sp,q,r (x) =: p(x) + p(Ax) + q(x) + r(Ax), x ∈XS(p∈~X, q ∈~XB, r ∈~YA).
(iii) Let A:D(A)⊆X→Ybe a single-valued linear operator that is XA×YA-closed,
let B:Y→P(Z) be an MLO that is YB×ZB-closed, and let YB→YA.Then the
MLO C=BA:X→P(Z) is XC×ZB-closed, where XC:= {x∈D(A) : Ax ∈YB}
and the topology on XCis induced by the system (sp,q) of fundamental seminorms,
defined as follows: sp,q(x) =: p(x) + p(Ax) + q(Ax), x ∈XC(p∈~X, q ∈~YB).
(iv) Let A:D(A)⊆X→Yand B:D(B)⊆X→Ybe two single-valued linear
operators. Set
A:= B−1A=(x, y) : x∈D(A), y ∈D(B) and Ax =By.
Then Ais an MLO in X, and the following holds:
(a) If one of the operators A, B is bounded and the other closed, then Ais closed.
(b) If Ais closed and Bis XB×Y-closed, then Ais [D(A)] ×XB-closed.
(c) If Bis closed and Ais XA×Y-closed, then Ais XA×[D(B)]-closed.
(d) If Ais XA×YA-closed and Bis XB×YB-closed, where YB→YA, then Ais
XC×XB-closed, where XCis defined as in (iii).
If A:X→P(Y) is an MLO, then we define the adjoint A∗:Y∗→P(X∗) of Aby
its graph
A∗:= ny∗, x∗∈Y∗×X∗:y∗, y=x∗, xfor all pairs (x, y)∈ Ao.
It is simpy verified that A∗is a closed MLO, and that hy∗, yi= 0 whenever y∗∈D(A∗)
and y∈ A0.Furthermore, A∗is single-valued provided that Ais densely defined, A∗=A∗
and the equations [FY, (1.2)-(1.6)] continue to hold for adjoints of MLOs acting on locally
convex spaces.
The following important lemma can be proved by using the Hahn-Banach theorem
and the argumentation from [B].
Abstract degenerate Volterra inclusions... 13
Lemma 1.5.Suppose that A:X→P(Y)is an MLO and Ais XA×YA-closed. Assume,
further, that x0∈X, y0∈Yand hx∗, x0i=hy∗, y0ifor all pairs (x∗, y∗)∈X∗
A×Y∗
A
satisfying that hx∗, xi=hy∗, yiwhenever y∈ Ax. Then y0∈ Ax0.
With Lemma 1.5 in view, we can simply prove the following extension of Theorem
1.3(iii) for relatively closed MLOs in locally convex spaces.
Theorem 1.6.Suppose that A:X→P(Y)is an MLO and Ais XA×YA-closed. Let
f: Ω →XAand g: Ω →YAbe µ-integrable, and let g(x)∈ Af(x), x ∈Ω.Then
RΩf dµ ∈D(A)and RΩg dµ ∈ ARΩf dµ.
In the remaining part of this section, we will analyze the C-resolvent sets of multival-
ued 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 and CA ⊆ AC(this is equivalent
to say that, for any (x, y)∈X×X, we have the implication (x, y)∈ A ⇒ (C x, Cy)∈ A;
by induction, we immediately get that CAk⊆ AkCfor all k∈N). Then the C-resolvent
set of A, ρC(A) for short, is defined as the union of those complex numbers λ∈Cfor
which
(i) R(C)⊆R(λ− A);
(ii) (λ− A)−1Cis a single-valued bounded operator on X.
The operator λ7→ (λ− A)−1Cis called the C-resolvent of A(λ∈ρC(A)); the resolvent
set of Ais defined by ρ(A) := ρI(A), R(λ:A)≡(λ− A)−1(λ∈ρ(A)). We can almost
trivially construct examples of MLOs for which ρ(A) = ∅and ρC(A)6=∅; for example,
let Ybe a proper closed linear subspace of X, let Abe an MLO in Y, and let λ∈Cso
that (λ− A)−1∈L(Y).Taking any injective operator C∈L(X) with R(C)⊆Y, and
looking A=AXas an MLO in X, it is clear that λ∈ρC(AX) and ρ(AX) = ∅.In general
case, if ρC(A)6=∅,then for any λ∈ρC(A) we have A0 = N((λI − A)−1C),as well as
λ∈ρC(A),A ⊆ C−1ACand ((λ− A)−1C)k(D(Al)) ⊆D(Ak+l), k, l ∈N0; here it is
worth noting that the equality A=C−1ACholds provided, in addition, that ρ(A)6=∅
(see e.g. the proofs of [DLYW, Proposition 2.1, Lemma 2.3]). The basic properties of
C-resolvent sets of single-valued linear operators ([MK1]-[MK2]) continue to hold in our
framework (observe, however, that there exist certain differences that we will not discuss
here). For example, if ρ(A)6=∅,then Ais closed; it is well known that this statement
does not hold if ρC(A)6=∅for some C6=I(cf. [DLYW, Example 2.2]). Arguing as in
the proofs of [FY, Theorem 1.7-Theorem 1.9], we can deduce the validity of the following
important theorem, which will be frequently used in the sequel.
Theorem 1.7.(i) We have
λ− A−1CA ⊆ λλ− A−1C−C⊆ Aλ− A−1C, λ ∈ρC(A).
The operator (λ− A)−1CAis single-valued on D(A)and (λ− A)−1CAx= (λ−
A)−1Cy, whenever y∈ Axand λ∈ρC(A).
(ii) Suppose that λ, µ ∈ρC(A).Then the resolvent equation
λ− A−1C2x−µ− A−1C2x= (µ−λ)λ− A−1Cµ− A−1Cx, x ∈X
holds good. In particular, (λ− A)−1C(µ− A)−1C= (µ− A)−1C(λ− A−1C.
14 Marko Kosti´c
By Theorem 1.7(i), it readily follows that the operator λ(λ− A)−1C−C∈L(X) is
a bounded linear section of the MLO A(λ− A)−1C(λ∈ρC(A)). Inductively, we can
prove that, for every x∈X, n ∈N0and λ∈ρC(A),we have card((λ− A)−nCx)≤1.
Having in mind this fact, as well as the argumentation already seen many times in our
previous research studies of C-resolvents of single-valued linear operators, we can prove
the following extension of [MK2, Proposition 2.1.14] for MLOs in locally convex spaces.
Proposition 1.8.Let ∅ 6= Ω ⊆ρC(A)be open, and let x∈X.
(i) The local boundedness of the mapping λ7→ (λ− A)−1Cx, λ ∈Ω,resp. the as-
sumption that Xis barreled and the local boundedness of the mapping λ7→ (λ−
A)−1C, λ ∈Ω,implies the analyticity of the mapping λ7→ (λ− A)−1C3x, λ ∈Ω,
resp. λ7→ (λ−A)−1C3, λ ∈Ω.Furthermore, if R(C)is dense in X, resp. if R(C)
is dense in Xand Xis barreled, then the mapping λ7→ (λ− A)−1Cx, λ ∈Ωis
analytic, resp. the mapping λ7→ (λ− A)−1C, λ ∈Ωis analytic.
(ii) Suppose that R(C)is dense in X. Then the local boundedness of the mapping λ7→
(λ− A)−1Cx, λ ∈Ωimplies its analyticity as well as Cx ∈R((λ− A)n), n ∈N
and
dn−1
dλn−1λ− A−1Cx = (−1)n−1(n−1)!λ− A−nC x, n ∈N.(1.7)
Furthermore, if Xis barreled, then the local boundedness of the mapping λ7→
(λ− A)−1C, λ ∈Ωimplies its analyticity as well as R(C)⊆R((λ− A)n), n ∈N
and
dn−1
dλn−1λ− A−1C= (−1)n−1(n−1)!λ− A−nC∈L(X), n ∈N.(1.8)
(iii) The continuity of mapping λ7→ (λ− A)−1Cx, λ ∈Ωimplies its analyticity and
(1.7). Furthermore, if Xis barreled, then the continuity of mapping λ7→ (λ−
A)−1C, λ ∈Ωimplies its analyticity and (1.8).
It is well known that ρC(A) need not be an open subset of Cif C6=Iand Ais a
single-valued linear operator (cf. [DLYW, Example 2.5]) and that ρ(A) is an open subset
of C,provided that Xis a Banach space and Ais an MLO in X(cf. [FY, Theorem 1.6]).
The regular C-resolvent set of A, ρr
C(A) for short, is defined as the union of those complex
numbers λ∈ρC(A) for which (λ− A)−1C∈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(Anx)≤cnq(x), x ∈X, n ∈N; the regular resolvent set of
A, ρr(A) for short, is then defined by ρr(A) := ρr
I(A).By the argumentation contained
in the proof of [FY, Theorem 1.6], it readily follows that ρr(A) is always an open subset
of C.
We close this section with the observation that 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.
Theorem 1.9.(i) Let x∈X, k ∈N0and λ, z ∈ρC(A)with z6=λ. Then the following
Abstract degenerate Volterra inclusions... 15
holds:
z− A−1Cλ− A−1Ckx
=(−1)k
(z−λ)kz− A−1Ck+1x+
k
X
i=1
(−1)k−i(λ− A)−1CiCk+1−ix
z−λk+1−i.
(ii) Let k∈N0, x, y ∈X,y∈(λ0− A)kxand λ0, z ∈ρC(A)with z6=λ0.Then the
following holds:
z− A−1Ck+1x=(−1)k
z−λ0kz− A−1Ck+1y
+
k
X
i=1
(−1)k−i(λ0− A)−1CiCk+1−iy
z−λ0k+1−i.
1.3. Laplace transform of functions with values in sequentially complete locally
convex spaces. In this section, we assume that µ=dt is the Lebesgue’s measure on
Ω = [0,∞) and f: [0,∞)→Xis 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 L1
loc([0,∞) : X); similarly we define the space L1([0, τ ] : X) for 0 < τ < ∞.
It is clear that (1.5) implies hx∗, f (·)i ∈ L1
loc([0,∞)) for x∗∈X∗.The first normalized
antiderivative t7→ f[1](t) := F(t) := Rt
0f(s)ds, t ≥0 of f(·) is continuous for t≥0,and
we have that Rt
0p(f)dµ < ∞for any p∈~and t≥0.Set f[n](t) := Rt
0gn(t−s)f(s)ds,
t≥0.
A few auxiliary results on integration in sequentially complete locally convex spaces
is collected in the following theorem, which might be known in the existing literature.
For the sake of completeness, we will include a proof.
Theorem 1.10.(i) Suppose that g∈C([0,∞)).Then gf ∈L1
loc([0,∞) : X).
(ii) If g∈L1
loc([0,∞)) and f∈C([0,∞) : X),then gf ∈L1
loc([0,∞) : X).
(iii) (The partial integration) Suppose that g∈ACloc([0,∞)).Then, for every τ≥0,
we have Zτ
0
g(t)f(t)dt =g(τ)F(τ)−Zτ
0
g0(t)F(t)dt. (1.9)
Proof. Fix a number τ∈(0,∞).Let (fn)n∈Nbe a sequence of simple functions in X[0,τ ]
such that limn→∞ fn(t) = f(t) a.e. t∈K= [0, τ] and for all > 0 and each p∈~there
is a number n0=n0(, p) such that (1.2) holds. Then Rτ
0f(t)dt = limn→∞ Rτ
0fn(t)dt
and the sequence (p(fn))n∈Nis convergent in L1[0, τ ].By the proof of [MS, Proposition
4.4.1], there exists a sequence (sn)n∈Nof simple functions in C[0,τ]such that limn→∞ ksn−
gkL∞[0,τ]= 0,supn∈NksnkL∞[0,τ]≤ kgkL∞[0,τ]and that for all > 0 and p=|·|there is
a number n0=n0(, p) such that (1.2) holds with the functions fn(·) and fm(·) replaced
respectively with sn(·) and sm(·).Clearly, (snfn)n∈Nis a sequence of simple functions
in X[0,τ]such that limn→∞ sn(t)fn(t) = g(t)f(t) a.e. t∈[0, τ].Furthermore, it can be
16 Marko Kosti´c
easily seen that
Zt
0
psn(t)fn(t)−sm(t)fm(t)dt
≤ ksnkL∞[0,τ]Zt
0
pfn(t)−fm(t)dt +ksn−smkL∞[0,τ]Zt
0
pfm(t)dt
≤ kgkL∞[0,τ]Zt
0
pfn(t)−fm(t)dt
+ksn−gkL∞[0,τ]+ksm−gkL∞[0,τ]Zt
0
pfm(t)dt, m, n ∈N.
This proves (i). To prove (ii), observe first that using Definition 1.1 we can directly prove
that a function g1f1(·) belongs to the space L1([0, τ ] : X),provided that f1: [0, τ]→X
is a simple function and g1∈L1[0, τ ].Using the arguments contained in the proof of
[MS, Proposition 4.4.1] once more, we can find a sequence (fn)n∈Nof simple functions
in X[0,τ]such that, for every p∈~,limn→∞ p(fn−f)L∞[0,τ]= 0,supn∈Np(fn)L∞[0,τ ]≤
p(f)L∞[0,τ]and that for all > 0 there is a number n0=n0(, p) such that (1.2) holds.
Therefore, (gfn)n∈Nis a sequence in L1([0, τ ] : X) and limn→∞ g(t)fn(t) = g(t)f(t) a.e.
t∈[0, τ ].Making use of the dominated convergence theorem (Theorem 1.3(i)), we get
that gf ∈L1
loc([0,∞) : X),as claimed. By (i) and (ii), the both integral in (1.9) are
well-defined. Let x∗∈X∗.Using the partial integration in the Lebesgue integral and
(1.5), we get that
Zτ
0
g(t)x∗, f (t)dt =g(τ)x∗, F (τ)−Zτ
0
g0(t)x∗, F (t)dt.
Since x∗was arbitrary, it readily follows from (1.5) that (1.9) 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
(Lf)(λ) := ˜
f(λ) := Z∞
0
e−λtf(t)dt := lim
τ→∞ Zτ
0
e−λtf(t)dt,
for λ∈C.If ˜
f(λ0) exists for some λ0∈C,then we define the abscissa of convergence of
˜
f(·) by
absX(f) := inf<λ:˜
f(λ) exists;
otherwise, absX(f) := +∞.It is said that f(·) is Laplace transformable, or equivalently,
that f(·) belongs to the class (P1)-X, iff absX(f)<∞.Assuming that there exists a
number ω∈Rsuch that for each seminorm p∈~there exists a number Mp>0
satisfying that p(f(t)) ≤Mpeωt, t ≥0,we define ωX(f)∈[−∞,∞) as the infimum of
all numbers ω∈Rwith the above property; if there is no such a number ω∈R,then
we define ωX(f) := +∞.Further on, we abbreviate ωX(f) (absX(f)) to ω(f) (abs(f)),
if there is no risk for confusion. Define
wabs(f) := inf(λ∈R: sup
t>0Zt
0
e−λsx∗, f (s)ds<∞for all x∗∈X∗),
F∞:= limτ→∞ F(τ),if the limit exists in X, and F∞:= 0,otherwise.
Abstract degenerate Volterra inclusions... 17
Keeping in mind Theorem 1.10, we can repeat literally the argumentation from
[ABHN, Section 1.4, pp. 27-30] in order to see that the following theorem holds good
(the only essential difference occurs on l. 6, p. 29, where we can use [MV, Mackey’s
theorem 23.15] in place of the uniform boundedness principle):
Theorem 1.11.Let f∈L1
loc([0,∞) : X).Then the following holds:
(i) The Laplace integral ˜
f(λ)converges if <λ > abs(f)and diverges if <λ < abs(f).If
<λ=abs(f),then the Laplace integral may or may not be convergent.
(ii) wabs(f) = abs(f).
(iii) Suppose that λ∈Cand the limit limτ→∞ Rt
0e−λsp(f(s)) ds exists for any p∈~.
Then ˜
f(λ)exists, as well.
(iv) We have
abs(f)≤abs(p(f)) ≤ω(f), p ∈~.
In general, any of these two inequalities can be strict.
(v) We have
abs(f) = ωF−F∞,(1.10)
˜
f(λ) = F∞+λZ∞
0
e−λtF(t)−F∞dt, <λ>ωF−F∞,(1.11)
˜
f(λ) = λ˜
F(λ),<λ > max(abs(f),0) (1.12)
and
abs(f)≤ω⇔ω(F)≤ω(if ω≥0).
In particular, f(·)is Laplace transformable iff ω(F)<∞.
Recall [XL], a function h(·) belongs to the class LT −Xiff there exist a function
g∈C([0,∞) : X) and a number ω∈Rsuch that ω(g)≤ω < ∞and h(λ) = (Lg)(λ) for
λ>ω; as observed in [MK2, Section 1.2], the assumption h∈LT −Ximmediately implies
that the function λ7→ 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 −Xwill be abbreviated to LTor −X. Keeping this observation and
the equations (1.10)-(1.11) in mind, we can simply prove that the mapping λ7→ ˜
f(λ),
<λ > abs(f) is analytic, provided that f∈(P1)−X. If this is the case, the following
formula holds:
dn
dλn˜
f(λ) = (−1)n∞
Z
0
e−λttnf(t)dt, n ∈N, λ ∈C,<λ > abs(f).(1.13)
In the following theorem, we will collect various operational properties of vector-valued
Laplace transform.
