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Author/s:
Williams, Paul Anthony
Title:
Unifying fractional calculus with time scales
Date:
2012
Citation:
Williams, P. A. (2012). Unifying fractional calculus with time scales. PhD thesis, Department
of Mathematics and Statistics, The University of Melbourne.
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http://hdl.handle.net/11343/37330
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Unifying fractional calculus with time scales
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Unifying fractional calculus with time scales
Paul Anthony Williams
Submitted in total fulfilment of the requirements
of the degree of Doctor of Philosophy
Date: July, 2012
Department of Mathematics and Statistics,
The University of Melbourne
Produced on archival quality paper
Abstract
Fractional calculus, the study of integration and differentiation of fractional order, has recently been extended to include its discrete analogues of fractional
difference calculus and fractional quantum calculus. Due to the similarities of
the three theories there has been research on whether there exist a single theory
that encapsulates them. One possible approach is via the theory of time scales
calculus, however it is still an open problem as whether there exist a consistent
theory for fractional calculus on time scales.
We propose an axiomatic approach to defining a Riemann-Liouville type
fractional integral on time scales. After evaluating the unifying properties of
power functions on various time scales, we posit axioms that the functions would
need to satisfy. We then use these functions as the kernel of an integral transform,
such that if the time scale was the real numbers our definition is equivalent to the
standard Riemann-Liouville fractional integral. Further if the time scale chosen was the integers we obtain backwards fractional summation. We incorporate
fractional derivatives by composing the fractional integral with the standard time
scales derivative.
Our choices of axioms are simple but we demonstrate a large number of desirable properties of the fractional calculus. We show integrability and continuity conditions of the fractional integral and after proving a Dirichlet formula for
nabla integration we also prove an index law. As for fractional derivatives, we
prove a very general Taylor type theorem. This is then applied to consider Opial
inequalities involving fractional derivatives. We start with a simple integral inequality involving the function and its fractional derivative, and then extend these
results by considering positive powers of the factors and also weighted integrals.
Opial inequalities have applications in studying fractional dynamic equations,
and we demonstrate how they are used in finding bounds on solutions of these
equations.
Declaration
This is to certify that:
1. the thesis comprises only my original work towards the PhD except where
indicated in the Preface,
2. due acknowledgement has been made in the text to all other material used,
3. the thesis is fewer than 100,000 words in length, exclusive of tables, maps,
bibliographies and appendices.
Paul A. Williams.
Preface
This work concerns unification of fractional calculus, fractional differences and
fractional quantum calculus using the methodology of time scales. We make
no assumptions on the reader’s knowledge of the theory of time scales calculus,
accordingly Chapter 2 contains an introductory overview of the theory.
This work is conducted under the supervision of J. J. Koliha, Craig D. Hodgson and J. Hyam Rubinstein of the Mathematics and Statistics Department of the
University of Melbourne, and also under Greg Hjorth during the first half of the
candidature.
Acknowledgements
Special thanks to my supervisors J. J. Koliha, Craig D. Hodgson, J. Hyam Rubinstein and Greg Hjorth, the latter whom sadly passed away during the candidature.
Contents
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14
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21
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31
33
36
3
Basic Definitions and Taylor’s Theorem
3.1 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Fractional Integrals and Basic Properties . . . . . . . . . . . . .
3.3 Fractional Derivatives and Taylor Theorems . . . . . . . . . . .
39
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59
4
Opial-type Inequalities
4.1 Background of Opial’s Inequality . . . . . . . . . . . . . . . . .
4.2 Opial’s Inequality and Generalizations . . . . . . . . . . . . . .
4.3 Applications of Opial’s Inequality . . . . . . . . . . . . . . . .
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88
5
A Blueprint for Further Work
95
2
Introduction and Literary Criticism
1.1 Real Fractional Calculus . . . .
1.2 Discrete Fractional Calculus . .
1.3 Quantum Fractional Calculus . .
1.4 Towards a Unified Theory . . .
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Overview of Time Scale Calculus
2.1 Basic Definitions . . . . . . . . . . . . .
2.2 The Lebesgue Nabla Integral . . . . . . .
2.3 Absolute Continuity . . . . . . . . . . . .
2.4 Fundamental Theorem of Nabla Calculus
2.5 Summary and Examples . . . . . . . . .
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Chapter 1
Introduction and Literary Criticism
Recently, there have been papers discussing the possibility of constructing a theory of fractional calculus which unifies the fractional calculus seen in continuum
calculus, discrete calculus and quantum calculus. Traditionally the term fractional calculus has referred to studying integration and differentiation of noninteger order, however we now see work involving the study of non-integer orders of the difference operator in discrete calculus. Similarly, there have been
papers which study non-integer orders of quantum calculus. These theories involve similar approaches, and have led to the open problem of whether there exist
a theory of fractional calculus unifying the three.
1.1
Real Fractional Calculus
Although it is true that historically Leibniz had written on the subject, fractional
calculus on the real numbers has its origin in the tautochrone problem. Predating
the theoretical studies of fractional calculus, in 1823 Abel [1] showed that the
generalized form of the problem could be modeled by a integral equation of form
1
T (x) = p
2g
Z
0
x
φ0 (t)
dt,
√
x−t
(1.1)
where the total time taken, T , is given, and the path the particle takes, φ, is
unknown. A derivation of this result can be found in Keller [84] or Simmons
5
6
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
[113]. The tautochrone problem itself is the special case of (1.1) where T is
constant. Abel went on to solve an even more general integral equation
x
Z
a
f (t)
dt = g(x),
(x − t)β
(1.2)
where 0 < β < 1 and x > a. Equation (1.2) is called Abel’s integral equation in
his honor.
In order to observe the relationship between Abel’s equation and fractional
calculus, we shall need to consider repeated integration. The equation
Z
x
a
(x − τ)n−1
f (τ) dτ,
(n − 1)!
(1.3)
can be evaluated by integration by parts
Z
a
x
(x − τ)n−1
f (τ) dτ =
(n − 1)!
Z
x
t1
Z
f (tn ) dtn := Ian f (x),
dt2 · · ·
dt1
a
tn−1
Z
a
(1.4)
a
where the operator Ian refers to n-fold integration. This result, called the Cauchy’s
integral formula, is well-defined for non-integer n. The resulting expression
ν
Ia+
f (t)
1
:=
Γ(ν)
t
Z
(t − τ)ν−1 f (τ) dτ,
(1.5)
a
for ν > 0, is called the Riemann-Liouville fractional integral, and is one of the
most common definitions of fractional integrals in the literature.
We see that the Riemann-Liouville fractional integral and the Abel integral
equation are related to each other. The fact that fractional calculus is not a theoretical curiosity but has applications, caused many mathematicians to study it
deeply. Interestingly, although it bears their name, it was the work of Krug [92]
which united Liouville’s work [97] based on Fourier series and Riemann’s work
[118] which was based on the Taylor formula.
The fractional derivative is defined though post-differentiation, that is, given
n−ν
ν > 0, let n = bνc + 1 and then define Dνa+ f (x) := (d/dx)n Ia+
f (x). This theory
has been extensively studied, with many textbooks on the topic. For introductory
texts see Miller and Ross [100], or the encyclopedic Samko et al [112]. Oldham
1.1. REAL FRACTIONAL CALCULUS
7
and Spanier [102] has a historical bibliography on the topic. For a more application oriented perspective see Podlubny [109], West et al [119] and Kilbas et al
[85].
However there are theories of fractional calculus which differ from the RiemannLiouville definition. One example is the Grünwald-Letnikov fractional derivative
[72], which is based on the observation that
n
1 X
(−1)k
f (t) = lim n
h→0+ h
k=0
n
X
1
= lim n
(−1)k
h→0+ h
k=0
(n)
!
n
f t + (n − k) h
k
!
n
f t − kh ,
k
(1.6)
(1.7)
The two equations correspond to considering the limits from the right and the left
respectively. Observing that the sum can be extended to infinity as the binomial
is 0 when k > n, we can replace the integer n with 0 < ν < 1 to define
!
∞
1 X
k ν
D f (t) := lim ν
(−1)
f (t − kh).
h→0+ h
k
k=0
ν
(1.8)
The existence of different definitions for fractional calculus begs the question
of whether these definitions of fractional calculus are equivalent to one another.
A partial result on the matter was shown by Cartwright and McMullen [54]. They
proved a uniqueness criterion for fractional integrals on continuous functions and
Lebesgue integrable functions.
Theorem 1.1.1 ([54]). Let E = C[0, 1] or E = L p [0, 1] for 1 ≤ p < ∞. Then
there is only one family of operators Iα : E → E, α > 0, which satisfy:
Rx
1. for all f ∈ E and x ∈ [0, 1], I1 f (x) = 0 f (t) dt,
2. for all α, β > 0, Iα Iβ = Ia+β ,
3. for all α > 0 and f ∈ E, f a positive function implies Iα f is also a positive
function,
4. the map α 7→ Iα is continuous (where the topology on L(E) is some Hausdorff topology weaker than the norm topology).
8
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
Importantly, the theorem does not prove uniqueness of fractional derivatives,
and so leaving the possibility of many definitions of Dα , which are not equivalent
α
with each other, and yet still being left-inverses of Ia+
. Indeed, a large number
of recent papers on fractional calculus are variations of the Riemann-Liouville
fractional calculus.
One of the first, and arguably the most important, is the work of Caputo [51]
(reprinted in [52]). For ν > 0 and n = dνe, the Caputo fractional derivative is den−ν (n)
fined as C Dνa+ f (x) := Ia+
f (x). This definition does have two important advantages. The first is that the fractional derivative of a constant is zero, which is not
true with the Riemann-Liouville fractional derivative. The second is that when
one wishes to consider the Laplace transform of the Caputo fractional derivative,
the initial values are in terms of integer order derivatives. See Podlubny [109] for
proofs, and a discussion of how these properties are indeed useful in applications.
However, the Caputo fractional derivative requires that the function be differentiable to begin with. As noted by West et al [119], one of the major uses of
fractional derivatives lies in the modeling of phenomena which is continuous but
not necessarily differentiable. Hence, Jumarie [79, 80, 81] has developed a theory of fractional calculus which causes the fractional derivative of constants to be
zero, and yet the function only be differentiable to order n − 1. This parallels the
theory of local fractional derivatives pioneered by Kolwankar and Gangal [91].
One of the major difficulties in applying fractional calculus lies in choosing the
definition in order to obtain the properties desired.
Lastly, Theorem 1.1.1 leaves open the possibility of a class of operators which
are indexed by more than one parameter. An example of two parameter fractional
operators are the Erdélyi-Kober operators [63, 64, 87, 88, 89] defined as
Z
2x−2α−2η x 2
(x − u2 )α−1 u2η+1 f (u) du,
Iη,α f (x) :=
Γ(α)
Z 0
2x2η ∞ 2
Kη,α f (x) :=
(u − x2 )α−1 u−α−η+1 f (u) du.
Γ(α) x
(1.9)
(1.10)
For a summary of the application to Hankel transforms see Sneddon [114]. These
operators were further extended in Kiryakova [86].
1.2. DISCRETE FRACTIONAL CALCULUS
1.2
9
Discrete Fractional Calculus
Just as the pioneer work of fractional calculus lies in differential equations, the pioneering work for fractional differences lies in the study of difference sequences
by Anderson [29], Kuttner [93, 94] and Cargo and Shisha [53]. These papers consider a certain class of operators defined on the sequences of complex numbers.
Given (an )n∈N ⊂ C and for a real number s, they define the difference sequence
of an as
!
∞
X
−s − 1 + m
s
an+m ,
(1.11)
∆ an :=
m
m=0
provided the series converges. Although (1.11) is related to the forward difference operator when s is an integer, this isn’t observed in the papers, and so,
theorems analogous to the Leibniz law of difference calculus for example were
not considered.
The first paper to investigate fractional difference calculus was Diaz and Osler
[59] using a discrete version of the Grünwald-Letnikov fractional derivative. By
making the observation of
∆ f (z) =
N
N
X
(−1)
−k
k=0
!
N
f (z + N − k),
k
N = 1, 2, . . . ,
(1.12)
the authors define the fractional difference of order α as
α
∆ f (z) :=
∞
X
!
α
f (z + α − k),
k
k
(−1)
k=0
(1.13)
where α could be rational, real or complex, provided the series converged.
The paper proves two important properties of (1.13), the first being a contour
integration representation and the second being a Leibniz rule. With regard to
the former, the authors observe that Cauchy’s integral formula
N!
D f (z) =
2πi
I
N
Γ
f (t)
dt,
(t − z)N+1
(1.14)
could be generalized to fractional differences by replacing (t − z)N+1 with the
falling factorial (t − z)(α+1) = Γ(t − z + 1)/Γ(t − z − α). The authors use the residue
10
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
theorem to show that (1.13) is equivalent to the integral
Γ(α + 1)
∆ f (z) =
2πi
α
Z
f (t)
C
Γ(t − z − α)
dt,
Γ(t − z + 1)
(1.15)
with C a contour which encircles the simple poles at t = z + α, z + α − 1, . . .. The
authors provide growth conditions on f so that the integral is defined. With regards to the latter, the paper derives an extension of the Leibniz rule to fractional
differences by showing
!
∞
X
α α−n
∆ f (z) ∆n g(z + α − n).
∆ f (z)g(z) =
n
n=0
α
(1.16)
By using these two results, the authors derive a large number formulas which can
be used in the study of infinite sequences.
The question of whether (1.13) satisfies an index law was examined in Isaacs
[78], and found that the law only holds in the form of a Cesàro summation. If
r + s < Z and r is negative if s is not a natural number, and further if ∆r+s f is
summable in Cesàro (C, λ) sense, then
∆r+s f (z) = ∆r ∆ s f (z),
(1.17)
where the right hand side is Cesàro summable (C, λ + s + ) sense. Here ≥ 0
is chosen so that λ + s + is an integer. This result is significant because (1.13)
itself does not use Cesàro summation. Isaacs shows that if one wishes to observe
an index law, one must be willing to extend the definition beyond the original
intentions.
However there is a larger problem associated with (1.13) in the arguments
of f . Because Diaz and Osler [59] are interested in functions with complex
argument, the existence of f (z + α − k) for non-integer α is guaranteed. However
in applications, usually the function f is only known when the argument is an
integer. Thus, when one wishes to use (1.13) as is, we need to assume that
z + α − k ∈ Z. Hence, one has ∆α : RZ → RZ−α , changing the domain of the
operand.
In contrast, the fractional difference based on the backward difference given
1.2. DISCRETE FRACTIONAL CALCULUS
11
in Hosking [76] and Granger & Joyeux [70] do not change the domain of the
functions. The definition given is
α
∇ f (t) :=
∞
X
(−1)
k=0
k
!
α
f (t − k).
k
(1.18)
Here, setting α = 1 we have f (t) − f (t − 1), and thus (1.18) is a direct generalization of the backward difference. Importantly, t − k is an integer for all t and all k,
and so the definition is applicable to applications in difference calculus. A disadvantage of the definition is that f needs to be known for all t − k, which may be
problematic in applications. None the less, both papers are applied mathematics
papers, and they show how fractional difference can be used in the study of long
memory time series.
In contrast to these definitions, Gray and Zhang [71] have defined a fractional
difference through summation. They observe that the n-fold summation has a
solution
kn−1
k1
t X
t
X
X
1 X
···
f (kn ) =
(t − k + 1)n−1 f (k),
(1.19)
Γ(n)
k =a k =a
k =a
k=a
1
2
n
where the rising factorial is defined as


Γ(α+β)


,
when α, α + β , 0, −1, −2, . . . ,

Γ(α)







