khaled thesis

7/29/2019 Khaled Thesis

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Generalized Time Scales and

Associated Difference Equations

by

Khaled Ahmad Ali Aldwoah

SUBMITTED FOR THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

AT

CAIRO UNIVERSITY

GIZA, EGYPT

2009

Supervisors

M.H.Annaby A.E.Hamza

Cairo University Cairo University
http://[email protected]/http://[email protected]/


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i

Approval Sheet

Title of the PhD thesis

Generalized Time Scales

and Associated Difference

Equations

Name of the candidate:Khaled Ahmad Ali Aldwoah

Submitted to:

Faculty of Science

Cairo University-Egypt

Supervision Committee:

Prof. Mahmud H.Annaby Assoc.Prof. Alaa Eldeen HamzaDept. of Mathematics, Dept. of MathematicsFaculty of Science, Faculty of Science,Cairo University. Cairo University.

Prof. Hamdy Ibrahim Abd-El-Gawad

Head of Dept. of Mathematics.


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Contents

Acknowledgements iv

Introduction v

List of Figures xi

Notations xiii

1 A Generalized Time Scale 1

1.1 Basic definitions and terminology . . . . . . . . . . . . . . . . . . . 1

1.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Limits and continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4 Induction principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 Differentiation 41

2.1 -Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Leibniz Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.5 -Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3 Integration 673.1 -Antiderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 -Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 -Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 First Order Dynamic Equations 94

4.1 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2 Existences and Uniqueness Theorems . . . . . . . . . . . . . . . . . 106

5 Difference Time Scales and Applications 112

5.1 A Difference Time Scale . . . . . . . . . . . . . . . . . . . . . . . . 112

ii


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iii

5.2 q-Jackson Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.3 Finite Difference Calculus and Norlund Sums . . . . . . . . . . . . 125

5.4 New Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6 Hahn Difference Calculus 133

6.1 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2 Jackson-Norlund Integration . . . . . . . . . . . . . . . . . . . . . . 139

6.3 Parametric q, -exponential and trigonometric functions . . . . . . . 146

Bibliography 150

Index 157


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Acknowledgements

First and foremost, I would like to thank God for giving me the strength to

complete this research study. It is a pleasure to thank the many people who

made this thesis possible. It is difficult to overstate my gratitude to my Ph.D.

supervisors, Assistant Prof. Alaa Eldeen Hamza, and Prof. Mahmud Annaby,

Cairo University, for their guidance throughout the preparation of this thesis,

under whose supervision I choose this topic and began the thesis. I cannot thank

them enough.

I want to express my gratitude to Prof. Mohammad Ali Alsyed and Dr. Hany

El-Hosseiny, Cairo university, for helping me, and for all the emotional support.

Special thanks are due to the college of science, department of mathematics, and

analysis group (especially Dr. Zeinab Mansour, Prof. Mohammad Zeidan and the

head of the department Prof. Hamdi Abd Algawad).

I would like to thank the many people who have taught me mathematics: my

graduate teachers at Baghdad university especially Prof. Adil G. Naoum, Prof.

Sakin Fathy and Prof. Basil Alhashimi.

I wish to thank my entire extended family for providing a loving environment for

me: my parents, my life and wife Om-Osama, my sons and daughter, big brother

Fahd, my brother Belal, and my sister Om-Mohammad. They bore me, raised me,

supported me, taught me, and loved me.

Lastly, and most importantly, I dedicate this thesis to the Prophet Mohammad

(peace be upon him).

Khaled Aldwoah, 2009.

iv


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Introduction

There are several attempts to unify continuous and discrete mathematics. One of

the main approaches is the time scale which is recently founded by Stefan Hilger in

his Ph.D. thesis [37], see also [13, 14]. It gives an efficient tool to unify continuous

and discrete problems in one theory. Since then hundreds of papers appeared in

the theory and its applications to dynamic equations, see e.g. the very interesting

monographs of Bohner and Peterson [19, 21]. At the beginning a time scale is

defined to be an arbitrary closed subset of the real numbers R, with the standard

topology. Examples of time scales include the real numbers R, the natural numbers

N, the integers Z, the Cantor set, and any finite union of closed intervals ofR.

Next, we review a few basics of the time scales calculus.For a time scale T, the forward jump operator : T T is defined by (t) :=inf{z T : z > t} and the backward jump operator : T T is defined by(t) := sup{z T : z < t}. Here, we put inf = supT and sup = infT. Also,the forward stepsize function, : T [0, ), is defined by (t) := (t) t andthe backward stepsize function : T [0, ), (t) := t(t), see [4, 19, 21]. In[19, 21, 39] the stepsize functions are called graininess functions. Here, we follow

the terminology of Ahlbrandt, Bohner and Ridenhour [5].

A point t T with (t) = t, (t) > t, (t) = t, (t) < t, (t) < t < (t) and(t) = t = (t) is called right dense, right scattered, left dense, left scattered,

isolated and dense respectively.

A real valued function f on T is called

(i) regulated on T if its right hand limits exist at all right dense points of T \{supT}, and left hand limits exist at all left dense points ofT\{infT} (herewe excluded the points which have no sense for the limit).

(ii) right dense continuous, or just rd-continuous, on T if it is regulated on T and

continuous at all right dense points ofT.

v


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Introduction vi

(iii) left dense continuous, or just ld-continuous, on T if it is regulated on T and

continuous at all left dense points ofT.

Differentiation: For any time scale T, we define subsets Tk and Tk of T as

follows: Tk := {t T : t = supT or t is left dense},Tk := {t T : t = infT or t is right dense}.

The delta derivative of f : T R at t Tk is then defined as the number f(t)(provided it exists) with the property that given any > 0, there is a neighborhood

U of t such that

|f((t)) f(r) f(t)((t) r)| |(t) r| for all r U,

cf. [19, 21, 38, 40]. Iff(t) exists for all t Tk, we say that f is -differentiableon Tk. The nabla derivative f(t) of f at a point t Tk (provided it exists) isthe number satisfying:

For any given > 0, there is a neighborhood U of t such that

|f((t)) f(r) f(t)((t) r)| |(t) r| for all r U.

If f(t) exists for all t Tk, we say that f is -differentiable on Tk, cf. [12, 21].Integration: There are many different ways to define integration on a time scale.

For example, we can use the Cauchy, Riemann or Lebesgue integrals, among other

concepts. Of most importance to this work is the Cauchy integral. First, for a

function f : T R, we say F is a -antiderivative of f on T, if F(t) = f(t)for all t Tk. Hilger, in [38], shows that any rd-continuous function f on T has a-antiderivative F on T. He then defined the -integral of f byb

a

f(t)t := F(b) F(a),see [19, 38].

Assume that p : T R is rd-continuous and 1 + (t)p(t) = 0 for all t Tk.Consider the initial value problem

y(t) = p(t)y(t), y(t0) = 1. (0.0.1)

Hilger showed that (0.0.1) has a unique solution. He denoted this solution by

ep(t, t0) and proved that it is given by

ep(t, t0) = exptt0 log(1 + (r)p(r))(r) r .


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Introduction vii

The classical time scales calculus includes three important special cases: T = R,

T = Z := {k : k Z}, > 0 and T = {qk : k Z}{0}, q > 1. One can check

that in these cases we have f

(t) = f

(t), f

(t) = f(t) and f

(t) = Dqf(t)respectively, where f(t) =

f(t + ) f(t)

is the forward difference operator

with stepsize [30, 49], and Dqf(t) =f(qt) f(t)

t(q 1) is the q-difference operator[11]. There are many examples and applications of time scales, see [ 19, 21].

In this thesis, we propose a generalization of the definition of time scale given by

Stefan Hilger. The natural question here is that why generalization? In fact the

theory outlined above does not include some important dynamical problems. The

first problem is Jackson q-difference operator and the associated Jackson integral.Jackson q-difference operator Dq is defined to be

Dqf(t) =f(qt) f(t)

t(q 1) , t R \ {0} (0.0.2)

where q is a fixed number, normally taken to lie in (0, 1). Here f is supposed to

be defined on a q-geometric set A, i.e. a subset ofR for which qt A whenevert A. The derivative at zero is normally defined to be f(0), provided that f(0)exists, see [3, 9, 24, 25, 41, 42]. In [64] Rajkovic, Marinkovic and Stankovic defined

Dqf(0) by Dqf(0) = limt0

Dqf(t). Abu-Risha, Annaby, Ismail, and Mansour, in

[3], defined it by

Dqf(0) = limk

f(qkt) f(0)qkt

, t = 0, (0.0.3)

provided this limit exists and does not depend on t, see [57, 58]. Jackson also

introduced q-integrals a0

f(t)dqt =

k=0aqk(1 q)f(aqk), (0.0.4)

and a

f(t)dqt =k=1

aqk(1 q)f(aqk), a R, (0.0.5)

provided that the series converge [6, 43]. He then defined

ba

f(t)dqt =

b0

f(t)dqt a0

f(t)dqt, a, b R. (0.0.6)


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Introduction viii

There is no unique canonical choice for the q-integral over [0, ) and the q-integralon R. Following Jackson we define the q-integral over [0, ) to be

0

f(t)dqt = (1 q) k=

qkf(qk), (0.0.7)

[2, 34, 50, 52]. Matsuo [59] defined the q-integral on more general sequences bys0

f(t)dqt = s(1 q)

k=

qkf(sqk), (0.0.8)

where s is a fixed positive number, see also [51, 52]. The bilateral q-integral is

defined in [51] by

ss

f(t)dqt = s(1 q)

k=

qkf(sqk) + qkf(sqk), s > 0, (0.0.9)

provided that the sums of (0.0.7), (0.0.8) and (0.0.9) converge. So, the definitions

in (0.0.8) and (0.0.9) are dependent on s. In [3] a q-calculus based on (0.0.2)-(0.0.6)

was introduced. For example it was proved that the propertyba

f(t)dqt

ba

|f(t)| dqt, 0 a < b < (0.0.10)

is not always true and it holds if a = 0 or a = qk

b, k N for 0 < q < 1.The second important problem which cannot be obtained from the calculus of time

scales, is the forward difference operator

f(t) =f(t + ) f(t)

, t R, (0.0.11)

where is a fixed positive number, see [17, 22, 45, 46, 49, 54]. The associated

integral of (0.0.11) is the well known Norlund sum

f(t)t =

j=0

f(t + j), (0.0.12)

see [32, 45, 61]. This sum is called in [46] indefinite sum and Carmichael [26] called

it just a sum.

