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CUBO A Mathematical Journal
Vol.17, No¯ 03, (15–41). October 2015

Degenerate k-regularized (C1,C2)-existence and uniqueness
families

Marko Kostić
1

Faculty of Technical Sciences,

University of Novi Sad,

Trg D. Obradovića 6, 21125 Novi Sad, Serbia.

marco.s@verat.net

ABSTRACT

In this paper, we consider various classes of degenerate k-regularized (C1,C2)-existence

and uniqueness families. The main purpose of the paper is to report how the techniques

established in a joint paper of C.-G. Li, M. Li and the author [32] can be successfully

applied in the analysis of a wide class of abstract degenerate multi-term fractional

differential equations with Caputo derivatives.

RESUMEN

En este art́ıculo, consideramos varias clases de familias k-regularizadas (C1,C2)-de

existencia y unicidad. El principal objetivo de este trabajo es mostrar como las técnicas

establecidas en un trabajo conjunto de C.-G. Li, M. Li y el autor [27], pueden ser

aplicadas satisfactoriamente en el análisis de una clase amplia de ecuaciones fracionarias

multi-término degeneradas con derivadas de Caputo.

Keywords and Phrases: Abstract multi-term fractional differential equations, degenerate differ-

ential equations, fractional calculus, Mittag-Leffler functions, Caputo time-fractional derivatives.

2010 AMS Mathematics Subject Classification: 47D06, 47D60, 47D62, 47D99.

1The author is partially supported by grant 174024 of Ministry of Science and Technological Development,

Republic of Serbia.



16 Marko Kostić CUBO
17, 3 (2015)

1 Introduction and preliminaries

During the past three decades, considerable interest in fractional calculus and fractional differential

equations has been stimulated due to their numerous applications in engineering, physics, chem-

istry, biology and other sciences. Basic information about fractional calculus and non-degenerate

fractional differential equations can be obtained by consulting [6], [15], [23]-[25], [42]-[44] and the

references cited therein. For the basic source of information on the abstract degenerate differential

equations, we refer the reader to [1], [3], [5], [7], [12], [17], [35], [40]-[41], [45]-[49] and [52]-[53].

The theory of abstract degenerate (multi-term) fractional differential equations is at its be-

ginning stage and we can freely say that it is a still-undeveloped subject. The most important

qualitative properties of abstract degenerate (multi-term) fractional differential equations have

been recently considered in the papers [19]-[20], [22], [27]-[30] and [34]. The existence and unique-

ness of solutions of the Cauchy and Showalter problems for a class of degenerate fractional evolution

systems have been analyzed by V. E. Fedorov and A. Debbouche in [19], while the necessary and

sufficient conditions for the relative p-boundedness of a pair of operators have been obtained by

V. E. Fedorov and D. M. Gordievskikh in [20]. In [27]-[28], the author has investigated degen-

erate Volterra integro-differential equations in locally convex spaces, as well as the generation of

degenerate fractional resolvent operator families associated with abstract differential operators and

the generation of various classes of exponentially equicontinuous k-regularized C-resolvent propa-

gation families associated with the degenerate multi-term problem (1.1) below. The hypercyclic

and topologically mixing properties of degenerate multi-term fractional differential equations with

Caputo derivatives have been analyzed in [29]-[30]. Among many other things, in a joint research

study with V. E. Fedorov [22], the author has analyzed the existence and uniqueness of regular-

ized solutions for a class of abstract degenerate multi-term fractional differential equations with

Caputo derivatives. The abstract degenerate multi-term fractional differential equations with clas-

sical Riemann-Liouville fractional derivatives have been recently investigated by the author in [34],

following the methods used in [33] and this paper.

The main subject under our consideration is the following degenerate multi-term problem:

BD
αn
t u(t) +

n−1∑

i=1

AiD
αi
t u(t) = AD

α
t u(t) + f(t), t ≥ 0;

u(j)(0) = uj, j = 0, · · ·,⌈αn⌉ − 1,

(1.1)

where n ∈ N \ {1}, A, B and A1, · · ·,An−1 are closed linear operators on a sequentially complete

locally convex space X, 0 ≤ α1 < · · · < αn, 0 ≤ α < αn, f(t) is an X-valued function, and D
α
t

denotes the Caputo fractional derivative of order α ([6], [25]). Define An := B, A0 := A, m := ⌈α⌉,

α0 := α and mi := ⌈αi⌉, i ∈ N
0
n, where Nn := {1,2, · · ·,n} and N

0
n := Nn ∪ {0}.

As mentioned in the abstract, the main purpose of this paper is to reconsider the various

notions of non-degenerate k-regularized (C1,C2)-existence and uniqueness families introduced in

the paper [32], whose organization is very similar to that of this paper. Without any doubt, this



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 17

causes the expositority of our paper in a certain sense. On the other hand, we will not be in wrong

if we say that our paper proposes an important theoretical novelty method capable of seeking of

solutions of some very atypical degenerate differential equations in Lp-spaces. In this place, it is

also worth noting that we initiate the analysis of existence of local solutions of abstract degenerate

differential equations in this paper; furthermore, we provide generalizations of [36, Theorem 2.3,

Theorem 3.1] for degenerate multi-term problems, and successfully apply the obtained theoretical

results in the analysis of some very interesting degenerate differential equations.

Before explaining the notation used in the paper, we would like to note that it is quite question-

able whether there exists any other significant reference which treats the existence and uniqueness

of various types of automorphic solutions to abstract degenerate multi-term fractional differential

equations (cf. [2], [8]-[9] and [13]-[14] for some results in the non-degenerate case). Unless specifed

otherwise, we assume that X is a Hausdorff sequentially complete locally convex space over the

field of complex numbers, SCLCS for short. If Y is also an SCLCS over the same field of scalars

as X, then we denote by L(Y,X) the space consisting of all continuous linear mappings from Y

into X; 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 Y is denoted by ⊛Y. The symbol I denotes the identity operator on X. Let

0 < τ ≤ ∞. A strongly continuous operator family (W(t))t∈[0,τ) ⊆ L(Y,X) is said to be locally

equicontinuous iff, for every T ∈ (0,τ) and for every p ∈ ⊛X, there exist qp ∈ ⊛Y and cp > 0

such that p(W(t)y) ≤ cpqp(y), y ∈ Y, t ∈ [0,T]; the notions of equicontinuity of (W(t))t∈[0,τ)
and the exponential equicontinuity of (W(t))t≥0 are defined similarly. Notice that (W(t))t∈[0,τ)
is automatically locally equicontinuous in case that the space Y is barreled ([39]).

By B we denote the family consisting of all bounded subsets of Y. Define pB(T) := supy∈B p(Ty),

p ∈ ⊛X, B ∈ B, T ∈ L(Y,X). Then pB(·) is a seminorm on L(Y,X) and the system (pB)(p,B)∈⊛X×B
induces the Hausdorff locally convex topology on L(Y,X). If X is a Banach space, then we denote

by ‖x‖ the norm of an element x ∈ X. Suppose that A is a closed linear operator acting on X. Then

we denote the domain, kernel space and range of A by D(A), N(A) and R(A), respectively. Since

no confusion seems likely, we will identify A with its graph. Set 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)]. If V is a general topological vector

space, then 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. We refer the reader to [4], [25, Section 1.1] and references cited there

for the basic information about vector-valued analytic functions. In our approach the space X is

sequentially complete, so that the analyticity of a mapping f : Ω → X is equivalent with its weak

analyticity. It is said that a function f : [0,∞) → E is locally Hölder continuous with the exponent

r ∈ (0,1] iff for each p ∈ ⊛ and T > 0 there exists M ≥ 1 such that p(f(t) − f(s)) ≤ M|t − s|r,

provided 0 ≤ t,s ≤ T.

Sometimes we use the following condition on a scalar-valued function K(·):



18 Marko Kostić CUBO
17, 3 (2015)

(P1) K(·) is Laplace transformable, i.e., it is locally integrable on [0,∞) and there exists β ∈ R so

that

K̃(λ) := L(K)(λ) := lim
b→∞

∫b

0

e−λtK(t)dt :=

∫
∞

0

e−λtK(t)dt

exists for all λ ∈ C with Reλ > β. Put abs(K) :=inf{Reλ : K̃(λ) exists}, and denote by L−1

the inverse Laplace transform.

We say that a function h(·) belongs to the class LT − E iff there exists a function f ∈ C([0,∞) : E)

such that for each p ∈ ⊛ there exists Mp > 0 satisfying p(f(t)) ≤ Mpe
at, t ≥ 0 and h(λ) =

(Lf)(λ),λ > a. The reader may consult [4], [51, Chapter 1] and [25, Section 1.2] for the basic

properties of vector-valued Laplace transform.

