One of the existence of solutions to fractional differential equations

HANOI PEDAGOGICAL UNIVERSITY 2 DEPARTMENT OF MATHEMATICS——————–o0o——————— NGUYEN THI HAO ON THE EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS GRADUATION THESIS Speciality:

HANOI PEDAGOGICAL UNIVERSITY 2 DEPARTMENT OF MATHEMATICS

——————–o0o———————

NGUYEN THI HAO

ON THE EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS

GRADUATION THESIS Speciality: Analysis

Hanoi - 2019

HANOI PEDAGOGICAL UNIVERSITY 2

DEPARTMENT OF MATHEMATICS

——————–o0o———————

NGUYEN THI HAO

ON THE EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS

GRADUATION THESIS Speciality: Analysis

Supervisor

Dr Hoang The Tuan

Hanoi - 2019

THESIS ASSURANCE

The thesis was written on the basis of my study under the guidance of Dr Hoang TheTuan and my effort I have studied and presented the results from bibliographies Thethesis does not coincide others

Hanoi, May 2019

Student

Nguyen Thi Hao

THESIS ACKNOWLEDGMENTS

This thesis is the final lesson after 4 years I have learned at Hanoi PedagogicalUniversity 2 I would like to express my sincere gratitude to Dr Hoang The Tuan forhis guidance and encouragement He always provides worth opinions and suggestionsthrough all stages of this thesis He guided and supported me by valuable knowledgeand explanation I also would like to express my gratitude to ThS Nguyen PhuongDong for his experience and suggestions that help me to complete my thesis

I also would like to thank all lecturers of Department of Mathematics - HPU2,especially Dr Tran Van Bang He inspires and finds the best opportunities for my class

to study English It is the most important chance for me to change my mind

My deepest gratitude goes to my family for their encouragement every time.They always believe and help me to pursue my dreams

Due to time, my capacity and conditions are limited, so the thesis can not avoiderrors Then, I am looking forward to receiving valuable comments from teachers andfriends to complete my thesis Once more times, thank you very much

Author

Page

Thesis Assurance 1

Thesis Acknowledgements 2

List of Symbols 3

Preface 5

1 Fractional Calculus 6

1.1 The basic idea 6

1.2 Riemann-Liouville Integral 9

1.3 Riemann - Liouville Derivative 15

1.4 Relations Between Riemann-Liouville Integral and Derivative 18

2 Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations 21

2.1 Main result 21

2.2 Existence of Solutions 22

2.3 Uniqueness of Solutions 25

Bibliography 28

LIST OF SYMBOLS

N Set of natural numbers.

R+ Set of strictly positive real numbers.

An, An[a, b] Set of functions with absolutely continuous derivative of order n − 1

C, C [a, b] Set of continuous functions (cf.Definition 1.1.2)

Ck, Ck[a, b] Set of functions with continuous kth derivative

Hµ, Hµ[a, b] Holder Space(cf Definition 1.1.2)

Lp, Lp[a, b] Lebesgue space (cf Definition 1.1.2)

k.k∞ Chebyshev norm; kf k∞ = supa≤x≤b|f (x)|

k.kp Lp norm (1 ≤ p < ∞);kf kp = Rab|f (x)|pdx

1/p

d.e Ceiling function, dxe = min {z ∈Z: z ≥ x}

b.c Floor function, bxc = max {z ∈Z: z ≤ x}

Γ Euler’s Gamma function, (cf Definition 1.1.1)

o, O Landau’s symbols

D Differential operator, Df (x) = f0(x)

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

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

I Identity operator

Ja Integral operator, Jaf (x) =Rx

a f (t) dt

Jan n ∈N: n-fold iterate of the integral operator Ja

n ∈R+N: Riemann- Liouville fractional integral operator

n = 0: identity operator

In the seventeenth century, Newton and Leibniz developed the foundations ofdifferential and integral calculus In particular, in a letter to de l’Hospital, Leibnizintroduced the symbol The question of de l’Hospital ”What does dxdnnf (x) mean if

n = 12?” was the first occurrence of fractional calculus In the last few decades, ematicians have studied and published the enormous numbers of very interesting andnovel applications of fractional differential equations in physics, chemistry, engineering,finance and psychology In addition, some applications of fractional calculus within var-ious fields of mathematics itself It turns out that many of these applications gave rise

math-to a type of equations, but they had not been covered in the standard mathematicalliterature Leibniz was not able to find the answer of de l’Hospital’s question, exceptfor the special case f (x) = x There are many possible generalizations of dxdnnf (x) tothe case n /∈N, while Riemann- Liuoville derivative and Caputo’derivative are popular