Theorem 1.12.Let f∈(P1)-X, z ∈Cand s≥0.
(i) Put g(t) := e−ztf(t),t≥0. Then g(·)is Laplace transformable, abs(g) = abs(f)−
<zand ˜g(λ) = ˜
f(λ+z),λ∈C,<λ > abs(f)− <z.
18 Marko Kosti´c
(ii) Put fs(t) := f(t+s),t≥0, hs(t) := f(t−s), t ≥sand hs(t) := 0, s ∈[0, t].Then
abs(fs) = abs(hs) = abs(f),e
fs(λ) = eλs(˜
f(λ)−Rs
0e−λtf(t)dt)and f
hs(λ) = e−λs ˜
f(λ)
(λ∈C,<λ>a).
(iii) Let T∈L(X, Y ).Then T◦f∈(P1)-Yand T˜
f(λ) = ^
(T◦f)(λ)for λ∈C,
<λ > abs(f).
(iv) Suppose that A:X→P(Y)is an MLO and Ais XA×YA-closed, as well as
f∈(P1) −XA, l ∈(P1) −YAand (f(t), l(t)) ∈ A for a.e. t≥0.Then (˜
f(λ),˜
l(λ)) ∈ A,
λ∈Cfor <λ > max(abs(f), abs(l)).
(v) Suppose, in addition, ω(f)<∞.Put
j(t) :=
∞
Z
0
e−s2/4t
√πt f(s)ds := lim
τ→∞
τ
Z
0
e−s2/4t
√πt f(s)ds, t > 0
and
k(t) :=
∞
Z
0
se−s2/4t
2√πt 3
2
f(s)ds := lim
τ→∞
τ
Z
0
se−s2/4t
2√πt 3
2
f(s)ds, t > 0.
Then j(·)and k(·)are Laplace transformable,
max(abs(j), abs(k)) ≤(max(ω(f),0))2,˜
j(λ) =
˜
f√λ
√λand ˜
k(λ) = ˜
f√λ
for all λ∈Cwith <λ > (max(ω(f),0))2.
(vi) Let f∈(P1)-X, h ∈L1
loc([0,∞)) and abs(|h|)<∞.Suppose, in addition, that
f∈C([0,∞) : X).Put
(h∗f)(t) := Zt
0
h(t−s)f(s)ds, t ≥0.
Then the mapping t7→ (h∗f)(t), t ≥0is continuous, h∗f∈(P1)-X, and
]
h∗f(λ) = ˜
h(λ)˜
f(λ), λ ∈C,<λ > maxabs(|h|), abs(f).
Proof. Keeping in mind Theorem 1.10, Theorem 1.3(ii) and Theorem 1.6, the asser-
tions (i)-(iv) can be proved as in the Banach space case (cf. [ABHN, Proposition 1.6.1-
Proposition 1.6.3] for more details). Consider now the part (v). Let λ∈Cwith <λ >
(max(ω(f),0))2be fixed. Then <(√λ)>max(ω(f),0) ≥max(ω(F),0) so that [MK2,
Theorem 1.2.1(v)] implies in combination with (1.12) that ˜
f(√λ) exists, as well as that
˜
f√λ=˜
F√λ
√λ=Z∞
0
e−λt Z∞
0
e−s2/4t
√πt f(s)ds dt.
On the other hand, we can use the dominated convergence theorem and an elementary
argumentation to prove that the mapping t7→ k(t), t > 0 is continuous as well as that for
each seminorm p∈~there exists a finite number mp>0 such that p(k(t)) ≤mpt(−1)/2,
t∈(0,1].This simply implies k∈L1
loc([0,∞) : X).Since
Z∞
0
e−λtx∗, k(t)dt =Z∞
0
e−√λtx∗, f (t)dt, x∗∈X∗,(1.14)
Abstract degenerate Volterra inclusions... 19
we obtain that
lim
τ→∞*x∗,Zτ
0
e−λtk(t)dt+=Dx∗,˜
f√λE, x∗∈X∗.
By Theorem 1.10(i), we get that the mapping τ7→ Rτ
0e−λtk(t)dt, τ ≥0 is continuous
so that the previous equality implies supτ≥0|hx∗,Rτ
0e−λtk(t)dti| <∞for all x∗∈X∗.
Therefore, Theorem 1.11(ii) shows that λ >wabs(k) = abs(k) and ˜
k(λ) exists. Using
again (1.14), it readily follows that ˜
k(λ) = f(√λ),as claimed. Similarly we can prove
that ˜
j(λ) = f(√λ)/√λ. Suppose, finally, that the requirements of (vi) hold. Then it is
very simple to prove that the mapping t7→ (h∗f)(t), t ≥0 is continuous as well as that
ω(1∗h∗f) = ω(h∗(1∗f)) <∞.An application of Theorem 1.11(v) yields that h∗f∈(P1)-
X. Fix now a number λ∈Cwith <λ > max(abs(|h|), abs(f)).Since abs(hx∗, f(·)i)≤
abs(f) for all x∗∈X∗,[ABHN, Proposition 1.6.4] implies that (L(h∗ hx∗, f (·)i))(λ)
exists. Using this fact, it readily follows that
sup
t>0Zt
0
e−s<λh∗x∗, f (·)(s)ds<∞, x∗∈X∗.
By Theorem 1.11(ii), we get that ]
h∗f(λ) exists. The equality ]
h∗f(λ) = ˜
h(λ)˜
f(λ) can
be proved in a routine manner.
For the sequel, we need the notion of a Lebesgue point of a function f∈L1
loc([0,∞) :
X).A point t≥0 is said to be a Lebesgue point of f(·) iff for each seminorm p∈~,we
have
lim
h→0
1
hZt+h
t
pf(s)−f(t)ds = 0.(1.15)
It is clear that any point of continuity of function f(·) is one of Lebesgue’s points of f(·),
as well as that the mapping t7→ F(t), t ≥0 is differentiable at any Lebesgue’s point of
f(·).Furthermore, a slight modification of the proof of [ABHN, Proposition 1.2.2; a)/b)]
shows that the following holds:
(Q1) For each seminorm p∈~there exists a set Np⊆[0,∞) of Lebesgue’s measure zero
such that
lim
h→0p 1
hZt+h
t
f(s)ds −f(t)!= 0, t ∈[0,∞)\Np
and that (1.15) holds for t∈[0,∞)\Np.
In the case that Xis a Fr´echet space, (Q1) immediately implies that almost every point
t > 0 is a Lebesgue point of f(·).
Using the proof of [ABHN, Theorem 1.7.7], Theorem 1.10(iii), as well as the equations
(1.4) and (1.13), we can simply prove that the Post-Widder inversion formula holds in
our framework:
Theorem 1.13.(Post-Widder) Suppose f∈(P1) −Xand t > 0is a Lebesgue point of
f(·).Then
f(t) = lim
n→∞(−1)n1
n!n
tn+1 ˜
f(n)n
t.
20 Marko Kosti´c
The situation is much more complicated if we consider the Phragm´en-Doetsch inver-
sion 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 1.14.Let f∈(P1)-Xand t≥0.Then the following holds:
f[2](t) = lim
λ→∞
∞
X
n=1
(−1)n−1n!−1enλt ˜
f(nλ)
nλ .
Proof. Due to Theorem 1.11(v), we have F∈C([0,∞) : X) and ω(F)<∞.The result
now follows easily from [MK2, Theorem 1.2.1(ix)].
Now we will state and prove the following uniqueness type theorem for the Laplace
transform.
Theorem 1.15.(The uniqueness theorem for the Laplace transform) Suppose f∈(P1)−
X, λ0> abs(f)and ˜
f(λ)=0for all λ>λ0.Then F(t)=0, t ≥0, f (t)=0if t > 0is
a Lebesgue point of f(·),and for each seminorm p∈~there exists a set Np⊆[0,∞)of
Lebesgue’s measure zero such that p(f(t)) = 0, t ∈[0,∞)\Np.In particular, if Xis a
Fr´echet space, then f(t)=0for a.e. t≥0.
Proof. The function t7→ F(t), t ≥0 is continuous and by Theorem 1.11(v) we get that
ω(F)<∞and ˜
F(λ) = 0, λ > max(λ0,0).Now we can apply Theorem 1.13 in order to
see that F(t)=0, t ≥0.The remaining part of proof is simple and therefore omitted.
Remark. Suppose that f∈L1
loc([0,∞) : X) and for each seminorm p∈~there exists a
set Np⊆[0,∞) of Lebesgue’s measure zero such that p(f(t)) = 0, t ∈[0,∞)\Np.Then
abs(f) =abs(p(f)) = −∞ (p∈~) and ˜
f(λ) = 0 for all λ∈C.
The following converse of Theorem 1.12(iv) simply follows from an application of
Theorem 1.14.
Proposition 1.16.Suppose that A:X→P(Y)is an MLO and Ais XA×YA-closed,
as well as f∈(P1) −XA, l ∈(P1) −YAand (˜
f(λ),˜
l(λ)) ∈ A, λ ∈Cfor <λ >
max(abs(f), abs(l)).Then Af[1](t) = l[1](t), t ≥0and Af(t) = l(t)for any t > 0which
is a Lebesgue point of both functions f(t)and l(t).
The method proposed by T.-J. Xiao and J. Liang in [XL1] 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. [XL, Section
1.1.1] and [MK2, 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 1.21 below). It is also worth noting that there exists a great number of
theoretical results from the monograph [ABHN], not mentioned so far, which can be
reconsidered for the Laplace transformable functions with values in SCLSCs; for example,
all structural results from [ABHN, 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.
Abstract degenerate Volterra inclusions... 21
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 0 < τ ≤ ∞
and F: [0, τ )→P(X).A single-valued function f: [0, τ )→Xis called a section of Fiff
f(t)∈ F(t) for all t∈[0, τ ).We denote the set of all sections, resp., all continuous sections,
of Fby sec(F),resp., secc(F).Suppose now that τ=∞and any function f∈sec(F)
belongs to the class (P1)-X. Then we define absX(F) := sup{absX(f) : f∈sec(v)};F(·)
is said to be Laplace transformable iff absX(F)<∞.
1.4. Abstract degenerate Volterra integro-differential inclusions. In the follow-
ing general definition, we introduce various types of solutions to the abstract degenerate
inclusions (1.1), (DFP)Rand (DFP)L.
Definition 1.17.Let 0 < τ ≤ ∞, α > 0, a ∈L1
loc([0, τ )), a 6= 0,F: [0, τ )→P(Y),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) iff (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
uB∈C([0, τ ) : Y) and ua,A∈C([0, τ) : Y) such that uB(t)∈ Bu(t) and ua,A(t)∈
ARt
0a(t−s)u(s)ds for t∈[0, τ ),as well as
uB(t)∈ua,A(t) + F(t), t ∈[0, τ ).
Strong solution of (1.1) is any function u∈C([0, τ) : X) satisfying that there exist
two continuous functions uB∈C([0, τ ) : Y) and uA∈C([0, τ) : Y) such that
uB(t)∈ Bu(t), uA(t)∈ Au(t) for all t∈[0, τ ),and
uB(t)∈(a∗uA)(t) + F(t), t ∈[0, τ ).
(ii) Let B=Bbe single-valued. By a p-solution of (DFP)R,we mean any X-valued
function t7→ u(t), t ≥0 such that the term t7→ Dα
tBu(t), t ≥0 is well-defined,
u(t)∈D(A) for t≥0,and the requirements of (DFP)Rhold; a pre-solution of
(DFP)Ris any p-solution of (DFP)Rthat is continuous for t≥0.Finally, a solution
of (DFP)Ris any pre-solution u(·) of (DFP)Rsatisfying additionally that there
exists a function uA∈C([0,∞) : Y) such that uA(t)∈ Au(t) for t≥0,and
Dα
tBu(t)∈uA(t) + F(t), t ≥0.
(iii) By a pre-solution of (DFP)L,we mean any continuous X-valued function t7→ u(t),
t≥0 such that the term t7→ Dα
tu(t), t ≥0 is well defined and continuous, as well
as that Dα
tu(t)∈D(B) and u(t)∈D(A) for t≥0, and that the requirements of
(DFP)Lhold; a solution of (DFP)Lis any pre-solution u(·) of (DFP)Lsatisfying
additionally that there exist functions uα,B∈C([0,∞) : Y) and uA∈C([0,∞) : Y)
such that uα,B(t)∈ BDα
tu(t) and uA(t)∈ Au(t) for t≥0,as well as that uα,B(t)∈
uA(t) + F(t), t ≥0.
Before proceeding further, we want to observe that the existence of solutions to (1.1),
(DFP)Ror (DFP)Limmediately implies that secc(F)6=∅,as well as that any strong
solution of (1.1) is already a solution of (1.1), provided that Aand Bare MLOs with
Abeing closed; this can be simply verified with the help of Theorem 1.6. The notion of
22 Marko Kosti´c
a (pre-)solution of problems (DFP)Rand (DFP)Lcan 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 problem (E) given on pp. 33-34 of [FY] (B=I, α = 1,F(t) = f(t) is continuous
single-valued). We refer the reader to [VFD] and [MK10] for some results about the
wellposedness of some special cases of problem (DFP)R.
In our further work, it will be assumed that Aand Bare multivalued linear opera-
tors. Observe that we cannot consider the qualitative properties of solutions of problems
(1.1), (DFP)Ror (DFP)Lin full generality by a simple passing to the multivalued linear
operators B−1Aor AB−1(see e.g. the definition of a solution of (1.1)). Concerning this
question, we have the following remark.
Remark. Suppose that 0 < τ ≤ ∞, α > 0,as well as that A:D(A)⊆X→Yand
B:D(B)⊆X→Yare two single-valued linear operators. Then B−1Ais an MLO in X,
and AB−1is an MLO in Y.
(i) Suppose that u(·) is a pre-solution (or, equivalently, solution) of problem (1.1)
with B=IX,A=B−1Aand F=f: [0, τ )→D(B) being single-valued. Then
u∈C([0, τ ) : X) and Bu(t) = A(a∗u)(t) + Bf (t), t ∈[0, τ ).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 t7→ Au(t), t ∈[0, τ ) and t7→ Bu(t),
t∈[0, τ ) are continuous, and (a∗Au)(t) = Bu(t)−Bf (t), t ∈[0, τ ).
(ii) Suppose that v(·) is a pre-solution (solution) of problem (DFP)Rwith B=IY,
A=AB−1,F=f: [0, τ )→Ybeing single-valued, and vj=Bxj(0 ≤j≤ dαe−1).
Let B−1∈L(Y, X ).Then the function u(t) := B−1v(t), t ≥0 is a pre-solution
(solution) of problem (DFP)Rwith B=Band A=A.
(iii) Suppose that F=f: [0, τ )→D(B) is single-valued and u(·) is a pre-solution
of problem (DFP)Lwith B=IXand A=B−1A. Then u(·) is a pre-solution
of problem (DFP)Lwith B=B, A=Aand F(t) = Bf (t), t ∈[0, τ ).If, in
addition to this, B∈L(X, Y ) and u(·) is a solution of problem (DFP)Lwith the
above requirements being satisfied, then u(·) is a solution of problem (DFP)Lwith
B=B, A=Aand F(t) = Bf (t), t ∈[0, τ ).
(iv) Suppose that u: [0,∞)→D(A)∩D(B).Then u(·) is a p-solution of problem
(DFP)Rwith B=Band A=Aiff v=Bu(·) is a pre-solution of problem
Dα
tv(t)∈AB−1v(t) + F(t), t ≥0,
v(j)(0) = Bxj,0≤j≤ dαe − 1.
(v) Suppose that CY∈L(Y) is injective and the closed graph theorem holds for the
mappings from Yinto Y. Then we define the set ρB
CY(A) := {λ∈C:λB −
Ais injective and (λB−A)−1CY∈L(Y)}.It can be simply checked that ρB
CY(A)⊆
ρCY(AB−1),as well as that
λ−AB−1−1CY=BλB −A−1CY, λ ∈ρB
CY(A).(1.16)
This is an extension of [FY, Theorem 1.14] and holds even in the case that the
operator CYdoes not commute with AB−1,when we define the CY-resolvent set of
the operator λ−AB−1in the same way as before. Observe also that the assumption
Abstract degenerate Volterra inclusions... 23
D(A)⊆D(B),which has been used in [FY, Section 1.6], is not necessary for the
validity of (1.16).
(vi) Suppose that X=Y, C ∈L(X) is injective, B∈L(X), CA ⊆AC and CB ⊆BC.
Define the set ρB
C(A) as above. Then we have ρB
C(A)⊆ρC(B−1A) and
λ−B−1A−1Cx =λB −A−1C Bx, x ∈X.
Furthermore, if C=I, X 6=Yand B∈L(X, Y ),then ρB(A)⊆ρ(B−1A) and the
previous equality holds.