when α = β = 0,
1,
(α)β = 



0,
when α = 0 and β , 0, −1, −2, . . . ,







undefined, otherwise.
(1.20)
Observing that (1.19) is well defined to complex α, they define the fractional
backwards difference.
Definition 1.2.1. Let α ∈ C, and f : {a − n, a − n + 1, . . . , t} → R. Then the α-th
order summation is defined as
t
S α f (t) :=
a
t
∇n X
(t − k + 1)n+α−1 f (k),
Γ(n + α) k=a
(1.21)
where n := max{0, n0 } with n0 ∈ Z such that 0 < Re(α + n0 ) ≤ 1. Also the α-th
12
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
order difference is defined as
t
t
a
a
∇α f (t) := S −α f (t).
(1.22)
To illustrate the connection between (1.21) and (1.5), consider α ∈ (0, ∞),
and n0 will be negative or zero, and in either case n is zero. We obtain
t
1 X
(t − k + 1)α−1 f (k).
Γ(α) k=a
(1.23)
It is now clear that (1.21) is analogous to the Riemann-Liouville fractional integral with the rising factorial playing a similar role as the power function. For negative α (i.e. fractional differences), n is non-zero, and we obtain post-differencing.
Thus (1.21) is satisfactorily the difference calculus analogue of the RiemannLiouville fractional differ-integral.
The paper proves two important properties of the fractional difference: an
extensive index law and a Leibniz rule. With respect to the former the authors
t
t
a
a
prove that for all complex α and all β not a positive integer one has ∇α ∇β f (t) =
t
∇α+β f (t).
a
An advantage of this definition is that one does not need to worry about
convergence when proving the index law. The Leibniz rule is the source of the
applications in the paper. The authors prove
!
t−a
X
ν ν−n
∇ f (t)g(t) =
∇ f (t − n) ∇n g(t) ,
n
n=0
ν
(1.24)
a generalization of the Leibniz rule for classical discrete calculus. The authors
then use (1.24) to solve a class of non-fractional difference equations.
The last of the definitions considered here will be that of Miller and Ross
[99], which has recently been studied by Atici and Eloe [32, 34]. Miller and
Ross’s paper uses a definition of fractional difference which is a forward difference analogue of the Riemann-Liouville fractional integral. Their analysis begins
by proving that for the n-fold forward difference operator, T = ∆n , the Green’s
1.2. DISCRETE FRACTIONAL CALCULUS
function is given as
H(t, s) =
Γ(t − s)
.
Γ(n) Γ(t − s − n + 1)
13
(1.25)
After a change of variables, the solution is given by
t
1 X Γ(t − s + ν)
∆ w(t) =
w(σ − n).
Γ(n) σ=a+n Γ(t − s + 1)
−n
(1.26)
As with other definitions, (1.26) is well-defined when n is not an integer, and so,
the definition is given as
t
1 X Γ(t − s + ν)
∆ w(t) :=
w(s − ν),
Γ(ν) s=ν Γ(t − s + 1)
−ν
(1.27)
where ν ∈ R. Here t = a + ν modulo 1, and thus we again have a domain shift.
The paper proves a number of properties. Firstly they show that the map
λ 7→ ∆λ is continuous when λ is a non-negative integer. Secondly they show that
the index law holds with respect to post-differencing ∆N ∆λ = ∆N+λ for any real
λ, and most importantly, they prove an extensive Leibniz rule
∆ w(t)u(t) =
−ν
∞
X
r=0
(−1)r
Γ(ν + r) r
(∆ w(t)) (∆−ν−r u(t + r)),
r! Γ(ν)
(1.28)
which is used to study the fractional difference of certain special functions. This
definition was further studied by Atici and Eloe [32, 34]. They prove an extensive
index law
∆−ν ∆−µ f (t) = ∆−ν−µ f (t), t = µ + ν mod 1,
(1.29)
for all positive µ and ν, and used the discrete Laplace transform from [46, 61] to
solve fractional difference equations.
In summary, there have been four major streams of research concerning the
discrete analogue to fractional derivatives. Two are analogues of the GrünwaldLetnikov fractional derivative, and the other two are analogues of the RiemannLiouville fractional integral. The latter two demonstrate an extensive index law,
while the other two do not. Further, the two fractional differences which generalize the forward difference modify the domain of the function, but the backwards
14
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
fractional differences do not.
1.3
Quantum Fractional Calculus
Quantum calculus is a generalization of calculus that can be interpreted as the
multiplicative analogue of the difference calculus. For a fixed q ∈ (0, ∞) such
that q , 1, we define q-derivative of a real function f as
Dq f (t) =
f (t) − f (qt)
.
t − qt
(1.30)
In this section we assume 0 < q < 1, and that f is a function of R → R. Observe
that by taking the limit q → 1 we are able to reproduce the standard definition
of the derivative. The study of quantum calculus then refers to discovering expressions and theorems that reduce to standard calculus when this limit is taken.
These expressions are called q-analogues in the literature.
As an example, the q-bracket is defined as
qx − 1
[x]q :=
.
q−1
(1.31)
Under the limit q → 1 we see that [x]q → x for any real x. Hence the function
f (x) = xn for n a natural number has q-derivative of
Dq f (x) =
(qx)n − xn qn − 1 n−1
=
x = [n]q xn−1 .
(q − 1)x
q−1
(1.32)
The q-analogue of the integral is the Jackson integral and is evaluated by the
summation
Z x
∞
X
f (t) dq t := x (1 − q)
f (xqk ) qk .
(1.33)
0
k=0
It is proved, that for a certain class of functions, the Jackson integral is the inverse
of the q-derivative
Theorem 1.3.1 ([82]). Suppose 0 < q < 1, 0 ≤ α < 1 and A > 0. Suppose that
f is a function with | f (x) xα | bounded over (0, A]. Then the Jackson integral of f
1.3. QUANTUM FRACTIONAL CALCULUS
15
converges to a function F, which is continuous at x = 0 with F(0) = 0. Further,
F is the q-anti-derivative of f .
Importantly, many special functions have q-analogues, which retain their useful properties even when q is not equal to 1. For example the q-polynomial is
defined as
(n)
(x − a)q := (x − a)(x − qa) . . . (x − qn−1 a),
(1.34)
(0)
when n is a natural number n ≥ 1, and (x − a)q := 1. Then
(n−1)
(n)
Dq (x − a)q = [n]q (x − a)q
,
(1.35)
generalizing this property of polynomials in the classical case. Interestingly, the
(∞)
limit as n → ∞ is well defined for 0 < q < 1, thus we can define (x − a)q
as the infinite product. The q-polynomials are extended to non-integer orders by
defining
(∞)
(1 − a/x)q
(ν)
ν
, ν ∈ R.
(1.36)
(x − a)q := x
(∞)
ν
(1 − q a/x)q
The analogue of the gamma function is defined as
Γq (t) :=
Z
∞
(∞)
xt−1 (1 + (1 − q)x)q dq x,
t > 0.
(1.37)
0
Like the gamma function, the q-gamma function satisfies a recurrence relation
Γq (t + 1) = [t]q Γq (t).
(1.38)
Also the q-gamma function satisfies
(x−1)
Γq (t) := (1 − q)1−x (1 − q)q
.
(1.39)
For proofs see Section 21 of Kac and Cheung [82].
The study of fractional quantum calculus was initiated by W. A. Al-Salam [5]
proving two q-analogues of the Cauchy integral formula for multiple integration:
x
Z
Z
dq xn−1
a
a
xn−1
dq xn−2 . . .
x1
Z
a
1
f (t) dq x1 =
Γq (n)
x
Z
(n−1)
(x−qt)q
a
f (t) dq t, (1.40)
16
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
and
Z ∞
Z
∞
dq xn−2 . . .
dq xn−1
x
Z
xn−1
∞
x1
q−n(n−1)/2
f (t) dq x1 =
Γq (n)
Z
x
(n−1)
(t − x)q
f (tq1−n ) dq t.
a
(1.41)
This was used by two papers to define a fractional quantum calculus. The first
is due to Al-Salam [6], which considers a q-analogue of the Sneddon fractional
integral [65] by defining
Kqη,α f (x)
q−η xη
:=
Γq (ν)
∞
Z
(α−1) −η−α
(y − x)q
f (yq1−α ) dq y,
y
(1.42)
x
for α , 0, −1, −2, . . .. The paper then proceeds to prove that the index law
Kqη,α Kqη+α,β f (x) = Kqη,α+β f (x) is valid for all α and β.
Meanwhile, Agarwal [2] developed a fractional integral based on the Kober
operator [87],
Iqη,α f (x)
x−η−α
:=
Γq (α)
x
Z
(α−1) η
(x − qt)q
t f (t) dq t,
(1.43)
0
for all α, and then proceeded to investigate its properties including an index law
Iqη,α Iqη+α,β f (x) = Iqη,α+β f (x) and a fractional integration by parts. The relationship
between the two definitions was explored in Al-Salam and Verma [7].
The recent paper Rajković et al [110] deserves special mention. The definition is given as
α
Iq,a
f (x)
1
:=
Γq (α)
Z
x
(α−1)
(x − qt)q
f (t) dq t,
(1.44)
a
where α > 0 and a ≥ 0. Observe that (1.44) and (1.43) are related by the equation
α
Iq,0
f (x) = x−α Iq0,α f (x)
(1.45)
The authors define the fractional q-derivative as
dαe−α
f (x),
Dαq,a f (x) = Ddαe
q Iq,a
(1.46)
for α > 0. As an application, the authors prove a general Taylor formula. For
1.4. TOWARDS A UNIFIED THEORY
17
0 < α < 1 and f defined on (0, b), with 0 < a < c < x < b, then
f (x) =
n−1
X
(Dα+k
q,a f )(c)
k=0
Γq (α + k + 1)
(α+k)
(x − c)q
+ Rn ( f ),
(1.47)
where Rn ( f ) is a remainder term.
1.4
Towards a Unified Theory
As one can see there are similarities to the definitions of fractional calculus, and
opens up the question of whether there exist a unified theory. In fact this question is not limited to fractional calculus. Making the observation that discrete
analogues to many theorems in calculus exist, Stefan Hilger introduced the theory of time scales calculus in his PhD thesis [75] and later published in [36]. The
theory studies functions whose domain is a closed and non-empty subset of the
real numbers called a time scale. It also defines an operator called the Hilger
derivative which has the property that if the domain chosen was R the Hilger
derivative is equal to the standard derivative. But also if the domain chosen was
Z then the Hilger derivative is the difference operator from discrete calculus.
Further if one chooses the time scale
{0, 1, q±1 , q±2 , . . .},
(1.48)
then the Hilger derivative becomes the quantum derivative. Different choices for
the time scale corresponds to different forms of calculus in the literature, and
a theorem proved using this theory is valid for all. Thus times scales calculus
allows one to unify and extend calculus. Indeed these two are considered its
main features in the literature.
The first papers which considered the unification question are the pair of papers by Atici and Eloe [31, 32]. The papers consider discrete fractional calculus
and quantum fractional calculus separately, however, the theory was framed in
the language of time scales calculus. The authors have published a number of
papers after this [33, 34, 35], all of which use the time scales theory. However,
these papers consider only specific time scales, and not a general theory.
18
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
This problem was considered in the papers of Anastassiou [21, 23]. The author considers properties which continuous fractional calculus and discrete fractional calculus have in common, and reframes the theory in time scales calculus.
However, it should be noted the theory does not consider quantum fractional calculus, and worse, there are a number of inconsistencies in the work. A secondary
goal of this thesis is to identify, correct and develop Anastassiou’s work.
A different approach is considered in the paper of Čermák and Nechvátal
[55]. They give a unification of discrete fractional calculus and quantum fractional calculus by the introduction of the (q, h) time scale, where q ≥ 1 and h ≥ 0
satisfy q + h > 1. Both the integers and quantum time scale are special cases
by setting (q, h) = (1, 1) and (q, h) = (q, 1) respectively. This paper gives a direct example of the unification process on time scale calculus. It is restricted to
discrete time scales and does not consider a union of closed intervals seen in applications. But none the less it is an important case study in fractional calculus of
discrete time scales. Interestingly, the authors consider both delta (forward) and
nabla (backward) fractional calculus. Their analysis shows that nabla fractional
calculus does not change the domain of the function operated on. This is not true
for the delta case and the authors give an indication as to how to rectify this for
arbitrary time scales. But this appears to be a difficult procedure in general. Thus
it appears that the nabla theory would be the simpler route to pursue.
A third approach via the Laplace transform has been suggested by Bastos et
al [38]. The Laplace transform on time scales was introduced by Bohner and
Peterson [47] (see also [46, Chapter 3]) and further developed in the paper by
Davis et al [57]. Given a time scale T and a function f : T → R with Laplace
transform F, the authors define the fractional integral of f of order α > 0 as
ITα f (t)
"
=L
−1
#
F(z)
(t),
zα
and the fractional derivative of order α > 0 as


dαe−1
X k


f (α) (t) = L−1 zα F(z) −
f ∆ (0+)zα−k−1  (t).
k=0
(1.49)
(1.50)
1.4. TOWARDS A UNIFIED THEORY
19
The authors then proceed to prove a number of properties of this definition, such
as an index law. It is interesting to observe that despite being a theory of right
fractional derivatives, the definition does not appear to change the domain of f .
Hence it not a generalization of (1.13), (1.27) or (1.42).
Unfortunately, despite being attractive, the theory also has a number of limitations. Most importantly, the inverse Laplace transform on time scales is a
difficult procedure. At the time of writing, there is no analogue to the contour
integration method used extensively in real calculus. Also the best inversion formula, proposed by Davis et al [57], requires that F have finitely many regressive
poles of finite order. But the definition of fractional integration (1.49) contains
a branch cut. Further, the analysis is limited to discrete time scales, and like the
Čermák-Nechvátal cannot be applied to collections of intervals, which is important in applications. Currently there is no answer to these problems and more
research is required.
In this thesis we present a theory of fractional calculus on time scales based
on the axiomatic properties of a power function on time scales. Rather than finding the power function for specific time scales, we concentrate on the universal
properties that the known examples possess and construct a theory of fractional
calculus with it. Thus this thesis concentrates on the unification aspect of time
scale calculus rather than extension. If one desires to consider fractional calculus
on other time scales, then one would need to find these functions and show that
they satisfy the axioms presented within.
The papers of Anastassiou [21, 23] are related to our work. Independently
of us, Anastassiou first asks which properties fractional calculus and fractional
differences share, and considers unifying the proofs of the literature using the
notation of time scales calculus. We differ in that in this thesis we are including
the quantum fractional calculus and focus only on left fractional calculus for
greater depth. Further, the inherent properties of power functions we consider
are stronger than those which are presented in Anastassiou, allowing us to obtain
simpler proofs. Lastly, we have identified errors in Anastassiou’s paper [21]
on left fractional calculus. Significantly, the index law proved contradicts not
only our results, but also the references given in Anastassiou’s paper itself. We
identify and correct the mistakes throughout the thesis, and also collate them in
20
CHAPTER 1. INTRODUCTION AND LITERARY CRITICISM
the conclusion.
The thesis is organized as follows: in Chapter 2, we give an overview of time
scale calculus, including details of the time scales mentioned above. Specific
attention is given to Lebesgue integration on time scales, and importantly the ∇Lebesgue integral as the thesis concerns nabla fractional calculus. At the time
of writing, the theory of Lebesgue integration for time scales is with respect to
delta calculus. This includes a number of important results which we shall use.
However, we shall be using the nabla analogues of these results, which have not
be established in the literature. Consequently we shall do so in the latter half of
the chapter.
The core of our work is Chapter 3, where we present the properties that power
functions need to satisfy in order to construct a consistent theory of fractional
calculus. We develop simple and yet important properties of fractional calculus,
including an index law and a general Taylor-type theorem. The former relies on
the Dirichlet law for nabla calculus, a result which we prove for the Lebesgue
nabla integral. This result has been misrepresented in the literature, and is an
important contribution to time scales calculus as a whole.
Our extension of the Taylor theorem is used in Chapter 4 to consider generalizations of Opial-type inequalities as an application. We first consider simple inequalities and build to a general inequality which is then used to study bounds on
solutions to fractional dynamic equations. We consider further work in Chapter
5, and the important question of whether there exist power functions for arbitrary
time scales.
Chapter 2
Overview of Time Scale Calculus
In this chapter we give an overview of elementary results from time scales calculus, intended for a reader not versed in the theory. We begin with the basic
definitions of time scales and their simple properties, then consider the calculus
defined on them. We give special attention to the time scales analogue of the
Lebesgue integral and absolute continuity, and illustrate the fundamental theorem of calculus for time scales. These results have only been proved with respect
to delta (right) calculus, hence we shall develop these with respect to nabla (left)
calculus. We end with a summary of how these concepts adapt to the individual
time scales which we are interested in the thesis.
2.1
Basic Definitions
In the literature a time scale T is defined as a non-empty closed subset of the real
numbers. However, there are situations where allowing either infinity or negative
infinity is useful, thus here we will define a time scale to be a non-empty subset
of the extended real numbers R. Examples include the real numbers and the
integers. An example which is important in applications is the set
P :=
∞
[
[2k, 2k + 1],
k=0
21
(2.1)
22
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
which can be used in the modeling of insect populations [56]. Another important
example is the quantum time scale which is defined as
qZ := {0, 1, q, q−1 , q2 , q−2 , . . .},
(2.2)
for a fixed q ∈ (0, ∞), q , 1. Time scale calculus is a methodology which aims to
unify and extend continuum and discrete calculus. By choosing T = R, the theory
reduces to real analysis, while choosing T = Z the theory reduces to discrete
analysis. However, the concepts extend to other time scales, and generalize the
two.
In order to define a derivative, we shall need to define the jumps operators.
The right jump operator is defined as
σ(t) := inf{s ∈ T | s > t}.
(2.3)
We shall use the convention inf ∅ = sup T. Thus if M := sup T is an element of
T, then σ(M) = M. We introduce the notation Tκ := T \ {M} in this situation, and
Tκ = T otherwise. In a similar fashion, we define the left jump operator as
ρ(t) := sup{s ∈ T | s < t},
(2.4)
and Tκ := T \ inf T analogously. On the reals, one has σ(t) = ρ(t) = t, while on
the integers one has σ(t) = t + 1 and ρ(t) = t − 1. On the quantum time scale, if
q > 1 we have σ(t) = qt and ρ(t) = q−1 t, and when 0 < q < 1 we have ρ(t) = qt
and σ(t) = q−1 t . Lastly, given a function f : T → R, we denote f σ (t) = f (σ(t))
and f ρ (t) = f (ρ(t)).
We now define the concept of graininess, which measures the distance to the
nearest point. The right graininess is defined as µ(t) = σ(t) − t. A point which
satisfies µ(t) = 0 is said to be right dense, while a point with µ(t) > 0 is said to be
right-scattered. Similarly, the left graininess operator is ν(t) = t − ρ(t). A point
which is both left and right dense is simply called dense, and a point which is
both left and right scattered is called discrete. On the reals, every point is dense
while on the integers every point is discrete.
We shall now move onto the time scale analogues of continuity and the
2.1. BASIC DEFINITIONS
23
derivative.
Definition 2.1.1. Let T be a time scale, a function f : T → R is called right
dense continuous if
1. f is continuous at every right-dense t ∈ T,
2. the limit of f (t) from below at every left-dense point t ∈ T is finite.
Similarly a function is left dense continuous if f is continuous for every left dense
point and f (t+) exists for every right-dense t ∈ T.
Sometimes the terms rd-continuous and ld-continuous are used in the literature.
Definition 2.1.2. A subset I ⊂ T is called a time scale interval, if it is of the
form I = A ∩ T for some real interval A ⊂ R. For a time scale interval I, a
function f : I → R is said to be right dense absolutely continuous if for all > 0
P
there exist δ > 0 such that nk=1 | f (bk ) − f (ak )| < whenever a disjoint finite
collection of sub-time scale intervals [ak , bk ) ∩ T ⊂ I, for 1 ≤ k ≤ n satisfies
Pn
k=1 |bk − ak | < δ.
Similarly, f is left dense absolutely continuous if for all > 0 there exist
P
δ > 0 such that nk=1 | f (bk ) − f (ak )| < whenever a disjoint finite collection of
P
sub-time scale intervals (ak , bk ] ∩ T ⊂ I, for 1 ≤ k ≤ n satisfies nk=1 |bk − ak | < δ.
Definition 2.1.3. For a function f : T → R and a point t ∈ Tκ , we define f ∆ (t)
to be the number such that for all > 0 there exist a neighborhood of t, U ⊂ T,
such that
σ
( f (t) − f (τ)) − f ∆ (t)[σ(t) − τ] ≤ |σ(t) − τ| ,
∀τ ∈ U.
(2.5)
If for all t ∈ Tκ , f ∆ (t) exists then it is called the delta derivative of f .
Definition 2.1.4. For a function f : T → R and a point t ∈ Tκ , we define f ∇ (t)
to be the number such that for all > 0 there exist a neighborhood U ⊂ T
containing t such that
ρ
( f (t) − f (τ)) − f ∇ (t)[ρ(t) − τ] ≤ |ρ(t) − τ| ,
∀τ ∈ U.
(2.6)
24
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
If for all t ∈ Tκ , f ∇ (t) exists then it is called the nabla derivative of f .
The space of all right-dense continuous functions on T is denoted Crd (T),
n
and for n ≥ 1, the space of functions with rd-continuous f ∆ , . . . , f ∆ is den
(T). Meanwhile, the space of rd-absolutely continuous functions is
noted Crd
n−1
denoted ACrd (T). The space of functions with rd-continuous f ∆ , . . . , f ∆ and
n
n
(T). The spaces Cld (T), ACld (T),
rd-absolutely continuous f ∆ is denoted ACrd
n
n
Cld (T) and ACld (T) are defined similarly.
2.2
The Lebesgue Nabla Integral
Integration on time scales is usually defined as the inverse of differentiation, that
is, given a time scale T and f : T → R one defines
t
Z
f (τ) ∆τ := F(t) − F(a),
a, t ∈ T,
(2.7)
a
where F ∆ = f . Similarly, the nabla integral is defined as
t
Z
f (τ) ∇τ = F̂(t) − F̂(a),
a, t ∈ T,
(2.8)
a
where F̂ ∇ = f . It is known that rd-continuity is a sufficient condition for the
existence of the delta integral, and ld-continuity is sufficient for nabla integration
[46, Theorem 1.74]. However, here we are interested in functions which may be
unbounded, and hence right dense continuity (and left dense continuity) maybe
too strong. There are two approaches to rectify this, the time scale analogue of
the improper Riemann integral or the time scale analogue of Lebesgue integral.
Both theories were introduced in Bohner and Guseinov [42].
We have chosen to apply the time scales analogue of Lebesgue integration
as this theory has been further developed. Bohner and Guseinov [43] have considered double Lebesgue integration on time scales. In [49], Cabada and Vivero
have considered the relationship between the time scales Lebesgue integral and
the Lebesgue integral on the reals by developing a formula which allows one to
express the former in terms of the latter. In [48] they have proved a time scales
2.2. THE LEBESGUE NABLA INTEGRAL
25
analogue of the Banach-Zarecki theorem and the fundamental theorem of calculus. However, in this thesis we shall develop fractional calculus using the nabla
integral, and the papers by Cabada and Vivero are written for the Lebesgue delta
integral. Unfortunately, there does not exist a nabla analogue to this work in
the literature, and one should not naively assume that these analogues are immediately true. Indeed, as Section 1.2 illustrates there can be instances where a
property will hold for delta calculus and fail for its nabla analogue. Accordingly,
Sections 2.2–2.4 will be devoted to creating this theory.
In order to study the Lebesgue theory of time scales we fix down an interval
[a, b] with b > a. Without loss of generality we shall assume a, b ∈ T, and denote
e
T := [a, b] ∩ T. Let F2 be the set of all time scale intervals of the form (a0 , b0 ] ∩ T
for a0 ≤ b0 and a0 , b0 ∈ e
T. We apply the convention (t, t] = ∅. We define a set
function m2 : F2 → [0, ∞] by
m2 ((a0 , b0 ] ∩ T) := b0 − a0 ,
(2.9)
we shall extend m2 to an outer measure on P(e
T), which we shall denote as m∗2 .
Let E ∈ P(e
T). If a ∈ E then we define m∗2 (E) = ∞. If not, let R ⊃ E be a
countable interval cover of the form
R=
[
(a0i , b0i ] ∩ T,
(2.10)
i∈I
for an index set I ⊂ N. We shall define
m∗2 (E) = inf
X
(b0 − a0 ),
(2.11)
i∈I
where the infimum is taken over all such countable covers R.
A set A ⊂ e
T such that, for all E ⊂ e
T, satisfies
m∗2 (E) = m∗2 (E ∩ A) + m∗2 (E ∩ (e
T \ A)),
(2.12)
is said to be ∇-measurable. The set of all ∇-measurable sets is denoted M(m∗2 ).
The restriction of m∗2 to M(m∗2 ) is the ∇-Lebesgue measure, denoted µ∇ . Observe
26
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
that for all t ∈ e
T with t , a, we have µ∇ ({t}) = ν(t), and for a0 , b0 ∈ e
T satisfying
0
0
0 0
0
0
a < a < b < b then µ∇ ((a , b ] ∩ T) = b − a .
Lemma 2.2.1. The set of all left scattered points of e
T is at most countable.
Proof. Define the function f : [a, b] → R by f (t) = t for t ∈ T and f (t) = ρ(t)
for ρ(s) < t < s for any s ∈ T. Clearly this function is monotone and therefore
has at most countable discontinuities [90], which are precisely the left scattered
points of e
T.
Because for any ∇-measurable E ⊂ e
T with a ∈ E we have µ∇ (E) = ∞, for the
rest of the thesis, unless otherwise stated we shall only consider the case when
E does not contain a. We define the set IE to be an index set of all left scattered
points of E ⊂ T. The set I x,y shall denote the set of all left scattered points in
(x, y] ∩ T.
Theorem 2.2.2. A set A ⊂ T is ∇-measurable if and only if it is Lebesgue measurable. Moreover,
X
µ∇ (A) = µ(A) +
ν(ti ),
(2.13)
i∈IA
if a < A.
Proof. We start by showing (2.13) for the ∇-outer measure, m∗2 . Denote µ∗ as
the outer measure for the Lebesgue measure, and let L to be the set of all left
scattered points of T. Suppose A ⊂ T with a < A. We observe
µ∗ (A) = µ∗ (A ∩ L) + µ∗ (A ∩ (T \ L)) = µ∗ (A ∩ (T \ L)),
(2.14)
because L is countable. Then because A ∩ (T \ L) has no left scattered points, and
a < A ∩ (T \ L), we have µ∗ (A) = m∗2 (A ∩ (T \ L)). Hence
m∗2 (A) = m∗2 (A ∩ L) + m∗2 (A ∩ (T \ L)) = µ∗ (A) +
X
ν(ti ),
(2.15)
i∈IE
as required. Observe that (2.15) is equivalent to (2.13) if A is both Lebesgue
measurable and Lebesgue ∇-measurable. We have proved the second part of the
theorem.
2.2. THE LEBESGUE NABLA INTEGRAL
27
As for the first part, suppose A ⊂ T is ∇-measurable, that is, for all E ⊂ T we
have (2.12). We shall show the inequality
µ∗ (E) ≤ µ∗ (E ∩ A) + µ∗ (E ∩ ([a, b] \ A)) ≤ µ∗ (E),
(2.16)
which proves Lebesgue measurability. The lower bound is clear. Suppose a < E,
then
µ∗ (E ∩ A) + µ∗ (E ∩ ([a, b] \ A)) ≤ µ∗ (E ∩ A) + µ∗ (E ∩ (T \ A)) + µ∗ (E ∩ ([a, b] \ T).
Apply the previous result to µ∗ (E ∩ A) and µ∗ (E ∩ (T \ A)), and the result holds.
Now assume the case when a ∈ E. Because a single point has Lebesgue
measure zero,
µ∗ (E ∩ A) + µ∗ (E ∩ ([a, b] \ A)) ≤ µ∗ ((E ∩ (a, b]) ∩ A) + µ∗ ((E ∩ (a, b]) ∩ ([a, b] \ A)).
This is bounded from above by µ∗ (E ∩ (a, b]), which in turn is bounded by µ∗ (E)
as required.
When a ∈ A, we observe that A \ {a} is Lebesgue measurable by the previous
argument, and because {a} is Lebesgue measurable, their union is also Lebesgue
measurable. The reverse case is proved similarly.
As an immediate consequence we see that every ∇-null set has no left scattered points and that the Lebesgue measure is absolutely continuous with respect
to µ∇ . In order to consider measurable functions, we shall need to define an auxiliary function. For a function f : e
T → R or a function f : [a, b] → R, we define
˜f : [a, b] → R by



