HANOI PEDAGOGICAL UNIVERSITY 2
DEPARTMENT OF MATHEMATICS
———o0o———

NGUYEN THI AN

A GENERALIZED DEFINITION OF
CAPUTO DERIVATIVES

GRADUATION THESIS

Hanoi – 2019


HANOI PEDAGOGICAL UNIVERSITY 2
DEPARTMENT OF MATHEMATICS
———o0o———

Nguyen Thi An

A GENERALIZED DEFINITION OF
CAPUTO DERIVATIVES

GRADUATION THESIS

Major: Analysis

SUPERVISOR:

Dr. Hoang The Tuan

Hanoi – 2019


Graduation thesis

Nguyen Thi An

Thesis Acknowledgement
I would like to express my gratitude to the teachers of the Department of Mathematics, Hanoi Pedagogical University 2, the teachers
in the analysis group as well as the teachers involved. The lecturers
have imparted valuable knowledge and facilitated for me to complete
the course and the thesis.
In particular, I would like to express my deep respect and gratitude
to Dr. Hoang The Tuan, who has direct guidance, help me complete
this thesis.
Due to time, capacity and conditions are limited, so the thesis can
not avoid errors. Then, I look forward to receiving valuable comments
from teachers and friends.

Ha Noi, May, 2019
Student

Nguyen Thi An

i


Graduation thesis

Nguyen Thi An


Thesis Assurance
This thesis is written based on the paper [3] A Generalized Definition of Caputo Derivatives and Its Application to Fractional ODEs of
Lei Li and Jian-Guo Liu.
I assure that the data and the results of this thesis are true and not
identical to other topics. I also assure that all the help for this thesis
has been acknowledged and that the results presented in the thesis
has been identified clearly.

Ha Noi, May, 2019
Student

Nguyen Thi An

ii


Contents

Thesis Acknowledgement

i

Thesis Assurance

ii

List of symbols

v


Preface

vi

Introduction

vii

1 Time-continuous groups and fractional calculus

1

1.1

A time-continuous convolution group . . . . . . . . . .

1

1.2

Time-continuous fractional calculus . . . . . . . . . . .

4

1.2.1

Fractional calculus for distributions . . . . . . .

4


1.2.2

Modified Riemann-Liouville calculus . . . . . .

9

1.3

Another group for right derivatives . . . . . . . . . . .

11

1.4

Regularities of the modified Riemann-Liouville operators 12

2 An extension of Caputo derivative
2.1

Generalized Caputo derivative . . . . . . . . . . . . . .

iii

14
14


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2.2

Nguyen Thi An

Some fundamental properties of generalized Caputo derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References

17
23

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List of symbols
N

Set of natural numbers.

R+

Set of non- negative real numbers.

C

Set of complex numbers.


Γ

Euler’s Gamma function, Γ(x) =

B

Euler’s Beta function, B(x, y) = Γ(x)Γ(y)/Γ(x + y)

D

Differential operator, Df (x) = f (x)

Dn

n ∈ N : n-fold iterate of the differential operator D.

Dan

n ∈ R+ : Riemann- Liouville fractional differential operator.

I

Identity operator.

Ja

Integral operator, Ja f (x) =

Jan


n ∈ N: n-fold iterate of the integral operator J a

x
a f

∞ x−1 −t
e dt
0 t

(t) dt

n ∈ R+ N: Riemann- Liouville fractional integral operator.
n = 0: Identity operator.
θ(t)

Heaviside step function

D

Class of distribution functions

supp (u) The support of u on G, supp(u) = {x ∈ G : u(x) = 0}.
.

Norm.

L1loc

Set of locally integrable functions


L

Laplace transform operator

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Nguyen Thi An

Preface
Fractional calculus is one of the most important areas of mathematics and many applications in the fields of science and technology.
In the seventeenth century, Newton and Leibniz developed the foundations of differential and integral calculus. In 1695, the first occurrence
of Leibniz introduced the notion fractional derivative in a letter to
dn
de l’Hospital to answer the question: ”What does
f (x) mean if
dxn
1
n = ?”. Since then, Fractional Differential Equations have impor2
tant implications in Mathematics and other sciences. It turned out
that many of these applications gave rise to a type of equations that
has not been covered in the standard mathematical literature. This
is connected to the fact that, in a certain sense, the answer to de
lHospital’s question is not unique. There are very many possible gendn
eralizations of
/ N. We shall only discuss
f (x) to the case n ∈

dxn
two of them, the Riemann-Liouville derivative ([2],cf. Chap.2) and
the Caputo derivative ([2],cf. Chap.3). However, They are defined for
n ∈ R+ or n ≥ 0 where both types of fractional derivatives mentioned
do not have the group property. This is one of reasons we propose
a generalized definition of Caputo derivatives from t = 0 of order in
(0, 1) using a convolution group. And we also build a define a modified Riemann-Liouville fractional calculus so that the group property
holds. Then, making use of this fractional calculus, we introduce the
generalized definition of Caputo derivatives.