In the first half of the nineteenth centery, in works of Abel, Riemann and Liouville, theformer concepts of fractional calculus was established Athough it leads to be difficultwhen applying it to ”real world” problems, the theories were studied and presentedcarefully The Riemann-Liouville idea is closed to others and is the basis of other re-sults Based on the monograph by Kai Diethelm, in this thesis, we focuse on equationswith Riemann-Liouville differential operators The structure of the thesis is arranged

in the following way:

• Chapter 1: Fractional Calculus, we introduce the fundamental concepts and nitions of fractional Riemann-Liouville differential and integral operators;

defi-• Chapter 2: Existence and Uniqueness results for Riemann-Liouville fractionaldifferential equations

Chapter 1

Fractional Calculus

In this chapter, the basic idea behind fractional calculus is related to a classical standardresult from differential and integral calculus, the fundamental theorem We also recalldefinitions of fractional integral and differential operators

1.1 The basic idea

Theorem 1.1.1 (Fundamental Theorem of Classical Calculus) Let f : [a, b] →

R be a continuous function, and let F : [a, b] →R be defined by

It has proven to be convenient to use the notational conventions introduced inthe following definition

Definition 1.1.1 (i) By D, we denote the operator that maps a differentiable tion onto its derivative, i.e

func-Df (x) := f0(x) (ii) By Ja, we denote the operator that maps a function f , assumed to be Riemannintegrable on the compact interval [a, b], onto its primitive centered at a, i.e

Jaf (x) :=

Z x a

f (t) dt

for a ≤ x ≤ b.

(iii) For n ∈ N we use the symbols Dn and Jan to denote the n-fold iterates of D and

Ja, respectively, i.e we set D1 := D, Ja1 := Ja, Dn := DDn−1 and Jan := JaJan−1for n ≥ 2

Note 1.1.1 The key question is ”How can we extend the concept of Definition 1.1.1 to

n /∈N?” Once we will have provided such an extension, we need to ask for the mappingproperties of the resulting operators and in particular, this includes the question fortheir domains and ranges Note that, from the Fundamental Theorem of ClassicalCalculus, we have a notation DJaf = f which implies that

DnJanf = ffor n ∈N, i.e Dn is the left inverse of Jan in a suitable space of functions We wish toretain this property However, the conditions of the theorem are not straightforward

to the fractional case n /∈N We give a generalization of the concepts in the definition1.1.1 in next part

Following the above outline, we begin with the integral operator Jan for n ∈ N.Two following lemmas are well known in [4, Theorem 2.16]

Lemma 1.1.2 Let f be Riemann integrable on [a, b] Then, for a ≤ x ≤ b and n ∈N,

we have

Janf (x) = 1

(n − 1)!

Z x a

Applying the operator Dn to both sides of this relation and using the fact that

DnDm−n = Dm, the statement follows

We recall here the definition of Gamma function

Definition 1.1.2 The function Γ : (0, ∞) →R, defined by

is called Euler’s Gamma function (or Euler’s integral of the second kind)

We have some properties of Gamma function

Theorem 1.1.4 (i) The functional equation

Γ (x + 1) = xΓ (x)holds for any x ∈ (0, ∞)

(ii) For n ∈N, Γ (n + 1) = n!

The proof of this theorem is presented in [2, P 192]

Before we start the main work, we introduce some function spaces in which weare going to discuss matters

Definition 1.1.3 Let 0 < µ ≤ 1, k ∈ Nand p ≥ 1 We define

L∞[a, b] := {f : [a, b] → R, f is measurable and bounded on [a, b]}

Hµ[a, b] :=f : [a, b] →R, ∃c > 0 such that ∀x, y ∈ [a, b] : |f (x) − f (y)| ≤ c|x − y|µ

Ck[a, b] :=f : [a, b] →R, f has a continuous kthderivative

C [a, b] := C0[a, b] and H0[a, b] := C [a, b]

Definition 1.1.4 By H∗ or H∗[a, b], we denote the set of functions f : [a, b] → Rwith the property that there exists some constants L > 0 such that

|f (x + h) − f (x)| ≤ L |h| ln |h|−1whenever |h| < 12 and x, x + h ∈ [a, b]

When working in a Lebesgue space rather than in a space of continuous functions,

we can still retain the main part of the statement of the fundamental theorem A proof

of this theorem can be found in [3]

Theorem 1.1.5 (Fundamental Theorem in Lebesgue Spaces) Let f ∈ L1[a, b].Then, Jaf is differentiable almost everywhere in [a, b] and DJaf = f also holds almosteverywhere on [a, b].