 f (t),
˜f (t) = 



 f (ti ),
if t ∈ e
T,
(2.17)
if t ∈ (ρ(ti ), ti ) for some left scattered ti ∈ e
T.
When the domain is e
T, the auxiliary function is the constant extension to [a, b].
When the domain is [a, b], the auxiliary function is an approximation. The definition of f¯ given in Cabada & Vivero [49] does not consider the case of f with
28
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
domain [a, b]. This will simplify a later proof.
Theorem 2.2.3. A function f : e
T → R is Lebesgue ∇-measurable if and only if
˜f is Lebesgue measurable.
Proof. Because the Borel σ-algebra on R is generated by open intervals it is
sufficient to consider an arbitrary open interval U. Suppose that f˜ is Lebesgue
measurable. Then due to the equality f˜−1 (U) = f −1 (U) ∩ T, f is Lebesgue ∇measurable.
Now suppose f is Lebesgue ∇-measurable. Due to the definition of f˜ the
pre-image splits into
f˜−1 (U) = f −1 (U) ∪
[
(ρ(ti ), ti ),
(2.18)
i∈I f −1 (U)
which is Lebesgue measurable as f −1 (U) is Lebesgue ∇-measurable.
We now show the equivalence of ∇-Lebesgue integrability and Lebesgue integrability. This is achieved in two parts: we first establish the result for simple
functions then extend the result by the monotone convergence theorem to L1 .
Theorem 2.2.4. Let E ⊂ T be a ∇-measurable set with a < E. Define Ẽ to be the
extension of E,
[
Ẽ := E ∪ (ρ(ti ), ti ).
(2.19)
i∈IE
P
If S : e
T → [0, ∞) is a simple ∇-measurable function with S = nj=1 α j χA j
for constants α j ∈ R, and S̃ is the extension of S to [a, b], then S̃ is Lebesgue
measurable on Ẽ and moreover
Z
Z
S (s) ∇s =
S̃ (s) ds.
(2.20)
E
Ẽ
Proof. Observe that
S̃ =
n
X
j=1
α j χà j ,
(2.21)
2.2. THE LEBESGUE NABLA INTEGRAL
29
where the sets à j are defined as
à j := A j ∪
[
(ρ(ti ), ti ).
(2.22)
i∈IA j
For each j = 1, . . . , n, Ã j is a countable union of ∇-measurable sets and hence
Lebesgue measurable. Hence S̃ is a Lebesgue measurable function.
Meanwhile, by standard arguments in measure theory and a < E, we observe
µ(Ã j ∩ Ẽ) = µ∇ (A j ∩ E). Then (2.20) follows from the definitions of S and S̃ . Theorem 2.2.5. Let E ⊂ T be ∇-measurable set with a < E. Let f : e
T → R and
f˜ its extension to [a, b]. Then f ∈ L∇1 (E) if and only if f˜ ∈ L1 (Ẽ).
Proof. Every integrable function can be written as the difference of two nonnegative integrable functions, f = f+ − f− . Thus without loss of generality we
will assume f is non-negative.
Assume that f is Lebesgue ∇-measurable, therefore f˜ is Lebesgue measurable. Thus there exist an increasing sequence of simple functions, S k , that converge to f˜ pointwise. Define S̃ k by (2.17). Further, let us define fk as the restriction of S̃ k to T, and hence f˜k , is equal to S̃ k . Then
Z
fk (s) ∇s =
Z
E
Z
f˜k (s) ds =
S̃ k (s) ds.
Ẽ
(2.23)
Ẽ
By the monotone convergence theorem we have
Z
fk (s) ∇s =
lim
k→∞
ZE
lim
k→∞
S̃ k (s) ds =
Z
f (s) ∇s,
(2.24)
f˜(s) ds,
(2.25)
ZE
Ẽ
Ẽ
and the conclusion follows. The converse is proved similarly.
As always, given E ⊂ e
T, we denote the L p -norm of ∇-measurable f : E → R
as
!1/p
Z
p
| f (t)| ∇t
k f k p :=
.
(2.26)
E
The set of all functions with finite k f k p is denoted L∇p (E). For simplicity, when
30
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
E = (c, d] ∩ T ⊂ e
T, we shall denote L∇p (c, d) = L∇p ((c, d] ∩ T). When there is no
confusion, k f k = k f k1 .
We have shown the equivalence of Lebesgue ∇-integration and Lebesgue integration. Compared with the delta analogue proved in [49], the proof of Theorem 2.2.5 is shorter. We shall now prove a formula to calculate the Lebesgue
nabla integral with respect to Lebesgue integration.
Theorem 2.2.6. Let E ⊂ T be ∇-measurable with a < E. If f : T → R is
∇-integrable on E, then
Z
f (s) ∇s =
Z
E
X
f (s) ds +
E
f (ti ) ν(ti ).
(2.27)
i∈IE
Proof. Since f is ∇-integrable on E, f˜ is Lebesgue integrable on Ẽ. Then from
the definition of Ẽ,
Z
Z
Z
X
f (s) ∇s =
f˜(s) ds =
f (s) ds +
f (ti ) ν(ti ),
(2.28)
E
Ẽ
E
i∈IE
as required.
Corollary 2.2.7. For all s, t ∈ e
T with s < t, and a Lebesgue integrable f :
[a, b] → R we have the expression
t
Z
f (τ) ∇τ =
s
t
Z
f (τ) dτ +
s
XZ
i∈I s,t
ti
ρ(ti )
f (ti ) − f (τ) dτ.
(2.29)
Proof. Observe that from countable additivity of the Lebesgue integral we have
Z
f (τ) dτ =
f (τ) dτ −
(s,t]∩T
s
By (2.27), and using f (ti ) ν(ti ) =
Z
(s,t]∩T
t
Z
f (τ) dτ =
i∈I s,t
R ti
ρ(ti )
f (τ) ∇τ −
s
ti
f (τ) dτ.
(2.30)
f (ti ) dτ.
(2.31)
ρ(ti )
f (ti ) dτ, we have
t
Z
XZ
XZ
i∈I s,t
ti
ρ(ti )
The result is obtained by equating the right sides of (2.30) and (2.31).
2.3. ABSOLUTE CONTINUITY
2.3
31
Absolute Continuity
In this section, we prove a nabla version of the Banach-Zarecki theorem, which
states a function g is absolutely continuous on a closed interval X if and only if
(i) g is continuous and of bounded variation on X and (ii) g maps every null set
of X into a null set.
We shall define a second auxiliary function f¯ : [a, b] → R as the linear
interpolation of f . When t ∈ T we let f¯(t) = f (t) and for each i ∈ I,
!
¯f (t) := f ρ (ti ) + ( f (ti ) − f ρ (ti )) t − ρ(ti ) ,
ν(ti )
if t ∈ (ρ(ti ), ti ).
(2.32)
Lemma 2.3.1. Assume that f : T → R and f¯ : [a, b] → R is the extension of f
to [a, b] defined above. The following statements are equivalent:
1. f maps every ∇-null set of T into a null set.
2. f¯ maps every null set of [a, b] into a null set.
Proof. Assume that 2 is valid. Observing that E ⊂ T a ∇-null set implies that it
has no left scattered points, we have f |E = f¯|E and 1 immediately follows.
Let E ⊂ [a, b] be a null set. Because µ({a}) = 0 we assume a < E. Let L be
the set of left scattered points, we can write E as