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Introduction
Fractional calculus in continuous time is one of the most important
areas of mathematics and many applications in physics, science and
technology, etc. The fractional integral from t = 0 with order γ > 0
of a function ϕ(t) is given by Abel’s formula,
1
Jγ ϕ(t) =
Γ(γ)

t

ϕ(s)(t − s)γ−1 ds, t > 0.


(1)

0

We can see that the integral operators Jγ form a semigroup.
Letting n − 1 < γ < n where n is a positive integer.
The Riemann-Liouville is defined by
γ
Drl
ϕ(t)

t

1
dn
=
Γ(n − γ) dtn

0

ϕ(s)
ds, t > 0,
(t − s)γ+1−n

(2)

The Caputo fractional derivatives is defined by
Dcγ ϕ(t)

1

=
Γ(n − γ)

t
0

ϕ(n) (s)
ds, t > 0.
(t − s)γ+1−n

(3)

Both types of fractional derivative mentioned do not have the semigroup property and they get limited by order γ > 0.
In ([1], Chap.1, sect. 5.5) by Gel’fand and Shilov, integrals and
derivative of arbitrary order for a distribution ϕ ∈ D (R) supported
on [0, ∞) are defined to be the convolution
ϕ

(α)

tα−1
= ϕ ∗ gα , gα (t) := + , α ∈ C.
Γ(α)

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Nguyen Thi An


where ϕ0 = ϕ. If α > 0, ϕ(α) = Jα ϕ is the fractional integral as
in Abel’s formula (1). When α < 0, they call so-defined ϕ(α) to be
derivative of ϕ. Then, the calculus (integrals and derivatives) {ϕ(α) :
α ∈ R} forms a group.
In this thesis, based on the paper [3], we find more convenient
(θ(t)t)α−1
expressions for the distributions gα (t) =
when α < 0. First,
Γ(α)
we define fractional calculus of a certain class of distributions E T ⊂
D (−∞, T ), T ∈ (0, ∞]:
ϕ → Iα ϕ ∀ϕ ∈ E T , α ∈ R.
We then make the distribution causal and define the modified RiemannLiouville calculus by
Jα ϕ := Iα (θϕ), α ∈ R.
which then places the foundation for us to generalize the Caputo
derivatives.
Although the Caputo derivatives do not have group property, they
are suitable for initial value problems and share many properties with
the ordinary derivative, so they can be defined of derivatives for functions with order smooth only. In [3], Allen, Caffarelli and Vasseur
introduced an alternative form of the Caputo derivative based on integration by parts for γ ∈ (0, 1):
Dcγ ϕ(t)

1
=
Γ(1 − γ)

ϕ(t) − ϕ(0)
+γ
tγ


viii

t
0

ϕ(t) − ϕ(s)
ds
(t − s)γ+1

(4)


Graduation thesis

Nguyen Thi An

−1
=
Γ(−γ)
where
ϕ(s) =

t

ϕ(t) − ϕ(s)
ds,
γ+1
(t
−

s)
−∞


ϕ(s), s ≥ 0,

ϕ(0), s < 0.

Gorenflo, Luchko and Yamamoto [5] used a functional analysis approach to extend the fractional caputo derivatives to certain Sobolev
spaces.
One of our main purposes is to use the convolution group {gα }
to generalize the Caputo derivatives from t = 0 of order γ ∈ (0, 1)
for initial value problems. The idea is to remove the singular term
ϕ0 θ(t) −γ
t , which corresponds to the jump of the causal function,
Γ(1 − γ)
from the modified Riemann-Liouville derivative J−γ ϕ
Dcγ ϕ := J−γ ϕ − ϕ0

θ(t) γ−1
t
= J−γ (ϕ − ϕ0 ), γ ∈ (0, 1).
Γ(1 − γ)

In this definition, the ϕ0 can be any real number, and this then allows
for the generalized definition (see Chapter 2).
In Chapter 2, an extension of Caputo derivatives is proposed so
that both the first derivative and Holder regularity of the function
are not needed in the definition. Some properties of the new Caputo
derivatives are proved, which may be used for fractional ODEs and

fractional PDEs.

ix


Chapter 1
Time-continuous groups and
fractional calculus
In this chapter, we defined the fractional calculus for a particular class
of distributions in D (−∞, T ) and then define the modified RiemannLiouville calculus.