Consider now the case in which the operator Ais closed, the operator B=Bis
single-valued and the function F(t) = f(t) is Y-continuous at each point t≥0.Then any
pre-solution u(·) of problem (DFP)Ris already a solution of this problem, and Theorem
1.6 in combination with the identity [EB, (1.21)] implies that
Bu(t)−dαe−1
X
k=0
gk+1(t)Bxj−gα∗f(t)∈ Agα∗u(t), t ≥0.
Suppose, conversely, that there exists a function uA∈C([0,∞) : Y) such that uA(t)∈
Au(t), t ≥0 and
Bu(t)−dαe−1
X
k=0
gk+1(t)Bxj−gα∗f(t) = gα∗uA(t), t ≥0.
Then it can be simply verified that u(·) is a solution of problem (DFP)R; it is noteworthy
that we do not need the assumption on closedness of Ain this direction. Even in the
case that A=Ais a closed single-valued linear operator, a corresponding statement for
the problem (DFP)Lcannot be proved. Suppose, finally, that the operators Aand Bare
closed, u(·) is a solution of problem (DFP)R,the function F(t) = f(t) is Y-continuous at
each point t≥0,as well as the functions uα,B∈C([0,∞) : Y) and uA∈C([0,∞) : Y)
satisfy the requirements stated in Definition 1.17(iii). Using again Theorem 1.6 and the
identity [EB, (1.21)], it readily follows that
B"u(t)−dαe−1
X
k=0
gk+1(t)xj#3gα∗uα,B(t)
=gα∗uA(t) + gα∗f(t)∈ Agα∗u(t) + gα∗f(t), t ≥0.
The proof of following important theorem can be deduced by using Theorem 1.6,
Theorem 1.12[(iv),(vi)], Theorem 1.14 and the argumentation already seen in the proof
of [K, Theorem 3.1] (cf. also [KN, 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).
Theorem 1.18.Suppose that A:X→P(Y)and B:X→P(Y)are MLOs, as
well as that Ais XA×YA-closed. Assume, further, that a∈L1
loc([0,∞)), a 6= 0,
abs(|a|)<∞, u ∈C([0,∞) : X), u ∈(P1) −X, as well as that u(t)∈D(B), t ≥0,
24 Marko Kosti´c
a∗u∈C([0,∞) : XA), a ∗u∈(P1) −XA,absYA(Bu)<∞,absYA(F)<∞,and
ω > max(0, ωX(u),absYA(Bu),absYA(F),absXA(a∗u)).Consider the following assertions:
(i) u(·)is a solution of (1.1) with τ=∞.
(ii) u(·)is a pre-solution of (1.1) with τ=∞.
(iii) For any section uB∈sec(Bu)there is a section f∈sec(F)such that
fuB(λ)−˜
f(λ)∈˜a(λ)A˜u(λ),<λ > ω, ˜a(λ)6= 0.
(iv) For any section uB∈sec(Bu)there is a section f∈sec(F)such that
fuB(λ)−˜
f(λ)∈˜a(λ)A˜u(λ), λ ∈N, λ > ω, ˜a(λ)6= 0.(1.17)
(v) For any section uB∈sec(Bu)there is a section f∈sec(F)such that
1∗uB(t)−(1 ∗f)(t)∈ A(1 ∗a∗u)(t), t ≥0.(1.18)
Then we have (i) ⇒(ii) ⇒(iii) ⇒(iv) ⇒(v). Furthermore, if B=Bis single-valued,
Bu ∈C([0,∞) : YA)and F=f∈C([0,∞) : YA)is single-valued, then the above is
equivalent.
Sketch of proof for (iv) ⇒(v). Suppose that for any section uB∈sec(Bu) there is
a section f∈sec(F) such that (1.17) holds. Let a number λ∈Nwith λ > ω and
˜a(λ) = 0 be temorarily fixed. Then there exists a sequence (λn)n∈Nin (λ, ∞) such that
˜a(λn)6= 0 and limn→+∞λn=λ. Since (˜a(λn)˜u(λn),fuB(λn)−˜
f(λn)) ∈ A, n ∈N,i.e.,
(]
a∗u(λn),fuB(λn)−˜
f(λn)) ∈ A, n ∈N,and Ais XA×YA-closed, it readily folows that
(]
a∗u(λ),fuB(λ)−˜
f(λ)) ∈ A; in other words, (0,fuB(λ)−˜
f(λ)) ∈ A.By the foregoing, we
have that (]
a∗u(λ),fuB(λ)−˜
f(λ)) ∈ A for all λ∈Nwith λ > ω. Using Theorem 1.6, we
get that R∞
0e−λt(uB−f)[2] (t)dt ∈ AR∞
0e−λt(a∗u)[2] (t)dt (λ∈N, λ > ω) and now we
can apply Theorem 1.14, along with the XA×YA-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 (1.18).
Remark. Observe that we do not require any type of closedness of the operator Bin the
formulation of Theorem 1.18. Even in the case that X=Yand B=B=I, we cannot
differentiate the equation (1.18) once more without making an additional assumption
that F=f∈C([0,∞) : YA) is single-valued (cf. [K, l. -1, p. 173; l. 1-3, p. 174], where
the author has made a small mistake in the consideration; speaking-matter-of-factly,
the equation [K, (3.1)] has to be valid for some f∈secc(F) in order for the proof of
implication (iii) ⇒(i) of [K, Theorem 3.1] to work).
If Ω is a non-empty open subset of Cand G: Ω →Xis an analytic mapping that it is
not identically equal to the zero function, then we can simply prove that for each zero λ0
of G(·) there exists a uniquely determined natural number n∈Nsuch that G(j)(λ0)=0
for 0 ≤j≤n−1 and G(n)(λ0)6= 0.Owing to this fact, we can repeat almost literally the
arguments given in the proof of [K, Theorem 3.2] to verify the validity of the following
Ljubich uniquness type theorem:
Theorem 1.19.Suppose A:X→P(Y)is an MLO, B=B:D(B)⊆X→Yis a single-
valued linear operator, Ais XA×YA-closed and Bis XB×YB-closed, where YA→YB.
Assume, further, that a∈L1
loc([0,∞)), a 6= 0,abs(|a|)<∞,F=f∈C([0,∞) : YA)is
Abstract degenerate Volterra inclusions... 25
single-valued, absYA(f)<∞,and there exist a sequence (λk)k∈Nof complex numbers and
a number ω > abs(|a|)such that limk→∞ <λk= +∞,˜a(λk)6= 0, k ∈N,and
1
˜a(λk)Bx /∈ Ax, k ∈N,06=x∈D(A)∩D(B).
Then there exists a unique pre-solution of (1.1), with τ=∞,satisfying that u∈(P1) −
XB, u(t)∈D(B), t ≥0, Bu ∈C([0,∞) : YA), a ∗u∈C([0,∞) : XA), a ∗u∈(P1) −XA
and absYA(Bu)<∞.
In the following extension of [MK2, Theorem 2.1.34], we will prove one more Ljubich’s
uniqueness criterium for abstract Cauchy problems with multivalued linear operators (cf.
also [KN, Theorem 3.5] and [MK2, Theorem 2.10.44]).
Theorem 1.20.Suppose α > 0, λ > 0,Ais an MLO in X, {(nλ)α:n∈N} ⊆ ρC(A)and,
for every σ > 0and x∈X, lim
n→∞
((nλ)α−A)−1Cx
enλσ = 0.Then, for every x0,···, xdαe−1∈X,
there exists at most one pre-solution of the initial value problem (DFP)Rwith B=I.
Proof. It suffices to show that the zero function is the only pre-solution of the problem
(DFP)Rwith B=Iand the initial values x0,·· ·, xdαe−1chosen to be zeroes. Let u(·) be
a pre-solution of such a problem. Set zn(t) := ((nλ)α− A)−1Cu(t), t ≥0, n ∈N.Then
it can be easily checked with the help of Theorem 1.7(i) that zn(·) is a solution of the
initial value problem:
zn∈Cdαe((0,∞) : X)∩Cdαe−1([0,∞) : X),
Dα
tzn(t)=(nλ)αzn(t)−C u(t), t > 0,
z(j)
n(0) = 0,0≤j≤ dαe − 1.
This implies zn(t) = −(u∗ ·α−1Eα,α((nλ)α·α−1))(t), t ≥0, n ∈Nand
lim
n→∞ e−nλσ
t
Z
0
sα−1Eα,α(nλ)αsαCu(t−s)ds = 0 (t > 0, σ > 0).
Now we can argue as in the second part of proof of [MK2, Theorem 2.1.34] so as to
conclude that u(t)=0, t ≥0 (in the case that α∈N,the assertion can be proved by a
trustworthy passing to the theory of abstract Cauchy problems of first order since [MK2,
Lemma 2.1.33(i)] admits an extension to multivalued linear operators).
Remark. Observe that, in the formulation of Theorem 1.20, we do not require any type
of closedness of the operator A.
The following theorem is very similar to [EB, Theorem 3.1, Theorem 3.3] and [MK2,
Theorem 2.4.2]. Because of its importance, we will include the most relevant details of
proof.
Theorem 1.21.(Subordination principle for abstract time-fractional inclusions) Suppose
that 0< α < β, γ =α/β, A:X→P(Y)is an MLO, B=B:D(B)⊆X→Yis a single-
valued linear operator, Ais XA×YA-closed and Bis XB×YB-closed, where XA→XB
and YA→YB.Assume, further, that fβ∈LTor −YAis single-valued and there exists a
pre-solution (or, equivalently, solution) u(t) := uβ(t)of (1.1), with τ=∞, a(t) = gβ(t)
and F=fβ,satisfying that uβ∈LTor −XB, Buβ∈LTor −YA, gβ∗uβ∈LTor −XA
26 Marko Kosti´c
and that for each seminorm p∈~XBthere exists ωp≥0such that p(uβ(t)) = O(eωpt),
t≥0, p ∈~XB.Define
uα(t) := Z∞
0
t−γΦγst−γuβ(s)ds, t > 0and uα(0) := uβ(0)
and
fα(t)x:= Z∞
0
t−γΦγst−γfβ(s)ds, t > 0and fα(0) := fβ(0).
Then uα(t)is a solution of (1.1), with τ=∞, a(t) = gα(t)and F(t) = fα(t)∈LTor −YA,
satisfying additionally that uα∈LTor −XB, Buα∈LTor −YA, gα∗uα∈LTor −XAand
puα(t)=Oexpω1/γ
pt, p ∈~XB, t ≥0.(1.19)
Let p∈~XBbe fixed. Then the condition
puβ(t)=O1 + tξpeωptfor some ξp≥0,(1.20)
resp.,
puβ(t)=Otξpeωpt, t ≥0 (1.21)
implies that
puα(t)=O1 + tξpγ1 + ωptξp(1−γ)expω1/γ
pt, t ≥0,(1.22)
resp.,
puα(t)=Otξpγ1 + ωptξp(1−γ)expω1/γ
pt, t ≥0.(1.23)
Furthermore, the following holds:
(i) The mapping t7→ uα(t), t > 0admits an extension to Σmin(( 1
γ−1) π
2,π)and the
mapping z7→ uα(z), z ∈Σmin(( 1
γ−1) π
2,π)is analytic.
(ii) Let ε∈(0,min(( 1
γ−1)π
2, π)).If, for every p∈~,one has ωp= 0,then for each
θ∈(0,min(( 1
γ−1)π
2, π)) the following holds: limz→0,z∈Σθuα(z) = uα(0).
(iii) If ωp>0for some p∈~,then for each θ∈(0,min(( 1
γ−1)π
2,π
2)) the following
holds: limz→0,z∈Σθuα(z) = uα(0).
Proof. The proofs of (i)-(iii) follows similarly as in that of [EB, Theorem 3.3], while the
proof that the condition (1.20), resp. (1.21), implies (1.22), resp. (1.23), follows similarly
as in that of [MK2, Theorem 2.4.2]. Furthermore, it can be easily seen that the estimate
(1.19) holds for solution uα(·).By Theorem 1.18, we should only show that uα∈LTor −
XB, Buα∈LTor −YA, fα∈LTor −YA, gα∗uα∈LTor −XAand
g
Buα(λ)−f
fα(λ)∈λ−αAfuα(λ), λ > ω suff. large.(1.24)
Since uβ∈LTor −XB,the proof of [EB, Theorem 3.1] immediately implies that uα∈
LTor −XB,as well as that fuα(λ) = λγ−1fuβ(λβ), λ > ω suff. large. Similarly, we have
that fα∈LTor −YAand ˜
fα(λ) = λγ−1˜
fβ(λβ), λ > ω suff. large. Keeping in mind that
XA→XBand gβ∗uβ∈LTor −XA,we can prove that (gα∗uα)(t) = R∞
0t−γΦγst−γ(gβ∗
uβ)(s)ds, t > 0 by performing the Laplace transform (the convergence of last integral
is taken for the topology of XA). This simply implies that gα∗uα∈LTor −XAand
Abstract degenerate Volterra inclusions... 27
(L(gα∗uα))(λ) = λγ−1(L(gβ∗uβ))(λγ), λ > ω suff. large. Since YA→YB,a similar line
of reasoning shows that Buα(t) = R∞
0t−γΦγst−γ(Buβ)(s)ds, t > 0 (the convergence
of this integral is taken for the topology of YA) and that g
Buα(λ) = Bfuα(λ), λ > ω suff.
large. The proof of (1.24) now follows from a simple computation.
We can similarly prove the following subordination principles for abstract degenerate
Volterra inclusions in locally convex spaces (cf. [JP, Section 4] and [MK2, Theorem 2.1.8,
Theorem 2.8.7] for more details concerning non-degenerate case and, especially, the case
in which b(t) = g1(t) or b(t) = g2(t)).
Theorem 1.22.Let b(t)and c(t)satisfy (P1)-C, let R∞
0e−βt|b(t)|dt < ∞for some
β≥0,and let
α= ˜c−11
βif
∞
Z
0
c(t)dt > 1
β, α = 0 otherwise.
Suppose that abs(|a|)<∞,˜a(λ) = ˜
b(1
˜c(λ)), λ > α, A:X→P(Y)is an MLO, B=B:
D(B)⊆X→Yis a single-valued linear operator, Ais XA×YA-closed and Bis XB×YB-
closed, where XA→XBand YA→YB.Assume, further, that fβ∈LTor −YAis single-
valued and there exists a pre-solution (or, equivalently, solution) u(t) := ub(t)of (1.1),
with τ=∞, a(t)replaced with b(t)therein, and F=fb,satisfying that ub∈LTor −XB,
Bub∈LTor −YA, b ∗ub∈LTor −XAand the family {e−ωbtub(t) : t≥0}is bounded in
XB(ωb≥0). Assume, further, that c(t)is completely positive and there exists a function
fa∈LTor −YAsatisfying
e
fa(λ) = 1
λ˜c(λ)e
fb1
˜c(λ), λ > ω0,e
fb1
˜c(λ)6= 0,for some ω0>0.
Let
ωa= ˜c−11
ωbif
∞
Z
0
c(t)dt > 1
ωb
, ωa= 0 otherwise.
Then, for every r∈(0,1],there exists a solution u(t) := ua,r(t)of (1.1), with τ=∞,
a(t)and F=fr:= gr∗fa,satisfying that ua,r ∈LTor −XB, Bua,r ∈LTor −YA,
a∗ua,r ∈LTor −XAand the set {e−ωatua,r(t) : t≥0}is bounded in XB,if ωb= 0 or
ωb˜c(0) 6= 1,resp., the set {e−εtua,r(t) : t≥0}is bounded in XBfor any ε > 0,if ωb>0
and ωb˜c(0) = 1.Furthermore, the function t7→ ua,r(t)∈XB, t ≥0is locally H¨older
continuous with the exponent r∈(0,1].
Remark. (i) In Theorem 1.21 and Theorem 1.22, we have only proved the existence of
a solution of the subordinated inclusion. The uniqueness of solutions can be proved,
for example, by using Theorem 1.19, Theorem 1.20 or [MK2, Theorem 2.2.6].
(ii) In Theorem 1.22, we have faced ourselves with a loss of regularity for solutions
of the subordinated problem. Even in the case that X=Yand B=I, it is not
so simple to prove the existence of a solution of problem (1.1), with τ=∞, a(t)
and F=fa,without imposing some additional unpleasant conditions. In the next
section, we will introduce various types of solution operator families for the abstract
28 Marko Kosti´c
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 1.28).
1.5. Multivalued linear operators as subgenerators of (a, k)-regularized C-
resolvent solution operator families. In [MK2, Section 2.8], the class of (a, k)-
regularized (C1, C2)-existence and uniqueness families has been introduced and analyzed
within the theory of abstract non-degenerate Volterra equations. The main aim of this sec-
tion is to consider multivalued linear operators in locally convex spaces as subgenerators
of (a, k)-regularized (C1, C2)-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 6= 0, a ∈L1
loc([0, τ )), a 6= 0,A:X→P(X)
is an MLO, C1∈L(Y, X ), C2∈L(X) is injective, C∈L(X) is injective and CA⊆AC.