Given θ ∈ (0,π] in advance, define Σθ := {λ ∈ C : λ 6= 0, | arg(λ)| < θ}. Further on, ⌈β⌉ :=

inf{n ∈ Z : β ≤ n} (β ∈ R). A scalar-valued function k ∈ L1loc[(0,τ)) is said to be a kernel

on [0,τ) iff for any scalar-valued continuous function t 7→ u(t), t ∈ [0,τ), the preassumption
∫t
0
k(t − s)u(s)ds = 0, t ∈ [0,τ) implies u(t) = 0, t ∈ [0,τ). If τ < ∞, then the Titchmarsh–

Foiaş theorem (see e.g. [24, Theorem 3.4.40]) states that the function k(t) is a kernel on [0,τ) iff

0 ∈ supp(k); on the other hand, if τ = ∞ and k 6= 0 in L1loc([0,∞)), then it is well known that

the function k(t) is automatically a kernel on [0,∞). The Gamma function is denoted by Γ(·) and

the principal branch is always used to take the powers; the convolution like mapping ∗ is given by

f ∗ g(t) :=
∫t
0
f(t − s)g(s)ds. Set gζ(t) := t

ζ−1/Γ(ζ), 0ζ := 0 (ζ > 0, t > 0) and g0(t) := the Dirac

δ-distribution. If f : [0,∞) → X is a continuous function, then we set g0 ∗ f ≡ f. The reader may

consult [43, Definition 4.5, p. 96] for the notion of a completely positive function on [0,∞) (cf.

also [37, Remark 3.6, (3.3)]). Denote by Sα,p(Rn) the fractional Sobolev space of order α (cf. [38,

Definition 12.3.1, p. 297]).

Let ζ > 0. Then the Caputo fractional derivative Dζtu ([6], [25]) is defined for those functions

u ∈ C⌈ζ⌉−1([0,∞) : E) for which

g⌈ζ⌉−ζ ∗ (u −
∑⌈ζ⌉−1

j=0 u
(j)(0)gj+1) ∈ C

⌈ζ⌉([0,∞) : E), by

D
ζ
tu(t) :=

d⌈ζ⌉

dt⌈ζ⌉

[
g⌈ζ⌉−ζ ∗

(
u −

⌈ζ⌉−1∑

j=0

u(j)(0)gj+1

)]
.

If the Caputo fractional derivative Dζtu(t) exists, then for each number ν ∈ (0,ζ) the Caputo

fractional derivative Dνt u(t) exists, as well, and the following equality holds:

D
ν
t u(t) =

(
gζ−ν ∗ D

ζ
tu(·)

)
(t) +

⌈ζ⌉−1∑

j=⌈ν⌉

u(j)(0)gj+1−ν(t), t ≥ 0. (1.2)

The Mittag-Leffler function Eβ,γ(z) (β > 0, γ ∈ R) is defined by

Eβ,γ(z) :=

∞∑

k=0

zk

Γ(βk + γ)
, z ∈ C.



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 19

In this place, we assume that 1/Γ(βk + γ) = 0 if βk + γ ∈ −N0. Set, for short, Eβ(z) := Eβ,1(z),

z ∈ C. Let β ∈ (0,1). Then the Wright function Φβ(·) is defined by

Φβ(t) := L
−1
(
Eβ(−λ)

)
(t), t ≥ 0.

For further information about the Mittag-Leffler and Wright functions, cf. [6], [25] and references

cited there.

2 Degenerate k-regularized (C1,C2)-existence and unique-

ness propagation families for (1.1)

We start this section by recalling that n ∈ N \ {1}, 0 ≤ α1 < · · · < αn, 0 ≤ α < αn, as well as

that A, B and A1, · · ·,An−1 are closed linear operators acting on X. Further on, An = B, A0 = A,

m = ⌈α⌉, α0 = α and mi = ⌈αi⌉, i ∈ N
0
n. Set Di := {j ∈ Nn−1 : mj − 1 ≥ i} (i ∈ N

0
mn−1

).

Let T > 0 and f ∈ C([0,T] : E). By a strong solution of problem (1.1) on the interval [0,T]

we mean any continuous function t 7→ u(t), t ∈ [0,T] satisfying that the term AiD
αi
t u(t) is

well-defined and continuous on [0,T] (i ∈ N0n), as well as that (1.1) holds identically on [0,T].

Convoluting both sides of (1.1) with gαn(t), we get that:

B

[
u(·)−

mn−1∑

k=0

ukgk+1
(
·
)
]
+

n−1∑

j=1

gαn−αj ∗ Aj

[
u(·) −

mj−1∑

k=0

ukgk+1
(
·
)
]

= gαn−α ∗ A

[
u(·) −

m−1∑

k=0

ukgk+1
(
·
)
]
+
(
gαn ∗ f

)
(·), t ∈ [0,T]. (2.1)

By a mild solution of (1.1) on [0,T] we mean any continuous X-valued function t 7→ u(t), t ∈ [0,T]

satisfying

B

[
u(·)−

mn−1∑

k=0

ukgk+1
(
·
)
]
+

n−1∑

j=1

Aj

(
gαn−αj ∗

[
u(·) −

mj−1∑

k=0

ukgk+1
(
·
)
])

= A

(
gαn−α ∗

[
u(·) −

m−1∑

k=0

ukgk+1
(
·
)
])

+
(
gαn ∗ f

)
(·), t ∈ [0,T].

Consider the following inhomogeneous integral equation:

Bu(t) +

n−1∑

j=1

(
gαn−αj ∗ Aju

)
(t) = f(t) +

(
gαn−α ∗ Au

)
(t), t ∈ [0,T]. (2.2)

Similarly to the above, we say that a function u ∈ C([0,T] : E) is:

(i) a strong solution of (2.2) iff Aju ∈ C([0,T] : E), j ∈ N
0
n−1 and (2.2) holds for every t ∈ [0,T].



20 Marko Kostić CUBO
17, 3 (2015)

(ii) a mild solution of (2.2) iff (gαn−αj ∗ u)(t) ∈ D(Aj), t ∈ [0,T], j ∈ N
0
n−1 and

Bu(t) +

n−1∑

j=1

Aj
(
gαn−αj ∗ u

)
(t) = f(t) + A

(
gαn−α ∗ u

)
(t), t ∈ [0,T].

A mild (strong) solution of problem (1.1), resp. (2.2), on [0,∞) is defined analogously.

We will be interested in the following notions.

Definition 2.1. (cf. [32, Definition 2.2] for the case B = I) Suppose 0 < τ ≤ ∞, k ∈ C([0,τ)),

C, C1, C2 ∈ L(X), C and C2 are injective.

(i) A sequence ((R0(t))t∈[0,τ), · · ·,(Rmn−1(t))t∈[0,τ)) of strongly continuous operator families in

L(X, [D(B)]) is called a (local, if τ < ∞) k-regularized C1-existence propagation family for

(1.1) iff, for every i = 0, · · ·,mn − 1, the following holds:

B
[
Ri(·)x −

(
k ∗ gi

)
(·)C1x

]

+
∑

j∈Di

Aj

[
gαn−αj ∗

(
Ri(·)x −

(
k ∗ gi

)
(·)C1x

)]

+
∑

j∈Nn−1\Di

Aj
(
gαn−αj ∗ Ri

)
(·)x

=






A
(
gαn−α ∗ Ri

)
(·)x, m − 1 < i, x ∈ X,

A
[
gαn−α ∗

(
Ri(·)x −

(
k ∗ gi

)
(·)C1x

)]
(·), m − 1 ≥ i, x ∈ X.

(ii) A sequence ((R0(t))t∈[0,τ), · · ·,(Rmn−1(t))t∈[0,τ)) of strongly continuous operator families in

L(X) is called a (local, if τ < ∞) k-regularized C2-uniqueness propagation family for (1.1) iff
[
Ri(·)Bx −

(
k ∗ gi

)
(·)C2Bx

]

+
∑

j∈Di

gαn−αj ∗
[
Ri(·)Ajx −

(
k ∗ gi

)
(·)C2Ajx

]

+
∑

j∈Nn−1\Di

(
gαn−αj ∗ Ri(·)Ajx

)
(·)

=






(
gαn−α ∗ Ri(·)Ax

)
(·), m − 1 < i,

gαn−α ∗
[
Ri(·)Ax −

(
k ∗ gi

)
(·)C2Ax

]
(·), m − 1 ≥ i,

for any x ∈
⋂

0≤j≤n D(Aj) and i ∈ N
0
mn−1

.

(iii) A sequence ((R0(t))t∈[0,τ), · · ·,(Rmn−1(t))t∈[0,τ)) of strongly continuous operator families in

L(X) is called a (local, if τ < ∞) k-regularized C-resolvent propagation family for (1.1), in

short k-regularized C-propagation family for (1.1), iff ((R0(t))t∈[0,τ), ···,(Rmn−1(t))t∈[0,τ)) is

a k-regularized C-uniqueness propagation family for (1.1), and for every t ∈ [0,τ), i ∈ N0mn−1
and j ∈ N0n, one has Ri(t)Aj ⊆ AjRi(t), Ri(t)C = CRi(t) and CAj ⊆ AjC.