1.1

A time-continuous convolution group

Let t ∈ (−∞, T ). Recall that θ(t) is the Heaviside step function, then
t+ = max(t, 0) = θ(t)t.
We consider α ∈ R, we define
tα−1
C := {gα : gα (t) = + , α ∈ R}
Γ(α)
We note that
tα−1
C+ := {gα : gα (t) = + , α ∈ R+ }
Γ(α)

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Nguyen Thi An

forms a semi-group of convolution. We can combine two distributions
as long as their supports are one-side bounded. This motivates us to
introduce the following set of distributions:
E := {v ∈ D (R) : ∃Mv ∈ R, supp(v) ⊂ [−Mv , +∞)}
is a linear vector space.
Definition 1.1.1. Given f, g ∈ E , we define
f ∗ (φi g), ϕ , ∀ϕ ∈ D = Cc∞ ,

f ∗ g, ϕ :=

(1.1)

i∈Z

where {φi }∞
i=−∞ is a partition of unity for R and f ∗ (φi g) is given
by the usual definition between two distributions when one of them is
compactly supported.
The following lemma is well known.
Lemma 1.1.2. (i)The definition is independent of {φi } and agrees
with the usual definition of convolution between distributions whenever one of the two distributions is compactly supported.
(ii) For f, g ∈ E , f ∗ g ∈ E
(iii)

f ∗ g = g ∗ f,
f ∗ (g ∗ h) = (f ∗ g) ∗ h.
(iv) We use D to mean the distributional derivative. Then, letting

f, g ∈ E , we have
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(Df ) ∗ g = D(f ∗ g) = f ∗ Dg.
Proof. See [3, Lemma 2.2].
Also well known is the following lemma.
Lemma 1.1.3. g0 = δ(t) is the convolution identity and for n ∈
N, g−n = Dn δ is the convolution inverse of gn .
Proof. See [3, Lemma 2.3].
Lemma 1.1.4. Letting 0 < γ < 1, the convolution inverse of gγ is
given
g−γ (t) :=

1
D(θ(t)t−γ ).
Γ(1 − γ)

where D is the distributional derivative.
Proof. We pick ϕ ∈ Cc∞ (R) and apply Lemma 1.1.2:
D(θ(t)t−γ ) ∗ [θ(t)t−γ ], ϕ = − θ(t)t−γ ∗ θ(t)t−γ , Dϕ
∞

= − B(1−γ, γ)θ(t), ϕ = −B(1−γ, γ)

ϕ (t)dt = B(1−γ, γ)ϕ(0).

0

where B(., .) is the Beta function. This computation verifies that the
claim is true.
We have another proof:
Proof. We apply Lemma 1.1.3: since 0 < γ < 1 so 1 − γ > 0, we have
1

g−γ (t) = g−1+1−γ (t) = D g1−γ (t) = D

=

θ(t)t( 1 − γ − 1)
Γ(1 − γ)

1
D(θ(t)t−γ ).
Γ(1 − γ)
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Similarly, for n < γ < n + 1, we define g−γ := Dn δ ∗ gn−γ .
We define
C− := {g−α : α ∈ R+ }
forms a convolution semi-group as well.
Proposition 1.1.5. C ⊂ E , and C is a convolution group under the

convolution on E
Proof. For any γ > 0, g−γ is the convolution inverse of gγ . Given the
fact that gγ ∗ g−γ = δ, the group property can then be verified using
the semi-group property.

1.2

Time-continuous fractional calculus

In this section, we use the group C to define the fraction calculus
for a certain class of distribution in D (−∞, T ) and the (modified)
Riemann-Liouville fractional calculus.
1.2.1

Fractional calculus for distributions

In E , the fractional derivatives can be easily generalized to distributions.
Definition 1.2.1. For φ ∈ E , the fractional calculus of φ, Iα : E →
E (α ∈ R), is given by
Iα φ := gα ∗ φ
4

(1.2)


Graduation thesis

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The following lemma is well known.