The following definition is an extension of [MK2, Definition 2.8.2] (X=Y, A is a
closed single-valued linear operator on X) and [FS, Definition 3.5] (X=Y, C =C1,
a(t) = k(t) = 1).
Definition 1.23.Suppose 0 < τ ≤ ∞, k ∈C([0, τ )), k 6= 0, a ∈L1
loc([0, τ )), a 6= 0,
A:X→P(X) is an MLO, C1∈L(Y, X ),and C2∈L(X) is injective.
(i) Then it is said that Ais a subgenerator of a (local, if τ < ∞) mild (a, k)-regularized
(C1, C2)-existence and uniqueness family (R1(t), R2(t))t∈[0,τ)⊆L(Y, X )×L(X) iff
the mappings t7→ R1(t)y, t ≥0 and t7→ R2(t)x, t ∈[0, τ ) are continuous for every
fixed x∈Xand y∈Y, as well as the following conditions hold:
t
Z
0
a(t−s)R1(s)y ds, R1(t)y−k(t)C1y!∈ A, t ∈[0, τ ), y ∈Yand (1.25)
t
Z
0
a(t−s)R2(s)y ds =R2(t)x−k(t)C2x, whenever t∈[0, τ ) and (x, y)∈ A.
(1.26)
(ii) Let (R1(t))t∈[0,τ)⊆L(Y, X ) be strongly continuous. Then it is said that Ais
a subgenerator of a (local, if τ < ∞) mild (a, k)-regularized C1-existence family
(R1(t))t∈[0,τ)iff (1.25) holds.
(iii) Let (R2(t))t∈[0,τ)⊆L(X) be strongly continuous. Then it is said that Ais a
subgenerator of a (local, if τ < ∞) mild (a, k)-regularized C2-uniqueness family
(R2(t))t∈[0,τ)iff (1.26) holds.
As an immediate consequence of definition, we have that R(R1(0) −k(0)C1)⊆ A0 as
well as that R2(t)Ais single-valued for any t≥0,and R2(t)y= 0 for any y∈ A0 and
t≥0.
Now we will extend the definition of an (a, k)-regularized C-resolvent family subgen-
erated by a single-valued linear operator (cf. [MK2, Definition 2.1.1]).
Definition 1.24.Suppose that 0 < τ ≤ ∞, k ∈C([0, τ )), k 6= 0, a ∈L1
loc([0, τ )), a 6= 0,
A:X→P(X) is an MLO, C∈L(X) is injective and CA⊆AC. Then it is said
that a strongly continuous operator family (R(t))t∈[0,τ)⊆L(X) is an (a, k)-regularized
Abstract degenerate Volterra inclusions... 29
C-resolvent family with a subgenerator Aiff (R(t))t∈[0,τ)is a mild (a, k)-regularized C-
uniqueness family having Aas subgenerator, R(t)C=CR(t) and R(t)A⊆AR(t) (t≥0).
An (a, k)-regularized C-resolvent family (R(t))t∈[0,τ)is said to be locally equicontin-
uous iff, for every t∈(0, τ),the family {R(s) : s∈[0, t]}is equicontinuous. In the case
τ=∞,(R(t))t≥0is said to be exponentially equicontinuous (equicontinuous) if there
exists ω∈R(ω= 0) such that the family {e−ωt R(t) : t≥0}is equicontinuous; the infi-
mum of such numbers is said to be the exponential type of (R(t))t≥0.The above notion
can be simply understood for the classes of mild (a, k)-regularized C1-existence fami-
lies and mild (a, k)-regularized C2-uniqueness families; a mild (a, k)-regularized (C1, C2)-
existence and uniqueness family (R1(t), R2(t))t∈[0,τ)⊆L(Y, X )×L(X) is said to be
locally equicontinuous (exponentially equicontinuous, provided that τ=∞) iff both op-
erator families (R1(t))t≥0and (R2(t))t≥0are. It would take too long to consider the no-
tion of q-exponential equicontinuity for the classes of mild (a, k)-regularized C1-existence
families and mild (a, k)-regularized C2-uniqueness families (cf. [MK2, Section 2.4] for
more details about non-degenerate case). If k(t) = gα+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 C1-existence families and
mild (a, k)-regularized C2-uniqueness families; in the case of consideration of convoluted
C-semigroups, it will be always assumed that the condition (1.25) holds with a(t) = 1
and the operator C1replaced by C. Let us mention in passing that the operator semi-
groups generated by multivalued linear operators have been analyzed by A. G. Baskakov
in [AB].
The following proposition can be simply proved with the help of Theorem 1.6 and
Theorem 1.10(ii).
Proposition 1.25.Suppose that (R1(t), R2(t))t∈[0,τ)⊆L(Y, X )×L(X)is a mild (a, k)-
regularized (C1, C2)-existence and uniqueness family with a subgenerator Aand
(R(t))t∈[0,τ)⊆L(X)is an (a, k)-regularized C-resolvent family with a subgenerator A.
Let b∈L1
loc([0, τ )) be such that a∗b6= 0 in L1([0, τ )) and k∗b6= 0 in C([0, τ )).Then
((b∗R2)(t))t≥0is a mild (a, k)-regularized C2-uniqueness family with a subgenerator A.
Furthermore, the following holds:
(i) Let Abe X1
A×X2
A-closed. Suppose that, for every y∈Y, the mapping t7→ (a∗
R1)(t)y, t ∈[0, τ )is continuous in X1
Aand the mapping t7→ R1(t)y, t ∈[0, τ )is
continuous in X2
A.Then ((b∗R1)(t))t≥0is a mild (a, k)-regularized C1-existence
family with a subgenerator A.
(ii) Let Abe X1
A×X2
A-closed. Suppose that, for every x∈D(A)and y∈R(A),the
mapping t7→ R(t)x, t ∈[0, τ )is continuous in X1
Aand the mapping t7→ R(t)y, t ∈
[0, τ )is continuous in X2
A.Then ((b∗R)(t))t≥0is 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 X1
A×X2
A-
closed subgenerators, the most important case in our further study will be that in which
X1
A=X2
A=X. This is primarily caused by the following fact: Let Abe a subgenerator of
30 Marko Kosti´c
a mild (a, k)-regularized C1-existence family (mild (a, k)-regularized C2-uniqueness fam-
ily; mild (a, k)-regularized C-resolvent family) (R1(t))t∈[0,τ)((R2(t))t∈[0,τ ); (R(t))t∈[0,τ )).
Then Ais likewise a subgenerator of (R1(t))t∈[0,τ)((R2(t))t∈[0,τ ); (R(t))t∈[0,τ ), provided
in addition that (R2(t))t∈[0,τ); (R(t))t∈[0,τ)is locally equicontinuous).
Suppose that (R1(t), R2(t))t∈[0,τ )is a mild (a, k)-regularized (C1, C2)-existence and
uniqueness family with a subgenerator A.Arguing as in non-degenerate case (cf. the
paragraph directly preceding [MK2, Definition 2.8.3]), we may conclude that
a∗R2(s)R1(t)y−R2(s)a∗R1(t)y
=k(t)a∗R2(s)C1y−k(s)C2a∗R1(t)y, t ∈[0, τ ), y ∈Y .
The integral generator of a mild (a, k)-regularized C2-uniqueness family (R2(t))t∈[0,τ)
(mild (a, k)-regularized (C1, C2)-existence and uniqueness family (R1(t), R2(t))t∈[0,τ )) is
defined by
Aint := ((x, y)∈X×X:R2(t)x−k(t)C2x=Zt
0
a(t−s)R2(s)y ds, t ∈[0, τ ));
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 Aint is an MLO in Xwhich is, in fact, the
maximal subgenerator of (R2(t))t∈[0,τ)((R(t))t∈[0,τ)) with respect to the set inclusion; fur-
thermore, the assumption R2(t)C2=C2R2(t), t ∈[0, τ ) implies that C−1
2AintC2=Aint
so that C−1AintC=Aint for resolvent families. The local equicontinuity of (R2(t))t∈[0,τ)
((R(t))t∈[0,τ)) immediately implies that Aint is closed. Observe that, in the above defini-
tion 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 Ais a closed subgenerator
of (R(t))t∈[0,τ).Then
Zt
0
a(t−s)R(s)x ds, R(t)x−k(t)Cx!∈ A, t ∈[0, τ ), x ∈D(A).
(ii) If Ais a subgenerator of (R(t))t∈[0,τ ),then C−1ACis a subgenerator of (R(t))t∈[0,τ ),
too.
(iii) Suppose that a(t) is a kernel on [0, τ ),Aand Bare two subgenerators of (R(t))t∈[0,τ),
and x∈D(A)∩D(B).Then R(t)(y−z)=0, t ∈[0, τ ) for each y∈ Axand z∈ Bx.
(iv) Let Abe a subgenerator of (R(t))t∈[0,τ ),and let λ∈ρC(A) (λ∈ρ(A)). Suppose
that x∈X, y = (λ− A)−1C x (y= (λ− A)−1x) and z∈ Ay. Then Theorem 1.7(i)
implies that λ(λ− A)−1Cx −C x ∈ A(λ− A)−1C x =Ay(λ(λ− A)−1x−x∈
A(λ− A)−1x=Ay), so that R(t)y−k(t)Cy ∈ A Rt
0a(t−s)R(s)[λ(λ− A)−1Cx −
Cx]ds =A{λ(λ− A)−1CRt
0a(t−s)R(s)x ds −Rt
0a(t−s)R(s)Cx ds}, t ∈[0, τ )
and Rt
0a(t−s)R(s)Cx ds ∈D(A), t ∈[0, τ ); from this, we may conclude that
R(t)Cx −k(t)C2x∈(λ− A)A(λ− A)−1CRt
0a(t−s)R(s)x ds, t ∈[0, τ ); similarly,
we have that Rt
0a(t−s)R(s)x ds ∈D(A) and R(t)x−k(t)Cx ∈(λ− A)A(λ−
A)−1Rt
0a(t−s)R(s)x ds, t ∈[0, τ ),provided that λ∈ρ(A).
Abstract degenerate Volterra inclusions... 31
The following extensions of [MK2, Theorem 2.8.5, Theorem 2.1.5] are stated without
proofs.
Theorem 1.26.Suppose Ais a closed MLO in X, C1∈L(Y, X), C2∈L(X), C2is
injective, ω0≥0and ω≥max(ω0,abs(|a|),abs(k)).
(i) Let (R1(t), R2(t))t≥0⊆L(Y, X )×L(X)be strongly continuous, and let the family
{e−ωtRi(t) : t≥0}be equicontinuous for i= 1,2.
(a) Suppose (R1(t), R2(t))t≥0is a mild (a, k)-regularized (C1, C2)-existence and
uniqueness family with a subgenerator A.Then, for every λ∈Cwith <λ>ω
and ˜
k(λ)6= 0,the operator I−˜a(λ)Ais injective, R(C1)⊆R(I−˜a(λ)A),
˜
k(λ)I−˜a(λ)A−1C1y=
∞
Z
0
e−λtR1(t)y dt, y ∈Y, (1.27)
(1
˜a(z):<z > ω, ˜
k(z)˜a(z)6= 0)⊆ρC1(A) (1.28)
and
˜
k(λ)C2x=
∞
Z
0
e−λtR2(t)x−a∗R2(t)ydt, whenever (x, y)∈ A.(1.29)
Here, ρC1(A)is defined in the obvious way.
(b) Let (1.28) hold, and let (1.27) and (1.29) hold for any λ∈Cwith <λ > ω
and ˜
k(λ)6= 0.Then (R1(t), R2(t))t≥0is a mild (a, k)-regularized (C1, C2)-
existence and uniqueness family with a subgenerator A.
(ii) Let (R1(t))t≥0be strongly continuous, and let the family {e−ωtR1(t) : t≥0}be
equicontinuous. Then (R1(t))t≥0is a mild (a, k)-regularized C1-existence family with
a subgenerator Aiff for every λ∈Cwith <λ > ω and ˜
k(λ)6= 0,one has R(C1)⊆
R(I−˜a(λ)A)and
˜
k(λ)C1y∈I−˜a(λ)A∞
Z
0
e−λtR1(t)y dt, y ∈Y.
(iii) Let (R2(t))t≥0be strongly continuous, and let the family {e−ωtR2(t) : t≥0}be
equicontinuous. Then (R2(t))t≥0is a mild (a, k)-regularized C2-uniqueness family
with a subgenerator Aiff for every λ∈Cwith <λ>ωand ˜
k(λ)6= 0,the operator
I−˜a(λ)Ais injective and (1.29) holds.
Theorem 1.27.Let (R(t))t≥0⊆L(X)be a strongly continuous operator family such that
there exists ω≥0satisfying that the family {e−ωtR(t) : t≥0}is equicontinuous, and let
ω0>max(ω, abs(|a|),abs(k)).Suppose that Ais a closed MLO in Xand CA⊆AC.
(i) Assume that Ais a subgenerator of the global (a, k)-regularized C-resolvent family
(R(t))t≥0satisfying (1.25) for all x=y∈X. Then, for every λ∈Cwith <λ>ω0
32 Marko Kosti´c
and ˜
k(λ)6= 0,the operator I−˜a(λ)Ais injective, R(C)⊆R(I−˜a(λ)A),
˜
k(λ)I−˜a(λ)A−1Cx =
∞
Z
0
e−λtR(t)x dt, x ∈X, <λ>ω0,˜
k(λ)6= 0,(1.30)
(1
˜a(λ):<λ>ω0,˜
k(λ)˜a(λ)6= 0)⊆ρC(A) (1.31)
and R(s)R(t) = R(t)R(s), t, s ≥0.
(ii) Assume (1.30)-(1.31). Then Ais a subgenerator of the global (a, k)-regularized C-
resolvent family (R(t))t≥0satisfying (1.25) for all x=y∈Xand R(s)R(t) =
R(t)R(s), t, s ≥0.
Remark. (i) Suppose that (R(t))t≥0is a degenerate exponentially equicontinuous
(a, k)-regularized C-resolvent family in the sense of [MK10, Definition 2.2], and
B∈L(X).Using Remark 1.4(iv)/(a), Remark 1.4(iv) and Theorem 1.27(ii), it can
be easily seen that (R(t))t≥0is an exponentially equicontinuous (a, k)-regularized
C-resolvent family with a closed subgenerator B−1A.
(ii) Suppose that n∈N, X and Yare Banach spaces, A:D(A)⊆X→Yis 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
[MF, Definition 1.5.3]. Then the arguments mentioned above show that (V(t))t≥0is
an exponentially bounded n-times integrated (g1, I)-regularized family (semigroup)
with a closed subgenerator B−1A.
(iii) Let n∈N0.Due to Theorem 1.27(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 [MF,
Definition 1.6.6, Definition 1.6.8].
(iv) Suppose that (R(t))t≥0⊆L(X, [D(B)]) is an exponentially equicontinuous (a, k)-
regularized C-resolvent family generated by A, B, in the sense of [MK11, Defini-
tion 2.5]. Then [MK11, Theorem 2.3(i)] in combination with Remark 1.4(v) and
Theorem 1.27(ii) implies that (BR(t))t≥0is an exponentially equicontinuous (a, k)-
regularized C-resolvent family generated by B−1A(recall that B−1Ais closed pro-
vided that C=I).
The proof of following extension of [MK2, Theorem 2.1.8(i), Theorem 2.8.7(i)] is
standard and therefore omitted; we can similarly reformulate Theorem 1.21 and [MK2,
Proposition 2.1.16] for the class of mild (a, k)-regularized (C1, C2)-existence and unique-
ness families ((a, k)-regularized C-resolvent families). Here it is only worth noting that
the existence of a mild (a, k1)-regularized C1-existence family (R0,1(t))t≥0in the second
part of theorem is not automatically guaranteed by the denseness of A(even in the case
that the operator A=Ais single-valued, it seems that the condition C1A⊆AC1is
necessary for such a mild existence family to exist).
Theorem 1.28.Suppose C1∈L(Y, X), C2∈L(X)is injective, Ais a closed MLO
in X, C ∈L(X)is injective and CA ⊆ AC. Let b(t)and c(t)satisfy (P1)-C, let
Abstract degenerate Volterra inclusions... 33
R∞
0e−βt|b(t)|dt < ∞for some β≥0,and let
α= ˜c−11
βif
∞
Z
0
c(t)dt > 1
β, α = 0 otherwise.
Suppose that abs(|a|)<∞and ˜a(λ) = ˜
b(1
˜c(λ)), λ ≥α. Let Abe a subgenerator of a
(b, k)-regularized C1-existence family (R1(t))t≥0((b, k)-regularized C2-uniqueness family
(R2(t))t≥0; (b, k)-regularized C-resolvent family (R0(t))t≥0with the property that (1.25)
holds for R1(·)replaced with R0(·)and each x=y∈X) satisfying that the family
{e−ωbtR1(t) : t≥0}({e−ωbtR2(t) : t≥0};{e−ωbtR(t) : t≥0}) is equicontinuous for
some ωb≥0.Assume, further, that c(t)is completely positive and that there exists a
scalar-valued continuous kernel k1(t)satisfying (P1)-Cand
˜
k1(λ) = 1
λ˜c(λ)˜
k1
˜c(λ), λ > ω0,˜
k1
˜c(λ)6= 0,for some ω0>0.