Another problem not included in Hilgers treatment is Hahn difference operator

Dq, which is defined by

Dq,f(t) =f(tq+ ) f(t)

t(q 1) + , (0.0.13)


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Introduction ix

where q (0, 1) and > 0. This operator was introduced by Hahn [35] in (1949),see also [8, 28, 29, 36, 53, 62]. Some recent literature have applied these operators

to construct families of orthogonal polynomials as well as to investigate someapproximation problems, cf. [8, 23, 27, 29, 33, 53, 55, 62]. However, no attention

was given to the construction of the right inverse of (0.0.13) and the associated

calculus, which we intend to do in Chapter 6. The Dq, includes as particular

cases the q-difference operator in (0.0.2) and the forward difference operator with

step in (0.0.11). For the best of our knowledge there is no associated integral

for (0.0.13), and in this work we investigate one, see Chapter 6.

Ahlbrandt et al., [5], gave a definition for a generalized time scale as follows:

A time scale is a pair T, such that

(i) T is a nonempty subset ofR such that every Cauchy sequence in T converges

to a point ofT with the possible exception of Cauchy sequences which con-

verge to a finite infimum or finite supremum ofT, and

(ii) is a function that maps T into T.

See also [18, 19, 21]. We denote by

T

k

:= T \ {t T : (t) = t and t is isolated}.Assume that f is a real valued function defined in T. If there exists a number

f(t), called alpha derivative off at t Tk, with the property that for every > 0there is a neighborhood U of t such that

|f((t)) f(r) f(t)((t) r)| |(t) r| for all r U,

then f is said to be alpha differentiable (or shortly -differentiable) at t. Assume

that T is any closed subset ofR and is the forward (backward) jump operator

(). Then f = f (f = f) where () are the -derivative (-derivative)in the classical time scales calculus. Also, this concept can also accommodate the

case of differential-difference equations by taking T = R with two jump operators

(t) = t and (t) = t + 1 which gives rise to differential-difference operator (f)

[19]. For the theory of differential-difference equations, see [15].

The linear alpha dynamic equation of the form

y(t) = p(t)y(t),


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Introduction x

was considered in [18], where p : T R with 1 +p(t)((t)t) = 0. Also, ep(t, t0)denotes the solution of the initial value problem

y(t) = p(t)y(t), y(t0) = 1,

if this solution exists and unique. ep(t, t0) is called here the generalized exponential

function, see [18, 19, 21].

In [5], Ahlbrandt et al. wrote

One gap in our development has been in generalizing the result that an rd-

continuous function has a -antiderivative. Perhaps some type of a class of -

continuous functions would have the analogous property of having -antiderivatives.

To the best of our knowledge, this question has not been resolved yet.

In this thesis, we address this problem from a different prospective. To achieve our

goal we generalize and extend time scales calculus to include the three mentioned

above problems together with the case of differential-difference equations. We

use the terminology and notations of the classical time scales calculus as far as

possible. Nonetheless, we opted to use forward and backward instead of right

and left. We generalize the concepts of limits and continuity in the classical real

analysis by introducing dense limit (d-limit) and dense continuity (d-continuity).

Summary of the thesis:

In what follows, we try to summarize the development and the organization of this

thesis.

In Chapter 1, the first section is devoted to our definition of a (generalized) time

scale. As in the classical definition, we assume that T is a nonempty closed subset

of R. In addition, T is equipped with a special type of equivalence relation E

which is called arranged. This equivalence relation defines a set of points, called

special. The set of special points is denoted by S. The triple T, E , is calleda time scale, where S {infT, supT}. The second section investigates theproperties of a time scale. In Section 1.3, we define and study the concepts of

dense limits, regularity, dense continuity, forward dense continuity and backward

dense continuity. We give the relation between limit and dense limit as well as the

continuity and dense continuity.

The first section of Chapter 2 develops the notion derivative with respect to

our (generalized) time scales and establishes the main properties of differentition.


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Introduction xi

The Leibniz formula, chain rules and a mean value theorem are studied in Sec-

tions 2.2, 2.3 and 2.4 respectively. Section 2.5 is devoted to the definition of

derivative with its main properties and results.In Chapter 3, we define the Cauchy and -integrals which generalize theCauchy and integrals in the classical time scales calculus.

In Chapter 4, we define the exponential function of a time sale T, E , andwe establish some of its properties. In Section 4.2, we study the existence and

uniqueness results for the equation y = f(t, y).

Chapter 5 includes some applications and examples of time scales. In the first

section, we introduce a general example of a time scale, we call this example

a difference time scale. The qJackson difference (on R), forward difference andHahn difference operators (on R) are included as special cases of the derivativeof this example. Some new examples are also given in Section 5.4.

The final chapter, Chapter 6, is devoted to investigate what we call the Hahn

difference calculus which is based on the Hahn difference operator, cf. (0.0.13).

The main results of this chapter are written in such a way to look independent

from the language of time scales we introduced in earlier chapters. However, these

results were obtained through the use of our time scales calculus.

The concepts and results of the time scale constructed in this work coincide with

the well known classical time scale if E is the universal equivalence relation and

= supT. So, we can say that the classical time scale is a special case of the

(generalized) time scale.


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List of Figures

1.1 The relations >, and . . . . . . . . . . . . . . . . . . . . . . . 31.2 An Arranged family F . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Classifications of points . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Forward dense and backward dense limits . . . . . . . . . . . . . . . 29

5.1 The iteration of(t) = qtn, t R, n 2N + 1, 0 < q < 1 . . . . . . 1306.1 The iteration ofh for 0 < q < 1, > 0 . . . . . . . . . . . . . . . . 135

6.2 The function f(x) for 0 < q < 14

, > 0 and x I . . . . . . . . . . 144

xii


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Notations

N := {1, 2, }N0 := N {0}Z := N N0R is the set of all real numbers.

Q is the set of all rational numbers.

T is a nonempty closed subset ofR.

A is the set of all limit points of A R.A is the closure set of A R.A B A is right contiguous to B, i.e. infA = sup B; A, B T.A > B A is to the right of B, i.e. infA sup B; A, B T.A B A is alternating with B; A, B T, if for every x A (x B), there exist

y, z B (y, z A) such that (y, z) (A B) = {x}.A B A is not alternating with B; A, B T.E is an equivalence relation on T.

[t] is the equivalence class that contains t T.[a, b] := {x T : a x b or b x a} for a, b T (a closed interval ofT).(a, b) :=

{x

T : a < x < b or b < x < a

}for a, b

T (an open interval ofT).

[a, b]c := [a, b] [c] for a,b,c T.(a, b)c := (a, b) [c] for a,b,c T.Tc :=

[t] T : c [t] for c T (the domain set of c T).

S :=

s T : Ts = [s]

(the set of special points ofT).

S {infT, supT}.c is the nearest point in [c] from .

c is the farthest point in [c] from .

I is the family of all nonsingular intervals ofT, where I

I iff

I = {[t] T : [t] = {t}, inf[t] = infI, and sup[t] = sup I}.xiii


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Notation xiv

[k]q :=1 qk1

1 q , 0 < q < 1, k N.O

{x

}:=

{n(x) : n

N0

} {n(x) : n

N0

}, x

T.

d-L(A) := {t T : t (A Tt)}, A T(the set of all dense limit points of A).

fd-L(A) := {t T : t (A Tt (, t))}, A T(the set of all forward dense limit points of A).

bd-L(A) := {t T : t (A Tt (t, t))}, A T(the set of all backward dense limit points of A).

is the forward jump operator on T.

is the backward jump operator on T.

is the forward stepsize function on T.

is the backward stepsize function on T.

f is the delta derivative (-derivative) of a function f.

f is the nabla derivative (-derivative) of a function f.


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Chapter 1

A Generalized Time Scale

The year 1988 had witnessed the birth of the time scales calculus. In order to

create a theory that unifies and extends the discrete and continuous analysis, Ste-

fan Hilger in [37] introduced the time scales calculus which will be referred in the

sequel as the classical time scales calculus. In this thesis, we attempt to widen

the scope of the time scales calculus to include some examples such as qJacksonoperator with q-difference equations on R, Norlund sum with difference equations

on R, and Hahn difference operator.

In this chapter, we introduce notations, concepts and terminology of a new ap-

proach of time scales. Roughly speaking, a time scale is a nonempty closed set

T R equipped with an equivalence relation E TT. We will see that one canobtain a variety of time scales from a single nonempty closed subset T by changing

E. We select a certain type of equivalence relations that suits our purpose. After

giving the definition of generalized time scales, we study some of their properties

in Section 1.2. In Section 1.3, we introduce new types of limits and continuity.

Some examples are given throughout this chapter. In the end of the chapter, we

summarize the main results of the chapter.

1.1 Basic definitions and terminology

In the following, we generalize the classical definition of a time scale [37], see

Definition 1.1.9. We start with some definitions and introduce notation and ter-

minology that will be used in the sequel.

1


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A Generalized Time Scale 2

Assume that T is a nonempty closed subset ofR and E is an equivalence relation

on T. Suppose also that T has the topology inherited from the standard topology

on R. A, as usual, denotes the closure of A and A

the set of all limit points of A,A T.For a, b T {infT, supT} and c T, let

Tc :=

[t] T : c [t], [a, b] := x T : a x b or b x a, [a, b]c := [a, b] [c],

S := s T : Ts = [s],

where [t] denotes the equivalence class that contains t T. The points ofS arecalled special and Tt is called the domain set of t. We define open and half-open

intervals similarly. Whenever we say a neighborhood ofc in a set A T we meanthe set A U, where U is a neighborhood of c in R. In particular, if A = T wesay a neighborhood of c without mentioning T.

One can see that:

Tc = [c] t T : c [t] \ [t] = t T : c [t] for c T. (1.1.1)Notice also that, in this setting, if a {, }, then a / [a, b] and in this case[a, b] = (a, b].

Definition 1.1.1. A class [t] is called singular if [t] = {t}, otherwise it is callednonsingular. An interval I of T is called nonsingular if I is the union of some

nonsingular classes which have the same infimum and supremum, i.e.

I = [x] T : [x] = {x}, inf[x] = infI and sup[x] = sup I.Finally, we denote by I the family of all nonsingular intervals ofT.

Note that I is a family of pairwise disjoint sets. Indeed, ifI, J I and x IJ,then [x] = {x}, [x] I and [x] J such that inf[x] = infI = infJ and sup[x] =sup I = sup J which implies that I = J. Note also that by (1.1.1), a point s Tis special iff there is t T such that s [t] \ [t]. That is S = tT[t] \ [t].