[

E =  E ∩ (ρ(ti ), ti ] ∪ (E ∩ (T \ L)).
(2.33)
i∈I
By subadditivity we observe that
µ( f¯(E)) ≤
X
µ( f¯(E ∩ (ρ(ti ), ti ])) + µ( f¯(E ∩ (T \ L))).
(2.34)
i∈I
Clearly, f¯ satisfies the requirements of the Banach-Zarecki theorem over [ρ(ti ), ti ]
for each fixed i. Hence we see that µ( f¯(E ∩ (ρ(ti ), ti ])) = 0 as E was null to begin
with. And further E ∩ (T \ L) does not have any left scattered points and its ∇Lebesgue measure is zero. The result immediately follows by restricting f¯ to T,
which is precisely f .
32
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
Definition 2.3.2. A partition of T is a finite ordered subset P = {x0 , . . . , xn } ⊂ e
T
such that a = x0 < x1 < . . . < xn−1 < xn = b. Given a function f : T → R and a
partition P we define the variation
V(P, f ) :=
n
X
| f (xk ) − f (xk−1 )| .
(2.35)
k=1
The total variation is defined as the supremum of the variations over all such
partitions P. It is denoted Vab ( f ). A function f is of bounded variation if Vab ( f ) <
∞.
Because f¯ is the linear interpolation of f , it is clear that f is of bounded
variation on a closed interval if and only if f¯ is also. We are now in a position to
prove the time scales nabla analogue of the Banach-Zarecki theorem.
Theorem 2.3.3. A function f : T → R is left dense absolutely continuous if and
only if
1. f is continuous and of bounded variation,
2. f maps every ∇-null subset of T into a null set.
Proof. First, assume that 1 & 2 hold. Then the definition of f¯ guarantees the
requirements of the Banach-Zarecki theorem for f¯, hence f¯ is absolutely continuous. But the restriction of f¯ to T is exactly f . Hence f is also absolutely
continuous on T.
We shall prove the converse. Suppose that f is left dense absolutely continuous. We see that f is continuous on T. We first show it is of bounded variation.
Let δ > 0 correspond to the = 1 in the definition of absolutely continuous.
For δ/2 choose a partition of T such that either xk − xk−1 < δ/2, or xk is left
scattered and ν(xk ) > δ/2. In the latter case choose xk−1 = ρ(xk ). In the former
case, observe that because V(P, f ) < = 1 for every partition [xk , xk−1 ] ∩ T, by
k
taking the supremum over all such partitions, we see V xxk−1
( f ) ≤ 1. Hence, f
k
restricted to [xk−1 , xk ] ∩ T is of bounded variation. In the latter case, V xxk−1
(f) =
| f (xk ) − f (xk−1 )|, and hence f restricted to [xk−1 , xk ] ∩ T is of bounded variation,
and therefore so is f¯. Hence f¯ is also of bounded variation on [xk−1 , xk ] for each
k, and over all of [a, b].
2.4. FUNDAMENTAL THEOREM OF NABLA CALCULUS
33
Now consider a ∇-null set E ⊂ T. Let > 0 and let δ > 0 (as per the definition
of left dense absolutely continuity), and choose a a time scale interval cover
E⊂
∞
[
(ak , bk ] ∩ T,
(2.36)
k=1
satisfying
P∞
k=1 (bk
− ak ) < δ. We observe that by subadditivity
µ( f (E)) ≤
∞
X
µ( f ((ak , bk ] ∩ T)).
(2.37)
k=1
For each k ∈ N, choose ck , dk ∈ [ak , bk ] such that
µ( f ([ak , bk ] ∩ T)) ≤ | f (dk ) − f (ck )| .
(2.38)
Existence of ck and dk are guaranteed by continuity of f . But because [ck , dk ] ⊂
P
[ak , bk ] and by the absolute continuity of f , we see nk=1 | f (dk ) − f (ck )|
P
P
≤ nk=1 | f (bk ) − f (ck )| < , for all n ∈ N. Hence ∞
k=1 | f (dk ) − f (ck )| ≤ . Then
we have
µ( f (E)) ≤
∞
X
k=1
µ( f ([ak , bk ) ∩ T)) ≤
∞
X
| f (dk ) − f (ck )| ≤ ,
(2.39)
k=1
for all > 0. Hence µ( f (E)) must be zero. This proves 2.
We have proved a time scales nabla analogue to the Banach-Zarecki theorem.
In fact the second half of the proof is shorter than the analogous form given in
[48].
2.4
Fundamental Theorem of Nabla Calculus
Because we shall be using the nabla calculus in the next chapter, we shall need
to guarantee the existence of nabla derivatives. Thus we shall need to prove a
Lebesgue nabla integral analogue of the fundamental theorem of calculus. We
shall need the following lemma which concerns the derivative of the interpolation
function f¯. Here L is the set of left scattered points and ρ(L) = {ρ(t) : t ∈ L}.
34
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
Lemma 2.4.1. The following is true
1. If f¯ is differentiable at t ∈ (a, b] ∩ T, then f is ∇-differentiable at t and we
have f ∇ (t) = f¯0 (t).
2. If t ∈ (a, b] such that t < (L ∪ ρ(L)), and if f is ∇-differentiable at t, then f¯
is differentiable at t and f¯0 (t) = f ∇ (t).
Proof. Assertion 1 is obtained by calculating f ∇ (t) for left scattered t and left
dense t separately, and observing that f ∇ = f¯0 .
We prove assertion 2. Observe if t ∈ (a, b] \ T, then by definition f¯ is linear
with derivative
f (ti ) − f ρ (ti )
f¯0 (t) =
,
(2.40)
ν(ti )
where t ∈ (ρ(ti ), ti ) for left scattered ti . Let f be ∇-differentiable at t ∈ ((a, b] ∩
T) \ (L ∪ ρ(L)). Using the observation ρ(t) = t, let > 0, then there exist a
neighborhood U of t such that
f (t) − f (s) − f ∇ (t) = f¯(t) − f¯(s) − f ∇ (t) < ,
t − s
t−s
(2.41)
for all s ∈ U. If there are no left scattered points in U, then the result is trivial.
Suppose that there exist a left scattered ti ∈ U with ρ(ti ) ∈ U. Choose s ∈
(ρ(ti ), ti ). Define
f (t) − f (ti )
f ρ (ti ) − f (t)
, B=
.
(2.42)
A=
t − ti
ρ(ti ) − t
Then it is easy to show
min{A, B} ≤
f¯(t) − f¯(s)
≤ max{A, B}.
t−s
Hence, (2.41) holds for all s, and so f¯ is ∇-differentiable at all t.
(2.43)
Theorem 2.4.2 (Fundamental Theorem of Nabla Calculus). A function f : T →
R is left dense absolutely continuous on e
T if and only if
1. f is ∇-differentiable µ∇ almost everywhere on (a, b] ∩ T and f ∇ ∈ L∇1 (T),
Rt
2. the equality f (t) = f (a) + a f ∇ (s) ∇s holds for all t ∈ e
T.
2.4. FUNDAMENTAL THEOREM OF NABLA CALCULUS
35
Proof. First suppose that f is left dense absolutely continuous on T. Thus f¯ is
also absolutely continuous on [a, b], and the fundamental theorem of calculus
holds for f¯. We wish to show that f is ∇-differentiable ∇-almost everywhere on
(a, b] ∩ T. First define E1 to the set of points such that f is not ∇-differentiable.
Similarly define E2 to be the set of points such that f¯ is not differentiable. Because f is continuous, f is ∇-differentiable at all left-scattered points and hence
E1 has no left scattered points. But then µ∇ (E1 ) = µ(E1 ). Moreover, because
f¯ is differentiable almost everywhere, from assertion 2 of Lemma 2.4.1 hence
E1 ⊂ E2 , we have µ∇ (E1 ) = µ(E1 ) ≤ µ(E2 ) = 0 and the result holds.
Because we only need the ∇-derivative up to ∇-a.e., assume that f ∇ = 0 on
E1 ∪ {a}. We shall need the function f
f ∇ , that is the extension of f ∇ to [a, b]. We
know that f
f ∇ = f¯0 over (a, b] \ E2 . But now µ(E2 ) = 0, so f¯0 ∈ L1 ([a, b]), and
further f
f ∇ ∈ L1 ([a, b]) with the equation
Z
f
f ∇ (s) ds =
Z
f¯0 (s) ds,
(a,t]
(2.44)
(a,t]
valid for all t ∈ [a, b]. However, due to the equivalence between the Lebesgue
integral and ∇-integral, we see f ∇ ∈ L∇1 (e
T) and
Z
t
f (s) ∇s =
Z
∇
a
t
f
f ∇ (s) ds =
a
t
Z
f¯0 (s) ds,
(2.45)
a
for all t ∈ e
T. And we obtain assertion 2 via
Z t
Z
0
f (t) − f (a) = f¯(t) − f¯(a) =
f¯ (s) ds =
a
t
f ∇ (s) ∇s,
(2.46)
a
for all t ∈ e
T.
Conversely, suppose that the two conditions hold. We shall show that f is left
dense absolutely continuous. Because f ∇ ∈ L∇1 (e
T) we know f
f ∇ ∈ L1 ([a, b]). We
now use
Z t
Z t
∇
f
f (t) − f (a) =
f (s) ∇s =
f ∇ (s) ds,
(2.47)
a
and hence
f¯(t) = f¯(a) +
Z
a
a
t
f
f ∇ (s) ds = f¯(a) +
Z
t
f¯0 (s) ds.
a
(2.48)
36
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
But this satisfies the requirements of the fundamental theorem of calculus of f¯,
and hence f¯ is absolutely continuous on [a, b]. And so f is left dense absolutely
continuous on e
T.
2.5
Summary and Examples
In this chapter, we have collated important known results of the literature concerning time scales calculus, and established nabla analogues to important results
in delta time scales analysis. We shall utilize these results in the next chapter. In
this section, we characterize important time scales seen in the literature. These
example will form the basis of our theory.
Example 2.5.1. The standard continuum calculus is obtained when one chooses
the time scale T = R. Clearly, every point is dense. Hence we have σ(t) =
ρ(t) = t, µ(t) = ν(t) = 0, and Tκ = Tκ = T. Right-dense continuity and rightdense absolute continuity correspond to the standard concepts of continuity and
absolute continuity respectively. Further we have f ∆ = f ∇ = f 0 when they exist.
The Lebesgue ∇-integral corresponds the standard Lebesgue integral on R.
Example 2.5.2. Discrete calculus can be obtained by choosing T = Z. We have
σ(t) = t+1, ρ(t) = t−1, µ(t) = ν(t) = 1 and so every point is discrete. The integers
are unbounded, thus Tκ = Tκ = T. The derivatives correspond to the right and left
difference operators: f ∆ (t) = f (t + 1) − f (t) = ∆ f (t) and f ∇ (t) = f (t) − f (t − 1) =
∇ f (t). Every function is by definition right dense continuous and right dense
absolutely continuous. Finally the ∇-integrals correspond to finite summation:
b
Z
f (t) ∆t =
a
b−1
X
k=a
b
Z
f (t) ∇t =
f (k),
a
b
X
f (k).
(2.49)
k=s+1
Example 2.5.3. Quantum calculus is obtained by choosing the time scale
T = qZ = {0, 1, q, q−1 , q2 , q−2 , q3 , . . .},
(2.50)
where we have fixed q ∈ (0, 1) ∪ (1, ∞). The choice of choosing q > 1 or 0 <
q < 1 is a matter of preference. The former leads to simpler expressions when
2.5. SUMMARY AND EXAMPLES
37
evaluating ∆-calculus, while the latter simplifies ∇-calculus. One can switch
between the two using the transformation q 7→ q−1 .
Let q > 1. Then for all t ∈ T, σ(t) = qt and ρ(t) = t/q. We have µ(0) = 0 and
ν(0) undefined. For t > 0, µ(t) = (q − 1)t and ν(t) = t(q − 1)/q. The time scale is
bounded on the left, hence Tκ = T \ {0}, while Tκ = T. For t > 0,
f ∆ (t) =
f (qt) − f (t)
,
qt
and t = 0,
f ∇ (t) =
q f (t) − f (tq−1 )
,
q−1
t
f (q−k ) − f (0)
f (0) = lim
.
k→∞
q−k
∆
(2.51)
(2.52)
Both right dense continuity and left dense continuity are equivalent to continuity
at t = 0. When 0 < a < b, the integrals are given by the formula
Z
b
f (t) ∆t = (q − 1)
a
b/q
X
Z
k f (k),
a
k=a
b
b
q−1 X
f (t) ∇t =
k f (k).
q k=qa
(2.53)
When a = 0, the integral is given by the limit obtained by a → 0.
Example 2.5.4. The (q, h)-time scale is an extension of the previous two time
scales. For a fixed t0 > 0, q ≥ 1 and h ≥ 0 satisfying q + h > 1, we define
) (
)
1 − qk
h
:= t0 q +
h : k∈Z ∪
,
1−q
1−q
(
0
Tt(q,h)
k
(2.54)
when q , 1. If q = 1 we define
0
Tt(1,h)
:= {t0 + kh : k ∈ Z} ∪ {−∞}.
(2.55)
Thus we have σ(t) = qt + h and ρ(t) = (t − h)/q. For q = 1, Tκ = T \ {−∞}, and
Tκ = T \ {h/(1 − q)} otherwise. In either situation, Tκ = T. The delta and nabla
derivatives are given by
f ∆ (t) =
f (qt + h) − f (t)
,
qt − t + h
f ∇ (t) =
f (t) − f ((t − h)/q)
.
(qt − t + h)/q
(2.56)
38
CHAPTER 2. OVERVIEW OF TIME SCALE CALCULUS
The ∆-integral is given by
Z
σn (t)
f (τ) ∆τ = (qt − t + h)
t
n−1
X
qk f (σk (t)).
(2.57)
k=0
Similarly the ∇-integral is given by
Z
t
ρn (t)
f (τ) ∇τ = (t − tq−1 + q−1 h)
n−1
X
k=0
q−k f (ρk (t)).
(2.58)
Chapter 3
Basic Definitions and Taylor’s
Theorem
In this chapter, we present a unified model of fractional calculus, such that, when
the time scale is chosen to be R, Z or qZ , one obtains the Riemann-Liouville
fractional derivative, the Gray-Zhang fractional difference and the Rajković qfractional derivative respectively. We first present axioms for defining power
functions of a time scale and use this to define the fractional integral and fractional derivative. After proving simple properties, we finish with a generalization
of the Taylor theorem.
3.1
Power Functions
From the introduction it is clear that there are many possible avenues to construct
a unified definition of fractional calculus. However, some appear to be simpler than others. For example, the different forms of right fractional difference
(which would generalize the delta derivative) consistently change the domain of
the function. That is, the domain of f and ∆α f are different. In the case of qcalculus, we see that the left q-fractional derivative (1.44) does not change the
domain, but the right q-fractional derivative does. This is a problem, as future
work on the subject may consider fractional calculus on other time scales (including the Cantor set), and in this situation it is not entirely clear how tractable
39
40
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
this problem is. Thus for this thesis we concentrate on nabla calculus.
In addition to the question of whether to pursue left or right fractional calculus, there remains the question of the whether to define fractional calculus
through integrals or derivatives. In the former situation, one would need to obtain a definition of power functions on time scales generalizing the monomials.
n
In the latter situation, one would need to obtain a closed form for f ∇ . Neither
approach is trivial. In this thesis we have chosen the former, and assume a priori
that a time scales power function is known. We posit the axioms that these functions would need to satisfy, and show that these properties characterize fractional
calculus.
To begin with, we must prove the Cauchy formula for multiple integration.
We begin by recalling the time scale monomials.
Definition 3.1.1. Let T be a time scale. Define h0 (t, s) = g0 (t, s) = 1 for all
t, s ∈ T. Then for k = 1, 2, . . . define
t
Z
hk (τ, s) ∆τ,
hk+1 (t, s) :=
t
Z
gk (σ(τ), s) ∆τ.
gk+1 (t, s) :=
s
(3.1)
s
Similarly, the equivalent functions for nabla calculus is denoted by ĥk (t, s)
and ĝk (t, s) respectively. The four functions are related via the relations
t ∈ T, s ∈ Tκ ,
n
hn (t, s) = (−1)n gn (s, t),
(3.2)
κn
t ∈ Tκ ∩ Tκ , s ∈ T ,
hn (t, s) = (−1)n ĥn (s, t),
(3.3)
proved in [46, 74] respectively. Combining the two leads to the simple corollary
n
ĝn (t, s) = (−1)n ĥn (s, t) for t ∈ Tκ ∩ Tκ and s ∈ Tκ . The monomials are used in
the Taylor theorem for nabla calculus.
Theorem 3.1.2 ([29]). Let α, t ∈ Tκn−1 . Let f be a function which is n times nabla
n
differentiable on Tκn with f ∇ ld-continuous over T. Then
f (t) =
n−1
X
k=0
∇k
ĥk (t, α) f (α) +
Z
α
t
n
ĥn−1 (t, ρ(τ)) f ∇ (τ) ∇τ.
(3.4)
We are interested in generalizing the Riemann-Liouville fractional integral,
3.1. POWER FUNCTIONS
41
so our first result shall be a time scale analogue of the Cauchy integral formula.
Theorem 3.1.3 (Cauchy Integral Formula). Fix n ∈ N. Let T be a time scale,
n
and let a ∈ Tκ , t ∈ Tκ , φ : T → R be rd-continuous (ld-continuous). Then
x
Z
∆t1
Z
a
∆t2 · · ·
Z
∇t1
tn−1
Z
a
x
Z
t1
a
t1
tn−1
Z
a
x
hn−1 x, σ(t) φ(t) ∆t,
(3.5)
ĥn−1 x, ρ(t) φ(t) ∇t.
(3.6)
a
∇t2 · · ·
a
φ(tn ) ∆tn =
Z
φ(tn ) ∇tn =
a
x
Z
a
Proof. We will prove the ∆-case, as the ∇-case is similar. The case n = 1 is
trivial. Assume it is true for n, we have for n + 1
x
Z
hn (x, σ(t))φ(t)∆t =
x
Z
a
(−1)n gn (σ(t), x) φ(t) ∆t.
(3.7)
a
Whence applying integration by parts formula we obtain
!
"
#t=x Z x
Z t
Z t
n
n ∆
(−1) gn (t, x)
φ(τ)∆τ
−
(−1) gn (t, x)
φ(τ)∆τ ∆t
a
a
a
t=a
!
Z x
Z t
n−1
=
(−1) gn−1 σ(t), x
φ(τ)∆τ ∆t
a
a
Z x
=
hn−1 (x, σ(t)) φ[1] (t) ∆t,
(3.8)
a
where we have let φ[1] (t) =
for n, so we have
Rt
a
x
Z
hn−1 (x, σ(t))φ (t)∆t =
[1]
a
φ(τ)∆τ. Then by assumption, the formula is true
x
Z
∆t1
Z
t1
tn−1
φ[1] (tn ) ∆tn
Za x
Za t1
Za tn−1
Z tn
=
∆t1
∆t2 · · ·
∆tn
φ(tn+1 ) ∆tn+1 .
a
a
∆t2 · · ·
Z
a
a
(3.9)
We also give an alternative proof of (3.6) with different initial conditions.
Theorem 3.1.4. Let T be a time scale. Let n ∈ N, α ∈ Tκn−1 and x ∈ T satisfying
α < x. Let φ : T → R be ld-continuous on [α, x] ∩ T. Then (3.6) is valid.
42
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
Proof. We choose f to be a function satisfying φ = f ∇ on Tκn . Then (3.4)
implies that (3.6) is equivalent to
n
Z
α
x
t1
Z
∇t1
α
∇t2 . . .
Z
tn−1
α
∇n
f (tn ) ∇tn = f (x) −
n−1
X
α
x
Z
∇t1
(3.10)
k=0
We prove using induction on n. If n = 1, then clearly
by ld-continuity of φ. Assume it is true for n. Then
Z
k
f ∇ (α) ĥk (x, α).
t1
Z
tn
R
x
α
f ∇ (t1 ) ∇t1 = f (x) − f (α)
n+1
∇t2 . . .
f ∇ (tn+1 ) ∇tn+1
α
α

Z x 
n−1
X

k
 ∇
=
( f ∇ )∇ (α) ĥk (t1 , α)  ∇t1
 f (t1 ) −
α
= f (x) − f (α) −
k=0
n−1
X
k+1
f ∇ (α) ĥk+1 (x, α)
k=0
= f (x) −
n
X
k
f ∇ (α) ĥk (x, α),
(3.11)
k=0
as required.
However, while the theory of time scale monomials has been extensively
studied, extending the monomials to ĥα for non-integer α is still an open problem.
Indeed, the major hurdle in defining fractional calculus on arbitrary time scales
is finding a suitable definition for power functions. As noted in the Chapter 1,
the majority of literature on the subject is devoted to considering specific time
scales and defining fractional integrals and derivatives for them.
In fact the process of extending monomials to non-integer order is different
for each time scale. This is a major cause of the difficulty of unification. For
example, on the time scale T = R, the monomials
(t − s)k
ĥk (t, s) =
,
k!
(3.12)
3.1. POWER FUNCTIONS
43
are extended via the logarithm
ĥα (t, s) =
eα log(t−s)
.
Γ(α + 1)
(3.13)
On the time scale T = Z, the rising factorial (α)β is defined using the gamma
function. That is, the monomial is defined as a special case of the power function,
and hence there is no extension procedure to invoke. Lastly, on the quantum time
scale qZ , the monomial
ĥk (t, s) =
(t − s)(t − qs) . . . (t − qk−1 s)
,
Γq (k + 1)
for 0 < q < 1,
(3.14)
is extended to non-integer order by a ratio of two infinite products
∞
ĥα (t, s) =
Y 1 − qi s/t
tα
.
Γq (α + 1) i=0 1 − qi+α s/t
(3.15)
The three cases each have a different extension procedure, and thus the core of
the problem is finding a procedure on time scales which unifies all three. We will
bypass this problem by considering the properties of power functions which are
consistent across all three cases.
Definition 3.1.5. Let T be a time scale. A collection of functions {ĥα (·, ·) : e
T×
e
T → R} for −1 < α < ∞ are called time scale power functions if they satisfy
P1 For all α > −1, ĥα (t, s) is a positive ld-continuous function in both variables
when t > s and ĥα (t, s) ≡ 0 whenever t ≤ s.
P2 Whenever α ∈ N0 and t ≥ s, ĥα (t, s) corresponds with the nabla time scale
monomials.
P3 For all α, β > −1 one has
t
Z
ĥα (t, ρ(τ)) ĥβ (τ, s) ∇τ = ĥα+β+1 (t, s),
s
for t, s ∈ e
T and s < t.
(3.16)
44
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
The second property states time scales power functions generalize time scales
monomials, and so its importance is straight forward enough. The remaining two
properties however are not so clear. The third property (3.16) gives a semi-group
property for power functions allowing us to evaluate convolutions of two power
functions. As we shall see in the sequel, it plays an integral part in many properties of fractional calculus, from the index law to studying Taylor-type theorems.
Also, as a simple corollary, by setting α = 0 and β = 0 in (3.16), we observe that
ĥβ (·, s) and ĥα (t, ρ(·)) are both elements of L∇1 (e
T).
Lastly, the significance of the requirement that the power functions must be
zero when t < s can be seen by comparison with Anastassiou [21], which does
not include this property. Anastassiou defines
ĥα (t, s) =
(t − s)α
,
Γ(α + 1)
(3.17)
if T = R. However, when t < s and α is a non-integer, we see that (3.17) is not
defined as a real function, despite the claim that ĥα must be ld-continuous over
all of T × T. The first property exists to mitigate this problem.
Before we consider examples of power functions for specific time scales, one
valid question is whether one can show that these functions do indeed exist for
arbitrary time scales. Because this thesis is mainly concerned about unifying preexisting forms of fractional calculus we shall defer this problem to Chapter 5.
Example 3.1.6. Let T = R. Then for ν > −1, the functions
ĥν (t, s) =
(t − s)ν
,
Γ(ν + 1)
t > s,
(3.18)
and ĥν (t, s) = 0 for t ≤ s, satisfies the requirements. To prove (3.16), we shall
use a well known argument. By definition
t
Z
ĥα (t, ρ(τ)) ĥβ (τ, s) ∇τ =
s
t
Z
s
(t − τ)α (τ − s)β
dτ.
Γ(α + 1) Γ(β + 1)
(3.19)
3.1. POWER FUNCTIONS
45
Let τ = s + θ(t − s). Then
t
Z
s
(t − s)α+β+1
(t − τ)α (τ − s)β
dτ =
Γ(α + 1) Γ(β + 1)
Γ(α + 1) Γ(β + 1)
1
Z
(1 − θ)α θβ dθ,
0
and the result holds using the properties of the beta function.
Example 3.1.7. Let T = Z. We define
ĥν (t, s) =
Γ(t − s + ν)
,
Γ(t − s) Γ(ν + 1)
t > s,
(3.20)
and ĥν (t, s) = 0 for t ≤ s. To prove (3.16), we shall use the rising factorial
xm = x(x + 1)(x + 2) . . . (x + m − 1) =
Γ(x + m)
,
Γ(x)
(3.21)
for positive integer m. This notation is from Graham et al [69]. Thus
ĥα+β+1 (t, s) =
Γ(t − s + α + β + 1)
(α + β + 2)t−s−1
=
.
Γ(t − s) Γ(α + β + 2)
Γ(t − s)
The proof follows [71]. We use the property (a + b)n =
Pn n
k=0 k
(3.22)
an−k bk we have
t−s−1
X t − s − 1!
(α + β + 2)t−s−1
1
=
(α + 1)t−s−1−k (β + 1)k
Γ(t − s)
Γ(t − s) k=0
k
=
t−s−1
X
k=0
1
Γ(α + t − s − k) Γ(β + 1 + k)
.
Γ(k + 1) Γ(t − s − k)
Γ(α + 1)
Γ(β + 1)
(3.23)
Now setting k = τ − s − 1, we have
t
X
Γ(α + t − τ + 1) Γ(β + τ − s)
=
Γ(τ
−
s)
Γ(t
−
τ
+
1)
Γ(α
+
1)
Γ(β
+
1)
τ=s+1
t
Z
ĥα (t, ρ(τ)) ĥβ (τ, s) ∇τ (3.24)
s
as required.
Example 3.1.8. As stated previously, we can choose either q > 1 or 0 < q < 1.
46
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
We shall choose the latter to be consistent with [2, 31]. We define
(ν)
(t − s)q
,
ĥν (t, s) =
Γq (ν + 1)
t > s,
(3.25)
and ĥν (t, s) = 0 for t ≤ s, where we have used the definitions of the q-power
function (1.36) and q-gamma function (1.37). We show (3.16) in two cases, the
first where the Jackson sum is infinite, and when the sum is finite. In the former
case s = 0. We follow Atici and Eloe [31]. We have
t
Z
ĥα (t, ρ(τ)) ĥβ (τ, 0) dq τ =
0
Z
t
0
(α)
(t − qτ)q
τβ
dq τ.
Γq (α + 1) Γq (β + 1)
(3.26)
Substituting in τ = tθ we have dq τ/dq θ = t and
t
Z
0
(α)
(t − qτ)q
τβ
dq τ =
Γq (α + 1) Γq (β + 1)
(α)
(t − tθq)q
(tθ)β
t dq θ
0 Γq (α + 1) Γq (β + 1)
Z 1
tα+β+1
(α)
=
(1 − qθ)q θβ dq θ. (3.27)
Γq (α + 1) Γq (β + 1) 0
Z
1
The result holds through the q-beta function [58], which satisfies
Bq (a, b) =
Z
1
(a−1)
ub−1 (1 − qu)q
0
dq u =
Γq (a) Γq (b)
.
Γq (a + b)
(3.28)
As for s , 0, we follow Agarwal [2]. We assume s , t hence there exist natural
n such that s = tqn . From the definition of the Jackson integral [82] we have
Rx
Pn−1
n
k
f
(t)
d
t
:=
x(1
−
q)
q
n
k=0 f (xq ) q and thus
xq
t
Z
ĥα (t, ρ(τ)) ĥβ (τ, s) ∇τ
s
(α)
(β)
(t − qτ)q (τ − tqn )q
=
dq t
tqn Γq (α + 1) Γq (β + 1)
n−1
X
t(1 − q)
(β)
(α)
=
qk (t − tqk+1 )q (tqk − tqn )q .
Γq (α + 1) Γq (β + 1) k=0
Z
t
(3.29)
3.1. POWER FUNCTIONS
47
(c)
(c)
Using the property (at − as)q = ac (t − s)q , we factor tα+β out of the summation
to obtain
X
(1 − q)tα+β+1
(β)
(α)
q(β+1)k (1 − qk+1 )q (1 − qn−k )q .
Γq (α + 1) Γq (β + 1) k=0
n−1
(3.30)
We see from the definition of the basic hypergeometric function (Gasper and
Rahman [68])
2 φ1 (q
−m
δ+1
,q
−m−γ
;q
; q, q) =
m
X
(γ)
(δ)
n(γ+1)
q
(1 − qn+1 )q (1 − qm+1−n )q
(δ)
(1 − q)q
n=0
(γ)
,
(3.31)
(1 − qm+1 )q
for γ, δ > 0 and positive integer m. Thus (3.30) is equal to
(β)
(α)
t
α+β+1
(1 − q) (1 − q)q (1 − qn )q 2 φ1 (q1−n , qα+1 ; q−β−n−1 ; q, q)
.
Γq (α + 1) Γq (β + 1)
(3.32)
By (1.39), and the identity
(n)
2 φ1 (q
−n
, b; c; q, q) =
(1 − c/b)q
(1 −
(n)
c)q
bn ,
(3.33)
(3.32) becomes
(α+β+1)
t
α+β+1
α+β+1
(1 − q)
(1 − qn )q
(α+β+1)
(1 − q)q
(α+β+1)
=
(t − qt)q
(α+β+1)
,
(3.34)
(1 − q)−α−β−1 (1 − q)q
as required.
Example 3.1.9. The fractional calculus for (q, h)-time scale was recently studied
in Čermák and Nechvátal [55]. When q = 1, we define the h-gamma function
[60] as
n! hn (nh)t/h−1
,
n→∞ t(t + h) . . . (t + (n − 1)/h)
Γh (t) := lim
t , 0, −h, −2h, . . . ,
(3.35)
48
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
and the power function as
ĥα (t, s) =
Γh (t − s + αh)
,
Γ(α + 1) Γh (t − s)
(3.36)
for t > s. For the case q , 1, we shall choose q > 1, and define t˜ := t + hq/(1 − q)
and s̃ := s + hq/(1 − q). Then
∞
Y
1 − s̃t˜−1 q− j
t˜α
ĥα (t, s) :=
,
Γq−1 (α + 1) j=0 1 − s̃t˜−1 q−α− j
(3.37)
when t > s. In either case, ĥα (t, s) = 0 whenever t ≤ s.
In the situation q = 1, observe that (3.36) is equivalent to (3.20) after a scaling
by h due to the property Γh (t) = ht/h−1 Γ(t/h). Thus we will omit the proof of
(3.16) for this case. Suppose q > 1, and suppose s > h/(1 − q). Hence there exist
a positive integer m such that s = ρm (t). We shall need [55, Lemma 1],
α−1
ĥα−1 (t, ρ(τ)) = ν(τ)