Let
ωa= ˜c−11
ωbif
∞
Z
0
c(t)dt > 1
ωb
, ωa= 0 otherwise.
Then, for every r∈(0,1],Ais a subgenerator of a global (a, k1∗gr)-regularized C1-
existence family (Rr,1(t))t≥0((a, k1∗gr)-regularized C2-uniqueness family
(Rr,2(t))t≥0; (a, k1∗gr)-regularized C-resolvent family (Rr,0(t))t≥0with the property that
(1.25) holds for R1(·)replaced with Rr,0(·)and each x=y∈X) such that the family
{e−ωatRr,i(t) : t≥0}is equicontinuous and that the mapping t7→ Rr,i(t), t ≥0is locally
H¨older continuous with exponent r, if ωb= 0 or ωb˜c(0) 6= 1 (i= 0,1,2), resp., for every
ε > 0,there exists Mε≥1such that the family {e−εtRr,i(t) : t≥0}is equicontinuous
and that the mapping t7→ Rr,i(t), t ≥0is locally H¨older continuous with exponent r,
if ωb>0and ωb˜c(0) = 1 (i= 0,1,2). Furthermore, if Ais densely defined, then Ais
a subgenerator of a global (a, k1)-regularized C2-uniqueness family (R0,2(t))t≥0((a, k1)-
regularized C-resolvent family (R0,0(t))t≥0with the property that (1.25) holds for R1(·)
replaced with R0,0(·)and each x=y∈X) such that the family {e−ωatRi(t) : t≥0}is
equicontinuous, resp., for every ε > 0,the family {e−εtRi(t) : t≥0}is equicontinuous
(i= 1,2).
Let (R1(t), R2(t))t∈[0,τ)be a mild (a, k)-regularized (C1, C2)-existence and uniqueness
family with a subgenerator A.Then it is straightforward to see that the function t7→
R1(t)y, t ∈[0, τ ) (y∈Y), resp. t7→ R2(t)x, t ∈[0, τ ) (x∈D(A)), is a solution of problem
(1.1) with B=Iand f(t) = k(t)C1y, t ∈[0, τ ), resp. a strong solution of (1.1) with B=I
and f(t) = k(t)C2x, t ∈[0, τ ), provided additionally in the last case that R2(t)x∈D(A),
t∈[0, τ ) and R2(t)Ax⊆ AR2(t)x, t ∈[0, τ ).Furthermore, it is very simple to transmit
the assertions of [MK2, Proposition 2.8.8, Proposition 2.8.9] to mild (a, k)-regularized
(C1, C2)-existence and uniqueness families subgenerated by multivalued linear operators:
Proposition 1.29.(i) Suppose that (R1(t), R2(t))t∈[0,τ)is a mild (a, k)-regularized
(C1, C2)-existence and uniqueness family with a subgenerator A,as well as that
34 Marko Kosti´c
(R2(t))t∈[0,τ)is locally equicontinuous and the functions a(t)and k(t)are kernels
on [0, τ ).Then C2R1(t) = R2(t)C1, t ∈[0, τ ).
(ii) Suppose that (R2(t))t∈[0,τ )is a locally equicontinuous mild
(a, k)-regularized C2-uniqueness family with a subgenerator A.Then every strong
solution u(t)of (1.1) with B=Iand F=f∈C([0, τ ) : X)satisfies
R2∗f(t) = kC2∗u(t),0≤t < τ. (1.32)
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.
The first part of following theorem is an extension of [MK2, Theorem 2.1.28(ii)] and
its validity can be verified with the help of proof of [CL, Theorem 2.7], Lemma 1.5 and
Theorem 1.6; the second part of theorem is an extension of [MK2, Proposition 2.1.31]
and can be shown by the arguments contained in the proof of [HO, Theorem 2.5], along
with Lemma 1.5.
Theorem 1.30.(i) Suppose that (R(t))t∈[0,τ )is a locally equicontinuous
(a, k)-regularized C-resolvent family generated by A, the equation (1.25) holds for
each y=x∈X, with R1(·)and C1replaced therein with R(·)and C, respectively,
k(t)is a kernel on [0, τ ), u, f ∈C([0, τ ) : X),and (1.32) holds with R2(·)and
C2replaced therein with R(·)and C, respectively. Then u(t)is a solution of the
abstract Volterra inclusion (1.1) with B=Iand F=f.
(ii) Suppose that the functions a(t)and k(t)are kernels on [0, τ),and Ais a closed
MLO in X. Consider the following assertions:
(a) Ais a subgenerator of a locally equicontinuous (a, k)-regularized C-resolvent
family (R(t))t∈[0,τ)satisfying the equation (1.25) for each y=x∈X, with
R1(·)and C1replaced therein by R(·)and C, respectively.
(b) For every x∈X, there exists a unique solution of (1.1) with B=Iand
F(t) = f(t) = k(t)Cx, t ∈[0, τ ).
Then (a) ⇒(b). If, in addition, Xis a Fr´echet 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 (DFP)L, where after integration we have a(t) = gα(t).Assume that
there exists a unique solution of problem (DFP)Lwith B=I, F(t)≡0, x0∈C(D(A))
and xj= 0,1≤j≤ dαe−1.If, in addition to this, Xis complete, Ais 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. [MK5, p. 304]) show that Ais a subgenerator of a locally equicontinuous
(gα, C)-resolvent family (Rα(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.
Theorem 1.31.Let ω0>max(0,abs(|a|),abs(k)),and let Abe a closed MLO in X.
Assume that, for every λ∈Cwith <λ > ω0and ˜
k(λ)6= 0,the operator I−˜a(λ)Ais
Abstract degenerate Volterra inclusions... 35
injective and R(C)⊆R(I−˜a(λ)A).If there exists a function Υ : {λ∈C:<λ>ω0} →
L(X)which satisfies:
(i) Υ(λ) = ˜
k(λ)(I−˜a(λ)A)−1C, <λ>ω0,˜
k(λ)6= 0,
(ii) the mapping λ7→ Υ(λ)x, <λ>ω0is analytic for every fixed x∈X,
(iii) there exists r≥ −1such that the family {λ−rΥ(λ) : <λ > ω0} ⊆ L(X)is equicon-
tinuous,
then, for every α > 1,Ais a subgenerator of a global (a, k ∗gα+r)-regularized C-resolvent
family (Rα(t))t≥0which satisfies that the family {e−ω0tRα(t) : t≥0} ⊆ L(X)is equicon-
tinuous. Furthermore, (Rα(t))t≥0is a mild (a, k ∗gα+r)-regularized C-existence family
having Aas subgenerator.
In the first part of following example, we will briefly explain how one can use multipli-
cation operators for construction of local integrated semigroups generated by multivalued
operators; in the second part of example, we will apply the complex characterization the-
orem for proving the existence of a very specific exponentially equicontinuous, convoluted
fractional resolvent family (cf. [MK7, Example 2.5] for an example of a locally defined
solution of an abstract degenerate multi-term fractional problem).
Example 1.32.(i) (cf. also [AEK, Example 4.4(c)]) Suppose that 1 ≤p≤ ∞, X :=
Lp(1,∞),1<a<b<∞, J := [a, b], mb(x) := χJ(x) and ma(x) := x+iex
(x > 1). Consider the multiplication operators A:D(A)→Xand B∈L(X),
where D(A) := {f(x)∈X: (x+iex)f(x)∈X}, Af(x) := (x+iex)f(x) and
Bf (x) := mb(x)f(x) (x > 1, f ∈X). Then it is very simple to prove that, for
every α∈(0,1),the resolvent set of the multivalued linear operator A:= B−1A
contains the exponential region E(α, 1) := {x+iy :x≥1,|y| ≤ eαx},as well as
that (λ− A)−1f(x) = (λB −A)−1Bf (x) = mb(x)f(x)/λmb(x)−ma(x) for x > 1,
f∈X. Furthermore, the operator Agenerates a local once integrated semigroup
(S1(t))t∈[0,1],given by
(S1(t)f)(x) = (x+iex−1het(x+iex)−1if(x), t ∈[0,1], x /∈J, f ∈X,
0, t ∈[0,1], x ∈J, f ∈X.
(ii) Put X:= {f∈C∞([0,∞)) : limx→+∞f(k)(x) = 0 for all k∈N0}and ||f||k:=
Pk
j=0 supx≥0|f(j)(x)|, f ∈X, k ∈N0.Then the topology induced by these norms
turns Xinto a Fr´echet space (cf. also [MK2, Example 2.4.6(ii)]). Let α∈(0,1)
and J= [a, b]⊆[0,∞) be such that Σαπ/2∩ {x+iex:x∈J}=∅,and
let mb∈C∞([0,∞)) satisfy 0 ≤mb(x)≤1, x ≥0, mb(x) = 1, x /∈Jand
mb(x) = 0, x ∈[a+, b −] for some > 0.As in the first part of this ex-
ample, we consider the multiplication operators A:D(A)→Xand B∈L(X),
where D(A) = {f(x)∈E: (x+iex)f(x)∈X}, Af(x) := (x+iex)f(x) and
Bf (x) := mb(x)f(x) (x≥0, f ∈X). In a recent research study with S. Pilipovi´c
and D. Velinov [MPV], we have shown that Acannot be the generator of any lo-
cal integrated semigroup in X, as well as that Agenerates an ultradistribution
semigroup of Beurling class. Set A:= B−1A. We will prove that there exists a suf-
ficiently large number ω > 0 such that for each s > 1 and d > 0 the operator family
36 Marko Kosti´c
{e−d|λ|1/s (λ−A)−1:<λ > ω, λ ∈Σαπ /2} ⊆ L(X) is equicontinuous, which imme-
diately implies by Theorem 1.31 that Agenerates an exponentially equicontinuous
(gα,L−1(e−d|λ|α/s ))-regularized resolvent family. It is clear that the resolvent of A
will be given by (λ−A)−1f(x) = (λB −A)−1Bf (x) = mb(x)f(x)/λmb(x)−ma(x)
for x≥0, f ∈X. Since mb(x)f(x)/λmb(x)−ma(x)=1/λ−(x+iex) for x /∈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+iex:x≥0},belongs to the resolvent set of Aand
λ−A−1f(x) = f(x)
λ−x+iex, λ ∈C\S, x ≥0.
Fix, after that, numbers s > 1, d > 0, a > 0, b > 1 satisfying that x−ln(((x−
b)/a)s+ 1) ≥1, x ≥b. 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∈Nthere exist complex polynomials Pj(z) = Pj
l=0 aj,lzl(1 ≤j≤n)
such that dg(Pj) = j, |aj,l| ≤ (n+ 1)! (1 ≤j≤n, 0≤l≤j) and
dn
dxnλ−x+iex−1=
n+1
X
j=1λ−x+iex−j−1Pjex, x ≥0, λ ∈C\S. (1.33)
Suppose λ∈Ω and x≥0.If |=λ−ex| ≥ 1,then we have the following estimate
e2jx
<λ−x2k+=λ−ex2k≤e2jx
=λ−ex2k
≤22j1 + |=λ|2j, k ∈N0,0≤j < k. (1.34)
If |=λ−ex|<1,then =λ > 0,0≤x < ln(=λ+ 1),and
e2jx
<λ−x2k+=λ−ex2k≤e2jx
<λ−x2k
≤=λ+ 1j
<λ−ln((<λ−b)/a)s+ 1≤=λ+ 1j, k ∈N0,0≤j < k. (1.35)
Let ω0>0 be such that {λ∈Σαπ/2:<λ > ω0} ⊆ Ω.Combining (1.33)-(1.35),
it can be simply proved that for each number n∈Nthere exists a finite constant
cn>0 such that
n
X
k=0
sup
x≥0,x /∈J
dn
dxnλ−x+iex−1≤cned|λ|1/s , λ ∈Σαπ/2,<λ>ω0.(1.36)
We can similarly prove an estimate of type (1.36) for the derivatives of function
(λmb(x)−(x+iex))−1on the interval J, which is well-defined for λ∈Σαπ/2due to
our assumption 0 ≤mb(x)≤1, x ≥0 and the condition Σαπ/2∩{x+iex:x∈J}=
∅.Speaking-matter-of-factly, an induction argument shows that for each number
n∈Nthere exist numbers aj,l1,···,lssuch that |aj,l1,···,ls| ≤ (n+ 1)! (1 ≤j≤n,
Abstract degenerate Volterra inclusions... 37
0≤l≤j) and that, for every x∈Jand λ∈Σαπ/2,
dn
dxnλmb(x)−x+iex−1=
n+1
X
j=1λmb(x)−x+iex−j−1
×
j
X
l=0
aj,l1,···,lsY
l1m1+···+lsms=nλm(lj)
b(x)−m(lj)
a(x)mj.(1.37)
Since d:=dist(Σαπ/2,{x+iex:x∈J}) is a positive real number and |(λm(lj)
b(x)−
m(lj)
a(x))mj| ≤ cmj|λ|mjfor all λ∈Σαπ/2with <λ > ω, where the number ω > ω0
is sufficiently large, (1.37) shows that for each number n∈Nthere exists a finite
number c0
n>0 such that
n
X
k=0
sup
x≥0,x∈J
dn
dxnλmb(x)−x+iex−1≤c0
ned|λ|1/s , λ ∈Σαπ/2,<λ > ω.
(1.38)
By (1.36) and (1.38), we have that the operator family {e−d|λ|1/s (λ− A)−1:λ∈
Σαπ/2,<λ>ω} ⊆ L(X) is equicontinuous, as claimed.
Now we would like to tell something more about the importance of condition k(0) 6=
0 in the part (ii) of subsequent theorem. If all the necessary requirements hold, the
arguments contained in the proof of [KN, Theorem 3.6] imply the existence of a global
(a, k∗g1)-regularized C-resolvent family (R1(t))t≥0subgenerated by A,which additionally
satisfies that for each t≥0 the operator R1(t)Ais single-valued on D(A).Then it is
necessary to differentiate the equality R1(t)x−(k∗g1)(t)Cx =Rt
0a(t−s)R1(s)Ax ds,
t≥0, x ∈D(A) and to employ the fact that ( d
dt R1(t)x)t=0 =k(0)Cx (x∈D(A))
(cf. the proof of [KN, Theorem 3.6], as well as the proofs of [FY, Proposition 2.1] and
[MK2, Proposition 2.1.7]) in order to see that the function R:D(R)≡ {˜a(λ)−1:λ >
b, ˜a(λ)˜
k(λ)6= 0} → L(D(A)),given by R(˜a(λ)−1) := (˜a(λ)−1− A)−1C, λ ∈D(R),
is a C-pseudoresolvent in the sense of [LS, Definition 3.1], satisfying additionally that
N(R(λ)) = {0}, λ ∈D(R).Only after that, we can use [LS, Theorem 3.4] with a view to
prove the existence of a single-valued linear operator A, with domain and range contained
in D(A),which satisfies the properties required in (ii): this consideration shows the full
importance of concepts introduced in Definition 1.23 and Definition 1.24 in integrated
and convoluted case k(0) = 0. Keeping in mind Theorem 1.7(i) and the argumentation
contained in the proofs of [KN, Theorem 3.6] and [MK2, Theorem 1.2.6], the remaining
parts of following theorem can be deduced, more or less, as in non-degenerate case.
Theorem 1.33.Suppose ω∈R,abs(k)<∞,abs(|a|)<∞,Ais a closed MLO in X,
λ0∈ρC(A), b ≥max(0, ω, abs(|a|),abs(k)),
(1
˜a(λ):λ > b, ˜
k(λ)˜a(λ)6= 0)⊆ρC(A),
the function H:D(H)≡ {λ>b: ˜a(λ)˜
k(λ)6= 0} → L(X),given by H(λ)x=˜
k(λ)(I−
˜a(λ)A)−1Cx, x ∈X, λ ∈D(H),satisfies that the mapping λ7→ H(λ)x, λ ∈D(H)is
infinitely differentiable for every fixed x∈Xand, for every p∈~,there exist cp>0and
38 Marko Kosti´c
rp∈~such that:
p l!−1(λ−ω)l+1 dl
dλlH(λ)x!≤cprp(x), x ∈X, λ ∈D(H), l ∈N0.(1.39)
Then, for every r∈(0,1],the operator Ais a subgenerator of a global (a, k∗gr)-regularized
C-resolvent family (Rr(t))t≥0satisfying that, for every p∈~,
pRr(t+h)x−Rr(t)x≤2cprp(x)
rΓ(r)maxeω(t+h),1hr, t ≥0, h > 0, x ∈X,
and that, for every p∈~and B∈ B,the mapping t7→ pB(Rr(t)), t ≥0is locally H¨older
continuous with exponent r;furthermore, (Rr(t))t≥0is a mild (a, k ∗gr)-regularized C-
existence family having Aas subgenerator, and the following holds:
(i) Suppose that Ais densely defined. Then Ais a subgenerator of a global (a, k )-
regularized C-resolvent family (R(t))t≥0⊆L(X)satisfying that the family
{e−ωtR(t) : t≥0} ⊆ L(X)is equicontinuous. Furthermore, (R(t))t≥0is a mild
(a, k)-regularized C-existence family having Aas subgenerator.