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_? _?

_? _? A > B

B A _?

_? A > B and A

B

B A _?

_? _? _? B > A and B A

A B

A B and A B A, B

(a) The relations and >

x A

y B z B

(y, z) (A B) = {x}

x B

y A z A

(y, z) (A B) = {x}

A, B

A B

A B B, A(b) The alternating relation

Figure 1.1: The relations >, and

Definition 1.1.2. Let A, B be subsets ofT. We say that:

A is right contiguous to B, and denote it by A B, if A = B and infA =sup B.

A is to the right ofB, and denote it by A > B, ifA = B and infA sup B.

A is alternating with B, and denote it by A B, if for every x A (x B)there exist y, z B (y, z A) such that (y, z) (A B) = {x}. We writeA B, if A is not alternating with B.

See Figures 1.1.

We note that in the previous definition the relations > and are neither sym-metric nor reflexive, > is transitive,

is not transitive, while is symmetric


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sup A = supT, B A _? _? _? _? _? _?A B

infA= infT, A

B _? _? _? _? _? _?

AB

(a) Condition (i) on A F

A > B _?

_? _?

_? _?

_?

B A

B > A _?

_? _?

_? _?

_?

A B

A B A, B

(b) Condition (ii) on A, B F, A = B

Figure 1.2: An Arranged family F

and transitive. For A = B and A B, there are no a1, a2 A such that(a1, a2) B = . One can check that

R0 R0, Z>0 > Z B, B > A or A B.

See Figure 1.2.

Example 1.1.4. Let T = R,

F1 :=

An := (n, n + 1) : n Z

and

F2 := Bm,n := m +n 1

n, m +

n

n + 1 : m, n N.


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Then, we can see that

An+1 An, n Z and An > Am, n, m Z, n > m.

That is F1 is arranged, but F2 is not. This is because for any m N, there are noB F2 such that Bm,1 B.

Definition 1.1.5. An equivalence relation E on T is called arranged if the family

of all equivalence classes is arranged, i.e. satisfies the following conditions:

(A1) For any class A with sup A

= supT (infA

= infT) there exists a class B

such that B A (A B).(A2) For any different classes A, B either A > B, B > A or A B.

The universal equivalence relation E = TT is arranged while the identity relationE = {(x, x) : x T} is not whenever T contains more than one element.In the following, we give more examples of equivalence relations which are not

arranged.

Example 1.1.6. Assume that T = A B C where

A =

0

, B =

1

, C =

1/22k

: k Z k/2 : k N+ 2.We define the equivalence relation E T T by

xEy {x, y} A, {x, y} B or {x, y} C.

Note that T0 = A

C, T1 = B

C and Tt = C for t

C. The relation E is not

arranged, since neither B > C, C > B nor B C.

Example 1.1.7. For fixed q R; 0 < q < 1. Let T = R0 and

xEy x = y and k Z such that x = qk y , x, y R.

Here x is the greatest integer function and x = x x. We claim that thisrelation is an equivalence relation on R0. For x,y,z R0, we have:

Reflexivity: xEx, since x = x and x = q0 x.


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Symmetry: ifxEy, then x = y and there exists k Z such that x = qk ywhich implies y = qk x, that is yEx.

Transitivity: if xEy and yEz, then x = y = z and there exist k1, k2 Zwith x = qk1 y and y = qk2 z, then x = qk1+k2 z, that is xEz.

Therefore, E is an equivalence relation. Easily, one can check that

[t] =

{s = t + qk t : k Z and s = t}, if t R0 \ N0,

{t}, if t N0,

Tt = [t], if t R0

\N

0,

[t, t + 1), if t N0.

In this example, we note that ifx, y T, x = y and x = y, then inf[x] = inf[y]and sup[x] = sup[y]. Thus, neither [x] > [y], [y] > [x] nor [x] [y]. Therefore, Eis not arranged.

Example 1.1.8. Assume that T = R. The relation E on T defined by

xEy x y Q,

is an equivalence relation. Easily, one can check that

[t] =

Q, if t Q,t + Q, if t R \Q.

For t1 Q and t2 R \Q, we have inf[t1] = inf[t2] = and sup[t1] = sup[t2] =, and hence neither [t1] > [t2] nor [t2] > [t1]. Also, [t1] [t2], since for t [t1],there are no y, z [t2] such that (y, z)([t1] [t2]) = {t}. Therefore (A2) propertyis not valid, and hence E is not arranged.

We are now in a position to give a definition that widens the scope of time scales

and preserves the classical ones as special cases.


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Definition 1.1.9. A triple T, E , is called a time scale ifT is a nonemptyclosed subset of R, E is an arranged equivalence relation on T and S {infT, supT}.

Throughout this work, one can see that all concepts (jump operators, derivatives,

Cauchy integrals, etc.) with respect to the triple T, E , where T R is anonempty closed subset, E is the universal equivalence relation and = supT co-

incide with the corresponding ones with respect to the classical time scale, defined

by Stefan Hilger [19, 21, 3739].

Remark 1.1.10. For more generality, we can replace the condition of closedness

of T by the following condition: T is a nonempty subset of R such that every

Cauchy sequence in T converges to a point of T with the possible exception of

Cauchy sequences which converge to a finite infimum or finite supremum of T.

See [5].

Example 1.1.11. Consider

T, E ,

, where

T = Z +1

N:=

z+1

n: z Z and n N,

E is defined by

xEy z Z, n,m N such that x = z+ 1n

, y = z+1

m(1.1.2)

or equivalently

xEy

zZ such that x, y

(z, z+ 1]

and Z {, }. One can see that T is closed and E is an equivalencerelation. For z Z we have

[z] =

z n 1n

: n N

=

z, z 12

, z 23

,

= (z 1, z],

Tz = [z]

[z+ 1] = (z

1, z+ 1],


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and Tt = [t] for all t T \ Z which imply S = Z and z = sup[z] = inf[z + 1].Consequently, for z Z we have

T =tZ

[t] and [z+ 1] [z].

For z1, z2 Z, z1 > z2 we note that [z1] > [z2]. Hence E is arranged. ThereforeT, E , is a time scale for any Z {, }.

Example 1.1.12. Let T be defined as in Example 1.1.11 and E be the equivalence

relation on T defined by

xEy z Z such that x, y [z, z+ 1),

and {, }. For z Z we have

[z] =

z, z+1

2, z+

1

3,

=

z, z+1

2

.

Here Tt = [t] for all t T, that is S = . For z1, z2 Z, z1 > z2, we have [z1] > [z2],that is E satisfies property (A2). Since z+ 1

2= sup[z] < inf[z+ 1] = z+ 1, z Z,

then there is no z Z such that [z] [z]. Thus, (A1) is not true and henceT, E , is not a time scale.

Note that on any nonempty closed subset T ofR, we can obtain as many time

scales as the number of arranged equivalence relations defined on T and the number

of values of S {infT, supT}. represents a point where the time changesits direction. For more examples, see Chapter 5.

One of the main properties of a time scale is the following.

Theorem 1.1.13. Assume that T, E , is a time scale and x, y T, x > y.Then, there is a sequence of nonsingular intervals {Ii}ni=1 such that

I1

I2

In, x

I1 and y

In. (1.1.3)


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In addition, either [x, y] S = , or

[x, y]S = {si}m

i=1, m N such that [x, s1] [s1, s2] [sm1, sm] [sm, y].(1.1.4)

We postpone the proof of this theorem to Section 1.2. The following lemma will

be needed in order to define jump operators. Also, it plays an essential role in this

work.

Lemma 1.1.14. In a time scale T, E , , either

[x] [, supT] or [x] [infT, ], x T. (1.1.5)

Proof. First, we can see that {inf[], sup[]}. Indeed, assume the contrary:i.e. (inf[], sup[]). Then / {infT, supT}, that is S \ {infT, supT}.Thus, there is c T\[], i.e. [c]\[c]. Since [c], then (inf[], sup[])[c] =. Hence, neither [] > [c] nor [c] > []. By property (A2), []

[c]. Then there

are y, z [c] such that (y, z) ([] [c]) = {}, which contradicts [c].Therefore {inf[], sup[]}.Let x T. We distinguish between the following three cases:First case: x = . Statement (1.1.5) is true, since {inf[], sup[]}.Second case: x > . If x [], then [x] = []. Hence (1.1.5) is true, by the firstcase. If inf[x] , thus [x] [, supT]. When inf[x] < , neither [] > [x] nor[x] > []. So, [x] [], by property (A2). Recall that {inf[], sup[]}. For = inf[] ( = sup[]), we have t [inf[x], )x (t (, sup[x] ]x). Then there areno y, z [] such that (y, z) ([] [x]) = {t}. This leads to a contradiction with[] [x]. Therefore [x] [, supT].Third case: x < . In a similar way, we can show that [x] [infT, ].

By the previous lemma, we can define the following operators.


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Definition 1.1.15. Let T, E , be a time scale. We define the jump operators, : T T as follows:The forward jump operator

(t) =

sup{z [t] : z < t}, if [t] [, supT], t = inf[t],inf{z [t] : z > t}, if [t] [infT, ], t = sup[t],t, otherwise.

The backward jump operator

(t) =

inf{z [t] : z > t}, if [t] [, supT], t = sup[t],sup{z [t] : z < t}, if [t] [infT, ], t = inf[t],t, otherwise.

We also define the following functions:

The forward stepsize function : T R:

(t) = (t)

t.

The backward stepsize function : T R:

(t) = t (t).

We can see that for t T \ {}:

(t) =

t inf{|t z| : z (t, )t}, if [t] [, supT],t + inf

{|t

z

|: z

(t, )t

}, if [t]

[infT, ],

(1.1.6)

(t) = r iff r [t, ]t and |t r| = inf{|t z| : z (t, )t}, (1.1.7)

and for t T we have

(t) =

t + inf{|t z| : z [t] \ [t, ]}, if [t] [, supT],t inf{|t z| : z [t] \ [t, ]}, if [t] [infT, ], (1.1.8)

(t) = r iff r [t] \ (t, ) and |t r| = inf{|t z| : z [t] \ [t, ]}. (1.1.9)


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t

(t)

t

(t)

t is forward scattered

t = (t) t = (t) t is forward dense

(t)

t

(t)

t

t is backward scattered

t = (t)

t = (t)

t is backward dense

Figure 1.3: Classifications of points

For a function f : T R, we denote by f

(t) := limn fn

(t) whenever thelimit exists. If f1 exists, then f(t) := (f1)(t). Here, fn denotes the n-th

iteration of f and we put f0(t) := t.