 α − 1 + logq

α−1
ν(t)
ν(τ)


 .
(3.38)
q
The right hand side of (3.16) is thus
ĥα+β+1 (t, s) = ν(ρ
m−1
α+β+1
(t))

ν(t)
 α + β + 1 + logq m−1
ν(ρ (t))

α+β+1


 .
(3.39)
q
To simplify, observe that ν(ρk (t)) = q−k ν(t), and further the q-binomial satisfies

 ν + k

ν



k 
−ν − 1
 = (−1)k q(ν+1)k+(2) 
k
q


 ,
ν ∈ R, k ∈ Z.
(3.40)
q
Hence we have
ĥα+β+1 (t, s) = (−1)
m−1
ν(t)
α+β+1
q
m−1+(m−1
2 )

 −α − β − 2

m−1


 .
q
(3.41)
3.2. FRACTIONAL INTEGRALS AND BASIC PROPERTIES
49
The left hand side of (3.16) is equal to
t
Z
s
ĥα (t, ρ(τ)) ĥβ (τ, s) ∇τ

Z t
 α + logq
α
=
ν(τ) 
α
ρm (t)
ν(t)
ν(τ)

 β + logq ν(τ)
ν(ρm−1 (t))

β

β + m − k − 1 
 .
β
q


 ν(ρm−1 (t))β
q

m−1
X
 α + k
−k−kα+β−βm
α+β+1 
q
ν(t)
=

α
k=0



q





 ∇τ
q
(3.42)
We shall use (3.40), and let p = m − 1, x = −β − 1 and y = −α − 1. Then
t
Z
α+β+1 p+(2p)
ĥα (t, ρ(τ)) ĥβ (τ, s) ∇τ = (−1) ν(t)
p
s
q
p
X
q
k2 +xk−kp
k=0
  
 y   x
  
k q p−k


 ,
q
which is equal to (3.41) after applying the summation formula for the q-binomial.
Lastly we have q > 1 and s = h/(1 − q). However, this follows from the
quantum time scale case with s = 0 shown above after we apply a change of
variable.
3.2
Fractional Integrals and Basic Properties
Definition 3.2.1. Let T be a time scale, a, b ∈ T and define e
T := [a, b] ∩ T. Let
c∈e
T. For α ≥ 0 and for a function f : T → R we define the fractional ∇-integral
of order α as (c ∇0t f )(t) = f (t) and
Z
(c ∇−α
t f )(t)
:=
t
ĥα−1 (t, ρ(τ)) f (τ) ∇τ,
(3.43)
c
for α > 0, t ∈ (c, b] ∩ T.
The definition is dependent on the choice of power functions for any given
time scale. For example, on the time scale T = R, by choosing (3.18) we obtain the Riemann-Liouville fractional integral (1.5). Similarly, if given T = Z,
we can choose (3.20) to obtain the Gray-Zhang fractional summation (1.21) for
α ∈ [0, ∞). On the quantum time scale T = qZ , we choose (3.25) and obtain the
50
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
Rajković variation of the Agarwal fractional q-integral (1.44). Lastly, our definition subsumes the (q, h) fractional nabla integral by choosing (3.36) and (3.37)
for the q = 1 and q > 1 cases respectively. By using our definition we are able to
study all four cases collectively.
Lemma 3.2.2. Let α > −1 and β > 0, then c ∇−β
t ĥα (t, c) = ĥα+β (t, c).
Rt
Proof. By (3.16), c ∇−β
ĥ
(t,
c)
=
ĥ (t, ρ(τ)) ĥα (τ, c) ∇τ = ĥα+β (t, c) as reα
t
c β−1
quired.
T), the fractional ∇-integral exists
Theorem 3.2.3. Let α > 0, then for f ∈ L∇1 (e
almost everywhere.
Proof. We shall use a well known argument. We have
b
Z
ĥα−1 (t, ρ(τ)) ∇t =
a
ρ(τ)
Z
ĥα−1 (t, ρ(τ)) ∇t +
a
Z
b
ρ(τ)
ĥα−1 (t, ρ(τ)) ∇t
= ĥα (b, ρ(τ)).
(3.44)
Hence
b
Z
Z
∇τ
a
b
ĥα−1 (t, ρ(τ)) | f (τ)| ∇t =
a
Z
b
ĥα (b, ρ(τ)) | f (τ)| ∇τ
a
≤ ĥα (b, a) k f k < ∞.
(3.45)
So then the map
Z
t 7→
b
ĥα−1 (t, ρ(τ)) f (τ) ∇τ,
(3.46)
a
is integrable by the Fubini-Tonelli theorem with respect to µ∇ . Hence (c ∇−α
t f )(t)
exists almost everywhere and is nabla integrable.
This theorem is also proved in Anastassiou [21, Lemma 3] for the specific
time scales T = R and T = Z. However, the proof presented is erroneous.
Anastassiou first defines




ĥα−1 (s, ρ(t)),
K(s, t) := 


 0,
if a ≤ t ≤ s ≤ b,
otherwise.
(3.47)
3.2. FRACTIONAL INTEGRALS AND BASIC PROPERTIES
51
And then in the first equation of the proof writes
Z
b
K(s, t) ∇s =
Z
K(s, t) ∇s +
[a,t)
a
b
Z
K(s, t) ∇s.
(3.48)
t
Rb
P
On the integers a f (t)∇t = bk=a+1 f (t), and so the right hand side of the equation
is missing the term K(t, t) = ĥα−1 (t, t − 1). This is a relatively minor error as the
proof only relies on the finiteness of the expression. Further, the proof we give
does not need one to define an auxiliary function K, as we require ĥα (t, s) = 0
for t ≤ s to begin with.
Theorem 3.2.4. If f ∈ L∇1 (e
T), then given α ≥ 1, (a ∇−α
t f )(t) is ld-continuous over
t∈e
T.
Proof. Let x, y ∈ e
T such that a ≤ x ≤ y ≤ b. Suppose x is right-dense, but not
left-dense. f ∈ L∇1 implies
y
Z
b
Z
| f (τ)| ∇τ = k f k < ∞.
| f (τ)| ∇τ ≤
a
(3.49)
a
So then a ∇αy f (y) ≤ k f k ĥα (y, a), and ĥα being ld-continuous implies
limy→x+ a ∇αy f (y) ≤ limy→x+ k f k1 ĥα (y, a) < ∞, as required.
Suppose that y is left-dense. Then
α
(a ∇y f )(y) − (a ∇αx f )(x)
Z x
Z y
= ĥα−1 (y, ρ(τ)) f (τ) ∇τ −
ĥα−1 (x, ρ(τ)) f (τ) ∇τ
Za y
Za x
= ĥα−1 (y, ρ(τ)) f (τ) ∇τ +
ĥα−1 (y, ρ(τ)) f (τ) ∇τ
a
x
Z x
−
ĥα−1 (x, ρ(τ)) f (τ) ∇τ
Z y
Z x h a
i
= ĥα−1 (y, ρ(τ)) − ĥα−1 (x, ρ(τ)) f (τ) ∇τ +
ĥα−1 (y, ρ(τ)) f (τ) ∇τ
Z ax Zx y
ĥα−1 (y, ρ(τ)) − ĥα−1 (x, ρ(τ)) | f (τ)| ∇τ +
≤
ĥα−1 (y, ρ(τ)) | f (τ)| ∇τ.
a
x
(3.50)
52
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
Then, f ∈ L∇1 implies that
y
Z
ĥα−1 (y, ρ(τ)) | f (τ)| ∇τ ≤ k f k
y
Z
x
ĥα−1 (y, ρ(τ)) ∇τ
x
= k f k ĥα (y, x),
(3.51)
which is 0 as x → y. As for the first integral, observe that ĥα−2 (·, s) being positive
implies that ĥα−1 (·, s) is non-decreasing, thus
ĥα−1 (y, ρ(τ)) − ĥα−1 (x, ρ(τ)) ≤ 2ĥα−1 (b, a).
(3.52)
So ĥα−1 (y, ρ(τ)) − ĥα−1 (x, ρ(τ)) | f (τ)| is dominated by 2ĥα−1 (b, a) | f (τ)| which is
integrable because f ∈ L∇1 . Applying the dominated convergence theorem, we
see
Z x
ĥ (y, ρ(τ)) − ĥ (x, ρ(τ)) | f (τ)| ∇τ
lim
α−1
α−1
x→y− a
Z x
=
lim ĥα−1 (y, ρ(τ)) − ĥα−1 (x, ρ(τ)) | f (τ)| ∇τ = 0.
(3.53)
x→y−
a
So, (a ∇αx f )(x) is left-continuous at y. If y is right-dense we can choose a ≤ y ≤
x ≤ b and repeat the argument. Thus (a ∇αx f )(x) is continuous at y.
The fractional integrals satisfy an index law. To prove this, we shall need a
time scale analogue of Dirichlet’s theorem.
Lemma 3.2.5. Let c, d ∈ e
T satisfy c < d. Suppose that f : e
T×e
T → R be a
function such that
Z d
Z τ
∇τ
(3.54)
| f (τ, θ)| ∇θ < ∞.
c
c
Then the order of integration can be changed and
Z
d
Z
∇τ
c
c
τ
f (τ, θ) ∇θ =
Z
d
Z
d
f (τ, θ) ∇τ.
∇θ
c
(3.55)
ρ(θ)
Proof. From Bohner and Guseinov [43], we see that the region in (3.54) corre-
3.2. FRACTIONAL INTEGRALS AND BASIC PROPERTIES
53
sponds to the region E ⊂ T2 ,
E = {(t, s) ∈ T2 | c < t ≤ d, c < s ≤ t}.
(3.56)
We define the extension of f by




 f (τ, θ),
F(τ, θ) = 


 0,
(τ, θ) ∈ E,
(3.57)
elsewhere.
Then because of (3.54), by the Fubini-Tonelli theorem
Z
d
Z
τ
| f (τ, θ)| ∇θ =
∇τ
c
Z Z
c
|F(τ, θ)| dµ∇ ⊗ µ∇ ,
(3.58)
E
where µ∇ ⊗ µ∇ is the product measure.
To prove (3.55) we shall use (2.27). In this section, τ and θ will be dummy
variables for time scale integrals and t and s for Lebesgue integration and summations. Further I x,y will mean I(x,y]∩T . Starting with the left hand side we see
Z
d
τ
Z
f (τ, θ) ∇θ


Z d Z τ
X



=
f (τ, s) ds +
ν(si ) f (τ, si ) ∇τ
∇τ
c
c
c
=
c
d
Z
i∈Ic,τ
Z
(c,τ]∩T
c
=
f (τ, s) ds +
∇τ
Z
Z
+
Z
c
f (t, s) ds +
dt
(c,d]∩T
(c,t]∩T
X
d
Z
Z
ν(si ) f (t, si ) dt +
ρ(θ)
X
j∈Ic,d
d
∇θ
c
ν(t j )
f (τ, θ) ∇τ
Z
j∈Ic,d
Meanwhile the right hand side expands to
d
ν(si ) f (τ, si ) ∇τ
(3.59)
i∈Ic,τ
X
(c,d]∩T i∈Ic,τ
Z
X
ν(t j )
(c, t j ]∩T
X
i∈Ic,t j
f (t j , s) ds
ν(si ) f (t j , si ).
(3.60)
54
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
=
d
Z
"Z
∇θ
ρ(θ)
θ
c
=
d
Z
Z
∇θ
ρ(θ)
c
However, because
Z
d
∇θ
ρ(θ)
c
Rt
ρ(t)
f (τ, θ) ∇τ +
f (τ, θ) ∇τ +
d
Z
#
f (τ, θ) ∇τ
θ
d
Z
Z
d
∇θ
f (τ, θ) ∇τ.
(3.61)
ν(si )2 f (si , si ).
(3.62)
θ
c
f (τ) ∇τ = ν(t) f (t), we have
θ
Z
θ
f (τ, θ) ∇τ =
Z
d
ν(θ) f (θ, θ) ∇θ
c
=
Z
X
ν(s) f (s, s) ds +
(c,d]∩T
i∈Ic,d
But the support of ν(s) is exactly the set of left-scattered points of (c, d] ∩ T,
which has Lebesgue measure zero. Thus (3.61) becomes
Z
d
Z
d
∇θ
ρ(θ)
c
f (τ, θ) ∇τ =
X
ν(si ) f (si , si ) +
d
Z
2
Z
∇θ
f (τ, θ) ∇τ.
(3.63)
θ
c
i∈Ic,d
d
Expanding the double integral in (3.63),
d
Z
d
Z
f (τ, θ) ∇τ
Z