(ii) Suppose that k(0) 6= 0.Then the operator C0:= C|D(A)∈L(D(A)) is injective, A0
is a closed subspace of X, D(A)∩A0 = {0},and we have the following: Define the
operator A:D(A)⊆D(A)→D(A)by
D(A) := nx∈D(A) : Cx =λ0− A−1C y for some y∈D(A)o
and
Ax := C−1ACx, x ∈D(A).
Then Ais a well-defined single-valued closed linear operator in D(A), and more-
over, Ais the integral generator of a global (a, k)-regularized C0-resolvent family
(S(t))t≥0⊆L(D(A)) satisfying that the family {e−ωtS(t) : t≥0} ⊆ L(D(A)) is
equicontinuous, ARt
0a(t−s)S(s)x ds =S(t)x−k(t)Cx, t ∈[0, τ ), x ∈D(A)and
R1(t)x=Rt
0S(s)x ds, t ≥0, x ∈D(A).
In the following proposition, which extends the assertions of [CL, Proposition 2.5] and
[MK2, Proposition 2.1.4(ii)], we will reconsider the condition k(0) 6= 0 from Theorem 1.33
once more. A straightforward proof is omitted.
Proposition 1.34.Let Abe a closed subgenerator of a mild (a, k)-regularized
C1-resolvent family (R1(t))t∈[0,τ)(mild (a, k)-regularized C2-uniqueness family
(R2(t))t∈[0,τ);(a, k)-regularized C-resolvent family (R(t))t∈[0,τ )). If k(t)is absolutely con-
tinuous and k(0) 6= 0,then Ais a subgenerator of a mild (a, g1)-regularized C1-resolvent
family (R1(t))t∈[0,τ)(mild (a, g1)-regularized C2-uniqueness family (R2(t))t∈[0,τ);(a, g1)-
regularized C-resolvent family (R(t))t∈[0,τ )).
Now we would like to present some illustrative applications of results obtained so far.
Example 1.35.Let α∈(0,1).
Abstract degenerate Volterra inclusions... 39
(i) ([FY]) Consider the following time-fractional analogon of homogeneous counterpart
of problem [FY, Example 2.1, (2.18)]:
(P)m,α :Dα
t[m(x)vα(t, x)] = −∂
∂x vα(t, x), t ≥0, x ∈R;
m(x)vα(0, x) = u0(x), x ∈R.
Let X=Y:= L2(R),and let the operator A:= −d/dx act on Xwith its maximal
distributional domain H1(R).
(a) Suppose first that (Bf )(x) := χ(−∞,a)∩(b,∞)(x)f(x), x ∈R(f∈X), where
−∞ <a<b<∞. Then B∈L(X), B =B∗, B2=Band (P)m,α is
formulated in Xin the following abstract form
(P)0
m,α :B∗Dα
tBvα(t) = Dα
tBvα(t) = Avα(t), t ≥0;
Bvα(0) = u0.
Further on, the multivalued linear operator A:= (B∗)−1AB−1is maximal
dissipative in the sense of [FY, Definition, p. 35] and k(λ−A)−1k ≤ λ−1, λ >
0.By the foregoing, we know that the operator Ais single-valued on D(A);
with a little abuse of notation, we will denote by T⊆ A the single-valued
linear operator which generates a bounded strongly continuous semigroup
(T(t))t≥0on D(A) (cf. Theorem 1.33(ii), where we have denoted this operator
by A). Using [KN, Theorem 3.6(a)] and the consideration from the paragraph
directly preceding the formulation of [FY, Theorem 2.8], it readily follows
that D(T) = D(A).Suppose now that u0=Bv0,where v0∈D(A) and
Av0∈R(B∗),i.e., that u0∈D(A) = D(T) (cf. the proof of [FY, Theorem
2.10]). Due to [FY, Theorem 2.8, Theorem 2.10], the problem (P)0
m,1,with
α= 1,has a unique solution v1(t) satisfying Bv1(t) = T(t)u0; moreover,
(d/dt)Bv1(t) = B∗(d/dt)Bv1(t) = Av1(t) = (d/dt)T(t)u0=T(t)T u0, t ≥0.
(1.40)
Since the condition [FY, (2.14)] holds, we get that there exists λ0>0 such
that (λ0B−A)−1∈L(X); hence, v1(·)=(λ0B−A)−1(λ0B−A)v1(·)∈
C([0,∞) : X) is bounded, as well as (d/dt)Bv1(t), Bv1(t) and Av1(t) are con-
tinuous and bounded for t≥0.Define vα(t) := R∞
0t−αΦα(st−α)v1(s)ds, t >
0 and vα(0) := v1(0).Using Theorem 1.21 and the arguments contained in
its proof, it readily follows that the function vα(·) is a bounded solution of
problem (P)0
m,α,satisfying in addition that the functions t7→ vα(·), t > 0 and
t7→ Avα(·), t > 0 can be analytically extended to the sector Σmin(( 1
α−1) π
2,π).
The uniqueness of solutions of problem (P)0
m,α can be proved with the help
of Theorem 1.20.
(b) Suppose now that (Bf)(x) := χ(a,∞,a)(x)f(x), x ∈R(f∈X), where
−∞ < a < ∞. Then B∈L(X), B =B∗, B2=Band the conclu-
sions established in the part (a) of this example, ending with the equation
(1.40), continue to hold. In our concrete situation, we have the validity of
condition [FY, (2.11)] but not the condition [FY, (2.14)], in general. Define
fα(t) := R∞
0t−αΦα(st−α)Bv1(s)ds, t > 0, fα(0) := Bv1(0) = u0, hα(t) :=
40 Marko Kosti´c
R∞
0t−αΦα(st−α)Av1(s)ds, t > 0 and hα(0) := Av1(0).By the foregoing,
we have that fα, hα∈C([0,∞) : X) are bounded and Dα
tfα(t) = hα(t),
t≥0,which simply implies Bhα(t) = hα(t), t ≥0.By (1.40), we have that
Av1(t) = T(t)T u0∈B−1[Av1(t)] and BAv1(t) = Av1(t) (t≥0), whence
we may conclude that Av1(t)∈ A[Bv1(t)] (t≥0). Since Ais closed, an ap-
plication of Theorem 1.6 yields that hα(t) = Bhα(t)∈AB−1fα(t) (t≥0);
consequently, the function t7→ fα(t), t ≥0 is a pre-solution of problem
(DFP)Rwith B≡I, F(t)≡0 and, by Remark 1.4(iv), the problem (P)0
m,α
has a bounded p-solution vα(·) satisfying, in addition, that the functions
t7→ Bvα(·), t > 0 and t7→ Avα(·), t > 0 can be analytically extended
to the sector Σmin(( 1
α−1) π
2,π).The uniqueness follows again from an essential
application of Theorem 1.20.
(ii) ([K]-[KN]) 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
∂
∂r vα(t, r) = a(r)Zt
0
gα(t−s)vα(s, r)ds +f(t, r), t ≥0, r ∈[0,1],
on the sector Σmin(( 1
α−1) π
2,π),where a∈C1[0,1] and the mapping t7→ f(t, ·), t ≥0
is continuous and exponentially bounded with the values in the Banach space C[0,1]
(cf. [K, Example 1] and Theorem 1.21); using Theorem 1.22 instead of Theorem
1.21, we can consider the well-posedness in C[0,1] for the equation
∂
∂r vc(t, r) = a(r)Zt
0
c(t−s)vc(s, r)ds +f(t, r), t ≥0, r ∈[0,1],
where c(·) is a completely positive function.
(iii) Fractional Maxwell’s equations have gained much attention in recent years (see
e.g. [D], [H], [MRE], [VT], [ZMN] and references cited therein for more details on
the subject). Here we want to briefly explain how we can use the analysis of A.
Favini and A. Yagi [FY, Exampe 2.2] for proving the existence and uniqueness
of analytical solutions of certain classes of inhomogeneous abstract time-fractional
Maxwell’s equations in R3; the time-fractional analogons of Poisson-wave equations
(see e.g. [FY, Example 2.3, Example 6.23]) will be considered somewhere else.
Consider the following abstract time-fractional Maxwell’s equations:
rotE=−Dα
tB, rotH=Dα
tD+J(1.41)
in R3,where E(resp. H) denotes the electric (resp. magnetic) field intensity, B
(resp. D) denotes the electric (resp. magnetic) flux density, and where Jis the
current density. It is assumed that the medium which fills the space R3is linear
but possibly anisotropic and nonhomogeneous, which means that D=E, B =µH
and J=σE +J0with some 3 ×3 real matrices (x), µ(x), σ(x) (x∈R3) and
J0being a given forced current density. Let any component of (x), µ(x), σ(x) be
a bounded, measurable function in R3,let the conditions [FY, (2.23)-(2.25)] hold,
and let f(t) = −(J0(·, t) 0)T. Then we can formulate the problem (1.41) in the
Abstract degenerate Volterra inclusions... 41
following abstract form
(P)1:B∗Dα
tBv1(t) = Av1(t) + f(t), t ≥0;
Bv1(0) = u0,
in the space X:= {L2(R3)}6,using the bounded self-adjoint operator Bof multipli-
cation by pc(x) acting in X, and Abeing the closed linear operator in Xgiven by
[FY, (2.27)]. In our concrete situation, the conditions [FY, (2.10) and (2.14)] hold,
so that the assumptions f∈C2([0,∞) : X) and u0=Bv0for some v0∈D(A)
satisfying Av0+f(0) ∈R(B∗) ensure by [FY, Corollary 2.11] that the problem
(P)1has a unique strict solution v1(·) in the sense of equation [FY, (2.13)]. Sup-
pose, additionally, that the function f00(t) is exponentially bounded. Then we can
use [FY, Theorem 2.5], the proof of [FY, Corollary 2.11] and the arguments from
the part (i)/(a) of this example in order to see that the solution v1∈C([0,∞) : X)
is exponentially bounded, as well as that H(t) := (d/dt)Bv1(t), Bv1(t) and Av1(t)
are continuous and exponentially bounded for t≥0.Define vα(t) and fα(t) as
before, Hα(t) := R∞
0t−αΦα(st−α)H(s)ds, t > 0 and Hα(0) := H(0).Perform-
ing the Laplace transform, it can be simply verifed that (g1−α∗(Bvα−u0))(t) =
Rt
0Hα(s)ds, t ≥0,so that Dα
tBvα(t) exists and equals to Hα(t).On the other
hand, we have B∗Bv1(t) = A(g1∗v1)(t) + B∗u0+Rt
0f(s)ds, t ≥0,so that
B∗Bvα(t) = A(gα∗vα)(t) + B∗u0+Rt
0fα(s)ds, t ≥0 by Theorem 1.21. This
implies Dα
tB∗Bvα(t) = Avα(t) + fα(t) and, since Dα
tBvα(t) exists, B∗Dα
tBvα(t) =
Avα(t) + fα(t), t ≥0.Clearly, Bvα(0) = u0so that vα∈C([0,∞) : X) is an
exponentially bounded solution of problem
(P)α:B∗Dα
tBvα(t) = Avα(t) + fα(t), t ≥0;
Bvα(0) = u0,
that is analytically extensible on the sector Σmin(( 1
α−1) π
2,π)and satisfies, in addition,
that the mapping Avα∈C([0,∞) : X) is exponentially bounded and analytically
extensible on the same sector, as well. The uniqueness of solutions of problem (Pα)
follows from Theorem 1.20.
We end this example with the observation that Theorem 1.21 and Theorem 1.22
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 ([FY], [FD], [VF], [NS], [SF], [TT]-[TT1]).
Concerning the adjoint type theorems, it should be noticed that the assertions of
[MK2, 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 Ais densely defined in the case of consid-
eration of [MK2, Theorem 2.1.12(i)].
Suppose now that Ais a subgenerator of an (a, k)-regularized C-resolvent family
(R(t))t∈[0,τ),n∈Nand xj∈ Axj−1for 1 ≤j≤n. Then we can prove inductively that,
42 Marko Kosti´c
for every t∈[0, τ),
R(t)x=k(t)Cx0+
n−1
X
j=1a∗,j ∗k(t)C xj+a∗,n ∗R(·)xn(t).(1.42)
Keeping in mind the identity (1.42), Theorem 1.6 and Proposition 1.29(ii), it is almost
straightforward to transfer the assertion of [MK2, Proposition 2.1.32] to degenerate case:
Proposition 1.36.(i) Suppose α∈(0,∞)\N, x ∈D(A)as well as C−1f, fA∈
C([0, τ ) : X), fA(t)∈ AC−1f(t), t ∈[0, τ)and Ais a closed subgenerator of a
(gα, C)-regularized resolvent family (R(t))t∈[0,τ).Set v(t) := (gdαe−α∗f)(t), t ∈
[0, τ ).If v∈Cdαe−1([0, τ) : X)and v(k)(0) = 0 for 1≤k≤ dαe − 2,then the
function u(t) := R(t)x+(R∗C−1f)(t), t ∈[0, τ )is a unique solution of the following
abstract time-fractional inclusion:
u∈Cdαe((0, τ ) : X)∩Cdαe−1([0, τ) : X),
Dα
tu(t)∈ Au(t) + ddαe−1
dtdαe−1gdαe−α∗f(t), t ∈[0, τ ),
u(0) = Cx, u(k)(0) = 0,1≤k≤ dαe − 1.
(ii) Suppose r≥0, n ∈N\ {1}, xj∈ Axj−1for 1≤j≤n, fj(t)∈ Afj−1(t)for
t∈[0, τ )and 1≤j≤n, fj∈C([0, τ) : X)for 0≤j≤n, and Ais a closed
subgenerator of a (g1/n, gr+1 )-regularized C-resolvent family (R(t))t∈[0,τ ).Then the
function v(t) := R(t)x+ (R∗C−1f)(t)x, t ∈[0, τ )is a unique solution of the
following abstract time-fractional inclusion:
v∈C1((0, τ ) : X)∩C([0, τ) : X),
v0(t)∈ Av(t) +
n−1
P
j=1
g(j/n)+r(t)Cxj
+
n−1
P
j=0g(j /n)+r∗fj(t) + d
dt gr+1(t)Cx, t ∈(0, τ ),
v(0) = gr+1(0)Cx.
Furthermore, v∈C1([0, τ ) : X)provided that r≥1or x= 0 and r≥0.
1.5.1. Differential and analytical properties of (a, k)-regularized C-resolvent
families. The main structural characterizations of differential and analytical
(a, k)-regularized C-resolvent families generated by single-valued linear operators con-
tinue to hold in our framework (cf. [FY, Chapter III], [MB]-[MSB] and [FF] for some
references on infinitely differentiable semigroups generated by MLOs).
We will use the following definition.
Definition 1.37.(cf. [MK2, Definition 2.2.1] for non-degenerate case)
(i) Suppose that Ais an MLO in X. Let α∈(0, π],and let (R(t))t≥0be an (a, k)-
regularized C-resolvent family which do have Aas a subgenerator. Then it is said
that (R(t))t≥0is an analytic (a, k)-regularized C-resolvent family of angle α, if there
exists a function R: Σα→L(X) which satisfies that, for every x∈X, the mapping
z7→ R(z)x, z ∈Σαis analytic as well as that:
(a) R(t) = R(t), t > 0 and
Abstract degenerate Volterra inclusions... 43
(b) limz→0,z∈ΣγR(z)x=k(0)Cx for all γ∈(0, α) and x∈X.
(ii) Let (R(t))t≥0be an analytic (a, k)-regularized C-resolvent family of angle α∈(0, π].
Then it is said that (R(t))t≥0is an exponentially equicontinuous, analytic (a, k)-
regularized C-resolvent family of angle α, resp. equicontinuous analytic (a, k)-
regularized C-resolvent family of angle α, if for every γ∈(0, α),there exists ωγ≥0,
resp. ωγ= 0,such that the family {e−ωγ<zR(z) : z∈Σγ} ⊆ L(X) is equicontin-
uous. Since there is no risk for confusion, we will identify in the sequel R(·) and
R(·).
In the following example, we will consider a time-fractional analogue of the linearized
Benney-Luke equation in L2-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 ([FY], [SF]). The
method employed by G. A. Sviridyuk and V. E. Fedorov [SF] 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 X0below (cf. also
[MK3, Example 2.2.49, Example 2.2.53] for our recent study of fractional analogons of
the abstract Barenblatt-Zheltov-Kochina equation in finite domains, where we have used
the pure Laplace transform techniques from [MK7]).
Example 1.38.Suppose that ∅ 6= Ω ⊆Rnis a bounded domain with smooth boundary,
and ∆ is the Dirichlet Laplacian in X:= L2(Ω),acting with domain H2(Ω) ∩H1
0(Ω).By
{λk}[= σ(∆)] we denote the eigenvalues of ∆ in L2(Ω) (recall that 0 <−λ1≤ −λ2··· ≤
−λk≤ · ·· → +∞as k→ ∞; cf. [HT, Section 5.6], [ABHN, Section 6] and [SF, Section
1.3] for more details) numbered in nonascending order with regard to multiplicities. By
{φk} ⊆ C∞(Ω) we denote the corresponding set of mutually orthogonal eigenfunctions.