Lemma 1.2.2. The operators {Iα : α ∈ R} form a group, and I−n φ =
Dn φ, (n = 1, 2, 3, ...).
By Definition 1.2.1 and Lemma 1.2.2, we can see
In Cc∞ (R) ⊂ E , we pick a function φ.
If α ⊂ Z, then Iα gives the usual integral (where the integral is from
−∞, if α > 0) and derivative (if α < 0), I0 φ = φ.
If α ⊂ R,then by the defined of gα , we only need consider α = −γ, 0 <
γ < 1, we have
1
d
I−γ φ(t) =
Γ(1 − γ) dt

t

φ(s)
1
ds
=
γ
Γ(1 − γ)
−∞ (t − s)

t

φ (s)
ds.
γ
−∞ (t − s)


Hence, I−γ gives the Riemann-Liouville derivatives from t = −∞.
We introduce the set
E T := {v ∈ D (−∞, T ) : ∃Mv ∈ (−∞, T ), supp(v) ⊂ (Mv , T )}
E T is not closed under the convolution. To define the fractional calculus for distributions supported in E T , we introduce the extension
operator KnT : E T → E ∞ given by
KnT v, ϕ = χn v, ϕ = v, χn ϕ , ∀ϕ ∈ Cc∞ (R),
where {χn } ⊂ Cc∞ (−∞, T ) is a sequence satisfying (i) 0 ≤ χn ≤ 1,
and (ii) χn = 1 on [−n, T − n1 )
Definition 1.2.3. For φ ∈ E T , we define the fractional calculus of φ
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to be IαT : E T → E T (α ∈ R):
IαT φ = lim RT (Iα (KnT φ)).
n→∞

(1.3)

Proposition 1.2.4. Fix φ ∈ E T .
(i) For any sequence {χn } satisfying the conditions given and

>

0, M > 0, there exists N > 0, such that for all n ≤ N and all ϕ ∈
Cc∞ (−∞, T ) with suppϕ ⊂ [−M, T − ],
KnT φ, ϕ = φ, ϕ .

(ii) The limit in Definition 1.2.3 exists under the topology of D (−∞, T ).
In particular, picking any partition of unity {φi } for R, we have
φ, (gα φi )(−.) ∗ ϕ ∀ϕ ∈ Cc∞ (−∞, T ), α ∈ R,

IαT φ, ϕ =
i

and the value on the right-hand side is independent of choice of the
partition of unit {φi }. When we have:
For

> 0, there exists N > 0, such that for all n ≥ N such that

and all ϕ ∈ Cc∞ (−∞, T ) with suppϕ ⊂ [−M, T − ],
KnT φ, ϕ = φ, ϕ .
Proof. For (i), we pick a sequence {χn } ⊂ Cc∞ (−∞, T ) satisfying (i)
0 ≤ χn ≤ 1, and (ii) χn = 1 on [−n, T − n1 ).
When we have:
Given

> 0, M > 0, there exists N > 0, such that for all n ≥ N

then t ∈ [−N ; T −

1
N)

i.e χn = 1 and all ϕ ∈ Cc∞ (−∞, T ) with
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Graduation thesis

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suppϕ ⊂ [−M, T − ], by definition previous
KnT φ, ϕ = φ, ϕ .
For (ii), we pick ϕ ∈ Cc∞ (−∞, T ). Then, for all n > 0,
RT (Iα (KnT φ)), ϕ = gα ∗ (KnT φ), ϕ =

(φi gα ) ∗ (KnT φ), ϕ .
i

There are only finitely many terms in the sum. Then, for each
term,
(φi gα ) ∗ (KnY φ), ϕ = KnT φ, ζi ∗ ϕ ,
where ζi (t) := (φi gα )(−t) is a distribution supported in [−N1 , 0], for
some N1 > 0. By (i), for ζi ∗ ϕ ∈ Cc∞ (−∞, T ),
IαT φ, ϕ = lim

n→∞

(φi gα ) ∗ (KnT φ), ϕ =
i

φ, ζi ∗ ϕ .
i

Lemma 1.2.5. IαT (α ∈ R) is independent off the choice of extension
operators {Kn }. For any T1 , T2 ∈ (0, ∞] and T1 < T2 ,

RT1 IαT2 φ = IαT1 RT1 φ ∀φ ∈ E T2 .
Proof. For any T1 , T2 ∈ (0, ∞] and T1 < T2 , let ϕ ∈ Cc∞ (−∞, T1 ).
then we need to show
lim gα ∗ (KnT2 φ, ϕ), ϕ = lim gα ∗ (KnT1 RT1 φ), ϕ .

n→∞

n→∞

We use the partition of unity {φi } for R.
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limn→∞ gα ∗ (KnT2 φ, ϕ), ϕ

n→∞

(φi gα ) ∗ (KnT −2 RT2 φ), ϕ .