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 21

In case k(t) = gζ+1(t), where ζ ≥ 0, it is also said that ((R0(t))t∈[0,τ), · · ·,(Rmn−1(t))t∈[0,τ))

is a ζ-times integrated C1-existence propagation family for (1.1); 0-times integrated C1-existence

propagation family for (1.1) is simply called C1-existence propagation family for (1.1). For a k-

regularized C1-existence propagation family ((R0(t))t∈[0,τ), · · ·,(Rmn−1(t))t∈[0,τ)), it is said that

is locally equicontinuous (exponentially equicontinuous) iff each single operator family

(R0(t))t∈[0,τ) ⊆ L(X, [D(B)]), · · ·, (Rmn−1(t))t∈[0,τ) ⊆ L(X, [D(B)]) is;

((R0(t))t≥0, · · ·,(Rmn−1(t))t≥0) is said to be an exponentially equicontinuous k-regularized C1-

existence propagation family for problem (1.1), of angle α ∈ (0,π/2], iff the following holds:

(a) For every x ∈ E and i ∈ N0mn−1, the mappings t 7→ Ri(t)x, t > 0 and t 7→ BRi(t)x, t > 0 can

be analytically extended to the sector Σα; since no confusion seems likely, we shall denote

these extensions by the same symbols.

(b) For every x ∈ E, β ∈ (0,α) and i ∈ N0mn−1, one has limz→0,z∈Σβ Ri(z)x = Ri(0)x and

limz→0,z∈Σβ BRi(z)x = BRi(0)x.

(c) For every β ∈ (0,α) and i ∈ N0mn−1, there exists ωβ ≥ max(0,abs(k)) (ωβ = 0) such that

the family {e−ωβzRi(z) : z ∈ Σβ} ⊆ L(E, [D(B)]) is equicontinuous.

The above terminological agreements and abbreviations can be also understood for the classes of k-

regularized C2-uniqueness propagation families for (1.1) and k-regularized C-resolvent propagation

families for (1.1).

The reader with a little experience can simply state a few noteworthy facts about the existence

and uniqueness of solutions of mild (strong) solutions of problem (2.2) provided that there exists

a k-regularized C1-existence propagation family for problem (1.1) (k-regularized C2-uniqueness

propagation family for problem (1.1)); because of that, the corresponding discussion is omitted.

The proof of following extension of [32, Proposition 2.3] is omitted, too.

Proposition 2.2. Let i ∈ N0mn−1, and let ((R0(t))t∈[0,τ), · · ·,(Rmn−1(t))t∈[0,τ)) be a locally

equicontinuous k-regularized C1-existence propagation family for (1.1). If Ri(t)Aj ⊆ AjRi(t)

(j ∈ N0n, t ∈ [0,τ)), Ri(t)C1 = C1Ri(t) (t ∈ [0,τ)), C1 is injective, k(t) is a kernel on [0,τ)

and C1Aj ⊆ AjC1 (j ∈ N
0
n), then the following holds:

(i) The equality

Ri(t)Ri(s) = Ri(s)Ri(t), 0 ≤ t, s < τ (2.3)

holds, provided that m − 1 < i and that the condition

(⋄) The assumption Bf(t)+
∑

j∈Di
Aj(gαn−αj ∗f)(t) = 0, t ∈ [0,τ) for some f ∈ C([0,τ) : E),

implies f(t) = 0, t ∈ [0,τ),

holds.

(ii) The equality (2.3) holds provided that m − 1 ≥ i, Nn−1 \ Di 6= ∅, and that the condition



22 Marko Kostić CUBO
17, 3 (2015)

(⋄⋄) If
∑

j∈Nn−1\Di
Aj(gαn−αj ∗f)(t) = 0, t ∈ [0,τ), for some f ∈ C([0,τ) : E), then f(t) = 0,

t ∈ [0,τ),

holds.

The assertions of [32, Proposition 2.5, Proposition 2.6] can be reformulated for degenerate

multi-term problems. This is also the case with the assertion of generalized variation of parameters

formula [32, Proposition 2.8]:

Theorem 2.3. Let C2 ∈ L(X) be injective. Suppose that

((R0(t))t∈[0,τ), ···,(Rmn−1(t))t∈[0,τ)) is a locally equicontinuous k-regularized C2-uniqueness prop-

agation family for (1.1), T ∈ (0,τ) and f ∈ C([0,T] : X). Then the following holds:

(i) If m − 1 < i, then any strong solution u(t) of (2.2) satisfies the equality:

(
Ri ∗ f

)
(t) =

(
k ∗ gi ∗ C2Bu

)
(t) +

∑

j∈Di

(
gαn−αj+i ∗ k ∗ C2Aju

)
(t),

for any t ∈ [0,T]. Therefore, there is at most one strong (mild) solution for (2.2), provided

that k(t) is a kernel on [0,τ) and (⋄) holds.

(ii) If m − 1 ≥ i, then any strong solution u(t) of (2.2) satisfies the equality:

(
Ri ∗ f

)
(t) = −

∑

j∈Nn−1\Di

(
gαn−αj+i ∗ k ∗ C2Aju

)
(t), t ∈ [0,T].

Therefore, there is at most one strong (mild) solution for (2.2), provided that k(t) is a kernel

on [0,τ), Nn−1 \ Di 6= ∅ and (⋄⋄) holds.

As explained in [25, Section 2.10], the notion of a k-regularized C-resolvent propagation family

is probably the best theoretical concept for the investigation of integral solutions of non-degenerate

abstract time-fractional equation (1.1) with Aj ∈ L(E), 1 ≤ j ≤ n − 1. If Aj /∈ L(E) for some

j ∈ Nn−1, then the vector-valued Laplace transform cannot be so easily applied, which certainly

implies that there exist some limitations to this class of propagation families. A similar problem

appears in the analysis od degenerate multi-term fractional differential equation (1.1) and, because

of that, we will leave the problem of restating [32, Theorem 2.9(i), Theorem 2.10-Theorem 2.12] in

our new framework to the reader’s own exploration. In contrast to the above, it is very simple to

reformulate the assertion of [32, Theorem 2.9(ii)] to degenerate equations, without imposing any

additional barriers at:

Theorem 2.4. Suppose k(t) satisfies (P1), ω ≥ max(0,abs(k)), (Ri(t))t≥0 is strongly continuous,

and the family {e−ωtRi(t) : t ≥ 0} ⊆ L(X) is equicontinuous (0 ≤ i ≤ mn − 1). Let C2 ∈ L(X) be

injective. Then ((R0(t))t≥0, ···,(Rmn−1(t))t≥0) is a global k-regularized C2-uniqueness propagation

family for (1.1) iff, for every λ ∈ C with Reλ > ω, and for every x ∈
⋂

0≤j≤n D(Aj), the following



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 23

equality holds:

∞∫

0

e−λt
[
Ri(t)Bx −

(
k ∗ gi

)
(t)C2Bx

]
dt

+
∑

j∈Di

λαj−αn

∞∫

0

e−λt
[
Ri(t)x −

(
k ∗ gi

)
(t)C2Ajx

]
dt

+
∑

j∈Nn−1\Di

λαj−αn

∞∫

0

e−λtRi(t)Ajxdt

=






λα−αn
∞∫

0

e−λtRi(t)Axdt, m − 1 < i,

λα−αn
∞∫

0

e−λt
[
Ri(t)Ax −

(
k ∗ gi

)
(t)C2Ax

]
dt, m − 1 ≥ i.

Now we would like to present an intriguing example of a local k-regularized I-resolvent prop-

agation family for (1.1):

Example 2.5. (cf. [32, Example 5.2] for non-degenerate case) Suppose 1 ≤ p ≤ ∞, E := Lp(R),

m : R → C is measurable, aj ∈ L
∞(R), (Ajf)(x) := aj(x)f(x), x ∈ R, f ∈ E (1 ≤ j ≤ n),

(Af)(x) := m(x)f(x), x ∈ R, with maximal domain, and α = 0. Assume s ∈ (1,2), δ = 1/s,

Mp = p!
s and kδ(t) = L

−1(exp(−λδ))(t), t ≥ 0. Denote by M(t) the associated function of

sequence (Mp) (cf. [24, Section 1.3] for more details) and put Λ
′
α′,β′,γ ′ := {λ ∈ C : Reλ ≥

γ′−1M(α′λ) + β′}, α′, β′, γ′ > 0. Clearly, there exists a constant Cs > 0 such that M(λ) ≤

Cs|λ|
1/s, λ ∈ C. Assume that the following condition holds

(CH): For every τ > 0, there exist α′ > 0, β′ > 0 and d > 0 such that τ ≤
cos( δπ

2
)

Cs(α′)1/s
and

∣∣∣∣∣

n∑

j=1

λαj−αaj(x) − m(x)

∣∣∣∣∣ ≥ d, x ∈ R, λ ∈ Λα
′,β′,1.