Definition 1.1.16. For t T, we say that:

t is forward scattered if (t) = t, t is forward dense if (t) = t,

t is backward scattered if (t) = t, t is backward dense if (t) = t, t is isolated if (t) = t = (t), and t is dense if (t) = t = (t).

Remark 1.1.17. Note that = inf[] ( = sup[]) when T and [] [, supT] ([] [infT, ]). Thus, we have always () = while it is not truein general for (). We should notice that and are not necessarily inverse to

each other. For example, ((t)) = t ift is backward dense and forward scattered.Notice also that (t) = t if t is forward dense and (t) = t if t is backward

dense. Finally, note that if t is isolated, dense, not isolated or not dense in the

time scale T, E , 0 for some 0 S {infT, supT}, then it is of the same typein any time scale T, E , , S {infT, supT}.

Examples 1.1.18. On the following time scales T, E , , we find and :


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(i) T is a closed subset ofR and E is the universal equivalence relation. In this

case, we have only one equivalence class T, and hence S = . Also, T is the

only nonsingular interval when T has more than one point.

For = supT, we have:

(t) = inf{z T : z > t} and (t) = sup{z T : z < t}

which are the forward and backward jump operators in the classical time

scale calculus respectively.

For = infT, we have:

(t) = sup{z T : z < t} and (t) = inf{z T : z > t}.

That is to say that and interchange their roles when changes between

infT and supT.

(ii) T = R, E is defined by

x E y z Z such that x, y (z, z+ 1],

and Z {, }. It can be seen that

[z] = (z 1, z], z Z and R =zZ

(z 1, z].

Easily, we can check that this relation is arranged, I = {(z, z+ 1] : z Z}and S = Z. Note that (t) = (t) = t for all t T, that is all points are

dense.

(iii) T = R, E is defined by

x E y k Z such that x = y + k,

and {, }. Here is a fixed positive real number. This equivalencerelation is arranged. In this time scale, T is the only nonsingular interval of

T, and S = .


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If = , then we have (t) = t + and (t) = t for all t T. Hence, allpoints are isolated. When = , then for any t T we have (t) = t,(t) = t + and t is isolated. We study this example in detail in Section 5.3.

(iv) T = A +Z := {x + z : x A and z Z} where A is a closed subset of [0, 1],E is defined by

x E y x y Z,

and {, }. E is an arranged equivalence relation, and I = {T},S = . If =

, then (t) = t + 1 and (t) = t

1. If =

, then

(t) = t 1 and (t) = t + 1. Thus, all points ofT are isolated.

Definition 1.1.19. Let T1, E1, 1 and T2, E2, 2 be time scales. Then, we saythey are equivalent ifT1 = T2 and 1 = 2, where 1 and 2 are the forward jump

operators in T1, E1, 1 and T2, E2, 2 respectively.

Examples 1.1.20. (i) Assume that T is a closed interval of R and E is the

universal equivalence relation on T. Then one can check that (t) = (t) = t for

both time scales T, E, infT and T, E, supT. So, they are equivalent.(ii) The time scale T, E , where T = R, E is the universal equivalence relation,and = infT ( = supT) is equivalent with the time scale of Example 1.1.18(ii).

1.2 Preliminary results

In this section, we study some properties of time scales that will be needed in

later sections and chapters. In Theorem 1.2.7, we give necessary and sufficient

conditions for a triple T, E , to be a time scale, where T is any closed subsetofR, E is an equivalent relation and S {infT, supT}. In the beginning, weprove some preliminary lemmas.

The following lemma gives us equivalent statements for a class [t] to be alternating

with itself.


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Lemma 1.2.1. Let T, E , be a time scale and t T. Then the followingstatements are equivalent:

(i) [t] [t].(ii) [t] = {inf[t], sup[t]} T and both inf[t], sup[t] / [t].

(iii) All points of [t] are isolated.

(iv) [t] = { , 2(t), (t), t , (t), 2(t), } whose elements are distinct.(v) [t] [d] for some d T.

Proof. (i)=(ii) Assume that [t] [t]. First, we show that [t] {inf[t], sup[t]}T.

Assume that inf[t] T\[t], then inf[t] = min[t] [t]. Hence, there are no y, z [t]such that (y, z) [t] = {inf[t]} which contradicts (i). That is {inf[t]} T [t].Similarly, {sup[t]}T [t]. Now, we show that [t] {inf[t], sup[t]}T. Supposefor the sake of contradiction that x [t] \ {inf[t], sup[t]}, then x (inf[t], sup[t]),in view of [t] [inf[t], sup[t]]. Then inf[x] < sup[t] and sup[x] > inf[t], andhence neither [x] > [t] nor [t] > [x]. Also, there are no y, z [t] such that

(y, z) ([x] [t]) = {x}, since x [t]

. That is [x] [t]. This contradictsthe arrangement of E. Hence, [t] = {inf[t], sup[t]} T. Now, if inf[t] (sup[t])is an element of [t], then there exist y, z [t] such that (y, z) [t] = {inf[t]}((y, z) [t] = {sup[t]}), which is impossible. Therefore inf[t] / [t] and sup[t] / [t].

(ii)=(iii) Assume the contrary, i.e. there is x [t] which is not isolated. Thenby definition of and , we have x [t] or x {min[t], max[t]}. But [t] [t] =

and [t] has neither maximum nor minimum. Then x [t]

which is a contradiction.

Therefore all points of [t] are isolated.

(iii)=(iv) Assume that all points of [t] are isolated. If inf[t] [t] (sup[t] [t]),then by the definition of (definition of ), we get inf[t] (sup[t]) is not isolated.

Hence, [t] (inf[t], sup[t]). We claim that (x), (x) [t] for every x [t]. For thesake of contradiction, assume that (x) / [t] for some x [t]. Then, (x) [t],


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that is (x) [inf[t], sup[t]], in view of [t] (inf[t], sup[t]). Then there is an infi-nite sequence of points in ((x), x) [t] which contradicts definition of . Hence,(x) [t]. Similarly, we can show that (x) [t]. Then, n(x), n(x) [t] forevery n N, x [t]. Therefore, O{x} := {n(x), n(x) : n N0} [t]. Clearly,all points of O{x} are distinct (if not, this leads to a contradiction with (iii)) and{(t), (t)} = {inf[t], sup[t]}. Finally, let z [t]. That is z (inf[t], sup[t]) =n=0

[n(t), n+1(t)] [n(t), n+1(t)]. But, for every n N0, z / (n(t), n+1(t))

and z / (n(t), n+1(t)) (by definition of and ), and thereby z O{t}. There-fore, [t] is equal to the set of distinct elements { , 2(t), (t), t , (t), 2(t), }.

(iv)=(v) It is enough to show that [t] [t]. First, for n N, we denoteby n := n. Let x [t]. Then there is n Z such that x = n(t). Thus,n1(t), n+1(t) [t] and (n1(t), n+1(t)) [t] = {x}. Therefore [t] [t].

(v)=(i) Since the relation is symmetric and transitive, then [t] [t].

Lemmas 1.2.2, 1.2.3, 1.2.4, 1.2.5 describe some properties of alternating classes,general classes, special points and nonsingular intervals of a time scale, respec-

tively.

Lemma 1.2.2. Let T, E , be a time scale and t T be such that [t] [t]. Thefollowing statements are true.

(i) If [t] [d] for d T, then [t] = [d].(ii) t /

[c] for every c

T.

(iii) Tt = [t].

(iv) (t) S {} and (t) S {infT, supT}.(v) If c (inf[t], sup[t]), then [c] [t].

Proof. (i) Since [t] [d], then [d] [d]. By Lemma 1.2.1, we need to show thatinf[t] = inf[d] and sup[t] = sup[d]. Suppose the contrary, i.e. inf[t] = inf[d]

or sup[t] = sup[d]. We may assume that inf[t] > inf[d], then there is a [d]


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such that inf[d] < a < inf[t]. Since [t] [d], there are y, z [t] such that(y, z) ([t] [d]) = {a} which is a contradiction. Similarly, we get a contradictionif sup[t] = sup[d]. Therefore [t] = [d].

(ii) By Lemma 1.2.1 (ii), t / [t]. Assume that t [c] for some c T. If t [c],then [t] = [c] and t [t] which is a contradiction, and thereby t / [c]. Also,we have t [inf[c], sup[c]], and again by Lemma 1.2.1 (ii), t / {inf[t], sup[t]}. If[c] > [t] ([t] > [c]), then t = inf[c] = sup[t] (t = inf[t] = sup[c]) (Contradic-

tion!). Hence by property (A2), [t]

[c]. Thus by part(i),

{inf[t], sup[t]

}= [t] =

[c] = {inf[c], sup[c]}. That is t {inf[c], sup[c]} = {inf[t], sup[t]} (Contradiction!).Consequently, t / [c] for every c T.

(iii) Assume the contrary, i.e. Tt \ [t] = . Then there is c Tt \ [t], that ist [c] \ [c]. This is a contradiction with part(ii) and hence Tt \ [t] = .

(iv) Assume that [t] [t] and a := (t) = (b := (t) / {infT, supT}).

Thus, a T (b T), since T is closed. Now, a [t] \ [t] (b [t] \ [t]). Therefore,a S (b S).

(v) Let c (inf[t], sup[t]). Ifc [t], then it is clear that [c] [t]. When c / [t],neither [c] > [t] nor [t] > [c]. By the arrangement of E, we get [c] [t].

The following lemma indicates possible forms and some properties of the classes

of E.

Lemma 1.2.3. Let T, E , be a time scale and t T. Then we have:(i) [t] is of the following types (here [t] can take two types at the same time):

1(T1) [t] is an interval ofT.

1(T2) [t] = {t}.1(T3) [t] = { , 2(t), (t), t , (t), 2(t), } whose points are distinct.

(ii) [t] = [(t)] = [(t)].


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(iii) If [t] is nonsingular, then there exists a unique nonsingular interval I such

that [t] I.(iv) If sup[t] = supT (inf[t] = infT), then sup[t] S (inf[t] S).(v) If [t] is nonsingular, then [t] =

[t] {inf[t], sup[t]} T and [t] = I

I {infI, sup I} where I is the nonsingular interval which contains t.

Proof. (i) Assume that [t] is neither a singleton nor an interval ofT. There exists

x T such that x (inf[t], sup[t]) \ [t]. Then neither [x] > [t] nor [t] > [x], andhence by property (A2), [x] [t]. By Lemma 1.2.2, [t] is of type (T3).