(x − t)n−1f (t) dtfor a ≤ x ≤ b, is called Riemann- Liouville fractional integral operator of order n.For n = 0, we set Ja0 := I, the identity operator

Proposition 1.2.1 Let f ∈ L1[a, b] and n > 0, then the integral Janf (x) exists foralmost every x ∈ [a, b] Moreover, the function Jnaf itself is also an element of L1[a, b].Proof We write the integral in question as

Z x a

J amJanφ = JanJamφhold almost everywhere on [a, b] If additionally φ ∈ C [a, b] and m + n ≥ 1, then theidentity holds everywhere on [a, b]

Proof We have

JamJanφ (x) = 1

Γ (m) Γ (n)

Z x a

(x − t)m−1

Z t a

Z x τ

(x − t)m−1(t − τ )n−1φ (τ ) dtdτ

Γ (m) Γ (n)

Z x a

φ (τ )

Z x τ

(x − t)m−1(t − τ )n−1dtdτ Substituting t = τ + s (x − τ ) , we yield

φ (τ )(x − τ )m+n−1

Z 1 0

(1 − s)m−1sn−1dsdτ Applying Euler’s integral of the first kind

Z 1 0

(1 − s)m−1sn−1ds =Γ (m) Γ (n)

T (m + n).Thus, we have

JamJanφ (x) = 1

Γ (m + n)

Z x a

φ (τ )(x − τ )m+n−1dτ = Jam+nφ (x)almost everywhere on [a, b]

By the classical theorems on parameter integrals, if φ ∈ C [a, b] then also Janφ ∈ C [a, b].Therefore, JamJanφ ∈ C [a, b] and Jam+nφ ∈ C [a, b] It follows that these are twocontinuous functions coincide almost everywhere, they must coincide everywhere.Finally, if φ ∈ C [a, b] and m + n ≥ 1, then we have

JamJanφ = Jam+nφ = Jam+n−1Ja1φalmost everywhere Since Ja1φ is continuous and Jam+nφ = Jam+n−1Ja1φ is also contin-uous, once again we may conclude that the two functions on either side of the equalityare continuous almost everywhere, that means they must be identical everywhere

Theorem 1.2.3 Let φ ∈ Hµ[a, b] for some µ ∈ [0, 1], and let 0 < n < 1 Then

Janφ (x) = φ (a)

Γ (n + 1)(x − a)

n

+ Φ (x)with some functions Φ This function Φ satisfies

(x − t)n−1dt + 1

Γ (n)

Z x a

φ (t) − φ (a)(x − t)1−n dt.

This yields the desired representation with

Φ (x) = 1

Γ (n)

Z x a

φ (t) − φ (x)(x − t)1−n dt.

In the view of φ ∈ Hµ,

|Φ (x)| ≤ 1

Γ (n)

Z x a

L|t − a|µ(x − t)1−ndt =

L

Γ (n)

Z x a

(t − a)µ(x − t)n−1dt

Γ (n)(x − a)

µ+nZ 10

sµ(1 − s)n−1ds = O (x − a)µ+n
.Now we set g (x) := (φ (x) − φ (a)) /Γ (n) and let h > 0 and x, x + h ∈ [a, b]

(g (t) − g (x))(x + h − t)n−1− (x − 1)n−1dt

=:K 1+

Z x+h x

(g (t) − g (x)) (x + h − t)n−1dt

=:K 2

+K3

where K3 contains the remaining terms An explicit calculation shows that

K3= g (x)

Z x a



(x + h − t)n−1− (x − t)n−1

dt +

Z x+h x

(x + h − t)n−1

!

We estimate the terms K1, K2 and K3 separately In the view of our assumption

φ ∈ Hµ, it is clear that φ ∈ Hµ Hence,

inte-x − a < h, the integral is bounded by R01tµ+n−1dt = (µ+n)1 and thus K1 = O hµ+n

in this case If x − a ≥ h, we find

tµ+n−2dt



O hµ+n
because of µ + n < 1 and x − a ≥ h For µ + n = 1, a similar calculation gives

|K1| ≤ O hµ+n
1 +

Z (x−a)/h 1

t−1dt

!