Then, for every ζ > 0,we define the spectral fractional power Cζ∈L(X) of −∆ by
Cζ·:= (−∆)−(ζ)/2·:= Pk≥1h·, φki(−λk)−(ζ/2) φk(cf. [SV] for more details). Then Cζis
injective and R(C) =: D((−∆)ζ/2) = {f∈L2(Ω) : Pk≥1|hf, φki|2(−λk)ζ<∞}.
Let λ∈σ(∆),let 0 < η ≤2,and let α, β > 0.Consider the following time-fractional
analogue of the linearized Benney-Luke equation:
(P)η,f :
(λ−∆)Dη
tu(t, x) = α∆−β∆2u(t, x) + f(t, x), t ≥0, x ∈Ω,
∂k
∂tku(t, x)t=0 =uk(x), x ∈Ω,0≤k≤ dηe − 1,
u(t, x)=∆u(t, x)=0, t ≥0, x ∈∂Ω,
for which is known that plays an important role in evolution modelling of some problems
appearing in the theory of liquid filtration. Denote by X0the vector space of those
functions from Xthat are orthogonal to the eigenfunctions φk(·) for λk=λ. Then X0
is a closed subspace of X, and therefore, becomes the Banach space equipped with the
topology inherited by the X-norm (cf. [SF, Example 5.3.1, Theorem 5.3.2] for the case
η= 1). On the other hand, the operators A:= α∆−β∆2and B:= λ−∆,acting
with maximal domains, are closed in L2(Ω).Set θ:= min((π/η)−(π/2), π).Using the
Parseval equality, the asymptotic expansion formulae [EB, (1.28)] and an elementary
44 Marko Kosti´c
argumentation, we can simply prove that the operator family (Tη(z))z∈Σθ∪{0}⊆L(X0),
given by
t7→ Tη(z)·:= X
k|λk6=λ
Eη αλk−βλ2
k
λ−λk
zη!D·, φkEφk, z ∈Σθ∪ {0},
is well-defined, provided η∈(0,2).If η= 2,then we define (T2(t))t≥0⊆L(X0) in the
same way as above; since E2(z2) = cosh(z),we have that, for every t≥0,
T2(t)·=1
2X
k|λk6=λheit(βλ2−αλk)/(λ−λk)1/2
−e−it(βλ2−αλk)/(λ−λk)1/2iD·, φkEφk,
and that (T2(t))t≥0is bounded in the uniform operator norm. Differentiating T2(t) term
by term, it can be easily seen that the mapping t7→ T2(t)f, t ≥0 is continuously differ-
entiable for any f∈D((−∆)1/2)∩X0,and therefore, continuous. Since D((−∆)1/2)∩X0
is dense in X0and (T2(t))t≥0is bounded, the usual arguments shows that (T2(t))t≥0is
strongly continuous. Now we can proceed as in the proof of Theorem 1.21 in order to
see that, for every η∈(0,2),(Tη(t))t≥0is an exponentially bounded, analytic (gη, I)-
regularized resolvent family of angle θ. A straightforward computation shows that, for
every η∈(0,2],the integral generator Aof (Tη(t))t≥0is a closed single-valued operator
in X0,given by A={(f, g)∈X0×X0: (λ−λk)hg, φki= (αλk−βλ2
k)hf, φkifor all k∈
Nwith λk6=λ}; in particular, Ais an extension of the operator B−1A|X0.It is also
clear that (Tη(t))t≥0is a mild (gη, I)-existence family generated by A.Keeping in mind
the identity [EB, (1.25)], we can carry out a direct computation showing that the homo-
geneous counterpart of problem (P)η,f ≡(P)η,0,with xj= 0 for 1 ≤j≤ dζe − 1,has
an exponentially bounded pre-solution uh,0(t) = Tη(t)x0, t ≥0 for any xk∈D(A)∩X0
(0 ≤k≤ dηe−1), which seems to be an optimal result in the case that η≤1.Concerning
the homogeneous counterpart of problem (P)η,0with x0= 0,its solution uh,1(t) has to
be find in the form uh,1(t) = Rt
0Tη(s)x1ds, t ≥0.Consider first the case η∈(1,2).Then
for each k∈Nwith λk6=λ, we have
d2
dt2"g2−η∗ Eη αλk−βλ2
k
λ−λk·η!−1!#(t)
=Dη
tEη αλk−βλ2
k
λ−λk
tη!=αλk−βλ2
k
λ−λk
Eη αλk−βλ2
k
λ−λk
tη!, t ≥0.
On the other hand, expanding the function Eη(αλk−βλ2
k
λ−λk·η)−1 in a power series we get
that
d
dt"g2−η∗ Eη αλk−βλ2
k
λ−λk·η!−1!#(t) = t∞
X
n=0 αλk−βλ2
k
λ−λktηn+1tnη
Γ(nη + 2) , t ≥0.
The previous two equalities together imply that d
dt [g2−η∗(Eη(αλk−βλ2
k
λ−λk·η)−1)](t) =
Abstract degenerate Volterra inclusions... 45
αλk−βλ2
k
λ−λkRt
0Eη(αλk−βλ2
k
λ−λksη)ds, t ≥0 and
Dη
tg1∗Tη(·)x1(t)
=X
k|λk6=λ
αλk−βλ2
k
λ−λkZt
0
Eη αλk−βλ2
k
λ−λk
sη!dsDx1, φkEφk
=X
k|λk6=λ
αλk−βλ2
k
λ−λk
tEη,2 αλk−βλ2
k
λ−λk
tη!Dx1, φkEφk, t ≥0.
Using again the asymptotic expansion formula [EB, (1.28)], we obtain that the above
series converges for any x1∈X0and belongs to D(B) provided, in addition, that x1∈
D(B)∩X0.In this case, the equality BDη
tuh,1(t) = Auh,1(t), t ≥0 readily follows, so that
the function uh(t) := uh,0(t)+uh,1(t), t ≥0 is a pre-solution of problem (DFP)Lprovided
that x0∈D(A)∩X0and x1∈D(B)∩X0(with X=Y=L2(Ω) in Definition 1.17(iii));
furthermore, the mappings t7→ uh(t)∈L2(Ω), t > 0 and t7→ Buh(t)∈L2(Ω), t > 0
can be analytically extended to the sector Σθ.The situation is slightly different in the
case that η= 2 since we cannot use the formula [EB, (1.28)]; then a simple computation
shows that, formally, for every t≥0,
Bu00
h,1(t) = Auh,1(t)
=1
2X
k|λk6=λhi(βλ2−αλk)/(λ−λk)1/2eit((β λ2−αλk)/(λ−λk))1/2
−i(βλ2−αλk)/(λ−λk)1/2e−it((β λ2−αλk)/(λ−λk))1/2iDx1, φkEφk.
Hence, the function uh(t) := uh,0(t) + uh,1(t), t ≥0 is a pre-solution of problem (DFP)L
with x0∈D(A)∩X0and x1∈D((−∆)3/2)∩X0.The range of any pre-solution of
problem (P)η,0must be contained in X0,so that the uniqueness of solutions of problem
(P)η,f follows from its linearity and Proposition 1.29(ii).
Before considering the inhomogeneous problem (P)η,f ,we would like to observe that
the assumptions (x, y)∈ A and x∈D(A) imply (x, y)∈B−1A|X0.Keeping in mind this
remark, Theorem 1.6, as well as the fact that the assertion of [JP, Proposition 2.1(iii)]
admits a reformulation in our framework, we can simply prove that for any function
h∈W1,1
loc ([0,∞) : X0) satisfying that
t7→ X
k|λk6=λαλk−βλ2
kDd
dt(gη∗h)(t), φkEφk∈L1
loc([0,∞) : X0),(1.43)
the function uBh(t) := Rt
0Tη(t−s)d
ds (gη∗h)ds, t ≥0 is a solution of problem (P)η,B h.
On the other hand, the operator Bannihilates any function from span{φk:k|λ=
λk}so that the function t7→ Pk|λk=λhf(t),φki
βλ2
k−αλkφk, t ≥0 is a pre-solution of problem
(P)η,Pk|λk=λhf(·),φkiφk,provided that the following condition holds:
(Q) : Dη
thf(t), φkiexists in L2(Ω) for k|λ=λk,hx0, φki= 0 for k|λ6=λk,hx1, φki= 0
for k|λ6=λk,1< η ≤2,hx0, φki=hf(0),φki
βλ2
k−αλkfor k|λ=λk,and hx1, φki=hf0(0),φki
βλ2
k−αλk
for k|λ=λk,1< η ≤2.
46 Marko Kosti´c
Summa summarum, we have the following:
(i) 0 < η < 2 : Suppose that x0∈D(A)∩X0, x1∈D(B)∩X0,if η > 1,
Pk|λk6=λhf(·),φki
λ−λkφk=h∈W1,1
loc ([0,∞) : X0) satisfies (1.43), and the condition (Q)
holds. Then there exists a unique pre-solution of problem (P)η,f .
(ii) η= 2 : Suppose x1∈D((−∆)3/2)∩X0and the remaining assumptions from (i)
hold. Then there exists a unique pre-solution of problem (P)η,f .
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 inhomo-
geneous Cauchy problem Pη,f in the case that η < 1 (cf. [FD, Theorem 4.2] for the first
result in this direction).
The following theorem can be deduced by making use of the argumentation contained
in the proof of [MK4, Theorem 2.16]. Here we would like to observe that the equality
Rλ,µ = 0,stated on [MK4, p. 12, l. 4], can be proved by taking the Laplace transform of
term appearing on [MK4, p. 12, l. 1-2] in variable µ, and by using the strong analyticity
of mapping λ7→ F(λ)∈L(X), λ ∈N, along with the equality Rλ,µ = 0 for <λ > ω,
˜a(λ)˜
k(λ)6= 0 (the repeated use of identity [MK4, (2.30)] on [MK4, p. 12, l. 4] is wrong
and makes a circulus vitiosus):
Theorem 1.39.(cf. [MK2, Theorem 2.2.4] for non-degenerate case) Suppose that α∈
(0, π/2],abs(k)<∞,abs(|a|)<∞,and ˜
k(λ)can be analytically continued to a function
g:ω+ Σπ
2+α→C,where ω≥max(0,abs(k),abs(|a|)).Suppose, further, that Ais a
closed subgenerator of an analytic (a, k)-regularized C-resolvent family (R(t))t≥0of angle
αsatisfying that the family {e−ωz R(z) : z∈Σγ} ⊆ L(X)is equicontinuous for all angles
γ∈(0, α),as well as that the equation (1.25) holds for each y=x∈X, with R1(·)and
C1replaced therein by R(·)and C, respectively. Set
N:= λ∈ω+ Σπ
2+α:g(λ)6= 0.
Then Nis an open connected subset of C.Assume that there exists an analytic function
ˆa:N→Csuch that ˆa(λ) = ˜a(λ),<λ > ω. Then the operator I−ˆa(λ)Ais injective for
every λ∈N, R(C)⊆R(I−ˆa(λ)C−1AC)for every λ∈N1:= {λ∈N: ˆa(λ)6= 0},the
operator (I−ˆa(λ)C−1AC)−1C∈L(X)is single-valued (λ∈N1), the family
n(λ−ω)g(λ)I−ˆa(λ)C−1AC−1C:λ∈N1∩(ω+ Σ π
2+γ1)o⊆L(X)
is equicontinuous for every angle γ1∈(0, α),the mapping
λ7→ I−ˆa(λ)C−1AC−1Cx, λ ∈N1is analytic for every x∈X,
and
lim
λ→+∞,˜
k(λ)6=0
λ˜
k(λ)I−˜a(λ)A−1Cx =R(0)x, x ∈X.
Keeping in mind Lemma 1.5, Theorem 1.6 and Theorem 1.27, we can repeat almost
literally the proof of [MK2, Theorem 2.2.5] in order to see that the following result holds.
Theorem 1.40.Assume that Ais a closed MLO in X, CA ⊆ AC, α ∈(0, π/2],abs(k)<
∞,abs(|a|)<∞and ω≥max(0,abs(k),abs(|a|)).Assume, further, that for every λ∈C
Abstract degenerate Volterra inclusions... 47
with <λ>ωand ˜
k(λ)6= 0,the operator I−˜a(λ)Ais injective with R(C)⊆R(I−˜a(λ)A).
If there exist a function q:ω+ Σπ
2+α→L(X)and an operator D∈L(X)such that, for
every x∈X, the mapping λ7→ q(λ)x, λ ∈ω+ Σ π
2+αis analytic as well as that:
q(λ)x=˜
k(λ)I−˜a(λ)A−1Cx, <λ > ω, ˜
k(λ)6= 0, x ∈X,
for every γ∈(0, α),the family {(λ−ω)q(λ) : λ∈ω+ Σπ
2+γ} ⊆ L(X)is equicontinuous
and
lim
λ→+∞λq(λ)x=Dx, x ∈X, if D(A)6=X,
then Ais a subgenerator of an exponentially equicontinuous, analytic (a, k)-regularized
C-resolvent family (R(t))t≥0of angle αsatisfying that R(z)A⊆AR(z), z ∈Σα,the
family {e−ωz R(z) : z∈Σγ} ⊆ L(X)is equicontinuous for all angles γ∈(0, α),as well
as that the equation (1.25) holds for each y=x∈X, with R1(·)and C1replaced therein
by R(·)and C, respectively.
Suppose that ∅ 6= Ω ⊆Rnis a bounded domain with smooth boundary. As explained
by M. V. Falaleev and S. S. Orlov in [FO], the equation
(α−∆)utt =β∆ut+ ∆u+Zt
0
g(t−s)∆u(s, x)ds, t > 0, x ∈Ω;
u(0, x) = φ(x), ut(0, x) = ψ(x),
(1.44)
where g∈L1
loc([0,∞)), α ∈Rand β∈R\ {0},appears in some models of nonlinear
viscoelasticity provided that n= 3.In the following illustrative example, we will consider
the wellposedness of equation (1.44) following the approaches from [MK7] and this paper.
Example 1.41.Let ∆ be the Dirichlet Laplacian in X:= L2(Ω),acting with domain
H2(Ω) ∩H1
0(Ω).As in Example 1.38, we will denote by {λk}[= σ(∆)] the eigenvalues of
∆ in L2(Ω) numbered in nonascending order with regard to multiplicities; {φk} ⊆ C∞(Ω)
denotes the corresponding set of mutually orthogonal eigenfunctions. Integrating (1.44)
twice with the respect to the time-variable t, we obtain the associated integral equation
(α−∆)u(t)=(α+ (β−1)∆)φ(x)
+t(α−∆)ψ+β∆g1∗u(t)+∆g2∗u(t)+∆g2∗g∗u(t), t ≥0.(1.45)
Set B:= α−∆, A2:= β∆, A1=A0:= ∆ (acting with the Dirichlet boundary condi-
tions), a2(t) := g1(t), a1(t) := g2(t), a0(t) := (g2∗g)(t),and
Pλ:= λ2+βλ + ˜g(λ)+1
λ2"αλ2
λ2+βλ + ˜g(λ)+1 −∆#.
Suppose that α=λk0∈σ(∆) for some k0∈Nand the function g(t) is Laplace trans-
formable (in [MK7, Example 3.15], we have considered the case α > 0,with the state
space being Lp(Ω) for some 1 ≤p < ∞). Then there exist finite constants M≥1 and
ω≥0 such that |Rt
0g(s)ds| ≤ Meωt , t ≥0 and λR∞
0e−λt Rt
0g(s)ds dt =R∞
0e−λtg(t)dt,
λ>ω, which simply implies that the set {˜g(λ) : λ>ω+ 1}is bounded. Define
D:L2(Ω) →L2(Ω) by Df := (−1)β−1P∞
k=1hφk, f iφk, f ∈L2(Ω).Using Parseval’s
equality, it can be simply verified that D, BD ∈L(L2(Ω)); furthermore, kR(λ: ∆)k=
O|α−λ|−1as λ→α(see e.g. [MK9, Example, pp. 57-58]). Using the resolvent equation
48 Marko Kosti´c
and these facts, we obtain the existence of a sufficiently large real number R > 0 such
that P−1
λ∈L(L2(Ω)) for |λ| ≥ R, as well as that
|λ|−2"
P−1
λ
+
BP−1
λ
+
2
X
j=0
˜aj(λ)AjP−1
λ
#≤M, |λ| ≥ R, (1.46)
lim|λ|→∞ λ−1P−1
λf=Df, lim|λ|→∞ λ−1BP−1
λf=BDf and lim|λ|→∞ λ−1˜aj(λ)P−1
λf=
0,0≤j≤2 (f∈L2(Ω)). Making use of [MK7, Theorem 3.9], we obtain that there
exists an exponentially bounded once integrated I-existence family (E1(t))t≥0for (1.45),
in the sense of [MK7, Definition 3.8(i)], satisfying additionally that for each f∈L2(Ω)
the mappings t7→ E1(t)f, t > 0, t 7→ BE1(t)f, t > 0 and t7→ Aj(aj∗E1)(t)f, t > 0 can
be analytically extended to the sector Σπ/2; furthermore, (E1(t))t≥0is an exponentially
bounded once integrated I-uniqueness family for (1.45), in the sense of [MK7, Definition
3.8(ii)]. Therefore, for every φ, ψ ∈H2(Ω) ∩H1
0(Ω),there exists a unique strong solution
of the associated once integrated problem (1.45)
(α−∆)u(t) = t(α+ (β−1)∆)φ(x) + t2
2(α−∆)ψ
+β∆g1∗u(t)+∆g2∗u(t)+∆g2∗g∗u(t), t ≥0,(1.47)
given by u(t) = E1(t)(α+(β−1)∆)φ+Rt
0E1(s)(α−∆)ψ ds, t ≥0.On the other hand, the
equation (1.46) taken together with the equality lim|λ|→∞ λ−1BP−1
λf=BDf, Theorem
1.40 and Remark 1.4(v) simply implies that for each θ∈(−π, π] the MLO eiθ AB−1
generates an exponentially bounded, analytic once integrated (b+g2(t)+(g2∗g)(t), I)-
regularized resolvent family (E1,B(t)≡BE1(t))t≥0of angle π/2.Since [MK2, Theorem
2.1.29(ii)] holds in our framework, this immediately yields some results on the existence
and uniqueness of analytical (possible, entire, cf. also [MK9, Theorem 2.2]) solutions of
the problem (1.47) with the term t(α+ (β−1)∆)φ+t2
2(α−∆)ψreplaced by a general
inhomogeneity f(t).