(ii) By part(i), we have three types of classes. For the second and third types the

proof of (ii) is trivial. Let [t] be an interval ofT. Clearly, either (t) [t, sup[t]]or (t) [inf[t], t]. Let (t) [t, sup[t]]. The case (t) = sup[t] / [t] leadsto a contradiction with the definition of . Consequently, (t) [t, sup[t]) or(t) = sup[t] [t] which implies that (t) [t], since [t] is an interval. Also, when(t) [inf[t], t] we get (t) (inf[t], t] or (t) = inf[t] [t]. Hence, (t) [t].Similarly, one can prove that (t) [t].

(iii) Let [t] = {t}. By part(i), we have two cases. First case: [t] is an interval.Putting, I = [t] and hence I I which is the unique nonsingular interval con-taining [t]. Second case: [t] [t]. In this case, we choose I = ((t), (t)).Therefore [t] I I and this nonsingular interval is unique.

(iv) Assume that sup[t] = supT. Applying property (A1), there is [y] T suchthat [y] [t], i.e. inf[y] = sup[t]. Hence, sup[t] [t] [y], and so [t] [y] Tsup[t].Therefore Tsup[t] = [sup[t]], that is sup[t] S. In a similar way, we can show thatinf[t] S when inf[t] = infT.

(v) By part(i), we see directly that [t] = [t] {inf[t], sup[t]}T. By part(iii), thereexists I I such that [t] I. Now, ifI = [t], the proof of this case follows

directly from part(i). Assume now that I = [t] and let r I\ [t], then by property


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(A2), [r] [t]. Thus [t] = [r] = I = {inf[r], sup[r]} T = {infI, sup I} T, inview of Lemma 1.2.1 (ii) and Lemma 1.2.2 (i).

Lemma 1.2.4. Let T, E , be a time scale. Assume thatS = . Then:(i) E is not the universal equivalence relation.

(ii) For s S, Ts = {I I : s I\ I} [s].(iii) For s S, eithers = infI = sup[s] or s = sup J = inf[s] for someI, J I.(iv) If infS = infT (sup S = supT), then either [infT, infS) or [infT, infS]

((sup S, supT] or [supS, supT]) is a nonsingular interval.

(v) If [t] = {t}, then t S \ I.

Proof. (i) Assume that s S. Then there is y T such that s [y] \ [y]. Thisimplies that E has more than one class, i.e. E is not the universal equivalence

relation.

(ii) Let s S and y T be such that s [y] \ [y]. Then [y] = {y}. Let I bethe nonsingular interval which contains [y], in view of Lemma 1.2.3 (iii). Now, we

show that s I\ I. If I = [y], then s I\ I. Let x I and [x] = [y]. It is easilyseen that neither [x] > [y] nor [y] > [x] and hence by property (A2), [x] [y]. ByLemma 1.2.2 (i) we have s [x] \ [x]. Consequently, s I\ I. On the other hand,let t I, I I and s I\I. Then, either s = infI = inf[t] or s = sup I = sup[t].Therefore, s [t] \ [t] and t Ts.

(iii) Let sS. First case: there is I

I such that s

I

\I and s = infI /

I.

Second case: there is J I such that s J \ J and s = sup J / J. In thefirst case, s = infI, by arrangement of E, [x] > [s] for every [x] I. That iss = infI sup[s] s which implies s = infI = sup[s]. Similarly, in the secondcase we see that s = sup J = inf[s].

(iv) Assume that infT = infS. Then (infT, infS) = . Let x (infT, infS). By

Lemma 1.2.3(i), we have three cases:


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First case: [x] is a nonsingular interval of T. We claim that inf[x] = infT and

sup[x] = infS. Note that if inf[x] > infT, then by Lemma 1.2.3 (iv), inf[x] S(Contradiction!). That is inf[x] = infT. Now, if sup[x] < infS, then by Lemma

1.2.3 (iv), sup[x] S (Contradiction!). Finally, if sup[x] > infS, then there iss S with s (inf[x], sup[x]) [x] and by part(ii) either s = infI or s = sup Ifor some I I. Hence, I [x] = which is impossible. Thus our claim istrue. Therefore [x] is one of the intervals [infT, infS) and [infT, infS] according

to whether infS belong to [x] or not.

Second case: [x] = {x}. This implies that x S, by Lemma 1.2.3 (iv). Thiscontradicts S (infT, infS) = . Hence, [x] is nonsingular.Third case: [x] [x]. By Lemma 1.2.3 (iv), either inf[x] S (infT, infS) orinf[x] = infT. But since S[infT, infS) = , then inf[x] = infT = . Similarly,we see that sup[x] = infS. Hence, (, infS) = [infT, infS) I.In a similar way, we show that either (sup S, supT] or [sup S, supT] is a nonsingular

interval when sup S = supT.(v) Since S = , then T = [t] = {t}. Either t = sup[t] = supT or t = inf[t] = infT,and thereby t S, in view of Lemma 1.2.3 (iv). By definition of I, t / I.Therefore t S \ I.

In the following lemma, a detailed description of the nonsingular intervals is given.

Lemma 1.2.5. Assume that T, E , is a time scale and I I. Then,(i) If all points of I are isolated, then I is the union of some alternating classes,

i.e. classes which are alternating with each other.

(ii) If s S I, then I = [s], s {infI, sup I} and s is not isolated.(iii) If infI I (sup I I), then I = [infI] (I = [sup I]).(iv) If infI = infT (sup I = supT), then

infI S and I TinfI (sup I S and I Tsup I).


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(v) S (infI, sup I) = .(vi) If T is not singleton, then I is an arranged countable family of pairwise

disjoint intervals and

T = I = I S.

Proof. (i) Suppose that I I and all points of I are isolated. By Lemma 1.2.1(iii), we have [x] [x] for any [x] I. I f [x] [y] I, then by property (A2),[x] [y].

(ii) Consider s S I. First, we show that s = infI or s = sup I. Assume thecontrary: s = infI and s = sup I. Since s S, there is [x] I, s [x] \ [x]. Thenby Lemma 1.2.1 (ii) and Lemma 1.2.2 (i), [x] [s]. Applying property (A2), either[x] > [s] or [s] < [x], but it contradicts the definition of a nonsingular interval.

Therefore s = infI or s = sup I. Finally, the assumption I\ [s] = leads us toa contradiction with property (A2). Indeed, if I\ [s] = , then there exists t Isuch that t /

[s] and neither [s] > [t], [t] > [s] nor [t]

[s].

(iii) Assume that infI I, I = [infI]. Then there is x I \ [infI]. By thedefinition of a nonsingular interval, [infI] I and inf[x] = inf[infI] = infI. Notethat neither [x] > [infI], [infI] > [x] nor [x] [infI]. This contradicts property(A2). Therefore, I = [infI]. Similarly, we can show that if sup I I, thenI = [sup I].

(iv) Let infI = infT. We have two cases. First case, infI I. By part(iii),I = [infI] TinfI. By (A1), there exists a nonsingular class [x] such that infI =inf[infI] = sup[x]. Then infI [x]\[x], and hence x TinfI\[infI]. Consequently,[x] TinfI \ [infI] and infI S. Second case, infI / I. One can see thatI TinfI \ [infI]. Therefore, infI S. In a similar way, one can prove that ifsup I = supT, then sup I S and I Tsup I.

(v) A direct consequence of part(ii).


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(vi) Since the family of all classes ofT is arranged, we can see easily that I is

arranged. I is a disjoint family, by the definition of a nonsingular interval and

hence it is countable. Any class ofT is either contained in a nonsingular interval or

it is singular, in view of Lemma 1.2.3 (iii) and Lemma 1.2.4 (v). By Lemma 1.2.3

(iv), we find T = I S. Finally, by Lemma 1.2.4 (iii), we have S (I).

The domain sets Tt, t T play an important role in the calculus of time scalessuggested in this thesis. Recall that Tt = [t] iff t / S. The following lemmacombines the most important properties for domain sets.

Lemma 1.2.6. For a time scale T, E , , t T we have the following:(i) ((t), t) Tt = ((t), t) Tt = .

(ii) If t = and [t] [, supT] ([t] [infT, ]), then Tt [, supT] (Tt [infT, ]).

(iii) If t S, then Tt is a neighborhood of t and Tt has only one of the followingforms:

(a) Tt = {t}I whent = infT (t = supT) and [t] is a singular class, whereI I with t = infI (t = sup I).

(b) Tt = {t} I J when infT = t = supT and [t] is a singular class,where I, J I with t = infI = sup J.

(c) Tt = [t] I when infT = t = supT and [t] is a nonsingular interval,where I I with t = sup I = inf[t] (t = infI = sup[t]).

(iv) If c Tt and V is a neighborhood of t in Tt, then V [c] is a neighborhoodof t in [c].

Proof. (i) We may assume that (t) = t, otherwise ((t), t) = . We have either[t] is of type (T1) or (T3), Lemma 1.2.3. Suppose [t] is of type (T1), that is an

interval ofT. To achieve a contradiction, we assume that ((t), t) Tt contains a

point x. By the definition of , we have x / [t] and hence t [x] \ [x]. Applying


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property (A2), we get [t] [x] which implies that t [x] = [t] (Lemma 1.2.2 (i)).This leads to a contradiction with Lemma 1.2.1 (ii). When [t] is of type (T3), then

Tt = [t], by Lemma 1.2.2 (iii), and consequently ((t), t) Tt = . The argumentof (t, (t)) Tt = is similar.

(ii) Let t (, supT] (t [infT, )) and y Tt. Then [y] [, supT] = ([y] [infT, ] = ). By (1.1.5), we have [y] [, supT] ([y] [infT, ]). ThereforeTt [, supT] (Tt [infT, ]).

(iii) Let t S. Consider three cases of [t]:Case 1: [t] is a nonsingular interval ofT. Then by Lemma 1.2.4 (ii),(iii), there

is I I such that either t = inf[t] = sup I or t = sup[t] = infI. Since [t] is anonsingular interval, then t / [c] for any c T \ ([t] I). Thus, Tt = [t] I.Case 2: [t] = {t}. Either t {infT, supT} or t / {infT, supT}. If t = infT(t = supT), then there exists I I such that t = infI(t = sup I) and Tt = {t}I,in view of Lemma 1.2.4 (ii),(iii). If t /

{infT, supT

}, then by property (A2) and

Lemma 1.2.4 (ii),(iii), the statement (c) holds.