= O h ln h−1

Finally, for µ + n > 1 and x ≥ b,

|K1| ≤ O hµ+n

Z (b−a)/h 1

|K2| ≤ L

Z x+h x

(t − x)µ(x + h − t)n−1dt

= Lh+n

Z 1 0

sµ(1 − s)n−1ds = O hµ+n
irrespective of µ and n Obviously, this bound is stronger than the above bound for

K3= O (1) (x − a)µh(x − a)n−1 = O (h) (x − a)µ+n−1Once again, we look at the three cases separately and find K3 = O (h) for µ + n = 1,

|K3| ≤ O (h) hµ+n−1 = O hµ+n
for µ + n < 1 and |K3| ≤ O (h) (b − a)µ+n−1 = O (h)for µ + n > 1 Again, these estimates are stronger than those we obtained for K1.Combining all the estimation, we derive

Remark 1.2.1 We can infer a similar result for the integrand φ to be in a suitableLebesgue class.

Lemma 1.2.4 Let n > 0, p > max {1, 1/n} and φ ∈ Lp[a, b] Then,

Proof See [1, Theorem 2.6]

Example 1.2.1 Let f (x) = (x − a)β for some β > −1 and n > 0 Then,

(t − a)β(x − t)n−1dt

Γ (n)(x − a)

n+βZ 10

sβ(1 − s)n−1ds = Γ (β + 1)

Γ (n + β + 1)(x − a)

n+β

.Next we discuss the interchange of limit operation and fractional integration.Theorem 1.2.5 Let n > 0 Assume that (fk)∞k=1 is a uniformly convergent sequence

of continuous functions on [a, b] Then we may interchange the fractional integral erator and the limit process, i.e.,

op-

Jan lim

k→∞fk(x) =lim

k→∞Janfk(x)

In particular, the sequence of functions (Janfk)∞k=1 is uniformly convergent

Proof The proof of this theorem is found in [1, P 21]

Corollary 1.2.6 Let f be analytic in (a − h, a + h) for some h > 0, and n > 0 Then,

kf (x)

for a ≤ x < a + h In particular, Janf is analytic in (a, a + h).

Example 1.2.2 Let f (x) = exp (λx) with some λ > 0 Compute J0nf (x) for n > 0

In the case n ∈N, we obviously have J0nf (x) = λ−nexp (λx)

For the case n /∈N, we find

n 0

1.3 Riemann - Liouville Derivative

Having established these fundamental properties of Riemann- Liouville integral tors, we come to the corresponding differential operators

opera-Definition 1.3.1 (See [1]) Let n ∈ R+ and m = dne The operator Dna defined by

Dnaf := DmJam−nf

is called the Riemann - Liouville fractional differential operator of order n

For n = 0, we set Da0:= I, the identity operator

Lemma 1.3.1 Let n ∈R+ and let m ∈N such that m > n Then,

Dan = DmJam−n.Proof Our assumptions on m imply that m ≥ dne Thus,

DmJam−n = DdneDm−dneJam−dneJadne−n= Dan

in the view of the semigroup property of fractional integration

Next, we present a very simple sufficient condition for the existence of Dnaf Lemma 1.3.2 Let f ∈ A1[a, b] and 0 < n < 1 Then, Danf exists almost everywhere

in [a, b] Moreover, Dnaf ∈ Lp[a, b] for 0 ≤ p < 1 and

Dnaf (x) = 1

Γ (1 − n)



f (a)(x − a)n +

Z x a

f0(t) (x − t)−ndt



Proof We use the definition of the Rienmann - Liouville differential operator and thefact that f ∈ A1 It follows that

Danf (x) = 1

Γ (1 − n)

ddx

Z x a

Z x a



f (a) +

Z t a



f (a)

Z x a

dt(x − t)n+

Z x a

Z t a

ddx

Z x a

Z t a

ddx

Z x a

Example 1.3.1 Let f (x) = (x − a)β for some β > −1 and n > 0 Then,

Danf (x) = DdneJa−ndnef (x) = Γ (β + 1)

Dan( − a)n−m(x) = 0for all n > 0, m ∈ {1, 2, , dne}

On the other hand, if n − β /∈N, we find

Theorem 1.3.3 Assume that n1, n2≥ 0 Moreover let φ ∈ L1[a, b] and f = Jn1 +n 2

Proof By our assumption on f and the definition of the Riemann-Liouville differentialoperator, we have

The proof that Dn1 +n 2

a f = φ goes along similar lines

The smoothness and zero condition in this theorem is not just a technicality Thefollowing examples show some cases where the condition is not satisfied They provethat an unconditional semigroup property of fractional differentiation in the Riemann-Liouville sense does not hold

Example 1.3.2 Consider example 1.3.1, let f (x) = x−1/2 and n1= n2= 1/2 Then,

Dn1

0 f (x) = Dn2

0 f (x) = 0and

0 f (x) =

√ π