Notice that the above condition holds provided n = 2, α2 = 2, α1 = 1, c1 ∈ L
∞(R), |c1(x)| ≥

d1 > 0 for a.e. x ∈ R, a2(x) ∈ L
∞(R), a2(x) = 0, x ∈ (−1,1), a1(x) = a2(x)c1(x) and m(x) =

1
4
c21(x)a2(x) −

1
16
c41(x)a2(x) − a2(x), x ∈ R (cf. [32, (5.7)]), and that the validity of condition

(CH) does not imply, in general, the essential boundedness of function m(·) or the injectivity of

the operator B. We will prove that there exists a global (not exponentially bounded, in general)

kδ-regularized I-resolvent propagation family ((R0(t))t≥0, · · ·,(Rmn−1(t))t≥0) for (1.1). Clearly, it

suffices to show that, for every τ > 0, there exists a local kδ-regularized I-resolvent propagation

family for (1.1) on [0,τ). Suppose τ > 0 is given in advance, and α′ > 0, β′ > 0 and d > 0

satisfy (CH), with this τ. Let Γ denote the upwards oriented boundary of ultra-logarithmic region



24 Marko Kostić CUBO
17, 3 (2015)

Λα′,β′,1. Put, for every t ∈ [0,τ), f ∈ E and x ∈ R,

(
Ri(t)f

)
(x) :=

1

2πi

∫

Γ

eλt−λ
δ

[
λαn−α−ian(x) +

∑
j∈Di

λαj−α−iaj(x)
]
f(x)

λαn−αan(x) +
n−1∑

j=1

λαj−αaj(x) − m(x)

dλ.

Then the analysis contained in [32, Example 5.2] shows that ((R0(t))t∈[0,τ), · · ·,(Rmn−1(t))t∈[0,τ))

is a local kδ-regularized I-resolvent propagation family for (1.1), as well as that, for every compact

set K ⊆ [0,∞), there exists hK > 0 such that

sup
t∈K,p∈N0,i∈N

0
mn−1

∥∥∥hpK
dp

dtp
Ri(t)

∥∥∥
p!s

< ∞.

We can similarly consider the existence of local k1/2-regularized I-resolvent propagation families

for (1.1) which obey slight modifications of the properties stated above with s = 2, and with

the operators Aj not belonging to the space L(E) for some indexes j ∈ Nn. Furthermore, we can

similarly construct some relevant examples of local k-regularized I-resolvent propagation families

for (1.1) in certain classes of Fréchet function spaces.

3 Degenerate k-regularized (C1,C2)-existence and unique-

ness families for (1.1)

In this section, we investigate the class of degenerate k-regularized (C1,C2)-existence and unique-

ness families for (1.1). Recall that Di = {j ∈ Nn−1 : mj − 1 ≥ i} (i ∈ N
0
mn−1

), as well as that A,

B and A1, · · ·,An−1 are closed linear operators acting on X. By Y we denote another SCLCS over

the same field of scalars as X.

In the following definition, we will generalize the notion introduced in our previous joint

research with C.-G. Li and M. Li (cf. [32, Definition 3.1], [31], R. deLaubenfels [10]-[11], and T.-J.

Xiao-J. Liang [54] for some other known concepts in the case that B = I).

Definition 3.1. Suppose 0 < τ ≤ ∞, k ∈ C([0,τ)), C1 ∈ L(Y,X), and C2 ∈ L(X) is injective.

(i) A strongly continuous operator family (E(t))t∈[0,τ) ⊆ L(Y,X) is said to be a (local, if τ < ∞)

k-regularized C1-existence family for (1.1) iff, for every y ∈ Y, the following holds: E(·)y ∈

Cmn−1([0,τ) : [D(B)]), E(i)(0)y = 0 for every i ∈ N0 with i < mn − 1, Aj(gαn−αj ∗

E(mn−1))(·)y ∈ C([0,τ) : X) for 0 ≤ j ≤ n, and

BE(mn−1)(t)y +

n−1∑

j=1

Aj
(
gαn−αj ∗ E

(mn−1)
)
(t)y

− A
(
gαn−α ∗ E

(mn−1)
)
(t)y = k(t)C1y, (3.1)

for any t ∈ [0,τ).



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 25

(ii) A strongly continuous operator family (U(t))t∈[0,τ) ⊆ L(X) is said to be a (local, if τ < ∞)

k-regularized C2-uniqueness family for (1.1) iff, for every τ ∈ [0,τ) and x ∈
⋂

0≤j≤n D(Aj),

the following holds:

U(t)Bx +

n−1∑

j=1

(
gαn−αj ∗ U(·)Ajx

)
(t)

−
(
gαn−α ∗ U(·)Ax

)
(t)y =

(
k ∗ gmn−1

)
(t)C2x. (3.2)

(iii) A strongly continuous family ((E(t))t∈[0,τ),(U(t))t∈[0,τ)) ⊆ L(Y,X)×L(X) is said to be a (lo-

cal, if τ < ∞) k-regularized (C1,C2)-existence and uniqueness family for (1.1) iff (E(t))t∈[0,τ)
is a k-regularized C1-existence family for (1.1), and (U(t))t∈[0,τ) is a k-regularized C2-

uniqueness family for (1.1).

(iv) Suppose Y = X and C = C1 = C2. Then a strongly continuous operator family (R(t))t∈[0,τ) ⊆

L(X) is said to be a (local, if τ < ∞) k-regularized C-resolvent family for (1.1) iff (R(t))t∈[0,τ)
is a k-regularized C-uniqueness family for (1.1), R(t)Aj ⊆ AjR(t), for 0 ≤ j ≤ n and t ∈ [0,τ),

as well as R(t)C = CR(t), t ∈ [0,τ), and CAj ⊆ AjC, for 0 ≤ j ≤ n.

If k(t) = gζ+1(t), where ζ ≥ 0, then it is also said that (E(t))t∈[0,τ) is a ζ-times integrated

C1-existence family for (1.1); 0-times integrated C1-existence family for (1.1) is also said to be a

C1-existence family for (1.1). A similar notion can be introduced for all other classes of uniqueness

and resolvent families introduced in Definition 3.1.

Albeit the choice of an SCLCS space Y different from X can produce a larger set of initial data

for which the abstract Cauchy problem (1.1) has a strong solution (see e.g. [54, Example 2.5]), in

our furher work the most important case will be that in which Y = X. Keeping in mind that the

operators A, B, A1, · · ·,An−1 are closed, we can integrate the both sides of (3.1) sufficiently many

times in order to see that:

BE(l)(t)y +

n−1∑

j=1

Aj
(
gαn−αj ∗ E

(l)
)
(t)y

− A
(
gαn−α ∗ E

(l)
)
(t)y =

(
k ∗ gmn−1−l

)
(t)C1y, (3.3)

for any t ∈ [0,τ), y ∈ Y and l ∈ N0mn−1.

Proposition 3.2. Suppose that ((E(t))t∈[0,τ),(U(t))t∈[0,τ)) is a k-regularized

(C1,C2)-existence and uniqueness family for (1.1), and let (U(t))t∈[0,τ) be locally equicontinuous.

Then C2E(t)y = U(t)C1y, t ∈ [0,τ), y ∈ Y.

Proof. The proof of proposition is almost the same as the corresponding proof of [32, Proposition

3.2]. Observe only that we can always assume, without loss of generality, that the number α is less

than or equal to α1.



26 Marko Kostić CUBO
17, 3 (2015)

Definition 3.3. (cf. [32, Definition 3.3]) Suppose 0 ≤ i ≤ mn − 1. Then we define D
′
i := {j ∈

N
0
n−1 : mj − 1 ≥ i}, D

′′
i := N

0
n−1 \ D

′
i and

Di :=

{

ui ∈
⋂

j∈D′′
i

D(Aj) : Ajui ∈ R(C1), j ∈ D
′′
i

}

.

It is not so predictable that the assertion of [32, Theorem 3.4] continues to hold in degenerate

case without any terminological changes, and that the operator B does not appear in the definition

of set Di, for which it is well known that represents, in non-degenerate case, the set which consists

of all initial values for which the homogeneous counterpart of abstract Cauchy problem (1.1), with

B = I and uj = 0, j ∈ N
0
mn−1

\ {i}, has a strong solution (provided that there exists a C1-existence

family for (1.1)). It is also worth nothing that we do not use the injectiveness of the operator B in

(ii):

Theorem 3.4. (i) Suppose (E(t))t∈[0,τ) is a C1-existence family for (1.1), T ∈ (0,τ), and ui ∈

Di for 0 ≤ i ≤ mn − 1. Then the function

u(t) =

mn−1∑

i=0

uigi+1(t) −

mn−1∑

i=0

∑

j∈Nn−1\Di

(
gαn−αj ∗ E

(mn−1−i)
)
(t)vi,j

+

mn−1∑

i=m

(
gαn−α ∗ E

(mn−1−i)
)
(t)vi,0, 0 ≤ t ≤ T,

is a strong solution of the problem (1.1) on [0,T], with f(t) ≡ 0, where vi,j ∈ Y satisfy

Ajui = C1vi,j for 0 ≤ j ≤ n − 1.