(φi gα ) ∗ (KnT2 φ), ϕ = lim

= lim

n→∞

i


i

Denote ζi (t) = (φi gα )(−t) is supported in (−∞, 0). Then it suffices to
show
limn→∞ (φi gα ) ∗ (KnT2 φ), ϕ
= limn→∞ KnT2 φ, ζ − i ∗ ϕ = limn→∞ KnT1 RT1 φ, ζi ∗ ϕ . By Proposition
1.2.4, this equality is valid.
Lemma 1.2.6. Verifies that if T = ∞, then IαT agrees with Iα in
Definition 1.2.3 and {IαT : α ∈ R} forms group.
T
Proof. By IαT agrees for T = ∞ we need to show that IαT (IβT φ) = Iα+β
φ

for all α, β ∈ R. Indeed,
T
IαT (IβT φ) = lim RT (Iα (RT (Iβ (KnT φ))) = lim RT (Iα+β (KnT φ))) = Iα+β
φ.
n→∞

n→∞

Definition 1.2.7. If φ ∈ E T , we can define the convolution between
g−α : α ∈ R and φ as
gα ∗ φ := IαT φ ∀φ ∈ E T .

(1.4)

By Lemma 1.2.6, we have
gα ∗ (gβ ∗ φ) = (gα ∗ gβ ) ∗ φ = gα+β ∗ φ.

Now, when we fix T ∈ (0, ∞], we can drop the super index T for
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convenience:
E T is denoted by E ; I T is denoted by I.
1.2.2

Modified Riemann-Liouville calculus

In section 1.2.1, the fractional calculus starting from t = −∞. In this
section, we will research the fractional calculus starting from t = 0.
In other words, for ϕ ∈ E (with T ∈ (0, ∞]), we hope to define the
fractional calculus from t = 0. We need to modify the distribution to
be supported in [0, T ), it is denoted by Gc .
We define
Gc := {φ ∈ E ⊂ D (−∞, T ) : suppφ ⊂ [0, T )}.
In D (−∞, T ) for any such sequence {un } satisfying (i) 0 ≤ un ≤ 1,
(ii) un (t) = 1 for t ∈ (−1/(2n), T − 1/(2n)). We introduce the space
G := {ϕ ∈ E : ∃φ ∈ Gc , un ϕ → φ}.
For ϕ ∈ G , the corresponding distribution φ is denoted as θϕ. If
ϕ ∈ L1loc (−∞, T ), θϕ can be understood as the usual multiplication.
Lemma 1.2.8. Gc ⊂ G . For all ϕ ∈ Gc , θϕ = ϕ.
Definition 1.2.9. The (modified) Riemann-Liouville operators Jα :
G → Gc are biven by
Jα ϕ := Iα (θϕ) = gα ∗ (θϕ),


9

(1.5)


Graduation thesis

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where gα ∗ (θϕ) is understood as in Definition 1.2.7.
Proposition 1.2.10. Fix ϕ ∈ E . For all α, β ∈ R, Jα Jβ ϕ = Jα+β ϕ
and for n = 0 we set J0 ϕ = θϕ. If we make the domain of them to be
Gc , then they form a group.
Proof. See [3, Proposition 2.9].
In Definition 1.2.9, α ∈ R Now, we checking some special cases.
Case 1: α > 0 and ϕ is a continuous function, we have Jα ϕ = Iα (θϕ) =
gα ∗ (θϕ) gives Abel’s formula of fractional integrals.
1
Γ(α)

Jα ϕ(t) :=

t

ϕ(s)(t − s)α−1 ds, t > 0
0

Case 2: α < 0 and smooth ϕ:
When −1 < α < 0, we have for any t < T

θ(t)
Jα ϕ(t) = J−γ ϕ(t) =
D
Γ(1 − γ)
=

t
0

ϕ(s)
ds
(t − s)γ

1
(θ(t)t−γ ) ∗ (θ(t)ϕ + δ(t)ϕ(0))
Γ(1 − γ)

1
=
Γ(1 − γ)

t
0

θ(t) −γ
1
ϕ
(s)ds
+
ϕ(0)

t ,
(t − s)γ
Γ(1 − γ)

(1.6)
where γ = −α, so 0 < γ < 1. This is the Riemann-Liouville fractional
derivative.
When α = −1, we have

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J−1 ϕ(t) = D (θ(t)ϕ(t)) = θ(t)ϕ (t) + δ(t)ϕ(0).
This is the derivative with order 1 of the function θ(t)ϕ for t > 0.
When α < −1, put γ = −1 − α > 0. By the group property, we
have for t < T
1
Jα ϕ(t) = J−1−γ ϕ(t) =
D θ(t)D
Γ(1 − γ)
1
=
D θ(t)D
Γ(2 − |α|)

t

0

t
0

ϕ(s)
ds
(t − s)γ

ϕ(s)
ds .
(t − s)γ

This is again the Riemann-Liouville derivative for t > 0.
We then call {Jα } the modified Riemann-Liouville operators.