Case 3: [t] [t]. By Lemma 1.2.2 (iii), we have Tt = [t] which contradicts ourassumption.

Therefore, statements (a), (b) and (c) hold. In any of the cases (a), (b) and (c),

one can see that Tt is a neighborhood of t. This completes the proof of this part.

(iv) Suppose that c

Tt and V is a neighborhood of t in Tt. Recall that [c]

Tt

and by Lemma 1.2.3 (i), we have [c] = [c] {inf[c], sup[c]}. If t [c], then V [c]is a neighborhood oft in [c]. Now, ift [c] \ [c], then t S and t {inf[c], sup[c]}.In the three cases of part(iii), we can see that our claim is true.

We are now in a position to prove Theorem 1.1.13.

Proof of Theorem 1.1.13. Assume that x, y T, x > y. By Lemma 1.2.5 (vi),

there are I, J I such that x I, y J. There are two cases: I = J or I > J.


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If I = J, then (1.1.3) is true and n = 1. If not, then I > J. We put I1 := I.

By Lemma 1.2.5 (iv), I is arranged, then by property (A1) there is I2 I with

I1 I2. Now, given {I1, , In}, n N, there exists In+1 such that In In+1.So, we construct a sequence {In} I such that I1 I2 . We have thefollowing two possibilities:

1 1) In > J for all n N.1 2) In = J for some n.

Assume, towards a contradiction, that In > J for all n. The sequence {infIn}nNis decreasing and bounded below by sup J, then it converges to a point ofT say z.

Now, there are two cases. Case 1: sup[z] = z. Then [z] = {z}. By using Lemma1.2.3 (iii), there is Iz I containing [z] and hence sup Iz = z. Now, we havez (infIz, sup Iz) and (infIz, sup Iz) ({infIn}nN \ {z}) = which contradictsour definition of z. Case 2: sup[z] = z. Now, by property (A1), there is [t] = {t},[t] [z]. Let It be the nonsingular interval which contains t, in view of Lemma1.2.3 (iii). Then infIt = z, this contradicts the definition of z. Therefore (1.1.3)

is true. Now, if [x, y] S = , then by using (1.1.3) and Lemma 1.2.5 (iv), (v), wehave (1.1.4) is true. .000000000000000000000000000000000000000000000000000

The following result establishes necessary and sufficient conditions for a triple

T, E , to be a time scale.

Theorem 1.2.7. Assume that T is a closed subset of R containing more than

one element, E is an equivalence relation onT and S {infT, supT}. ThenT, E , is a time scale (i.e. E is arranged) iff the familyI of nonsingular in-tervals ofT satisfies:

(C1) I has property (A1).

(C2) For any x T, there exists I I such that [x] I.(C3) For I I, either I consists of one class only or all classes of I are alter-nating.

Proof. The left-to-right direction: By Lemma 1.2.5 (vi), (C1) is true. Let x T.

If [x] is nonsingular, then by Lemma 1.2.3 (iii), there is a unique I I such that


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[x] I. In the case where [x] = {x}, Lemma 1.2.4 (iii),(v), insures property (C2).Property (C3) follows directly from (A2), since for any two different classes A and

B; A, B I I, neither A > B nor B < A.For the right-to-left direction, let F be the family of all classes ofT. To show that

F has property (A1), suppose that A F and infA = infT. We have two cases.First case: A is singular, i.e. A = {a} for some a T. Then by definition ofIand condition (C2), there exists I I such that a = sup I or a = infI. We mayassume that a = sup I and the other case can be treated similarly. Then for any

class B I we have a = inf[a] = sup I = sup B, that is A B. Second case:A is a nonsingular class. By condition (C2), there is I I such that A I,infA = infI and sup A = sup I. Since infI = infA = infT = infI and Isatisfies property (A1), there is J I such that I J. Hence, for any classB of J, A B. Similarly when sup A < supT, there exists B F such thatB A. Therefore F satisfies property (A1). To show that F has property (A2),

let A, B F, A = B. If A and B are singular, then the proof is trivial. If Ais singular and B is not, then by condition (C2), there exists I I such thatB I. By definition of nonsingular intervals, A I. That is A > B or B > A.If A and B are nonsingular, then there are I, J I such that A I and B J.We have the following three cases: case 1: I = J; case 2: I > J; case 3: J > I.

Case 1, by condition (C3), A B. In case 2 and case 3, it is clear that A > Band B > A respectively. Consequently, F satisfies property (A2). Therefore E is

arranged.

Example 1.2.8. For a fixed positive integer n > 1. Assume that the triple

T, E , with T = R0, E is defined by

xEy k Z such that x = ynk for x, y R0,

and {0, 1, }. For x,y,z R0 we have:Reflexivity: xEx, since x = xn0.


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Symmetry: if xEy, then there exists k Z such that x = ynk , that is xnk =

ynk

nk

= y. Thus, yEx.

Transitivity: if xEy and yEz, then there exist k1, k2 Z with x = ynk1 andy = zn

k2 . Hence, x = (znk2 )n

k1 = znk1+k2 , that is xEz.

Therefore, E is an equivalence relation. Note that

[t] = {tnk : k Z} and Tt = [t] for all t T \ {0, 1},

and[0] =

{0

}, [1] =

{1

}, T0 = [0, 1) and T1 = (0,

).

Furthermore, 0 and 1 are the special points. Also, the intervals (0, 1) and (1, )are the nonsingular intervals ofT, since

inf[t] =

0, if 0 < t < 1,1, if t > 1, sup[t] = 1, if 0 < t < 1,, if t > 1.

To show that E is arranged, we use Theorem 1.2.7. First, we have I is arranged.

For x [0, 1], [x] (0, 1) = [0, 1] and for x [1, ], [x] (1, ) = [1, ), i.e.I satisfies condition (C2). All the classes which containing in (0, 1) or in (1, )are alternating, and hence I satisfies condition (C3). Applying Theorem 1.2.7,

T, E , is a time scale.If = , then we have:

(t) =

n

t, if 0 t 1,

tn

, if t 1,(t) =

tn, if 0 t 1,n

t, if t 1.So, that

(t) =

0, if t = 0,

1, if 0 < t 1,, if t > 1,

(t) =

0, if 0 t < 1,1, if t 1.If we choose = 1, then

(t) =n

t, (t) = tn

, for all t T.


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This implies that

(t) = t, ift = 0,1, otherwise, (t) = 0, if 0

t < 1,

1, if t = 1,

, if t > 1.Finally, when = 0, we get

(t) =

tn, if 0 t 1,n

t, if t 1,(t) =

n

t, if 0 t 1,tn, if t 1,

and hence

(t) =

0, if 0 t < 1,1, if t 1, (t) =

0, if t = 0,

1, if 0 < t 1,, if t > 1.

We can see that and are continuous on T.

1.3 Limits and continuity

In this section, we generalize the classical concepts of limits and continuity. Fur-

thermore, we give analogous concepts of right dense and left dense continuity in

the classical time scale calculus. Throughout the remainder of this work, we sup-

pose that T, E , is a time scale. Often, if there is no confusion concerning Eand , we denote T, E , simply by T. When we write limrt we mean t, r T.

Definition 1.3.1. Let A be a subset ofT. A point tT is called:

(i) a forward dense limit point, or shortly a fd-limit point, of A if t T \ {}and it is a limit point of A Tt (, t),

(ii) a backward dense limit point, or shortly a bd-limit point, of A if t T \{, infT, supT} and it is a limit point of A Tt \ [, t], and

(iii) dense limit point, or shortly d-limit point, ofA if it is a limit point of A Tt.


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The sets fd-L(A), bd-L(A) and d-L(A) denote all fd-limit, bd-limit, and d-limit

points of A respectively.

Note that d-L(A) = fd-L(A) bd-L(A), A T.The following lemma investigates some properties of the dense limit points.

Lemma 1.3.2. LetT {t}. Then the following statements hold:(i) If t d-L(T), then Tt is a neighborhood of t.

(ii) A point t /

d-L(T) iff one of the following statements hold:

(a) t is isolated.

(b) t {infT, supT} is forward scattered.

(c) t = {infT, supT} is backward scattered.

Proof. (i) Assume that t d-L(T), then by Lemma 1.2.2 (ii), (iii), we have [t] [t].By Lemma 1.2.3 (i), either [t] = {t} or [t] is an interval ofT. Let [t] = {t}. Then

Tt = {t}, since t Tt . Hence t S, and by Lemma 1.2.6 (iii), Tt is a neighborhood

of t. When [t] is an interval ofT, there are two cases. First case: t S, thenagain by Lemma 1.2.6 (iii), our claim is true. Second case: t / S, that is [t] = Tt.Either t {infT, supT} or t / {infT, supT}. Let t {infT, supT}. Hence, Ttis a neighborhood of t. Consider t / {infT, supT}. If t {inf[t], sup[t]}, thenby Lemma 1.2.3 (iv), t S which contradicts t / S. Thus, t / {inf[t], sup[t]}.Therefore Tt is a neighborhood of t, since Tt = [t] is an interval ofT.

(ii) In the left-to-right direction, let t / d-L(T). Assume that t is not isolated.First, ift is dense, then either t [t] Tt or [t] = {t}, and consequently [t] = {t}.Then either t = infT or t = supT. By property (A1), there is [y] T such thatt [y] Tt (Contradiction!). Hence, t is not dense. Thus, either t is backwarddense and forward scattered, or forward dense and backward scattered, in view of

t is not isolated. Let t be backward dense and forward scattered. We claim that


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t {infT, supT}. Let us assume the contrary, i.e. t / {infT, supT}. There aretwo cases: t {inf[t], sup[t]} and t / {inf[t], sup[t]}. In the first case: by property(A1), there is y T such that either [y] [t] or [t] [y]. Thus, [y] Tt, andt = inf[y] or t = sup[y]. This implies that t [y] Tt (Contradiction!). Secondcase: we have t (inf[t], sup[t]) and (t) = t. That is t [t] Tt (Contradiction!).Thus, t {infT, supT} and by assumption t is forward scattered. Now, let usassume that t is forward dense and backward scattered. We have either t =

or t = . When t = , then t {infT, supT}, since if not, then by property

(A2), we get T. Finally, suppose t = . Then either t {inf[t], sup[t]} ort (inf[t], sup[t]). In case oft {inf[t], sup[t]}, by property (A1), we arrive atcontradiction with t / Tt . Suppose that t (inf[t], sup[t]). Since (t) = t, wehave t [t] Tt (Contradiction!). This ends the proof of this direction.For the right-to-left direction, we have three cases: t is isolated, t {infT, supT}is forward scattered, and t = {infT, supT} is backward scattered, we can see

directly that t / d-L(T).