(ii) Suppose (U(t))t∈[0,τ) is a locally equicontinuous k-regularized C2-uniqueness family for (1.1),

T ∈ (0,τ) and 0 ∈ supp(k). Then there exists at most one strong (mild) solution of (1.1) on

[0,T], with ui = 0, i ∈ N
0
mn−1

.

Proof. We will provide all the relevant details for the sake of completeness. Making use of (3.3),



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 27

it can be easily verified that:

B

[
u(·) −

mn−1
∑

i=0

uigi+1
(
·
)
]
+

n−1
∑

j=1

Aj

(
gαn−αj ∗

[
u(·) −

mj−1
∑

i=0

uigi+1
(
·
)
])

= −

mn−1
∑

i=0

∑

j∈Nn−1\Di

(
gαn−αj ∗ BE

(mn−1−i)
)
(·)vi,j

+

mn−1
∑

i=m

(
gαn−α ∗ BE

(mn−1−i)
)
(·)vi,0

+

n−1
∑

j=1

Aj

(
gαn−αj ∗

{

mn−1
∑

i=mj

gi+1
(
·
)
ui −

mn−1
∑

i=0

∑

l∈Nn−1\Di

(
gαn−αl ∗ E

(mn−1−i)
)
(·)vi,l

+

mn−1
∑

i=m

(
gαn−α ∗ E

(mn−1−i)
)
(·)vi,0

})

= −

mn−1
∑

i=0

∑

j∈Nn−1\Di

(
gαn−αj ∗ BE

(mn−1−i)
)
(·)vi,j

+

mn−1
∑

i=m

(
gαn−α ∗ BE

(mn−1−i)
)
(·)vi,0

+

n−1
∑

j=1

mn−1
∑

i=mj

C1vi,jgαn−αj+i+1
(
·
)
−

mn−1
∑

i=0

∑

l∈Nn−1\Di

gαn−αl ∗
[
−BE

(mn−1−i)
(
·
)
vi,l

+ A
(
gαn−α ∗ E

(mn−1−i)
)
(·)vi,l + gi+1

(
·
)
C1vi,l

]

+

mn−1
∑

i=m

gαn−α ∗
[
−BE

(mn−1−i)
(
·
)
vi,0

+ A
(
gαn−α ∗ R

(mn−1−i)
)
(·)vi,0 + gi+1

(
·
)
C1vi,0

]

= gαn−α ∗ A

[
u(·) −

m−1
∑

i=0

uigi+1
(
·
)
]
,

since
n−1
∑

j=1

mn−1
∑

i=mj

C1vi,jgαn−αj+i+1(·) =

mn−1
∑

i=0

∑

j∈Nn−1\Di

C1vi,jgαn−αj+i+1(·).

This implies that u(t) is a mild solution of (1.1) on [0,T]. In order to complete the proof of (i), it
suffices to show that Dαnt u(t) ∈ C([0,T] : X) and AiD

αi
t u ∈ C([0,T] : X) for all i ∈ N

0
n. Towards

this end, notice that the partial integration implies that, for every t ∈ [0,T],

gmn−αn ∗

[
u(·) −

mn−1
∑

i=0

uigi+1(·)

]
(t)

=

mn−1
∑

i=m

(
gmn−α+i ∗ E

(mn−1)
)
(t)vi,0 −

mn−1
∑

i=0

∑

j∈Nn−1\Di

(
gmn−αj+i ∗ E

(mn−1)
)
(t)vi,j.



28 Marko Kostić CUBO
17, 3 (2015)

Therefore, Dαnt u ∈ C([0,T] : X) and, for every t ∈ [0,T],

D
αn
t u(t) =

dmn

dtmn

{

gmn−αn ∗

[
u(·) −

mn−1
∑

i=0

uigi+1(·)

]
(t)

}

=

mn−1
∑

i=m

(
gi−α ∗ E

(mn−1)
)
(t)vi,0 −

mn−1
∑

i=0

∑

j∈Nn−1\Di

(
gi−αj ∗ E

(mn−1)
)
(t)vi,j, (3.4)

whence we may directly conclude that BDαnt u ∈ C([0,T] : X). Suppose, for the time being,
i ∈ N0n−1. Then Aiuj ∈ R(C1) for j ≥ mi. Moreover, the inequality l ≥ αj holds provided

0 ≤ l ≤ mn −1 and j ∈ Nn−1 \Dl, and Aj(gαn−αj ∗E
(mn−1))(·)y ∈ C([0,T] : X) for 0 ≤ j ≤ n−1

and y ∈ Y. Using (1.2) and (3.4), it is not difficult to prove that:

AiD
αi
t u(·)

=

mn−1
∑

j=mi

gj+1−αi(·)Aiuj −

mn−1
∑

l=0

∑

j∈Nn−1\Dl

[
gl−αj ∗ Ai

(
gαn−αi ∗ E

(mn−1)
)]
(·)vl,j

+

mn−1
∑

l=m

[
gl−α ∗ Ai

(
gαn−αi ∗ E

(mn−1)
)]
(·)vl,0 ∈ C([0,T] : X),

finishing the proof of (i). The second part of theorem can be proved as follows. Suppose u(t) is
a strong solution of (1.1) on [0,T], with ui = 0, i ∈ N

0
mn−1

. Making use of (3.2) and the equality

t
∫

0

t−s
∫

0

gαn−αj(r)U(t − s − r)Aju(s)drds =

t
∫

0

s
∫

0

gαn−αj(r)U(t − s)Aju(s − r)drds,

holding for any t ∈ [0,T] and j ∈ N0n−1, we have that
(
UB ∗ u

)
(t) =

(
k ∗ gmn−1C2 ∗ u

)
(t)

+

t∫

0

t−s∫

0

[
gαn−αj(r)U(t − s − r)Aju(s) − gαn−α(r)U(t − s − r)Au(s)

]
drds

=
(
k ∗ gmn−1C2 ∗ u

)
(t) +

(
U ∗ Bu

)
(t), t ∈ [0,T].

Therefore, (k ∗ gmn−1C2 ∗ u)(t) = 0, t ∈ [0,T] and u(t) = 0, t ∈ [0,T].

The standard proof of following theorem is omitted.

Theorem 3.5. Suppose k(t) satisfies (P1), (E(t))t≥0 ⊆ L(Y,X), (U(t))t≥0 ⊆ L(X), ω ≥ max(0,

abs(k)), C1 ∈ L(Y,X) and C2 ∈ L(X) is injective. Set Pλ := B +
∑n−1

j=1 λ
αj−αnAj − λ

α−αnA,

Reλ > 0.

(i) (a) Let (E(t))t≥0 be a k-regularized C1-existence family for (1.1), let the family {e
−ωtE(t) :

t ≥ 0} be equicontinuous, and let the family

{e−ωtAj(gαn−αj ∗ E)(t) : t ≥ 0} be equicontinuous (0 ≤ j ≤ n). Then the following

holds:

Pλ

∞∫

0

e−λtE(t)ydt = k̃(λ)λ1−mnC1y, y ∈ Y, Reλ > ω.



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 29

(b) Let the operator Pλ be injective for every λ > ω with k̃(λ) 6= 0. Suppose, addition-

ally, that there exist strongly continuous operator families (W(t))t≥0 ⊆ L(Y,X) and

(Wj(t))t≥0 ⊆ L(Y,X) such that {e
−ωtW(t) : t ≥ 0} and {e−ωtWj(t) : t ≥ 0} are equicon-

tinuous (0 ≤ j ≤ n) as well as that:

∞∫

0

e−λtW(t)ydt = k̃(λ)P−1λ C1y

and
∞∫

0

e−λtWj(t)ydt = k̃(λ)λ
αj−αnAjP

−1
λ C1y,

for every λ > ω with k̃(λ) 6= 0, y ∈ Y and j ∈ N0n. Then there exists a k-regularized C1-

existence family for (1.1), denoted by (E(t))t≥0. Furthermore, E
(mn−1)(t)y = W(t)y,

t ≥ 0, y ∈ Y and Aj(gαn−αj ∗ E
(mn−1))(t)y = Wj(t)y, t ≥ 0, y ∈ Y, j ∈ N

0
n−1.

(ii) Suppose (U(t))t≥0 is strongly continuous and the operator family

{e−ωtU(t) : t ≥ 0} is equicontinuous. Then (U(t))t≥0 is a k-regularized C2-uniqueness family

for (1.1) iff, for every x ∈
⋂n

j=0 D(Aj), the following holds:

∞∫

0

e−λtU(t)Pλxdt = k̃(λ)λ
1−mnC2x, Reλ > ω.