The classes of exponentially equicontinuous, analytic (a, k)-regularized C1-existence
families and (a, k)-regularized C2-uniqueness families can be introduced and analyzed, as
well. For the sequel, we need the following notion.
Definition 1.42.Let X=Y, and let Abe a subgenerator of a C1-existence family
(R1(t))t≥0(cf. Definition 1.23(i) with a(t)≡1 and k(t)≡1). Then (R1(t))t≥0is said to
be entire if, for every x∈X, the mapping t7→ R1(t)x, t ≥0 can be analytically extended
to the whole complex plane.
Using the arguments contained in the proof of [MK6, Theorem 3.15], we can deduce
the following result.
Theorem 1.43.Suppose r≥0, θ ∈(0, π/2),Ais a closed MLO and −A is a subgenerator
of an exponentially equicontinuous, analytic r-times integrated C-semigroup (Sr(t))t≥0of
angle θ. Then there exists an operator C1∈L(X)such that Ais a subgenerator of an
entire C1-existence family in X.
Remark. (i) It ought to be observed that we do not require the injectivity of oper-
ator C1here. The operators Tα(z) and Sα,z0(z),appearing in the proof of [MK6,
Abstract degenerate Volterra inclusions... 49
Theorem 3.15], annulates on the subspace A0.
(ii) Theorem 1.43 is closely linked with the assertions of [MK5, Theorem 2.1, Theorem
2.2]. These results can be extended to abstract degenerate fractional differential
inclusions, as well.
Example 1.44.In a great number of research papers, many authors have considered
infinitely differentiable semigroups generated by multivalued linear operators of form
AB−1or B−1A, where the operators Aand Bsatisfy the condition [FY, (3.14)], or its
slight modification, with certain real constants 0 < β ≤α≤1, γ ∈Rand c, C > 0 (in
our notation, we have A=Land B=M). The validity of this condition with α= 1 (see
e.g. [FY, Example 3.3, 3.6]) immediately implies by Theorem 1.40 and Remark 1.4(v)
that the operator AB−1generates an exponentially bounded, analytic σ-times integrated
semigroup of angle Σarcctan(1/c),provided that σ > 1−β; in the concrete situation of
[FY, Example 3.4, 3.5], the above holds with the operator AB−1replaced by B−1A.
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 [FY, Chapter III] have a removable singularity at zero
(cf. the proof of [EB, Theorem 3.1], [MK12] and the forthcoming monograph [MK3] for
more details). On the other hand, from the point of view of possible applications of
Theorem 1.43, it is very important to know that the operators AB−1or B−1Agenerate
exponentially bounded, analytic integrated semigroups. This enables us to consider the
abstract degenerate Cauchy problems that are backward to those appearing in [FY,
Example 3.3-Example 3.6]. For example, we can consider the following modification of
the backward Poisson heat equation in the space Lp(Ω):
(P) :
∂
∂t [m(x)v(t, x)] = −∆v+bv, t ≥0, x ∈Ω;
v(t, x)=0,(t, x)∈[0,∞)×∂Ω,
m(x)v(0, x) = u0(x), x ∈Ω,
where Ω is a bounded domain in Rn, b > 0, m(x)≥0 a.e. x∈Ω, m∈L∞(Ω) and
1< p < ∞.Let Bbe the multiplication in Lp(Ω) with m(x),and let A= ∆ −bact
with the Dirichlet boundary conditions. Then Theorem 1.43 implies that there exists
an operator C1∈L(Lp(Ω)) such that A=−AB−1is a subgenerator of an entire C1-
existence family; hence, for every u0∈R(C1),the problem (P) has a unique solution
t7→ 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 u0for which there exists a unique solution
of problem (P) is dense in Lp(Ω) provided that there exists a constant d > 0 such that
|m(x)| ≥ da.e. x∈Ω.
In the following example, we consider the existence and uniqueness of solutions of
abstract degenerate relaxation Cauchy problems that are not subordinated to those of
first order.
Example 1.45.It is clear that the examples presented in [FY, Chapter III] can serve
one for consideration of a wide class of abstract degenerate relaxation equations that
50 Marko Kosti´c
are not subordinated to the problems of first order (a fairly complete analysis of such
equations is quite non-trivial and we shall skip all related details for convenience): Sup-
pose that the condition [FY, (3.1)] holds with certain real constants 0 < β ≤α≤1,
c, M > 0,as well as that θ∈(π/2,0), ζ ∈(0,1) and π
2> π −arctan 1
c+θ > 1
2πζ. Then
Σπ−arctan 1
c+θ⊆ρ(eiθA) and, in general, ρ(eiθ A) does not contain any right half-plane.
An application of Theorem 1.40 shows that the operator eiθAgenerates an exponen-
tially bounded, analytic (gζ, gr+1)-regularized resolvent family of angle θ0:= min((π−
arctan(1/c) + θ−(πζ/2))/ζ , π/2),where r > ζ(1 −β),if Ais not densely defined, and
r=ζ(1 −β),otherwise.
Suppose now that x∈E, 1−ζ > η > 1−ζ β, δ > 0,0< γ < θ0, t > 0 is
fixed temporarily, Γ1:= {rei((π/2)+γ):r≥t−1}∪{t−1eiθ :θ∈[0,(π/2) + γ]},Γ2:=
{re−i((π/2)+γ):r≥t−1}∪{t−1eiθ :θ∈[−(π/2) −γ, 0]}and Γ := Γ1∪Γ2is oriented
counterclockwise. Define u(0) := 0 and
u(t) := 1
2πi Z
Γ
eλtλ−ηλζ−eiθ A−1x dλ.
Arguing as in [ABHN, Theorem 2.6.1, Theorem 2.6.4], it readily follows that
u∈C([0,∞) : E),ku(t)k=O(tη+ζβ−1), t ≥0 and that the mapping t7→ u(t), t > 0 can
be analytically extended to the sector Σθ0.Keeping in mind Theorem 1.6 and Theorem
1.7(i), we obtain that there exists a continuous section t7→ uA,θ,ζ (t), t > 0 of the
multivalued mapping t7→ eiθ A(gζ∗u)(t), t > 0,with the meaning clear, such that
u(t) = uA,θ,ζ (t) + gη+ζ(t)x, t > 0.
Observe, finally, that the Riemann-Liouville fractional derivative Dζ
tu(t) need not be
defined here.
For the sequel, we need the following notion. Suppose that a sequence (Mn)n∈N0of
positive real numbers satisfies M0= 1,as well as the following conditions:
(M.1) M2
p≤Mp+1Mp−1, p ∈N,
(M.2) Mp≤AHpmin
p1,p2∈N,p1+p2=pMp1Mp2, n ∈N,for some A > 1 and H > 1,
(M.3)’ P∞
p=1
Mp−1
Mp<∞.
Set
ωL(t) := ∞
X
n=0
tn
Mn
, t ≥0.
The most important results concerning differential properties of non-degenerate (a, k)-
regularized C-resolvent families remain true, with almost minimal reformulations, in our
new setting. The proofs of following extensions of [MK2, Theorem 2.2.15, Theorem 2.2.17]
are omitted.
Theorem 1.46.Suppose that Ais a closed MLO in X, abs(k)<∞,abs(|a|)<∞, r ≥ −1
and there exists ω≥max(0,abs(k),abs(|a|)) such that, for every z∈ {λ∈C:<λ >
ω, ˜
k(λ)6= 0},we have that the operator I−˜a(z)Ais injective and R(C)⊆R(I−˜a(z)A).
If, additionally, for every σ > 0,there exist Cσ>0and an open neighborhood Ωσ,ω of
Abstract degenerate Volterra inclusions... 51
the region
Λσ,ω := λ∈C:<λ≤ω, <λ≥ −σln |=λ|+Cσ∪ {λ∈C:<λ≥ω},
and a function hσ: Ωσ,ω →L(X)such that, for every x∈X, the mapping λ7→ hσ(λ)x,
λ∈Ωσ,ω is analytic as well as that hσ(λ) = ˜
k(λ)(I−˜a(λ)A)−1C, <λ > ω, ˜
k(λ)6= 0,
and that the family {|λ|−rhσ(λ) : λ∈Λσ,ω}is equicontinuous, then, for every ζ > 1,A
is a subgenerator of an exponentially equicontinuous (a, k ∗gζ+r)-regularized C-resolvent
family (Rζ(t))t≥0satisfying that the mapping t7→ Rζ(t), t > 0is infinitely differentiable
in L(X).
Theorem 1.47.Let (Mn)n∈N0satisfy (M.1), (M.2) and (M.3)’.
(i) Suppose that abs(k)<∞,abs(|a|)<∞,Ais a closed subgenerator of a (local)
(a, k)-regularized C-resolvent family (R(t))t∈[0,τ ), ω > max(0,abs(k),abs(|a|)) and
m∈N.Denote, for every ε∈(0,1) and a corresponding Kε>0,
Fε,ω := λ∈C:<λ≥ −ln ωLKε|=λ|+ω.
Assume that, for every ε∈(0,1),there exist Kε>0,an open neighborhood Oε,ω of
the region Gε,ω := {λ∈C:<λ≥ω, ˜
k(λ)6= 0}∪{λ∈Fε,ω :<λ≤ω},a mapping
hε:Oε,ω →L(E)and analytic mappings fε:Oε,ω →C, gε:Oε,ω →Csuch that:
(a) fε(λ) = ˜
k(λ),<λ>ω;gε(λ) = ˜a(λ),<λ > ω,
(b) for every λ∈Fε,ω ,the operator I−gε(λ)Ais injective and R(C)⊆R(I−
gε(λ)A),
(c) for every x∈X, the mapping λ7→ hε(λ)x, λ ∈Gε,ω is analytic, hε(λ) =
fε(λ)(I−gε(λ)A)−1C, λ ∈Gε,ω,
(d) the family {(1 + |λ|)−me−ε|<λ|hε(λ) : λ∈Fε,ω ,<λ≤ω} ⊆ L(X)is equicon-
tinuous and the family {(1 + |λ|)−mhε(λ) : λ∈C,<λ≥ω} ⊆ L(X)is
equicontinuous.
Then the mapping t7→ R(t), t ∈(0, τ )is infinitely differentiable in L(X)and, for
every compact set K⊆(0, τ ),there exists hK>0such that the set {hn
Kdn
dtnR(t)
Mn:t∈
K, n ∈N0}is equicontinuous.
(ii) Suppose that abs(k)<∞,abs(|a|)<∞,Ais a closed subgenerator of a (local)
(a, k)-regularized C-resolvent family (R(t))t∈[0,τ ), ω > max(0,abs(k),abs(|a|)) and
m∈N.Denote, for every ε∈(0,1), ρ ∈[1,∞)and a corresponding Kε>0,
Fε,ω,ρ := nλ∈C:<λ≥ −Kε|=λ|1/ρ +ωo.
Assume that, for every ε∈(0,1),there exist Kε>0,an open neighborhood Oε,ω
of the region Gε,ω,ρ := {λ∈C:<λ≥ω, ˜
k(λ)6= 0} ∪ {λ∈Fε,ω,ρ :<λ≤ω},a
mapping hε:Oε,ω →L(X)and analytic mappings fε:Oε,ω →Cand gε:Oε,ω →
Csuch that the conditions (i)(a)-(d) of this theorem hold with Fε,ω,resp. Gε,ω,
replaced by Fε,ω,ρ,resp. Gε,ω,ρ.Then the mapping t7→ R(t), t ∈(0, τ )is infinitely
differentiable in L(X)and, for every compact set K⊆(0, τ),there exists hK>0
such that the set {hn
Kdn
dtnR(t)
n!ρ:t∈K, n ∈N0}is equicontinuous.
52 Marko Kosti´c
Let us recall that the case ρ= 1 in Theorem 1.47 is very important because it gives
a sufficient condition for an (a, k)-regularized C-resolvent family to be real analytic.
Suppose now that n∈N,|a|(t) satisfies (P1)-Cand abs(a)=0.Following [JP,
Definition 3.3, p. 69], we say that a(t) is n-regular iff there exists c > 0 such that
|λmˆa(m)(λ)| ≤ c|ˆa(λ)|, λ ∈C+,1≤m≤n. Set a(−1)(t) := Rt
0a(s)ds, t ≥0 and suppose
that a(t) and b(t) are n-regular for some n∈N.Then we know that ˆa(λ)6= 0, λ ∈C+,
as well as that (a∗b)(t) and a(−1)(t) are n-regular, and that a0(t) is n-regular provided
that abs(a0)=0.
Following [JP, Definition 3.1, p. 68] and [MK2, Definition 2.1.23], it will be said that
the abstract Volterra inclusion (1.1) with B=I(denoted henceforth by the same symbol)
is (kC)-parabolic iff the following holds:
(i) |a|(t) and k(t) satisfy (P1)-Cand there exist meromorphic extensions of the func-
tions ˜a(λ) and ˜
k(λ) on C+,denoted by ˆa(λ) and ˆ
k(λ).Let Nbe the subset of C+
which consists of all zeroes and possible poles of ˆa(λ) and ˆ
k(λ).
(ii) There exists M≥1 such that, for every λ∈C+\N, 1/ˆa(λ)∈ρC(A) and ||ˆ
k(λ)(I−
ˆa(λ)A)−1C|| ≤ M/|λ|.
If k(t)≡1,resp. C=I, then it is also said that (1.1) is C-parabolic, resp. k-parabolic.
Now we are ready to formulate the following extension of [MK2, Theorem 2.1.24]:
Theorem 1.48.Assume n∈N, a(t)is n-regular, (X, k · k)is a Banach space, Ais
a closed MLO in X, the abstract Volterra inclusion (1.1) is C-parabolic, and the map-
ping λ7→ (I−˜a(λ)A)−1C, λ ∈C+is continuous. Then, for every α∈(0,1],Ais
a subgenerator of an (a, gα+1)-regularized C2-resolvent family (Sα(t))t≥0which satisfies
suph>0,t≥0h−α||Sα(t+h)−Sα(t)|| <∞, Dα
tSα(t)Ck−1∈Ck−1((0,∞) : L(X)),1≤k≤n
as well as:
tjDj
tDα
tSα(t)Ck−1
≤M, t ≥0,1≤k≤n, 0≤j≤k−1,(1.48)
tkDk−1
tDα
tSα(t)Ck−1−skDk−1
sDα
sSα(s)Ck−1
≤M|t−s|1 + ln t
t−s,0≤s < t < ∞,1≤k≤n, (1.49)
and, for every T > 0, ε > 0and k∈Nn,there exists Mε
T,k >0such that
tkDk−1
tDα
tSα(t)Ck−1−skDk−1
sDα
sSα(s)Ck−1
≤Mε
T,k (t−s)1−ε,0≤s < t ≤T , 1≤k≤n. (1.50)
Furthermore, if Ais densely defined, then Ais a subgenerator of a bounded (a, C2)-
regularized resolvent family (S(t))t≥0,satisfying additionally that the mapping
t7→ S(t)Ck−1, t > 0is in class Ck−1((0,∞) : L(X)),1≤k≤nand that (1.48)-(1.50)
hold with Dα
tSα(t)Ck−1replaced by S(t)Ck−1(1≤k≤n) therein.
The representation formula [JP, (3.41), p. 81] and the assertions of [JP, Corollary
3.2-Corollary 3.3, pp. 74-75] can be extended to exponentially bounded (a, C )-regularized
resolvent families subgenerated by multivalued linear operators, as well. For more details
Abstract degenerate Volterra inclusions... 53
about parabolicity of abstract non-degenerate Volterra equations, we refer the reader to
[JP, Chapter I, Section 3].
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