Remark 1.3.3. By Lemma 1.2.6 and Lemma 1.3.2, for t T and T = {t}, wecan summarize the previous results as follows:

(i) t d-L(T) iff one of the following cases holds:

(a) t is dense.

(b) t / {

infT, supT}

is forward dense.

(c) t / {infT, supT} is backward dense.

(ii) Ift = , then t is a forward dense point iff t fd-L(T).

(iii) If t / {, infT, supT}, then t is a backward dense point iff t bd-L(T).

Definition 1.3.4. Let f : T R. Assume that t fd-L(T) (t bd-L(T)). We

say that the forward (backward) dense limit; or just fd-limit (bd-limit); off exists


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Ttrt roo

tr //

rTt

(a) fd-limrt f(r) when t fd-L(T

)

rTtr t//

t rooTtr

(b) bd-limrt

f(r) when t bd-L(T)

Figure 1.4: Forward dense and backward dense limits

at t, if there is a real number l such that for every > 0 there is a neighborhood

U of t in Tt such that:

|f(r) l| < for all r U (, t) r U\ [, t].In this case, we write fd-lim

rtf(r) = l

bd-lim

rtf(r) = l

, see Figure 1.4.

For t d-L(T), we say that the dense limit; or just d-limit; of f exists at t, ifthere is a real number l such that for every > 0 there is a neighborhood U of t

in Tt such that:|f(r) l| < for all r U\ {t}.

Here, we write d-limrt

f(r) = l. When t / d-L(T), then we put d-limrt

f(r) = f(t),

and when t / fd-L(T) (t / bd-L(T)) and t = , then we put fd-limrt

f(r) := f(t)

(bd-limrt

f(r) = f(t)).

The following remark gives us some relations between the usual limit and the

d-limit.

Remark 1.3.5. Let f be a real-valued function defined on T. We have the fol-

lowing:

(i) For t fd-L(T) (t bd-L(T)):(a) If d-lim

rtf(r) or fd-lim

rtf(r)

bd-lim

rtf(r)

exists, then it is unique.

(b) If d-limrtf(r) exists, then fd-limrt f(r) = bd-limrt f(r) = d-limrtf(r).


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(c) Ift > and limrt

f(r) ( limrt+

f(r)) exists, then fd-limrt

f(r) = limrt

f(r)

bd-limrt f(r) = limrt+ f(r) .(d) Ift < and lim

rt+f(r) ( lim

rtf(r)) exists, then fd-lim

rtf(r) = lim

rt+f(r)

bd-limrt

f(r) = limrt

f(r)

.

(ii) Ift d-L(T) and limrt

f(r) exists, then by Lemma 1.3.2 (i):

(a) limrt

f(r) = d-limrt

f(r).

(b) If (t) < t ((t) > t), then d-limrt

f(r) = limrt+

f(r)

d-limrt

f(r) =

limrt f(r).(iii) Let t d-L(T) and d-lim

rtf(r) exists:

(a) If t S, then d-limrt

f(r) = limrt,rI

f(r) for any nonsingular interval

I Tt, t I, see Lemma 1.2.6 (iii).

(b) Ift / S, then d-limrt

f(r) = limrt,r[t]

f(r).

The following lemma gives us d-limit of sum, product and quotients of two func-

tions. The proof is quite similar to the one in the classical calculus.

Lemma 1.3.6. Assume that f, g : T R and d-limrt

f(r) = l1, d-limrt

g(r) = l2.

Then for t T, the following statements are true:

(i) d-limrt

f(r) g(r) = l1 l2.

(ii) d-limr

tcf(r) = c l1 for any constant c

R.

(iii) d-limrt

f(r)g(r) = l1l2.

(iv) d-limrt

f(r)

g(r)=

l1l2

if l2 = 0.

The previous properties (i)-(iv) are true if we replace d-limrt

by fd-limrt

bd-lim

rt

.

Example 1.3.7. Consider the time scale T, E , which is defined in Example1.2.8. Recall that T1 = (0,

), T0 = [0, 1), and Tt = [t], t

T

\ {0, 1

}. Note that


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t / Tt for t T \ {0, 1}, 0 T0 and 1 T1, i.e. d-L(T) = S = {0, 1}. Assume thatf : T R, then

d-limr1

f(r) = limr1

f(r), d-limr0

f(r) = limr0+

f(r),

when these limits exist. Now, if we define f by

f(t) =

t, if t T Q,t, otherwise,

then limrt

f(r) does not exist for any t T, but d-limrt

f(r) = f(t) for t T\{0, 1}.

Definition 1.3.8. Let f be a real valued function on T. Then f is called:

(i) Regulated on A T if the limits fd-limrt

f(r) for t fd-L(A) and bd-limrt

f(r)

for t bd-L(A) exist, and if d-L(A) then d-limr

f(r) exist, i.e. f satisfies

the following:

(R1) its fd-limits exist at all forward dense points in A \ {},(R2) its bd-limits exist at all backward dense points in A \ {, infT, supT},

and

(R3) its d-limit exists at when d-L(A).

(ii) Dense continuous; or just d-continuous; at a point t T if for any > 0,there is a neighborhood U of t in Tt such that:

|f(r) f(t)| < whenever r Uor equivalently

d-limrt

f(r) = f(t),

and f is said to be d-continuous on A T if it is d-continuous at all pointsof A.

(iii) Forward dense continuous; or just fd-continuous; on A T provided it is

regulated on A and d-continuous at all forward dense points in A.


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(iv) Backward dense continuous, or just bd-continuous, on A T provided it isregulated on A and d-continuous at all backward dense points in A.

Remark 1.3.9. For any real valued function f defined on T, we note the following:

(i) f is d-continuous on A T iff it is continuous at all points of d-L(A), inview of Lemma 1.3.2 (i).

(ii) f is d-continuous on A T iff it is d-continuous at all forward dense pointsin A and at all backward dense points in A, i.e. f is d-continuous on A ifff

is fd-continuous and bd-continuous on A.

(iii) In general, fd-continuity (bd-continuity) does not imply d-continuity.

(iv) Continuity implies d-continuity, the converse is not necessarily true. In Ex-

ample 1.3.7, f is d-continuous at points ofT \ {0, 1} and discontinuous atall points ofT.

(v) f is fd-continuous on A T ifff is d-continuous at all forward dense points inA and all its bd-limits exist at all backward dense points in A\{infT, supT}.

Theorem 1.3.10. The following statements are true:

(i) and are nondecreasing.

(ii) and are fd-continuous and bd-continuous respectively.

(iii) If f : T R is a regulated (fd-continuous) function, then so is f .(iv) Let{fn}nN be a sequence of functions which is locally uniformly convergent

to f. If fn is a regulated, fd-continuous or d-continuous for all n N, thenso is f.

Proof. (i) Let x, y T and x > y. We have two cases: [x] = [y] or [x] = [y]. Thefirst case is trivial. In the second case: since x > y, by property (A2), we have

[x] > [y] or [x] [y]. Assume that [x] > [y]. By Lemma 1.2.3 (ii), (x) [x]and (y) [y]. Consequently, (x) inf[x] sup[y] (y). Finally, suppose[x] [y]. Assume, on the contrary, that (x) < (y). We have two cases:

First case: [x] [, supT]. Then [y] [, supT]. In this case, we can see that


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(x) < (y) < y < x. Thus, ((x), x)x = , and hence there are no y, z [x]

satisfying (y, z) ([x] [y]) = {y} (Contradiction!).Second case: [x] [infT, ], then also [y] [infT, ]. Here, y < x < (x) 0, there is k N such that yn (t , t)for all n k. We claim that there is k1 N such that:

(yk1) (t , t). (1.3.1)We have two cases:

Case 1: t / S. Then {yn}nN [t] = Tt. By definition of , taking k1 = k + 1,then yk (yk+1) yk+1, i.e. (1.3.1) is true.Case 2: t S. By Lemma 1.2.6 (iii), there is I I such that {yn}nN I andsup I = t. By condition (C3) of Theorem 1.2.7, either I consists of one class or

all classes of I are alternating. If I = [y] for some y T, then {yn}nN [y], andchoosing k1 = k +1, (1.3.1) holds. Assume now that all classes ofI are alternating,

that is [yn] [ym], n, m N. In this case, we assume the contrary: i.e. (yn) /

(t , t) for all n N. Note that [yn] = [ym] for any n, m > k, m > n (because,


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if [yn] = [ym], then (ym) [yn, t) (Contradiction!)). Since t = sup I = sup[yn] forall n N, then (yn) < t, n N. Thus, we have

(ym) < t < yn < (yn) < ym < t, m, n > k, m > n.

Thus, ((ym), ym)ym = , and hence there are no y, z [ym] such that (y, z)([yn][ym]) = {yn}. This contradicts [yn] [ym]. Hence, (1.3.1) is true in all cases. Bypart(i), |(yn) t| < for all n > k1, and thereby lim

rt(r) = t. Similarly, we

can show that limrt+

(r) = t when [t] [infT, ]. Then fd-limrt

(r) = t, i.e. is

fd-continuous at any forward dense point in T

\ {

}.

Now, suppose that d-L(T), we need to show that d-limr

(r) = . Either

(T[, supT]) or (T[infT, ]). It suffices to show that limr+

(r) =

when (T[, supT]), and limr

(r) = when (T[infT, ]). Assumethat (T [, supT]) ( (T [infT, ])) and {yn}nN is strictly decreasing(increasing) sequence to . As above, we can show that {(yn)}nN is convergentto , i.e. lim

r+(r) = ( lim

r(r) = ). Therefore, is d-continuous at .

With similar argument we can see that bd-limrt

(r) exists at any backward dense

point t T \ {infT, supT} and this limit is equal to t. This implies that isfd-continuous. The bd-continuity of can be proved similarly.

(iii) Suppose that f : T R is regulated. Let t T \ {} be forward dense.Then t fd-L(T). We consider the two cases: [t] [, supT] and [t] [infT, ].First, assume that [t]

[, supT] and fd-lim

rt

f(r) = l. By Lemma 1.3.2 (i), Tt is

a neighborhood of t. Let > 0, there exists > 0 such that

|f(s) l| < wherever s (t , t). (1.3.2)

Then there is 1 > 0 such that

t (r) = |(r) t| < wherever r (t 1, t), (1.3.3)

because fd-limrt (r) = t and is nondecreasing ((r) < (t) = t for any r < t).