The assertion of [32, Theorem 3.7], concerning the inhomogeneous Cauchy problem (1.1), can

be stated for degenerate equations without any terminological changes, as well:

Theorem 3.6. Suppose (E(t))t∈[0,τ) is a locally equicontinuous C1-existence family for (1.1),

T ∈ (0,τ), and ui ∈ Di for 0 ≤ i ≤ mn − 1. Let f ∈ C([0,T] : X), let g ∈ C([0,T] : Y) satisfy

C1g(t) = f(t), t ∈ [0,T], and let G ∈ C([0,T] : Y) satisfy (gαn−mn+1 ∗g)(t) = (g1 ∗G)(t), t ∈ [0,T].

Then the function

u(t) =

mn−1∑

i=0

uigi+1(t) −

mn−1∑

i=0

∑

j∈Nn−1\Di

(
gαn−αj ∗ E

(mn−1−i)
)
(t)vi,j

+

mn−1∑

i=m

(
gαn−α ∗ E

(mn−1−i)
)
(t)vi,0 +

t∫

0

E(t − s)G(s)ds, 0 ≤ t ≤ T, (3.5)

is a mild solution of the problem (2.1) on [0,T], where vi,j ∈ Y satisfy Ajui = C1vi,j for 0 ≤ j ≤

n − 1. If, additionally, g ∈ C1([0,T] : Y) and

(E(mn−1)(t))t∈[0,τ) ⊆ L(Y,X) is locally equicontinuous, then the solution u(t), given by (3.5), is a

strong solution of (1.1) on [0,T].



30 Marko Kostić CUBO
17, 3 (2015)

Contrary to the assertion of [32, Theorem 3.7], the final conclusions of [32, Remark 3.8] cannot

be proved for degenerate equations without imposing some additional conditions. Details can be

left to the interested reader.

Concerning the action of subordination principles, we can state the following analogue of [32,

Theorem 4.1] for degenerate multi-term problems (the final conclusions of [32, Remark 4.2] can be

restated in our new setting, as well).

Theorem 3.7. Suppose C1 ∈ L(Y,X), C2 ∈ L(X) is injective and γ ∈ (0,1).

(i) Let ω ≥ max(0,abs(k)), and let the assumptions of Theorem 3.5(i)-(b) hold. Put

Wγ(t) :=

∞∫

0

t−γΦγ
(
t−γs

)
W(s)yds, t > 0, y ∈ Y and Wγ(0) := W(0). (3.6)

Define, for every j ∈ N0n and t ≥ 0, Wj,γ(t) by replacing W(t) in (3.6) with Wj(t). Suppose

that there exist a number ν > 0 and a continuous kernel kγ(t) on [0,∞) satisfying (P1) and

k̃γ(λ) = λ
γ−1k̃(λγ), λ > ν. Then there exists an exponentially equicontinuous kγ-regularized

C1-existence family (Eγ(t))t≥0 for (1.1), with αj replaced by αjγ therein (0 ≤ j ≤ n).

Furthermore, the family {(1 + t⌈αnγ⌉−2)−1e−ω
1/γtEγ(t) : t ≥ 0} is equicontinuous.

(ii) Suppose (U(t))t≥0 is a k-regularized C2-uniqueness family for (1.1), and the family {e
−ωtU(t) :

t ≥ 0} is equicontinuous. Define, for every t ≥ 0, Uγ(t) by replacing W(t) in (3.6) with U(t).

Suppose that there exist a number ν > 0 and a continuous kernel kγ(t) on [0,∞) satisfying

(P1) and k̃γ(λ) = λ
γ(2−mn)−2+⌈αnγ⌉k̃(λγ), λ > ν. Then there exists a kγ-regularized C2-

uniqueness family for (1.1), with αj replaced by αjγ therein (0 ≤ j ≤ n). Furthermore, the

family {e−ω
1/γtUγ(t) : t ≥ 0} is equicontinuous.

Of importance is the following abstract degenerate Volterra equation:

Bu(t) = f(t) +

n−1∑

j=0

(
aj ∗ Aju

)
(t), t ∈ [0,τ), (3.7)

where 0 < τ ≤ ∞, f ∈ C([0,τ) : X), a0, · · ·,an−1 ∈ L
1
loc([0,τ)), and A = A0, · · ·,An−1,B are

closed linear operators on X. We define the notion of a mild (strong) solution of problem (3.7) in

the same way as it has been done before for the problem (2.2).

The following definition plays a crucial role in our investigation of problem (3.7).

Definition 3.8. (cf. [32, Definition 4.3] for the case B = I) Suppose 0 < τ ≤ ∞, k ∈ C([0,τ)),

C1 ∈ L(Y,X), and C2 ∈ L(X) is injective.

(i) A strongly continuous operator family (E(t))t∈[0,τ) ⊆ L(Y, [D(B)]) is said to be a (local, if

τ < ∞) k-regularized C1-existence family for (3.7) iff

BE(t)y = k(t)C1y +

n−1∑

j=0

Aj
(
aj ∗ E

)
(t)y, t ∈ [0,τ), y ∈ Y.



CUBO
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Degenerate k-regularized (C1,C2)-existence . . . 31

(ii) A strongly continuous operator family (U(t))t∈[0,τ) ⊆ L(X) is said to be a (local, if τ < ∞)

k-regularized C2-uniqueness family for (3.7) iff

U(t)Bx = k(t)C2x +

n−1∑

j=0

(
aj ∗ AjU

)
(t)x, t ∈ [0,τ), x ∈

n⋂

j=0

D(Aj).

As in non-degenerate case, we have the following:

(i) Suppose (E(t))t∈[0,τ) is a k-regularized C1-existence family for (3.7). Then, for every y ∈ Y,

the function u(t) = E(t)y, t ∈ [0,τ) is a mild solution of (3.7) with f(t) = k(t)C1y, t ∈ [0,τ).

(ii) Let (U(t))t∈[0,τ) be a locally equicontinuous k-regularized C2-uniqueness family for (3.7).

Then there exists at most one mild (strong) solution of (3.7).

The most important structural properties of k-regularized C1-existence families for (3.7) and

k-regularized C2-uniqueness families for (3.7) are stated in the following analogue of Theorem 3.5.

Theorem 3.9. Suppose that k(t) and a0(t), · · ·,an−1(t) satisfy (P1), (E(t))t≥0 ⊆ L(Y,X),

(U(t))t≥0 ⊆ L(X), ω ≥ max(0,abs(k),abs(a0), · · ·,abs(an−1)), C1 ∈ L(Y,X) and C2 ∈ L(X) is

injective. Set Pλ := B −
∑n−1

j=0 ãj(λ)Aj, Reλ > ω.

(i) (a) Let (E(t))t≥0 be a k-regularized C1-existence family for (3.7), let the family {e
−ωtE(t) :

t ≥ 0} ⊆ L(Y, [D(B)]) be equicontinuous, and let the family {e−ωtAj(aj ∗E)(t) : t ≥ 0} ⊆

L(Y,X) be equicontinuous (0 ≤ j ≤ n − 1). Then the following holds:

Pλ

∞∫

0

e−λtE(t)ydt = k̃(λ)C1y, y ∈ Y, Reλ > ω.

(b) Let the operator Pλ be injective for every λ > ω with k̃(λ) 6= 0. Suppose, additionally,
that there exist strongly continuous operator families (E(t))t≥0 ⊆ L(Y,X), (EB(t))t≥0 ⊆
L(Y,X), and (Ej(t))t≥0 ⊆ L(Y,X) such that the operator families {e

−ωtE(t) : t ≥ 0},
{e−ωtEB(t) : t ≥ 0}, and {e

−ωtEj(t) : t ≥ 0} are equicontinuous (0 ≤ j ≤ n − 1) as well
as that:

∞
∫

0

e
−λt
E(t)ydt = k̃(λ)P

−1
λ C1y,

∞
∫

0

e
−λt
EB(t)ydt = k̃(λ)BP

−1
λ C1y

and
∞
∫

0

e
−λt
Ej(t)ydt = k̃(λ)ãj(λ)AjP

−1
λ C1y,

for every λ > ω with k̃(λ) 6= 0, y ∈ Y and j ∈ N0n−1. Then (E(t))t≥0 is a k-regularized

C1-existence family for (3.7). Furthermore, BE(t)y = EB(t)y, t ≥ 0, y ∈ Y and Aj(aj ∗

E)(t)y = Ej(t)y, t ≥ 0, y ∈ Y, j ∈ N
0
n−1.



32 Marko Kostić CUBO
17, 3 (2015)

(ii) Suppose (U(t))t≥0 is strongly continuous and the operator family

{e−ωtU(t) : t ≥ 0} ⊆ L(X) is equicontinuous. Then (U(t))t≥0 is a k-regularized C2-

uniqueness family for (3.7) iff, for every x ∈
⋂n

j=0 D(Aj), the following holds:

∞∫

0

e−λtU(t)Pλxdt = k̃(λ)C2x, Reλ > ω.