1.3

Another group for right derivatives

In this section, we consider another group C generated by
gα (t) :=

θ(−t)
(−t)α−1 , α > 0.
Γ(γ)

1
D(θ(−t)(−t)−γ ).
Γ(1 − γ)

This derivative is sometimes called the right Riemann-Liouville derivaFor −1 < α < 0, gα (t) = g−γ (t) = −

tive.
This group is actually the adjoint of C in the following sense:
gα ∗ φ, ϕ = φ, gα ∗ ϕ ,
where both φ and ϕ are in Cc∞ (R). This dual identity actually provides

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a type of integration by parts.
It is interesting to explicitly write out the case α = −γ:
∞

∞

1
gα ∗ φ(t)ϕ(t)dt =
−∞
−∞ Γ(1 − γ)
∞

=−

1
D

−∞ Γ(1 − γ) Ds

1.4

t

(t − s)−∞ Dφ(s)dsϕ(t)dt
−∞

∞

∞

(t−s)−∞ ϕ(t)dtφ(s)ds =

gα ∗ϕ(s)φ(s)ds.
−∞

s

Regularities of the modified Riemann-Liouville
operators

In this section, we check the definition of Jα by considering their actions on a specific class of Sobolev spaces.
Review H s (0, T ) is the closure of C ∞ [0, T ] then H0s (0, T ) is the closure
of Cc∞ (0, T ) under the norm of H s (0, T ).
We introduce the space H s (0, T ), which is the closure of Cc∞ (0, T ]
under the norm of H s (0, T ).
Lemma 1.4.1. Let s ∈ R, s ≥ 0.
The restriction mapping is bounded from H s (R) to H s (0, T ), i.e., for

all v ∈ H s (R), then v ∈ H s (0, T ), and there exists C = C(s, T ) such
that v

H s (0,T )

≤ v

H s (R) .

For v ∈ H s (0, T ), there exists vn ∈ Cc∞ (R) such that the following
conditions hold: (i) suppvn ⊂ (0, 2T ). (ii) vn

H s (R)

≤ C vn

H s (0,T ) ,

where C = C(s, T ). (iii) vn → v in H s (0, T ).
Lemma 1.4.2. If vn → f in H s (0, T ) (s ≥ 0), then Jα vn → Jα f in
D (0, T ) for all α ∈ R.
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Nguyen Thi An

Proof. See [3, Lemma 2.17].
Theorem 1.4.3. If min{s, s + α} ≥ 0, then Jα is bounded from

H s (0, T ) to H s+α (0, T ). In other words, if f ∈ H s (0, T ), then Jα f ∈
H s+α (0, T ) and there exists a constant C depending on T , s, and α
such that
Jα f

H s+α (0,T )

≤C f

H s (0,T ) ∀f

Proof. See [3, Theorem 2.18].

13

∈ H s (0, T ).


Chapter 2
An extension of Caputo derivative
In this chapter, we introduce a generalized definition of the Caputo
derivative.

2.1

Generalized Caputo derivative

Recall that L1loc [0, T ) is the set of locally integrable functions on [0, T ).
Lemma 2.1.1. Suppose f, g ∈ L1loc [0, T ), where T ∈ (0, ∞]. Then
x


f (x − y)g(y)dy is defined for almost every x ∈ [0, T ) and

h(x) =
0

h ∈ L1loc [0, T ).
Proof. Fix M ∈ (0, T ). Denote Ω = {(x, y) : 0 ≤ y ≤ x ≤ M }.
F (x, y) = |f (x − y||g(y)| is measurable and nonnegative on Ω. By
Tonelli’s Theorem
M

|g(y)|

F (x, y)dA =
D

M

0

|f (x − y)|dxdy ≤ C(M )
y

for some C(M ) ∈ (0, ∞). F (x, y) is integrable on D. Hence, h(x) is
14