Z d
X


=
∇θ 
f (t, θ) dt +
ν(t j ) f (t j , θ)
∇θ
c
θ
(θ,d]∩T
c
=
d
Z
Z
j∈Iθ,d
f (t, θ) dt +
∇θ
(θ,d]∩T
c
Z
d
c
X
ν(t j ) f (t j , θ) ∇θ
(3.64)
j∈Iθ,d
We finally obtain
X
ν(si ) f (si , si ) +
Z
Z
X
(c,d]∩T i∈I
s,d
f (t, s) ds +
ds
(c,d]∩T
i∈Ic,d
+
Z
2
(s,d]∩T
ν(t j ) f (t j , s) ds +
X
i∈Ic,d
X
ν(si )
Z
j∈Ic,d
ν(si )
X
ν(t j ) f (t j , si ),
(si ,d]∩T
f (t, si ) dt
(3.65)
j∈I si ,d
for a total of five terms. In order to prove that (3.60) is equal to (3.65) we shall
3.2. FRACTIONAL INTEGRALS AND BASIC PROPERTIES
55
need to prove the following four equations:
Z
Z
f (t, s) ds =
dt
(c,d]∩T
Z
ν(t j )
(c,d]∩T
j∈Ic,d
i∈Ic,d
Z
f (t j , s) ds =
(c,t j ]∩T
j∈Ic,d
X
Z
ds
Z
X
X
ν(si ) f (t, si ) dt =
ν(si )
(c,t]∩T
(c,d]∩T i∈Ic,t
X
Z
ν(t j )
X
Z
f (t, s) ds,
(3.66)
f (t, si ) dt,
(3.67)
ν(t j ) f (t j , s) ds,
(3.68)
(s,d]∩T
(si ,d]∩T
X
(c,d]∩T j∈I
s,d
ν(si ) f (t j , si ) =
i∈Ic,t j
X
ν(si )
i∈Ic,d
X
ν(t j ) f (t j , si ) +
j∈I si ,d
X
ν(si )2 f (si , si ).
i∈Ic,d
(3.69)
Firstly, we see that (3.66) is a standard Lebesgue integral. Integrating f over
(c, d] ∩ T is the same is integrating f χ over (c, d] where is χ is characteristic
function for (c, d] ∩ T. But this is exactly the definition of F on one variable.
Thus
Z
Z
Z
Z
dt
f (t, s) ds =
dt
F(t, s) ds,
(3.70)
(c,d]∩T
(c,t]∩T
[c,d]
[c,t]
because we have specified a , c. Then we apply the standard Dirichlet theorem
to (3.70) to obtain our result.
The proof of (3.67) and (3.68) are similar so we shall only prove the former.
Observe that if we repeat the work to derive (3.67), but with f replaced with | f |
we obtain
XZ
ν(si ) | f | (t, si ) dt.
(3.71)
i∈Ic,d
(si ,b]∩T
Set χ(t) = χ(si ,b]∩T (t) the characteristic function over (si , b] ∩ T, then (3.71) is
equal to
XZ
ν(si ) χ(t) | f | (t) dt.
(3.72)
i∈Ic,d
[c,d]
However, by (3.58), we know that (3.72) is finite, and from sums of Lebesgue
integrals, it converges and we can interchange the sum and integral. By the fact
P
P
that i∈Ic,d ν(si ) χ(t) = i∈Ic,t ν(si ), we obtain (3.67).
56
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
Lastly, we prove (3.69). We have
X
ν(t j )
j∈Ic,d
X
ν(si ) f (t j , si )
i∈Ic,t j
=
X
ν(t j ) ν(s j ) f (t j , si ) +
X
i∈Ic,d
ν(si ) f (t, si ) ν(t j )
j∈Ic,d i∈I(c,t j )
j∈Ic,d
=
X X
ν(si )2 f (si , si ) +
X X
ν(si ) ν(t j ) f (t j , si ),
(3.73)
i∈Ic,d j∈I si ,d
completing the proof.
We are now in a position to prove the Cauchy formula for Lebesgue integrable
functions.
Corollary 3.2.6 (Cauchy Formula, Lebesgue version). Let T be a time scale, and
α, x ∈ T̃ with α < x. Let φ ∈ L∇1 (e
T). Then (3.6) is valid.
Proof. We shall begin by showing the left hand side of (3.6) is well-defined.
R x
Rx
Observe that α ĥ1 (x, ρ(t)) φ(t) ∇t ≤ α |x − ρ(t)| |φ(t)| ∇t ≤ |b − a| kφk < ∞.
Indeed with induction on n,
Z x
Z x Z x
ĥn (x, ρ(t)) φ(t) ∇t ≤
ĥn−1 (τ, ρ(t)) ∇τ |φ(t)| ∇t
α
α
ρ(t)
Z x
≤ |b − a|
ĥn−1 (x, ρ(t)) |φ(t)| ∇t < ∞.
(3.74)
α
This also guarantees that we can change the order of integration. As before, (3.6)
is trivial for n = 1. Assume it is true for n then
Z x
Z t1
Z tn
Z x
Z t1
∇t1
∇t2 . . .
φ(tn+1 ) ∇tn+1 =
∇t1
ĥn−1 (t1 , ρ(t)) φ(t) ∇t
α
α
α
α
α
Z x
Z x
=
φ(t) ∇t
ĥn−1 (t1 , ρ(t)) ∇t1
α
ρ(t)
Z x
=
ĥn (x, ρ(t)) φ(t) ∇t,
(3.75)
α
as required.
3.2. FRACTIONAL INTEGRALS AND BASIC PROPERTIES
57
Theorem 3.2.7 (Index Law). Let α, β > 0 and suppose that f ∈ L∇1 . Then
−β
−(α+β)
(c ∇−α
f )(t),
t c ∇t f )(t) = (c ∇t
(3.76)
for µ∇ -almost all t ∈ e
Tκ .
Proof. By definition we have
−β
(c ∇−α
t c ∇t f )(t)
=
t
Z
ĥα−1 (t, ρ(τ)) ∇τ
c
τ
Z
ĥβ−1 (τ, ρ(θ)) f (θ) ∇θ,
(3.77)
ĥα−1 (t, ρ(τ)) ĥβ−1 (τ, ρ(θ)) ∇τ,
(3.78)
c
we now apply (3.55) to obtain
−β
(c ∇−α
t c ∇t f )(t)
=
Z
t
Z
t
f (θ) ∇θ
ρ(θ)
c
and now use (3.16)
−β
(c ∇−α
t c ∇t f )(t)
=
Z
t
ĥα+β−1 (t, ρ(θ)) f (θ) ∇θ = (c ∇−(α+β)
f )(t).
t
(3.79)
c
The index law we have presented is different to that in Anastassiou [21]. In
fact the index law presented therein is erroneous.
Theorem 3.2.8 (incorrect, [21], Theorem 4). Let Tκ = T, a, b ∈ T, f ∈ L1 ([a, b]∩
T); let α, β > 1; ĥα−1 (s, ρ(t)) be continuous on ([a, b] ∩ T) × ([a, b] ∩ T) for any
α > 1. Then
Jaα Jaβ f (t) + D( f, α, β, T, t) = Jaα+β f (t),
∀t ∈ [a, b] ∩ T.
(3.80)
Here Jaα is notation used to denote the Riemann-Liouville fractional integral.
Anastassiou also defines
Z t
D( f, α, β, T, t) =
f (u) ν(u) ĥα−1 (t, ρ(u)) ĥβ−1 (u, ρ(u)) ∇u,
(3.81)
a
and names it the backward graininess deviation functional of f . If we consider
58
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
the integers then ν(u) = 1 and ĥβ−1 (u, ρ(u)) = 1. Thus we have
D( f, α, β, Z, t) = Jaα f (t),
(3.82)
which would imply that the index law is given as
Jaα Jaβ f (t) + Jaα f (t) = Jaα+β f (t),
(3.83)
a result which directly contradicts [33, Theorem 2.1] and [71, Property 2]. Both
of these papers were cited in Anastassiou [21], which is unfortunate. The source
of the error is due to the paper making the assumption that
Z
t
Z
τ
∇τ
a
f (τ, θ) ∇θ =
t
Z
a
t
Z
f (τ, θ) ∇τ.
∇θ
(3.84)
θ
a
Anastassiou claims this is valid by Fubini’s theorem, although no proof is given.
We on the other hand, have derived a form of the Dirichlet law (3.55) that is
consistent with discrete calculus, and an index law (3.76) that is consistent with
the literature.
Theorem 3.2.9. Let c, t ∈ e
Tκ with c < t. If f : e
T → R is ld-absolutely continuous,
then for α > 0 one has
−(α+1) ∇
(c ∇−α
f )(t) + f (c) ĥα (t, c).
t f )(t) = (c ∇t
(3.85)
Proof. Because f is ld-absolutely continuous, there exist f ∇ ∈ L∇1 , and so the
fractional integral exists for all t ∈ e
Tκ . We have
(c ∇t−(α+1) f ∇ )(t)
=
Z
t
Z
t
∇
f (τ) ∇τ
ρ(τ)
c
ĥα−1 (t, ρ(θ)) ∇θ,
(3.86)
after using (3.16) with β = 0. Then applying (3.55) we have
(c ∇t−(α+1) f ∇ )(t)
=
t
Z
ĥα−1 (t, ρ(θ)) ∇θ
c
=
(c ∇−α
t f )(t)
θ
Z
f ∇ (τ) ∇τ
c
− f (c) ĥα (t, c),
(3.87)
3.3. FRACTIONAL DERIVATIVES AND TAYLOR THEOREMS
59
as required.
Theorem 3.2.10. Suppose that for all t, s ∈ e
T, t > s, one has ĥβ (t, s) → 1 as
β → 0+. Then for f ∈ ACld (e
T) we have
lim (c ∇αt f )(t) = f (t).
α→0+
(3.88)
Proof. From (3.85) we have
(c ∇−α
t f )(t)
=
Z
t
ĥα (t, ρ(τ)) f ∇ (τ) ∇τ + f (c) ĥα (t, c).
(3.89)
c
Because α > 0, ĥα−1 (t, s) is positive thus
ĥα (t, ρ(τ)) f ∇ (τ) ≤ ĥα (b, a) f ∇ (τ) ,
(3.90)
which is integrable as f ∈ ACld (e
T) implies f ∇ ∈ L∇1 (e
T) which in turn implies
∇
1
f ∈ L∇ (e
T), as the Lebesgue integral is an absolute integral. Thus the dominated
convergence theorem allows limit passage and
Z
ĥα (t, ρ(τ)) f (τ) ∇τ =
t
Z
lim ĥα (t, ρ(τ)) f ∇ (τ) ∇τ
∇
lim
α→0+
t
c
=
α→0+
Zc t
∇
f (τ) ∇τ = f (t) − f (c).
(3.91)
c
And the result follows from f (c) ĥα (t, c) → f (c) as α → 0+.
3.3
Fractional Derivatives and Taylor Theorems
We shall now investigate fractional derivatives and their properties. Recall that
in time scales calculus, the nabla derivative maps a function with a domain of T
to a function with a domain of Tκ . Similarly, in this section the left base point,
denoted c, will need to be chosen similarly.
Definition 3.3.1. Let α > 0, n = bαc + 1 and f : e
T → R. For c, t ∈ e
Tκn with
c < t, we define the Riemann-Liouville fractional ∇-derivative of order α to be
60
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
the expression
(c ∇αt f )(t)
n
:= (∇
−(n−α)
f )(t)
c ∇t
=
#∇n
t
"Z
ĥn−α−1 (t, ρ(τ)) f (τ) ∇τ
,
(3.92)
c
if it exists.
Definition 3.3.2. Let α > 0, m = dαe and c, t ∈ e
Tκm with c < t. For f : e
T → R,
we define the Caputo fractional ∇-derivative of order α is defined as
(Cc ∇αt f )(t)
m
(c ∇t−(m−α) f ∇ )(t)
:=
t
Z
=
m
ĥn−α−1 (t, ρ(τ)) f ∇ (τ) ∇τ,
(3.93)
c
if it exists.
In this paper, the notation (c ∇αt f )(c) will denote the limit of (c ∇αt f )(t) as
t → c+. When c is right scattered, this limit is clearly zero. It is clear that the
Riemann-Liouville fractional derivative generalizes the standard ∇-derivative.
We shall give conditions for the Caputo case.
Theorem 3.3.3. Let α > 0, n = dαe and suppose that ĥβ (t, s) → 1 as β → 0+ for
all s, t ∈ e
T satisfying t > s. Let f ∈ ACldn+1 (e
T). Then
lim (Cc ∇αt f )(t) = f ∇ (t).
n
(3.94)
α→n
n+1
Proof. We have f ∇ ld-absolutely continuous over T ∩ [a, t] for each t ∈ e
Tκn ,
thus the fractional integral of order n − α + 1 is defined. So
Z
c
t
n+1
ĥn−α (t, ρ(τ)) f ∇ (τ) ∇τ
Z t
Z t
∇n+1
=
f (τ) ∇τ
ĥn−α−1 (t, ρ(θ)) ∇θ
ρ(τ)
c
=
=
Z
t
ĥn−α−1 (t, ρ(θ)) ∇θ
c
C α
(c ∇t f )(t)
Z
θ
n+1
f ∇ (τ) ∇τ
c
∇n
− f (c) ĥn−α (t, c).
(3.95)
3.3. FRACTIONAL DERIVATIVES AND TAYLOR THEOREMS
61
Thus
(Cc ∇αt f )(t)
∇n
= f (c) ĥn−α (t, c) +
Z
t
n+1
ĥn−α (t, ρ(τ)) f ∇ (τ) ∇τ.
(3.96)
c
Because ĥn−α−1 (·, ρ(·)) is positive, ĥn−α (t, ρ(τ)) is non-decreasing in t and nonincreasing in τ. So
ĥn−α (t, ρ(τ)) f ∇n+1 (τ) ≤ ĥn−α (b, a)
n+1 f ∇ (τ) ,
(3.97)
which is integrable due to ld-absolute continuity. The dominated convergence
theorem allows to apply limit passage
Z
t
lim
α→n
c
n+1
ĥn−α (t, ρ(τ)) f ∇ (τ) ∇τ
Z t
n+1
=
lim ĥn−α (t, ρ(τ)) f ∇ (τ) ∇τ
α→n
Zc t
n+1
=
f ∇ (τ) ∇τ,
(3.98)
c
and the result holds.
A note on the notation. We have denoted fractional integrals with a negative
sign in the index: ∇−α where α is a positive real number, and the reader is encouraged to use the negative sign as a visual indicator of fractional integration.
However, in discussing Taylor-type theorems, we will at times need to discard
this, so for a negative real α, the fractional integral of order −α is written ∇α .
Note that this is the reverse of Samko et al [112] who use fractional derivatives
as negative powers of integration.
Extending Taylor’s theorem to fractional derivatives has a long history dating
back to
∞
X
hm+r
f (x + h) =
(Dm+r
f )(x),
(3.99)
a
Γ(m
+
r
+
1)
m=−∞
established by Riemann (see Hardy [73] for a detailed investigation). For further
work see [20, 37, 79, 101, 105, 110, 116, 117]. Special mention should be given
to the Taylor theorem given in Anastassiou [21], which is incorrect.
62
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
Theorem 3.3.4 (incorrect, [21], Theorem 8). Let µ > 2, m−1 < µ < m, ν̃ = m−µ,
f ∈ Cldm (T), a, b ∈ T, Tκ = T. Suppose ĥµ−2 (s, ρ(t)), ĥν̃ (s, ρ(t)) to be continuous
on ([a, b] ∩ T)2 . Then
f (t) =
m−1
X
∇k
ĥk (t, a) f (a) +
Z
t
m
f ∇ (u) ν(u) ĥµ−2 (t, ρ(u)) ĥν̃ (u, ρ(u)) ∇u
a
k=0
+
t
Z
ĥµ−2 (t, ρ(τ)) ∇µ−1
a∗ f (τ) ∇τ.
(3.100)
a
The error in the proof of (3.100) is due to the incorrect (3.80). Anastassiou
uses the index law to derive the expression
m
(Jaµ−1 Jaν̃+1 f ∇ )(t)
=
m
(Jaµ+ν̃ f ∇ )(t)
Z
−
t
m
f ∇ (u) ν(u) ĥµ−2 (t, ρ(u)) ĥν̃ (u, ρ(u)) ∇u,
a
(3.101)
and hence
Jaµ−1 ∇µ−1
a∗ f (t)
+
=
t
Z
m
f ∇ (u) ν(u) ĥµ−2 (t, ρ(u)) ĥν̃ (u, ρ(u)) ∇u
Za t
m
ĥm−1 (t, ρ(τ)) f ∇ (τ) ∇τ.
a
This result is not consistent with the established literature, for example, Anastassiou [22, Theorem 4].
In this section of the thesis, we shall derive a time scale analogue of the
expression obtained in Džrbašjan and Nersesyan [62], namely given a sequence
0 = µ0 < µ1 < . . . < µn+1 with 0 < µk+1 − µk ≤ 1, define D(µ0 ) f (x) = f (x) and
D(µk ) f (x) :=
dµk −µk−1 −1 d (µk−1 )
D
f (x).
dxµk −µk−1 −1 dx
(3.102)
Then for x ∈ [0, b), the generalized Taylor theorem is given as
Z x
n
X
D(µk ) f (0) µk
1
f (x) =
x +
(x − t)µn+1 −1 D(µn+1 ) f (t) dt.
Γ(1 + µk )
Γ(µn+1 ) 0
k=0
(3.103)
Unfortunately, this result has been misquoted in the literature. For example
3.3. FRACTIONAL DERIVATIVES AND TAYLOR THEOREMS
63
1−(αk −αk−1 ) 1+αk−1
Samko et al [112] define D(αk ) f = I0+
D0+ f and then state
Z x
m−1
X
D(αk ) f (0) αk
1
f (x) =
x +
(x − t)αm −1 (D(αm ) f )(t) dt.
Γ(1
+
α
)
Γ(1
+
α
)
k
m
0
k=0
(3.104)
This error has been repeated in [116]. We shall begin with a simple Taylor theorem for the Caputo fractional ∇-derivative.
Theorem 3.3.5. Let α > 0 and set n = dαe and c ∈ e
Tκn . Suppose that f : e
T→R
C α
is a function such that (c ∇t f )(t) exists and is integrable of order α. Then
f (t) =
n−1
X
∇k
f (c) ĥk (t, c) +
Z
t
ĥα−1 (t, ρ(τ)) (Cc ∇ατ f )(τ) ∇τ.
(3.105)
c
k=0
Proof. We observe
Z
c
t
ĥα−1 (t, ρ(τ)) (Cc ∇ατ f )(τ) ∇τ
Z t
Z τ
n
=
ĥα−1 (t, ρ(τ)) ∇τ
ĥn−α−1 (τ, ρ(θ)) f ∇ (θ) ∇θ,
c
(3.106)
c
We apply (3.55) and (3.16)
Z
t
ĥα−1 (t, ρ(τ)) (Cc ∇ατ f )(τ) ∇τ
c
=
Z
t
n
f ∇ (θ) ĥn−1 (t, ρ(θ)) ∇θ.
(3.107)
c
Then by the Taylor theorem for ∇-calculus (3.4) we obtain
Z
c
as required.
t
∇n
f (θ) ĥn−1 (t, ρ(θ)) ∇θ = f (t) −
n−1
X
k
f ∇ (c) ĥk (t, c),
(3.108)
k=0
We shall now consider the case when of a fractional integral of the Caputo
∇-integral.
Theorem 3.3.6. Let 0 < α < β and let n := dαe, m := dβe. Suppose that
64
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
f ∈ AC m (e
T). Then for n = m
C β
(Cc ∇αt f )(t) = (c ∇−(β−α)
t
c ∇t f )(t),
(3.109)
and if m > n
(Cc ∇αt f )(t)
=
m−n−1
X
C β
f ∇ (c) ĥk+n−α (x, c) + (c ∇−(β−α)
t
c ∇t f )(t).
k
(3.110)
k=0
Proof. We shall begin with (3.109). Evaluating the right hand side we obtain
(c ∇t−(β−α) Cc ∇βt f )(t)
Z t
Z τ
m
=
ĥβ−α−1 (t, ρ(τ)) ∇τ
ĥm−β−1 (τ, ρ(θ)) f ∇ (θ) ∇θ.
c
(3.111)
c
Using the Dirichlet theorem we obtain
Z
t
t
(θ) ∇θ
ĥβ−α−1 (t, ρ(τ)) ĥm−β−1 (τ, ρ(θ)) ∇τ
ρ(θ)
Z t
m
=
f ∇ (θ) ĥm−α−1 (t, ρ(θ)) ∇θ = (Cc ∇αt f )(t),
f
c
Z
∇m
(3.112)
c
because n = m. As for (3.110), evaluating the fractional integral and after (3.55)
we obtain
(c ∇t−(β−α)Cc ∇βt f )(t)
Z t
Z τ
∇m
=
f (θ) ∇θ
ĥβ−α−1 (t, ρ(τ)) ĥm−β−1 (τ, ρ(θ)) ∇τ
c
ρ(θ)
Z t
m
=
f ∇ (θ) ĥm−α−1 (t, ρ(θ)) ∇θ,
(3.113)
c
as before. Using m > n, we expand ĥm−α−1 (t, ρ(θ)) using (3.16) to obtain
Z
t
τ
(θ) ∇θ
ĥn−α−1 (t, ρ(τ)) ĥm−n−1 (τ, ρ(θ)) ∇τ
ρ(θ)
Z t
Z τ
n
m−n
=
ĥn−α−1 (t, ρ(τ)) ∇τ
ĥm−n−1 (τ, ρ(θ)) [ f ∇ (θ)]∇ ∇θ.
f
c
Z
∇m
c
c
(3.114)
3.3. FRACTIONAL DERIVATIVES AND TAYLOR THEOREMS
65
n
m
Observing f ∇ is ld-absolutely continuous, we can apply (3.4) to f ∇ and observe
t
Z
c


m−n−1
X
 ∇n

k
∇
ĥn−α−1 (t, ρ(τ))  f (τ) −
ĥk (τ, c) f (c) ∇τ
k=0
= (Cc ∇αt f )(t) −
m−n−1
X
k
ĥk+n−α (t, c) f ∇ (c),
(3.115)
k=0
as required.
Theorem 3.3.7. For m ∈ N, let 0 = µ0 < µ1 < µ2 < . . . < µm be an increasing
sequence and define νk := dµk+1 − µk e. Let c ∈ e
Tκdµm e . Define the operator
µ
−µ
T → R has
D(µ0 ) f (t) = f (t) and D(µk+1 ) f (t) = Cc ∇t k+1 k D(µk ) f (t). Suppose that f : e
(µm )
(µi )
ld-continuous (D f )(t) and for each 0 ≤ i ≤ m − 1, D f has ld-continuous
nabla derivatives up to order νi − 1. Then
f (t) =
νi −1 h
m−1 X
X
(µi )
D
f
i∇k
(c) ĥµi +k (t, c) +
t
Z
ĥµm −1 (t, ρ(τ)) (D(µm ) f )(τ) ∇τ. (3.116)
c
i=0 k=0
Proof. Suppose m = 1, then ν0 = dµ1 e and (D(µ1 ) f )(t) = (Cc ∇µt 1 f )(t). Then by
1
1 C µ1
(3.105) the integral is equal to (c ∇−µ
D(µ1 ) f )(t) = (c ∇−µ
t
t
c ∇t f )(t) = f (t) −
Pν0 −1 ∇k
k=0 f (c) ĥk (t, c).
For m + 1 case we repeat the previous argument
m+1
(c ∇−µ
D(µm+1 ) f )(t)
t
−(µm+1 −µm ) C µm+1 −µm (µm )
m
= (c ∇−µ
D f )(t)
t
c ∇t
c ∇t