Theorem 3.10. (Subordination principle)

(i) Suppose that the requirements of Theorem 3.9(i)-(b) hold. Let c(t) be completely positive, let

c(t), k(t), k1(t), a0(t), · · ·,an−1(t) and b0(t), · · ·,bn−1(t) satisfy (P1), and let ω0 > 0 be

such that, for every λ > ω0 with c̃(λ) 6= 0 and k̃(1/c̃(λ)) 6= 0, the following holds:

ãj(1/c̃(λ)) = b̃j(λ), j ∈ N
0
n−1 and k̃1(λ) =

1

λc̃(λ)
k̃(1/c̃(λ)). (3.8)

Then for each r ∈ (0,1] there exists a locally Hölder continuous (with exponent r), exponen-

tially equicontinuous (k1 ∗ gr)-regularized C1-existence family for

Bu(t) = f(t) +

n−1∑

j=0

(
bj ∗ Aju

)
(t), t ∈ [0,τ). (3.9)

(ii) Suppose that the requirements of Theorem 3.9(ii) hold. Let c(t) be completely positive, let c(t),

k(t), k1(t) a0(t), ···,an−1(t) and b0(t), ···,bn−1(t) satisfy (P1), and let ω0 > 0 be such that,

for every λ > ω0 with c̃(λ) 6= 0 and k̃(1/c̃(λ)) 6= 0, (3.8) holds. Then for each r ∈ (0,1] there

exists a locally Hölder continuous (with exponent r), exponentially equicontinuous (k1 ∗ gr)-

regularized C2-uniqueness family for (3.9).

The interested reader may try to transfer the final conclusions of [36, Theorem 2.1, Theorem

2.2, Remark 2.1, Proposition 2.1] to degenerate multi-term fractional differential equations. In

order to do the same with the perturbation result [36, Theorem 2.3], we need to introduce the

following notion.

Definition 3.11. A strongly continuous operator family (U(t))t∈[0,τ) ⊆ L(X) is said to be a (local,

if τ < ∞) (k,C2)-uniqueness family for (1.1) iff, for every t ∈ [0,τ) and x ∈
⋂

0≤j≤n D(Aj), the

following holds:

U(t)Bx +

n−1∑

j=1

(
gαn−αj ∗ U(·)Ajx

)
(t) −

(
gαn−α ∗ U(·)Ax

)
(t)x = k(t)C2x.

Then it is clear that for any strongly continuous operator family (U(t))t∈[0,τ) the following

equivalence relation holds: (U(t))t∈[0,τ) is a (local) (k ∗ gmn−1,C2)-uniqueness family for (1.1) iff

(U(t))t∈[0,τ) is a (local) k-regularized C2-uniqueness family for (1.1).



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 33

Consider now the perturbed equation:

BD
αn
t u(t) +

n−1∑

i=1

(Ai + Fi)D
αi
t u(t) = (A + F)D

α
t u(t) + f(t), t ≥ 0;

u(j)(0) = uj, j = 0, · · ·,⌈αn⌉ − 1,

(3.10)

where Fi ∈ L(X) for 0 ≤ i ≤ n − 1 and F0 ≡ F. A similar line of reasoning as in the proof of [36,

Theorem 2.3] shows that the following result about the C-wellposedness of problem (3.10) holds

good (observe that the employed method is based on the arguments contained in the proof of [43,

Theorem 6.1], which does not work any longer if we replace the term BDαnt u(t), in (3.10), with

(B + Fn)D
αn
t u(t)):

Theorem 3.12. (i) Suppose Y = X, (E(t))t∈[0,τ) ⊆ L(X) is a (local) C1-existence family for

(1.1), Ej ∈ L(X) and Fj = C1Ej (j ∈ N
0
n−1). Suppose that the following conditions hold:

(a) For every p ∈ ⊛X and for every T ∈ (0,τ), there exists cp,T > 0 such that

p
(
E(mn−1)(t)x

)
≤ cp,Tp(x), x ∈ X, t ∈ [0,T].

(b) For every p ∈ ⊛X, there exists cp > 0 such that

p
(
Ejx
)
≤ cpp(x), j ∈ N

0
n−1, x ∈ X.

(c) αn − αn−1 ≥ 1 and αn − α ≥ 1.

Then there exists a (local) C1-existence propagation family (R(t))t∈[0,τ) for (3.10). If τ = ∞

and if, for every p ∈ ⊛X, there exist M ≥ 1 and ω ≥ 0 such that

p
(
E(mn−1)(t)x

)
≤ Meωtp(x), t ≥ 0, x ∈ X, (3.11)

respectively (3.11) and

p
(
BE(mn−1)(t)x

)
≤ Meωtp(x), t ≥ 0, x ∈ X, (3.12)

then (R(t))t≥0 is exponentially equicontinuous, and moreover, (R(t))t≥0 also satisfies the

condition (3.11), repectively (3.11) and (3.12), with possibly different numbers M ≥ 1 and

ω > 0.

(ii) Suppose Y = X, (U(t))t∈[0,τ) ⊆ L(X) is a (local) (1,C2)-uniqueness family for (1.1), Ej ∈

L(E) and Fj = EjC2 (j ∈ N
0
n−1). Suppose that (b)-(c) hold, and that (a) holds with

(E(mn−1)(t))t∈[0,τ) replaced by (U(t))t∈[0,τ) therein. Then there exists a (local) (1,C2)-

uniqueness family (W(t))t∈[0,τ) for (3.10). If τ = ∞ and if, for every p ∈ ⊛X, there exist

M ≥ 1 and ω ≥ 0 such that (3.11) holds, then (W(t))t≥0 is exponentially equicontinuous,

and moreover, (W(t))t≥0 also satisfies the condition (3.11), with possibly different numbers

M ≥ 1 and ω > 0.



34 Marko Kostić CUBO
17, 3 (2015)

For some other results concerning perturbation properties of abstract degenerate differential

equations, one may refer e.g. to [18], [21] and [50]. Concerning the existence of strong solutions of

(1.1), we can prove the following slight extension of [36, Theorem 3.1]; this result can be viewed

of some independent interest.

Theorem 3.13. Suppose A, B, A1, · · ·, An−1 are closed linear operators on X, ω > 0, L(X) ∋ C

is injective and u0, · · ·,umn−1 ∈ X. Set Pλ := λ
αn−αB +

∑n−1
j=1 λ

αj−αAj − A, λ ∈ C \ {0}. Let the

following conditions hold:

(i) The operator Pλ is injective for λ > ω and D(P
−1
λ C) = X, λ > ω.

(ii) If 0 ≤ j ≤ n, 0 ≤ k ≤ mn − 1, m − 1 < k, 1 ≤ l ≤ n, ml − 1 ≥ k and λ > ω, then

Cuk ∈ D(P
−1
λ Al),

Aj

{

λαj

[
λαn−α−k−1P−1λ BCuk +

∑

l∈Dk

λαl−α−k−1P−1λ AlCuk

]

−

mj−1∑

l=0

δklλ
αj−1−lCuk

}

∈ LT − X (3.13)

and

λαn

[
λαn−α−k−1P−1λ BCuk +

∑

l∈Dk

λαl−α−k−1P−1λ AlCuk

]

− λαn−1−kCuk ∈ LT − X. (3.14)

(iii) If 0 ≤ j ≤ n, 0 ≤ k ≤ mn − 1, m − 1 ≥ k, Nn−1 \ Dk 6= ∅, s ∈ Nn−1 \ Dk and λ > ω, then

Cuk ∈ D(As),
∑

l∈Nn−1\Dk
λαl−α−k−1AlCuk ∈ D(P

−1
λ ),

Aj

{

λαj

[
λ−k−1Cuk−P

−1
λ

∑

l∈Nn−1\Dk

λαl−α−k−1AlCuk

]

−

mj−1∑

l=0

δklλ
αj−1−lCuk

}

∈ LT − X (3.15)

and

λαn

[
λ−k−1Cuk−P

−1
λ

∑

l∈Nn−1\Dk

λαl−α−k−1AlCuk

]

− λαn−1−kCuk ∈ LT − X. (3.16)

Then the abstract Cauchy problem (1.1) has a strong solution, with uk replaced by Cuk (0 ≤ k ≤

mn − 1).



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 35

Remark 3.14. Let 0 ≤ k ≤ mn − 1 and m − 1 < k. Then Theorem 3.13 continues to hold if we

replace the term

λαn−α−k−1P−1λ BCuk +
∑

l∈Dk

λαl−α−k−1P−1λ AlCuk

i.e., the Laplace transform of uk(t), in (3.13)-(3.14) by

λ−k−1Cuk −
∑

l∈Nn−1\Dk

λαl−α−k−1P−1λ AlCuk + λ
−k−1P−1λ ACuk;

in this case, it is necessary to assume that Cuk ∈ D(P
−1
λ Al), provided 0 ≤ l ≤ n − 1, k > ml − 1

and λ > ω. Let us also observe that a similar modification can be made in the case 0 ≤ k ≤ mn −1

and m − 1 ≥ k. Strictly speaking, one can replace the term

λ−k−1Cuk − P
−1
λ

∑

l∈Nn−1\Dk

λαl−α−k−1AlCuk

i.e., the Laplace transform of uk(t), in (3.15)-(3.16) by

λαn−α−k−1P−1λ BCuk +
∑

l∈Dk

λαl−α−k−1P−1λ AlCuk − λ
−k−1P−1λ ACuk;

in this case, one has to assume that Cuk ∈ D(P
−1
λ Al), provided 0 ≤ l ≤ n, ml −1 ≥ k and λ > ω.