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Hence by (1.3.3), (r) (t , t) for every r (t 1, t). Using (1.3.2), we have|f((r)) l| < . That is lim

rt(f)(r) = l. In the case of [t] [infT, ], one can

show that limrt+

(f)(r) = l. Thus, fd-limrt

(f)(r) = l. By a similar argument itcan be shown that bd-lim

rt(f)(r) exists when t T\{infT, supT, } is backward

dense. Also, it can be shown that fd-limr

(f )(r) exists whenever d-L(T).Since is fd-continuous, then d-lim

r(r) = . Consequently, d- lim

rf((r)) =

f(). Therefore f is regulated.Assume that f is fd-continuous and t is forward dense. We need to show that f

is d-continuous at t. Let > 0. Since f is d-continuous at t, there is > 0 such

that |f(s) t| < wherever s (t , t + ). (1.3.4)By part(ii), is fd-continuous. Hence there is 1 > 0 such that

|(r) t| < wherever r (t 1, t + 1). (1.3.5)

Combining (1.3.4) and (1.3.5), and taking s = (r), we find that:

|f((r)) t| < whenever r (t 1, t +

1).

Therefore d-limrt

(f )(r) = t, that is f is fd-continuous.

(iv) Let {fm}mN be a sequence of regulated functions that is locally uniformlyconvergent to f. We want to prove that f is regulated. Assume that t T \ {}is forward dense (t T \ {infT, supT} is backward dense). Our aim now is toshow that fd-lim

rtf(r) = f(t), for the case [t] [, supT] (the other case [t]

[infT, ] can be treated similarly). Let [t] [, supT]. Assume that {sn}nN Tis an increasing (decreasing) sequence to t. Since {fm}mN is locally uniformlyconvergent to f, then we have

limn

f(sn) = limn

limm

fm(sn) = limm

limn

fm(sn) = limm

fm(t) = f(t)

limn

f(sn) = limn

limm

fm(sn) = limm

limn

fm(sn) = limm

fm(t+) = f(t+)

.

Then limrt f(r) ( limrt+ f(r)) exists. Hence, by Remark 1.3.5 (i), fd-limrt f(r) exists.


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Similarly, we use Remark 1.3.5, to show that d-limr

f(r) exists. Thus, f is reg-

ulated. Finally, if t is a forward dense point (t is a point ofT) and {fm}nN isa sequence of fd-continuous (d-continuous) functions which is locally uniformly

convergent to f. Then easily, we can prove that f is fd-continuous (d-continuous),

since fd-limrt

fm(r) = fm(t) (d-limrt

fm(r) = fm(t)), m N.

1.4 Induction principle

In what follows, we give two analogues of the induction principle in the classical

time scale. Our proofs follow the proofs of [19, Theorem 1.7] and [39, Theorem

1.4]. In the first one, we give sufficient conditions for a statement S(t) to be

true for any t in [c]. While the second induction principle is used to show that a

statement S(t) is true for all t [a, b], where a S and b [, a). From now on,to simplify the discussion, we provide the following notation:

c := inf [c], if [c] [, supT],sup[c], if [c] [infT, ],

and c := sup[c], if [c] [, supT],inf [c], if [c] [infT, ].

(1.4.1)

That is c and c are the nearest and the farthest points respectively in [c] from

.

Lemma 1.4.1. (First Induction Principle). Letc T, a [c] and b (a, c].Assume that{S(t) : t [a, b] c} is a family of statements satisfying:

(I1) S(a) is true,

(I2) if t [a, b)c is forward scattered and S(t) is true, then S((t)) is true,(I3) if t [a, b)c is forward dense and S(t) is true, then there is a neighborhood

U of t such that S(s) is true for all s U (t, b)c, and(I4) if t (a, b]c is backward dense and S(s) is true for all s [a, t)c , then S(t)

is true.

Then S(t) is true for all t

[a, b]c.


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Proof. Recall that [c] [, supT] or [c] [infT, ], Theorem 1.1.14. Now, letS =: {t [a, b]c : S(t) is not true}. We prove S = . Assume the contrary, i.e.S = . Let t := sup S if [c] [, supT] and t := infS if [c] [infT, ]. Since[c] [c], then t [a, b)c. To prove the theorem, first we show that S(t) is true.Indeed, if t = a, then S(t) is true from (I1). If t = a, then either (t) = t

or (t) = t. First assume that (t) = t. Then from the definition of t, S(s)is true for all s [a, t)c. Therefore S(t) is true from (I4). Second assume that(t) = t. By Lemma 1.2.3 (ii) and (v), (t) [a, b)c. From the definition of t,

S((t)) is true. Hence by (I2), S(t) = S(((t))) is true. Then t S.Now, we have three cases:

Case 1: t is forward dense, then from (I3), S(s) is true for all s in U (t, b)c fora neighborhood U of t, which is a contradiction with the definition of t.

Case 2: t is forward scattered, then S((t)) is true from (I2) and (t) [a, b]cwhich also contradicts the assumption.

Case 3: t

= b, then S

= which is a contradiction with our assumption. Hence,

S = .

A dual version of the previous induction principle is the following.

Lemma 1.4.2. (First Dual Induction Principle) Letc T, a [c] andd [c, a).Assume that the family of statements {S(t) : t [d, a]c} satisfies:(I1) S(a) is true.

(I2) If t (d, a]c is backward scattered and S(t) is true, then S((t)) is true.(I3) If t (d, a]c is backward dense and S(t) is true, then there is a neighborhood

U of t such that S(s) is true for all s U (d, t)c.(I4) If t [d, a)c is forward dense and S(s) is true for all s [d, t)c then S(t) is

true.

Then S(t) is true for all t

[d, a]c.


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Lemma 1.4.3. (Second Induction Principle). Let a S and b [, a). Assumethat {S(t) : t [a, b]} is a family of statements satisfying:(I1) S(a) is true,

(I2) if t [a, b) is forward scattered and S(t) is true, then S((t)) is true,(I3) if t [a, b) is forward dense and S(t) is true, then there is a neighborhood

U of t such that S(s) is true for all s U (t, b), and(I4) if t (a, b] is backward dense and S(s) is true for all s [a, t), then S(t) is

true.

Then S(t) is true for all t [a, b].

Proof. Suppose that S =: {t [a, b] : S(t) is not true}. We claim that S = .Assume, for the sake of contradiction, that S = . There are two cases. Firstcase: a > b . Second case: b > a. Let t := sup S in the first case, andt := infS in the second case. Then t [a, b]. We show that S(t) is true. Lett = a, then S(t) is true by (I1). Ift = a, then either (t) = t or (t) = t.Assume that (t) = t. Then S(s) is true for all s [a, t), and hence by (I4),S(t) is true. Assume now that (t) = t. Thus, (t) (t, a], and by definitionoft, S((t)) is true. Since (t) is forward scattered (((t)) = t < (t)), then

by (I2), S(t) is true. That is t S.There are three cases; Case A: t is forward dense. By (I3), we have S(s) is true for

all s in U(t, b) for a neighborhood U oft (Contradiction!). Case B: t is forward

scattered, and so by (I2) and (t) [a, b], S((t)) is true (Contradiction!). CaseC: t = b, so that S = . This contradiction completes the proof.

Lemma 1.4.4. (Second Dual Induction Principle). Assume thata S [, supT](a S [infT, ]) and c (a, supT] (c [infT, a)). If the family of statements{S(t) : t [c, a]} satisfying:(I1) S(a) is true,

(I2) if t (c, a] is backward scattered and S(t) is true, then S((t)) is true,


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(I3) if t (c, a] is backward dense and S(t) is true, then there is a neighborhoodU of t such that S(s) is true for all s U (c, t), and

(I4) if t [c, a) is forward dense and S(s) is true for all s (t, a], then S(t) istrue,

then S(t) is true for all t [c, a].

The second induction principle is a useful tool, that helps us to prove theorems

on T.

Summary:

(i) We defined the new generalized concept of time scales. An equivalent defi-

nition of a time scale is:

Assume that T is a closed subset ofR, E is any equivalence relation and

S {infT, supT}. The triple T, E , is a time scale iff:

(a) T = I.(b) I = {Ik}kJ, J {Zn,Z} such that Ik Ik+1 for k, k + 1 J.(c) For I I, either I = [t] for some t T or I = {[t] T : [t] [t]}.

(ii) For a time scale T, E , , we have:

T1. For t T, either [t] [, supT] or[t] [infT, ].T2. If x, y T and x > y, then

{Ii

}ni=1

I s.t. x

I1, y

In and I1

I2

In.

T3. If x, y T and x > y, theneither [x, y] S = , or [x, y] S = {si}ni=1 such that

[x, s1] [s1, s2] [sn1, sn] [sn, y].

T4. If I I, then infI, sup I S {infT, supT}.T5. If t S (t d-L(T)), then Tt is a neighborhood of t.

T6. For I I, infI S {infT} and sup I S {supT}.


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(iii) We defined and studied the concepts of d-limits and d-continuity of functions

on time scales. Some properties of this type of continuity are:

(a) A function f : T R is d-continuous at t T iff the restrictionfunction f : Tt R is continuous at t.

(b) f is d-continuous on T iff it is continuous at all points of d-L(T).

(iv) We proved two analogues of the induction principle for our time scales.


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Chapter 2

Differentiation

In this chapter, we define the -derivative and -derivative on a time scaleT, E , and we study their main properties. Moreover, we obtain analoguesfor the chain rules in the classical time scales calculus and Leibniz formula. In

this setting, the -derivative generalizes both the and -derivatives of the clas-sical time scales calculus. The same is true for the

-derivative. Throughout this

chapter, we assume that T is a time scale.

2.1 -Derivative

We denote by

Tk

:= T \ {}, if {infT, supT} is backward scattered,T, otherwise.For a subset A T, we denote the set A Tk by Ak.

Definition 2.1.1. (The delta derivative). A function f : T R is called deltadifferentiable (-differentiable) at t Tk if f is d-continuous at t and

f(t) := d-limrt

f((t)) f(r)(t) r

41


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Differentiation 42

exists. We say that f(t) is the delta derivative (-derivative) of f at t. For

A T, iff is -differentiable at every t Ak, then f is said to be -differentiableon Ak.

Now, we investigate some relationships concerning the -derivative, see [19, 38].

Theorem 2.1.2. Assume that f : T R and t Tk. The following statementsare true:

(i) If f is -differentiable at t, then f is d-continuous at t.

(ii) If f is d-cont

khaled thesis

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