νX
m −1
 (µ )

k
−µm 
(µ
)
∇
m
m
= c ∇t (D f )(t) −
(D f ) (c) ĥk (t, c)
m
= (c ∇−µ
D(µm ) f )(t) −
x
k=0
νX
m −1
k
(D(µm ) f )∇ (c) ĥµm +k (t, c).
k=0
Assuming the formula holds for m and expanding
νi −1
νX
m−1 X
m −1
X
k
(µi ) ∇k
f (t) −
(D f ) (c) ĥµi +k (t, c) −
(D(µm ) f )∇ (c) ĥµm +k (t, c)
i=0 k=0
k=0
(3.117)
66
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
νi −1
m X
X
k
(D(µi ) f )∇ (c) ĥµi +k (t, c),
= f (t) −
(3.118)
i=0 k=0
as required.
Interestingly, (3.116) implies that the removal of the restriction µi+1 − µi ≤ 1
has little bearing on the Taylor expansion, as the same expansion can be generated with the sequence
0 = µ0 < µ1 < µ1 + 1 < µ1 + 2 < . . . < µ2 < µ2 + 1 < . . . < µm ,
(3.119)
where the difference of successive terms is bounded by 1. Thus we will assume
this restriction in our considerations of a Taylor formula for Riemann-Liouville
fractional ∇-derivatives. First, we shall need the following lemma.
Lemma 3.3.8. Let c ∈ e
Tκ . For 0 < β ≤ 1 we have
β
−(1−β)
(c ∇−β
f )(c) ĥβ−1 (t, c).
t c ∇t f )(t) = f (t) − (c ∇t
(3.120)
Proof. We let ψ(t) = c ∇−(1−β)
f (t). The ∇-integral is ld-absolutely continuous,
t
Rt
∇
f ]∇ (t) = (c ∇βt f )(t) and
so c ψ (τ) ∇τ = ψ(t) − ψ(c). Then ψ∇ (t) = [c ∇−(1−β)
t
ψ(t) − ψ(c) = (c ∇−(1−β)
f )(t) − (c ∇−(1−β)
f )(c), meanwhile
t
t
Z
t
β
−(1−β)
−β
β
ψ∇ (τ) ∇τ = (c ∇−1
t c ∇t f )(t) = (c ∇t
c ∇t c ∇t f )(t).
(3.121)
c
−(1−β)
f = f we have
Thus, differentiating by order 1 − β, and using c ∇1−β
t c ∇t
−β
β
c ∇t c ∇t f (t)
µ∇ -almost everywhere.
= f (t) − c ∇−(1−β)
f (c) ĥβ−1 (t, c),
t
(3.122)
Theorem 3.3.9. (Taylor Theorem for RL-Fractional Calculus) Let c ∈ e
Tκn−1 . Let
0 = µ0 < µ1 < . . . < µn be a real sequence satisfying 0 < µk+1 − µk ≤ 1 for
k = 0, . . . , n − 1. Let αk := 1 − (µk+1 − µk ). Define the operator L(µk ) by L(µ0 ) f := f
T → R be a function such that for each k,
and L(µk+1 ) f := c ∇µxk+1 −µk L(µk ) f . Let f : e
3.3. FRACTIONAL DERIVATIVES AND TAYLOR THEOREMS
67
L(µk ) f exists and is integrable of order αk . Then
f (t) =
n−1
X
k
(c ∇−α
t
L
(µk )
f )(c) ĥµk+1 −1 (t, c) +
Z
t
ĥµn −1 (t, ρ(τ)) (L(µn ) f )(τ) ∇τ. (3.123)
c
k=0
Proof. The proof is analogous as before. Let n = 1. Then
µ1
1
1
(c ∇−µ
L(µ1 ) f )(t) = (c ∇−µ
t
t
c ∇t f )(t)
1)
= f (t) − (c ∇−(1−µ
f )(a) ĥµ1 −1 (t, c)
t
0
= f (t) − (c ∇−α
f )(c) ĥµ1 −1 (t, c),
t
(3.124)
using µ0 = 0. Assume the theorem is true for n. Then evaluating n + 1 we have
−(µn+1 −µn )
µn+1 −µn (µn )
n+1
n
(c ∇−µ
L(µn+1 ) f )(t) = (c ∇−µ
L f )(t)
t
t
c ∇t
c ∇t
h
i
n
n+1 −µn ))
= c ∇−µ
L(µn ) f (t) − (c ∇−(1−(µ
L(µn ) f )(c) ĥµn+1 −µn −1 (t, c)
t
t
n (µn )
n
L(µn ) f )(c) ĥµn+1 −1 (t, c)
= (c ∇−µ
f )(t) − (c ∇−α
t
t L
n
X
k
= f (t) −
(c ∇−α
L(µk ) f )(c) ĥµk+1 −1 (t, c),
t
(3.125)
k=0
as required.
The theorem has a number of special cases.
Definition 3.3.10. Let 0 < α < 1. Define the Miller-Ross sequential fractional
derivative of f as
(c ∇αt )n f (t) := c ∇αt · · · · · · c ∇αt f (t).
(3.126)
|
{z
}
n
Corollary 3.3.11. Let 0 < α < 1. For each k = 0, . . . , n, suppose that f is a
function with (c ∇αt )n f existing and integrable of order 1 − α. Then we have
f (t) =
n−1
X
(c ∇−(1−α)
(c ∇αt )k f )(c) ĥ(k+1)α−1 (t, c) + R f (t)
t
(3.127)
k=0
where
R f (t) =
Z
c
t
ĥnα−1 (t, ρ(τ)) (c ∇αt )n f (τ) ∇τ.
(3.128)
68
CHAPTER 3. BASIC DEFINITIONS AND TAYLOR’S THEOREM
Proof. Let µk = k α, then µk+1 − µk = (k + 1)α − k α and αk = 1 − α. So
L(µk+1 ) f = c ∇αx L(µk ) f = (c ∇αx )k+1 f . And apply the previous theorem.
Corollary 3.3.12. Let 0 < α < 1, and f having all the relevant properties. Then
f (t) =
f )(c) ĥα−1 (t, c)
(c ∇−(1−α)
t
+
n−2
X
j
(c ∇α+
f )(c) ĥα+ j (t, c) + R f (t),
t
(3.129)
j=0
where
R f (t) =
Z
t
ĥα+n−2 (t, ρ(τ)) (c ∇α+n−1
f )(τ) ∇τ.
τ
(3.130)
c
Proof. Choose µ0 = 0 and µk = α + k − 1 for k = 1, . . . , n. Then α0 = 1 −
(µ1 − µ0 ) = 1 − α and ĥµ1 −1 (t, c) = ĥα−1 (t, c). When k > 0, we have αk = 0
and ĥµk+1 −1 (t, c) = ĥα+k−1 (t, c). Also L(µ0 ) f = f , L(µ1 ) f = c ∇αt f and in general
L(µk ) f = ∇L(µk−1 ) f = c ∇α+k−1
f . And so the summation equals
t
n−1
X
k (µk )
(c ∇−α
f )(c) ĥµk+1 −1 = (c ∇−(1−α)
f )(c) ĥα−1 (t, c)
t L
t
k=0
+ (c ∇αt f )(c) ĥα (t, c) + (c ∇α+1
f )(c) ĥα+1 (t, c) + . . .
t
+ (c ∇α+n−2
f )(c) ĥα+n−2 (t, c).
t
(3.131)
The result immediately follows.
The first corollary (when the time scale is chosen to be the reals) is the result
of Trujillo et al [116], while the second is the Taylor expression in Samko et al
[112]. Observe that by setting γ = α + n − 1 we can rewrite (3.129) as
f (x) =
m
X
(c ∇γ−k
t
f )(c) ĥγ−k (t, c) +
Z
t
ĥγ−1 (t, ρ(τ)) (c ∇γt f )(τ) ∇τ,
(3.132)
c
k=1
where m = bγc + 1. This leads to the following corollary which will be used in
studying Opial inequalities.
Corollary 3.3.13. Let β > α > 0 and let f : e
T → R have integrable fractional
β−k
α
derivative c ∇t f (t), and further c ∇ x f (c) = 0 for k = 1, . . . , bβc + 1. Then
α
c ∇ s f (s)
=
Z
c
s
ĥβ−α−1 (s, ρ(τ)) (c ∇βτ f )(τ) ∇τ,
(3.133)
3.3. FRACTIONAL DERIVATIVES AND TAYLOR THEOREMS
69
for s ∈ [c, b] ∩ T.
Proof. The right hand side of (3.133) exists due to Theorem 3.2.3. From (3.132)
we know there exist φ satisfying f (t) = (c ∇−β
t φ)(t). Let m = bαc + 1. Then
α
c ∇ s f (s)
−β
m
−m
−(β−α)
= ∇m c ∇−(m−α)
φ(s)
s
c ∇ s φ(s) = ∇ c ∇ s c ∇ s
β
= c ∇−(β−α)
φ(s) = c ∇−(β−α)
s
s
c ∇ s f (s).
(3.134)
In this chapter we have considered an axiomatic approach to fractional nabla
calculus. We have defined a time scale analogue to the Riemann-Liouville fractional integral by considering the properties which are universal across the different time scales and using them as axioms. Then we demonstrated that the
three properties are sufficient to construct a theory of fractional calculus and thus
unified important special cases. We have shown integrability and an index law,
through the Dirichlet theorem. The Dirichlet theorem will be useful not just in
fractional calculus but to the theory of time scales calculus as a whole. Lastly
we have defined fractional nabla derivatives and considered various extensions
to Taylor’s theorem.
Chapter 4
Opial-type Inequalities
In this chapter we consider Opial type inequalities for the fractional nabla derivatives given in Chapter 3. We first give an overview of Opial inequalities for continuum and discrete cases. We prove a relatively simple generalization of the
inequality to fractional calculus, and then progressively increase the complexity
by considering the case when the factors are raised to various powers and also
the case when the integrals are weighted. Lastly we show how these inequality can be used to bound certain dynamical equation which uses the fractional
derivatives.
4.1
Background of Opial’s Inequality
In 1960, Opial proved an integral inequality relating a function and its first derivative.
Theorem 4.1.1 ([104]). Let f ∈ C 1 [0, h] satisfy f (0) = f (h) = 0 and further
f (x) > 0 for all 0 < x < h. Then
Z
h
h
| f (t) f (t)| dt ≤
4
Z
0
0
h
f (t)2 dt.
(4.1)
0
Moreover, the constant is best possible.
The novelty of Opial’s approach was the establishment of the best possible
constant. Since then, both the statement of the theorem and the proof have been
71
72
CHAPTER 4. OPIAL-TYPE INEQUALITIES
subsequently simplified by Olech [103], Beesack [39], Levinson [96], Mallows
[98], Hua [77] and Pederson [108]. Their statement is given as follows.
Theorem 4.1.2. Let x : [0, h] → R be absolutely continuous on (0, h) with x(0) =
0. Then
Z b
Z
b b 0 2
0
(4.2)
|x(t)| |x (t)| dt ≤
|x (t)| dt,
2 0
0
holds.
These inequalities, and their generalizations, have been used in studying the
properties of solutions to differential equations, and thus the theory have become
significant in its own right. See the monograph Agarwal and Pang [4] for a
survey of these applications and also a collection of the generalizations of Opial’s
inequality.
Indeed the recent interest in difference calculus has prompted the study of
discrete analogues of these inequalities. For example, the discrete version of
(4.1) is given as
Theorem 4.1.3 ([94]). Let xn be a real sequence with 0 ≤ n ≤ h such that
x0 = xh = 0, and ∆xi be the forward difference operator. Then
h
X
h−1
1h + 1 X
|∆xn |2 ,
|xn | |∆xn | ≤
2
2
n=0
n=1
(4.3)
holds.
Meanwhile a discrete version of (4.2) is given in [4, Theorem 5.2.2] as
Theorem 4.1.4 ([4]). Let xn be a sequence of real numbers, 0 ≤ n ≤ h with
x0 = 0. Then the following holds:
h−1
X
h−1
h−1X
|∆xi |2 .
|xn | |∆xn | ≤
2
i=0
n=1
(4.4)
Naturally, this has motivated the study of Opial’s inequality on time scales.
The first paper in this direction is due to Bohner and Kaymakçalan [45]. The
4.1. BACKGROUND OF OPIAL’S INEQUALITY
73
authors show that for delta differentiable x : [0, b] ∩ T → R with x(0) = 0 we
have
Z h
Z h
∆ x∆ (t)2 ∆t.
σ
(4.5)
|x(t) + x (t)| x (t) ∆t ≤ h
0
0
Further, if r and q are positive rd-continuous functions on [0, b] satisfying
Rb
1/r(t)∆t < ∞ and q is non-increasing then
0
Z
b
Z
∆ q (t) |x(t) + x (t)| x (t) ∆t ≤
σ
σ
0
0
b
Z
1
∆t
r(t)
b
2
r(t) q(t) x∆ (t) ∆t.
(4.6)
0
An inequality similar to (4.6) was proved by Karpuz et al [83], where qσ was
replaced with q. Thus for positive rd-continuous function on [0, b], and x :
[0, b] ∩ T → R is delta differentiable with x(a) = 0. Then
b
Z
Z b
∆ x∆ (t)2 ∆t,
q(t) |x(t) + x (t)| x (t) ∆t ≤ Kq
σ
a
(4.7)
a
where the constant is
Kq = 4
Z
b
!2
q (u) (σ(u) − a)∆u .
2
(4.8)
a
However, as noted in Saker [111], the inequality (4.5) does not generalize
(4.4) because when one chooses T = Z one obtains
h−1
X
|xi + xi+1 | |∆xi | ≤ (h − 1)
h−1
X
i=1
|∆xi |2 .
(4.9)
i=0
Saker [111] proved the following inequality.
Theorem 4.1.5 ([111]). Let T be a time scale with a, b ∈ T and p, q > 0 such
that p + q > 1. Let r and s be non-negative rd-continuous functions on (a, x) ∩ T
Rt
such that a r(τ)−1/(p+q−1) ∆τ < ∞. If f : [a, x] ∩ T → R with f (a) = 0 and f ∆
does not change sign on (a, x) then
Z
x
Z x
∆ q
p+q
s(t) | f (t)| f (t) ∆t ≤ K1 (a, x, p, q)
r(t) f ∆ (t)
∆t,
p
a
a
(4.10)
74
CHAPTER 4. OPIAL-TYPE INEQUALITIES
where
q
K1 (a, x, p, q) =
p+q
!q/(p+q)
.
(4.11)
Their results have been extended by Wong et al [120], and Srivastava et al
[115]. These results not only unify those of discrete and continuum calculus,
but also quantum Opial inequalities, such as Anastassiou [25]. Interestingly, the
majority of these papers only consider delta calculus. Bohner and Duman [41]
unifies both delta and nabla versions.
The study of Opial’s inequality to higher orders by FitzGerald [67], Fink [66]
and Pang and Agarwal [107] has led to the study of Opial’s inequality using
fractional derivatives. This line of research was initiated by Anastassiou [8].
The paper follows Canavati [50] in its use of a non-standard definition of the
fractional derivative of order ν > 0, Dν g := DJ1−α Dn g, where n := bνc, α = ν − n
and J1−α is the standard Riemann-Liouville fractional integral of order 1 − α. The
main theorem of the paper is a very general Opial inequality for this definition of
fractional derivative.
Theorem 4.1.6 ([8]). Let γi ≤ 1, ν ≥ 2 such that ν − γi ≥ 1; i = 1, . . . , l and
f ∈ C νx0 ([a, b]) with f ( j) (x0 ) = 0, j = 0, 1, . . . , n − 1, n := bνc. Here x, x0 ∈
[a, b] with x ≥ x0 . Let q1 , q2 > 0 be continuous functions on [a, b] and ri > 0
P
and li=1 ri = r. Let s1 , s01 > 1 and 1/s1 + 1/s01 = 1 and s2 , s02 > 1 such that
1/s2 + 1/s02 = 1 and p > s2 . Furthermore, assume that
x
Z
Q1 :=
s01
!1/s01
q1 (w) dw
< +∞,
(4.12)
x0
and
x
Z
Q2 :=
−s02 /p
q2 (w)
!r/s01
dw
< +∞.
(4.13)
x0
Set σ := (p − s2 )/(ps2 ). Then
Z
x
x0
l Y
Dγi f (w)ri dw
q1 (w)
x0
i=1
≤ Q1 Q2
l
Y
i=1
σri σ
Γ(ν − γi )ri (ν − γi − 1 + σ)ri σ
!
4.2. OPIAL’S INEQUALITY AND GENERALIZATIONS
75
(x − x0 )( i=1 (ν−γi −1)ri +σr+1/s1 )
× Pl
(( i=1 (ν − γi − 1)ri s1 ) + rs1 σ + 1)1/s1
!
Z x
p r/p
ν
.
×
q2 (w) D x0 f (w)
Pl
(4.14)
x0
Anastassiou then published a series of papers for other definitions of fractional derivatives [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and coauthored others
[26, 27, 28]. These results have been catalogued in the books [19] and [24].
4.2
Opial’s Inequality and Generalizations
In this chapter e
T will denote [a, b] ∩ T for a, b ∈ T and c ∈ e
Tκn for a determined
n. We shall let x ∈ e
Tκn such that x > c. Also p, q > 1 will have Hölder conjugates
p0 and q0 respectively. We shall first prove an Opial inequality analogous to
Theorem 2.1 of [28]. The theorem in question is given below.
Theorem 4.2.1 ([28]). Let f ∈ L(0, x) have an integrable fractional derivative
Dν f ∈ L∞ (0, x) such that Dν− j f (0) = 0 for j = 1, . . . , bνc + 1. For k = 1, 2 let
sk > 1 and p ∈ R satisfy
αs2 < 1,
p>
s2
1 − αs2
(4.15)
and let σ = 1/s2 − 1/p. Finally let
Q1 =
Z
x
!1/s02
ω1 (τ) dτ
,
s01
Q2 =
Z
0
x
−s02 /p
ω2 (τ)
!r/s02
dτ
.
(4.16)
0
Then,
Z
x
ω1 (τ)
0
where ρ =
l
Y
µi
ri
|D f (τ)| dτ ≤ Q1 Q2C1 x
i=1
x
Z
!r/p
ω2 (τ) |D f (τ)| dτ
(4.17)
ν
p
0
i=1
Pl
ρ+1/s1
αi ri + σr and
σrσ
.
ri
ri σ
1/s1
i=1 Γ(ν − µi ) (αi + σ) (ρs1 + 1)
C 1 = Ql
(4.18)
76
CHAPTER 4. OPIAL-TYPE INEQUALITIES
Here we will not prove an inequality of this generality, but focus on the special case l = 2, which has important applications to dynamic equations.
Theorem 4.2.2. Let β > α > 0 and n := bβc + 1. Assume that there exist power
functions for e
T, such that the function defined as
t
Z
0
ĥβ−α−1 (t, ρ(τ))q ∇τ,
A(t) :=
(4.19)
c
is defined for all t ∈ (c, x] ∩ T and is integrable over (c, t] ∩ T for each t. Let
f :e
T → R have integrable fractional derivative of order β, and c ∇β−k
f (c) = 0
t
for all 0 < k ≤ n. Then the Opial inequality
Z x
Z
∇α f (t) ∇β f (t) ∇t ≤
c t
c t
c
x
!1/q0 Z x
!2/q
β
q
c ∇t f (t) ∇t
A(t) ∇t
c
(4.20)
c
is valid.
Proof. From the properties of f and (3.133), we have the equation
(c ∇αt f )(t)
=
Z
t
ĥβ−α−1 (t, ρ(τ)) (c ∇βτ f )(τ) ∇τ,
(4.21)
c
for all c < t ≤ x. Thus, using Hölder’s inequality with the parameters q0 and q to
obtain
α
Z t
c ∇t f (t) ≤
ĥβ−α−1 (t, ρ(τ)) c ∇βτ f (τ) ∇τ
c
!1/q0 Z t
!1/q
Z t
β
q
q0
c ∇τ f (τ) ∇τ
≤
ĥβ−α−1 (t, ρ(τ)) ∇τ
,
(4.22)
c
c
under the assumption the right hand side is valid. Define the function y(t) :=
q
β
R t β
= y∇ (t)1/q for almost all t. Further, we shall
∇
f
(τ)
∇τ,
and
so
∇
f
(t)
τ
c t
c c
Rt
0
define A(t) := c ĥβ−α−1 (t, ρ(τ))q ∇τ. Thus (4.22) is equivalent to
α
0
c ∇t f (t) ≤ A(t)1/q y(t)1/q .
(4.23)
We now multiply both sides of (4.23) with c ∇βt f (t), integrate over t ∈ (c, x]∩
4.2. OPIAL’S INEQUALITY AND GENERALIZATIONS
77
T, and apply Hölder’s inequality again to obtain
Z x
∇α f (t) ∇β f (t) ∇t
c t
c t
c
Z x
0
≤
A(t)1/q y(t)1/q y∇ (t)1/q ∇t
c
!1/q0 Z x
!1/q
Z x
∇
≤
A(t) ∇t
y(t) y (t) ∇t
.
c
(4.24)
c
Now, because c ∇βt f (t) = y∇ (t)1/q is a positive function, we have
y y∇ ≤ (y + yρ ) y∇ = [y2 ]∇ ,
(4.25)
and further y(c) = 0, so
Z
x
Z
∇
(yy )(t) ∇t ≤
c
x
[y2 (t)]∇ ∇t = y(x)2 .
(4.26)
c
Applying (4.26) to (4.24) we finally have (4.20).
However, upon comparison with [28], we see that (4.20) is weaker than the
Opial inequality given there. To see this, we let r = β−α−1 with β ≥ α+1−1/q0 .
Then
Z t
Z t
0
(t − τ)(β−α−1)/q
1/q0
dτ,
(4.27)
A(t) =
ĥβ−α−1 (t, ρ(τ)) ∇τ =
Γ(β − α)1/q0
0
0
which is integrable because r > −1. Thus
tr/q +1
,
(r/q0 + 1) Γ(r + 1)1/q0
0
A(t) =
(4.28)
and hence
Z
!1/q0
x
A(t) ∇t
0
0
xr+2/q
=
.
[(rq0 + 1)(rq0 + 2)]1/q0 Γ(r + 1)
(4.29)
Compare this to the result in [28] which gives the coefficient term of
Ω(x) =
x(rp+2)/p
.
21/q Γ(r + 1)((rp + 1)(rp + 2))1/p
(4.30)
78
CHAPTER 4. OPIAL-TYPE INEQUALITIES
The authors have used p where we have q0 , however, the major difference is the
presence of a 1/21/q factor. The discrepancy exists because on the reals, one
has y = yρ and so [y2 ]0 = 2yy0 . On general time scales we cannot achieve this.
Nonetheless, we can modify (4.20) to achieve the same factor term.
Theorem 4.2.3. As with Theorem 4.2.2, except now define
ρ(t)
Z
ĥβ−α−1 (ρ(t), ρ(τ)) ∇τ,
B(t) :=
(4.31)
c
and let β > α > 0 such that B(t) exists for all t ∈ (c, x] ∩ e
T and integrable over
(c, x] ∩ e
T. Then
Z x
∇α f (ρ(t)) ∇β f (t) ∇t
c ρ(t)
c t
c
≤
1
21/q
Z
x
!1/q0 Z x
!2/q
β
q
c ∇t f (t) ∇t
B(t) ∇t
.
c
Proof. As before we choose y(t) =
we have
(4.32)
c
q0
R t β
∇τ. Then for c < ρ(t) ≤ t ≤ x,
∇
f
(τ)
τ
c
c
α
c ∇ρ(t) f (ρ(t))
!1/q0 Z
Z ρ(t)
q0
≤
ĥβ−α−1 (ρ(t), ρ(τ)) ∇τ
c
ρ(t)
β
q
c ∇τ f (τ) ∇τ
!1/q
.
(4.33)
c
α
When ρ(t) = c, we have both sides equal to zero, and so we have c ∇ρ(t) f (ρ(t)) ≤
0
B(t)1/q yρ (t)1/q . Integrating as before
Z
Z x
∇α f (ρ(t)) ∇β f (t) ∇t ≤
c ρ(t)
c t
c
!1/q0 Z
x
B(t) ∇t
c
x
ρ
!1/q
∇
y (t) y (t) ∇t
. (4.34)
c
Observing y is non-decreasing, yρ y∇ ≤ (y + yρ )y∇ /2 = [y2 ]∇ /2. And the result
follows.
We shall also consider the limiting case q = ∞. Define the L∇∞ -norm as
kgk∞ = kgkL∇∞ (c,x) := ess sup |g(t)| .
t∈[c,x]∩e
T
(4.35)
4.2. OPIAL’S INEQUALITY AND GENERALIZATIONS
79
Theorem 4.2.4. For 0 < α < β, n := bβc + 1. Let f : e
T → R have bounded
β−k
β
∞
fractional derivative c ∇t f ∈ L∇ (c, x) such that c ∇t f (c) = 0 for all 0 < k ≤ n.
Then one has
Z x
β 2
∇α f (t) ∇β f (t) ∇t ≤ ĥ
c ∇t f ∞ .
(4.36)
(x,
c)
β−α+1
c t
c t
c
Proof. Due to the properties of f , we have
α
Z t
c ∇t f (t) ≤
ĥβ−α−1 (t, ρ(τ)) c ∇βτ f (τ) ∇τ
c
Z t
β
ĥβ−α−1 (t, ρ(τ)) ∇τ
≤ ess sup c ∇τ f (τ) ·
τ∈[c,t]∩e
T
c
= ess sup c ∇βτ f (τ) ĥβ−α (t, c).
(4.37)
τ∈[c,t]∩e
T
Thus
Z x
∇α f (t) ∇β f (t) ∇t
c t
c t
c
Z x
≤
ess sup c ∇βτ f (τ) · ĥβ−α (t, c) c ∇βt f (t) ∇t
c
τ∈[c,t]∩e
T