Now we would like to illustrate the obtained results with some examples.

Example 3.15. Suppose 1 ≤ p < ∞, ∅ 6= Ω ⊆ Rn is an open bounded domain with smooth

boundary, and X := Lp(Ω). Consider the equation

(α − ∆)utt = β∆ut + ∆u +

∫t

0

g(t − s)∆u(s,x)ds = 0, t > 0, x ∈ Ω;

u(0,x) = φ(x), ut(0,x) = ψ(x),

(3.17)

where g ∈ L1loc([0,∞)) satisfies (P1), α > 0 and β ∈ R \ {0}. As explained by M. V. Falaleev and

S. S. Orlov in [16], the equation (3.17) appears in the models of nonlinear viscoelasticity provided

n = 3. Integrating (3.17) 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

)
+ ∆

(
g2 ∗ g ∗ u

)
(t), (3.18)

which is of the form (3.7) with B := α − ∆, A2 := β∆, A1 = A0 := ∆ (acting with the Dirichlet

boundary conditions) and a2(t) := g1(t), a1(t) := g2(t), a0(t) := (g2 ∗ g)(t). Then

Pλ =
λ2 + βλ + g̃(λ) + 1

λ2

[
αλ2

λ2 + βλ + g̃(λ) + 1
− ∆

]
.



36 Marko Kostić CUBO
17, 3 (2015)

We assume that g(t) is of the following form:

g(t) =

l∑

j=0

cjgβj (t) + f(t), t > 0,

where l ∈ N, cj ∈ C (0 ≤ j ≤ l), 0 < β1 < · · · < βl < 1 and the function f(t) satisfies the
requirements of [26, Theorem 3.4(i)-(a)] with α = π/2 and ω > 0 sufficiently large. Using the
resolvent equation and the fact that the operator ∆ generates a bounded analytic C0-semigroup of
angle π/2, it can be simply verified that

1

λ
P

−1
λ ∈ LT − L(X),

1

λ
BP

−1
λ ∈ LT − L(X) and

ãj(λ)

λ
P

−1
λ ∈ LT − L(X), j = 0,1,2.

This implies by Theorem 3.9 that there exists an exponentially bounded I-existence family

(E(t))t≥0 for (3.18), satisfying additionally that for each f ∈ X the mappings t 7→ E(t)f, t > 0,

t 7→ BE(t)f, t > 0 and t 7→ Aj(aj ∗ E)(t)f, t > 0 can be analytically extended to the sector Σπ/2;

furthermore, (E(t))t≥0 is an exponentailly bounded I-uniqueness family for (3.18). This implies

that for each φ, ψ ∈ W2,p(Ω) ∩ W
1,p
0 (Ω), there exists a unique strong solution

u(t) = E(t)(α + (β − 1)∆)φ(x) +

∫t

0

E(s)(α − ∆)ψds

of the integral equation (3.18), and that u(t) can be analytically extended to the sector Σπ/2.

Example 3.16. Suppose 1 < p < ∞, X := Lp(Rn), 1/2 < γ ≤ 1, Q ∈ N\{1}, P1(x) =
∑

|η|≤Q aηx
η,

P2(x) = −1 − |x|
2 (x ∈ Rn), P1(x) is positive, σ ≥ 0, the estimate

∣∣∣∣∣D
η

(
P1(x)

P2(x)

)∣∣∣∣∣ ≤ cη
(
1 + |x|

)|η|(σ−1)
, x ∈ Rn

holds for each multi-index η ∈ Nn0 with |η| > 0, V2 ≥ 0 and for each η ∈ N
n
0 there exists Mη > 0

such that |Dη(P2(x)
−1)| ≤ Mη(1 + |x|)

|η|(V2−1), x ∈ Rn. Set A2 := ∆ − I, A0f :=
∑

|η|≤Q aηD
ηf

with maximal distributional domain, where Dη ≡ (−i)|η|fη, and C1 := (I − ∆)
− n

2
| 1
p
− 1

2
| max(σ,V2).

Let Ei ∈ L(X) and Fi = C1Ei (i = 0,1). Then we know (cf. [27]-[28]) that λ(λ
2A2 − A0)

−1C1 ∈

LT −L(X) and λA2(λ
2A2 −A0)

−1C1 ∈ LT − L(X), which implies by Theorem 3.9(i)-(b) that there

exists an exponentially bounded C1-existence family (E(t))t≥0 for the following degenerate second

order Cauchy problem:
{ (

∆ − I
)
utt(t,x) =

∑
|η|≤Q aηD

ηu(t,x),

u(0,x) = u0(x) = φ(x), ut(0,x) = u1(x) = ψ(x),

obeying the properties (3.11)-(3.12) stated in the formulation of Theorem 3.12. Applying Theorem

3.12, we get there exists an exponentially bounded

C1-existence family (R(t))t≥0 for the following degenerate second order Cauchy problem:

(P) :






(
∆ − I

)
utt(t,x) + F1ut(t,x) =

(∑
|η|≤Q aηD

η + F0

)
u(t,x),

u(0,x) = u0(x) = φ(x), ut(0,x) = u1(x) = ψ(x).



CUBO
17, 3 (2015)

Degenerate k-regularized (C1,C2)-existence . . . 37

Then Theorem 3.4(i) shows that there exists a strong solution of problem (P) provided that

φ, ψ ∈ SQ+n|
1
p
− 1

2
| max(σ,V2),p(R

n
),
(
A0 + F

)
φ ∈ Sn|

1
p
− 1

2
| max(σ,V2),p(R

n
),

F1ψ ∈ S
n| 1

p
− 1

2
| max(σ,V2),p(R

n
) and

(
A0 + F

)
ψ ∈ Sn|

1
p
− 1

2
| max(σ,V2),p(R

n
).

If we denote by U(t,x), resp. V(t,x), the corresponding strong solution of problem (P) with the
initial values φ(x) and ψ(x) ≡ 0, resp. φ(x) ≡ 0 and ψ(x), then one can simply verify that the
function

u(t,x) :=

∞
∫

0

t
−γ
Φγ
(
st

−γ
)
U(s,x)ds +

t
∫

0

g1−γ(t − s)

∞
∫

0

s
−γ
Φγ
(
rs

−γ
)
V(r,x)drds,

is a strong solution of the following integral equation

A2
[
u(t,x) − φ(x) − tψ(x)

]
+ F1

∫t

0

gγ(t − s)
[
u(s,x) − φ(x)

]
ds

=

∫t

0

g2γ(t − s)
(
A0 + F

)
u(s,x)ds, t ≥ 0, x ∈ Rn; (3.19)

furthermore, the function t 7→ u(t, ·) ∈ X can be analytically extended to the sector Σ( 1
γ
−1) π

2
. In the

present situation, we can only prove that there is at most one strong solution of the integral equation

(3.19) provided that p = 2. Speaking-matter-of-factly, suppose that u(t,x) is a strong solution of

(3.19) with φ(x) ≡ ψ(x) ≡ 0. Then A−12 ∈ L(X), C1 = I and the function v(t,x) := A2u(t,x) is a

strong solution of the following non-degenerate integral equation

u(t,x) +

∫t

0

gγ(t − s)F1A
−1
2 u(s,x)ds

=

∫t

0

g2γ(t − s)
(
A0A

−1
2 + FA

−1
2

)
u(s,x)ds, t ≥ 0, x ∈ Rn. (3.20)

Since λ(λ2 − A0A
−1
2 )

−1 = λA2(λ
2A2 − A0)

−1 ∈ LT − L(X), the operator A0A
−1
2 generates a

cosine operator function and we can apply Theorem 3.12(ii) in order to see that there exists an

exponentially bounded (1,I)-uniqueness family for (3.20), with the meaning clear. This proves the

claimed assertion on the uniqueness of strong solutions of problem (3.19). In general case p 6= 2,

it is not clear how we can prove that there is at most one strong solution of the integral equation

(3.19) without assuming that F1 and F take some specific forms.

Acknowledgments. The author would like to convey a special vote of thanks to Prof. V. E.

Fedorov, Chelyabinsk State University (Russia), and Prof. R. Ponce, Universidad de Talca (Chile),

for many stimulating discussions and valuable suggestions.

Received: June 2014. Accepted: March 2015.



38 Marko Kostić CUBO
17, 3 (2015)

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	Introduction and preliminaries
	Degenerate k-regularized (C1,C2)-existence and uniqueness propagation families for (1.1)
	Degenerate k-regularized (C1,C2)-existence and uniqueness families for (1.1)