β
2

≤ ess sup c ∇ x f (t) ĥβ−α+1 (x, c).
(4.38)
t∈[c,x]∩e
T
We shall now consider the case when the factors are raised to positive powers.
We shall need an integration by substitution rule, but not in the form provided in
[46, Theorem 1.98]. We will prove a time scales nabla analogue of the formulation given in Lang [95].
Lemma 4.2.5. Let e
T1 be a time scale and suppose that f : e
T1 → R have positive
ld-continuous ∇-derivative. Suppose e
T2 := [a, b] ∩ f (e
T1 ) is also a time scale and
g:e
T2 → R be ld-continuous as well. Let c, d ∈ e
T1 with c < d and f (c), f (d) ∈
[a, b]. Then
Z
Z
f (d)
d
g(τ) ∇τ =
f (c)
(g ◦ f )(τ) f ∇ (τ) ∇τ.
c
(4.39)
80
CHAPTER 4. OPIAL-TYPE INEQUALITIES
Proof. First, observe that positive ∇-derivative implies that f (d) > f (c) and so
the integrals do not change sign. We shall initially suppose that g is constant.
Then clearly one has
Z
f (d)
k ∇τ = k[ f (d) − f (c)] =
d
Z
k f ∇ (τ) ∇τ.
f (c)
(4.40)
c
Now suppose that g is ld-continuous, and hence Darboux integrable [42]. Thus
given > 0, there exist a partition of (c, d] ∩ T into time scale intervals S j , such
that the difference between the supremum of g over S j , M j , and the infimum of
R f (d)
P R
g over S j , m j , is less than for each j. Then using f (c) g(τ) ∇τ = j S g(τ) ∇τ,
j
and replacing g with M j and m j in (4.39) proves the result.
Lemma 4.2.6. Let f : [r, t] → R be a non-decreasing function. Then
Z
Z
ρ
f (τ) ∇τ ≤
f (s) ds.
(r,t]∩T
(4.41)
(r,t]
Proof. Clearly, f non-decreasing guarantees Lebesgue integrability and hence
Lebesgue nabla integrability. We have from (2.29)
Z
ρ
f (τ) ∇τ =
(r,t]∩T
Z
ρ
f (s) ds +
(r,t]
XZ
i∈Ir,t
f ρ (ti ) − f ρ (s) ds.
(4.42)
(ρ(ti ),ti ]
But the Lebesgue integration is not affected by ρ because it is integrating over a
real interval. So (4.42) reduces to
Z
ρ
f (τ) ∇τ =
(r,t]∩T
Z
f (s) ds +
(r,t]
XZ
i∈Ir,t
f (ρ(ti )) − f (s) ds.
(4.43)
(ρ(ti ),ti ]
Because we know that f is non-decreasing f (ρ(ti )) − f (s) ≤ 0 for all s ∈ (ρ(ti ), ti ]
and each i ∈ Ir,t , the result holds.
Theorem 4.2.7. Let 0 < α < β and n := bβc+1. For positive constants p, q, s such
that 1 < p, q < s, let s0 be the Hölder conjugate for s. Let µ = p(s − q)/(s − q).
Assume that there exist time scale power functions such that the function
Z
B(t) :=
c
ρ(t)
0
ĥβ−α−1 (ρ(t), ρ(τ)) s ∇τ,
(4.44)
4.2. OPIAL’S INEQUALITY AND GENERALIZATIONS
81
is well defined for all t ∈ [c, x] ∩ T. Further assume B ∈ L∇µ (c, x). Let f : e
T→R
β
s
have integrable fractional ∇-derivative c ∇t f (t) ∈ L∇ (c, t) for all t ∈ [c, x] ∩ T
which does not change sign. Then
Z x
∇α f (ρ(t)) p ∇β f (t)q ∇t
c ρ(t)
c t
c
!(s−q)/s
!q/s Z x
!(p+q)/s
Z x
β
s
q
µ
c ∇t f (t) ∇t
≤
B(t) ∇t
.
p+q
c
c
(4.45)
q
s
Rt
Proof. Let y(t) = c c ∇βτ f (τ) ∇τ. Then for µ∇ -almost all t we have c ∇βt f (t) =
y∇ (t)q/s . Then using Hölder’s inequality with indices s and s0 ,
ρ(t)
α
Z
c ∇ρ(t) f (t) ≤
ĥβ−α−1 (ρ(t), ρ(τ)) c ∇βτ f (τ) ∇τ
c
Z
ρ(t)
s0
ĥβ−α−1 (ρ(t), ρ(τ)) ∇τ
≤
!1/s0 Z
c
β
s
c ∇τ f (τ) ∇τ
!1/s
c
1/s0
= B(t)
Thus we have
over (c, x] ∩ T
we have
Z
ρ(t)
y(t)1/s .
(4.46)
p ∇α f (ρ(t)) ∇β f (t)q ≤ B(t) p/s0 yρ (t) p/s y∇ (t)q/s . Integrating t
c ρ(t)
c t
and applying Hölder’s inequality with indices s/(s − q) and s/q
x
c
α
p q
c ∇ρ(t) f (ρ(t)) c ∇βt f (t) ∇t
Z x
0
≤
B(t) p/s yρ (t) p/s y∇ (t)q/s ∇t
c
!(s−q)/s Z x
!q/s
Z x
ps
ρ
p/q ∇
0 (s−q)
s
y (t) y (t) ∇t
.
∇t
≤
B(t)
(4.47)
c
c
Observing that y is increasing, and so y ◦ ρ = ρ ◦ y, we define z(t) = ρ(t) p/q . Then
by the substitution rule,
Z
x
z ◦ y(t) y (t) ∇t =
Z
y(x)
z(t) ∇t =
∇
c
y(c)
y(x)
Z
ρ(t) p/q ∇t
0
y(x) p/q+1
q
≤
=
y(x)(p+q)/q .
p/q + 1
p+q
(4.48)
82
CHAPTER 4. OPIAL-TYPE INEQUALITIES
Thus we have
Z x
∇α f (ρ(t)) ∇β ∇t ∇t
c τ
c ρ(τ)
c
!(s−q)/s
!q/s Z t
!(p+q)/s
Z x
β
s
q
µ
c ∇τ f (τ) ∇τ
≤
B(t) ∇t
.
p+q
c
c
(4.49)
We shall now consider the case when c ∇βt f (t) does not appear in the original
integral.
Theorem 4.2.8. Let 0 < α1 < α2 < β and let n := bβc + 1. For q > 1, let q0 be
its Hölder conjugate. Assume that there exist time scale power functions with the
property that the function
t
Z
0
ĥβ−αi −1 (t, ρ(τ))q ∇τ,
Ai (t) :=
i = 1, 2,
(4.50)
c
0
exists for all t ∈ [c, x] ∩ T. Let r1 , r2 > 1 and assume that A1 (t)r1 A2 (t)r2 ∈ L∇q (c, t)
for all t ∈ [c, x] ∩ T. Let f : e
T → R be a function with integrable fractional
β
fractional derivative c ∇t f (t) ∈ L∇q (c, t) for all t ∈ [c, x] ∩ T such that c ∇β−k
f (c) =
t
0 for all 0 < k ≤ n. Then
Z x
∇α1 f (t)r1
c t
α
r
c ∇t 2 f (t) 2 ∇t
c
1/q0
≤ Ω(x)
1/q
(x − c)
!(r1 +r2 )/q
Z x
∇β f (t)q ∇t
,
c t
(4.51)
c
where Ω(x) =
x
R
c
A1 (t)r1 A2 (t)r2 ∇t.
q
Rt
Proof. Again we let y(t) = c c ∇βτ f (τ) ∇τ, and so y∇ (t) =
everywhere. For i = 1, 2 we have
β
q
c ∇τ f (t) µ∇ -almost
!1/q0 Z t
!1/q
Z t
α
β
q
0
q
i
c ∇t f (t) ≤
c ∇τ f (τ) ∇τ
ĥβ−αi −1 (t, ρ(τ)) ∇τ
.
c
c
(4.52)
4.2. OPIAL’S INEQUALITY AND GENERALIZATIONS
83
by Hölder’s inequality. Let the first integral be Ai (t). Hence
Z x
∇α1 f (t)r1 ∇α2 f (t)r2 ∇t
c t
c t
c
Z x
0
0
≤
A1 (t)r1 /q y(t)r1 /q A2 (t)r2 /q y(t)r2 /q ∇t.
(4.53)
c
Applying Hölder once more with q and q0 we have the bound
Z
!1/q0 Z
x
r1
x
r2
A1 (t) A2 (t) ∇t
r1 +r2
y(t)
c
!1/q
∇t
.
(4.54)
c
The first integral we shall define as Ω(x). As for the second, y(t) is non-decreasing
and absolutely continuous, so it attains its maximum at t = x. So we have the
bound
Z x
y(t)r1 +r2 ∇t ≤ (x − c) y(x)r1 +r2 .
(4.55)
c
And this gives
Z x
∇α1 f (t)r1
c t
α
r
c ∇t 2 f (t) 2 ∇t
c
1/q0
≤ Ω(x)
1/q
(x − c)
!(r1 +r2 )/q
Z x
∇β f (t)q ∇t
,
c τ
(4.56)
c
as required.
While the previous Opial type inequalities are interesting in their own right,
as will be seen in the sequel, their usefulness in application increases once one
considers weighted inequalities [40, 121]. Here we shall consider weights which
are strictly positive and essentially bounded. We shall prove analogues of (4.20)
and (4.51), and consider their extremal cases.
Theorem 4.2.9. Let β > α > 0 and n := bβc + 1. Let ω1 , ω2 : e
T → R be positive
and essentially bounded with respect to µ∇ . Suppose that the function
Z
B(t) :=
c
t
ĥβ−α−1 (t, ρ(τ))q ω2 (τ)−q /q ∇τ,
0
0
(4.57)
84
CHAPTER 4. OPIAL-TYPE INEQUALITIES
is defined for each t ∈ (c, x] ∩ T, and such that
x
Z
0
0
ω1 (t)q ω2 (t)−1/q B(t)1/q ∇t < ∞.
(4.58)
c
Let f : e
T → R have integrable fractional derivative of order β, and c ∇β−k
f (c) = 0
t
for all 0 < k ≤ n. Then
x
Z
!2/q
Z x
α
β
q
β
1/q
,
ω2 (t) c ∇t f (t) ∇t
ω1 (t) c ∇t f (t) c ∇t f (t) ∇t ≤ Ω(x)
c
(4.59)
c
where
Ω(x) :=
Z
x
ω1 (t)q ω2 (t)−q /q B(t)∇t.
0
0
(4.60)
c
q
q
Rt
Proof. Define y(t) = c ω2 (τ) c ∇βt f (τ) ∇τ. Then y∇ (t) = ω2 (t) c ∇βt f (t) µ∇ -a.e.
and hence c ∇βt f (t) = ω2 (t)−1/q y∇ (t)1/q . From the properties of f , we have
α
Z t
c ∇t f (t) ≤
ĥβ−α−1 (t, ρ(τ)) c ∇βτ f (τ) ∇τ
Zc t
=
ĥβ−α−1 (t, ρ(τ)) ω−1/q
(τ) ω1/q
2
2
β
c ∇τ f (τ) ∇τ
c
t
Z
q0
ĥβ−α−1 (t, ρ(τ))
≤
c
1/q0
= B(t)
1/q
y(t)
0 /q
ω−q
(τ) ∇τ
2
!1/q0 Z
t
q
ω2 (τ) c ∇βτ f (τ) ∇τ
!1/q
c
,
(4.61)
where we have used Hölder’s inequality with q0 and q. Then
0
ω1 (t) c ∇αt f (t) c ∇βt f (t) ≤ ω1 (t) B(t)1/q y(t)1/q ω2 (t)−1/q y∇ (t)1/q .
(4.62)
Integrating over (c, x] ∩ T and using Hölder’s inequality once more,
Z
c
x
0
ω1 (t) ω2 (t)−1/q B(t)1/q y(t)1/q y∇ (t)1/q ∇t
!1/q0 Z x
!1/q
Z x
q0
−q0 /q
∇
≤
ω1 (t) ω2 (t)
B(t)∇t
y(t) y (t) ∇t
,
c
(4.63)
c
Then the result follows after one sets the first factor to Ω(x) and observing the
4.2. OPIAL’S INEQUALITY AND GENERALIZATIONS
85
second factor is bounded by y(x)2/q as before.
Theorem 4.2.10. Let β > α > 0 and n := bβc+1. Let ω1 : e
T → R be positive and
essentially bounded with respect to µ∇ . Let f : e
T → R have fractional derivative
β
β−k
∞
c ∇t ∈ L∇ (c, x), such that c ∇ x f (c) = 0 for all 0 < k ≤ n. Then one has
Z
c
x
ω1 (t) c ∇αt f (t) c ∇βt f (t) ∇t ≤ kω1 k∞ ĥβ−α+1 (x, c) ∇β f ∞ .
(4.64)
Proof. As with (4.37) we have
α
c ∇t f (t) ≤ ess sup c ∇βτ f (τ) ĥβ−α (t, c).
(4.65)
τ∈[c,t]∩e
T
Thus we have
Z x
ω1 (t) c ∇αt f (t) c ∇βt f (t) ∇t
c

 
Z x
β
2 


ĥβ−α (t, c) ∇t,
≤ ess sup c ∇ x f (t) ess sup ω1 (t)
t∈[c,x]∩e
T
(4.66)
c
t∈[c,x]∩e
T
as required.
Theorem 4.2.11. Let 0 < α1 < α2 < β and let n := bβc + 1. Let ω1 (t) and ω2 (t)
be essentially bounded positive functions over e
T. Assume that there exist time
scale power functions with the property that the function
Bi (t) =
t
Z
ĥβ−αi −1 (t, ρ(τ))q ω2 (τ)q /q ∇τ.
0
0
(4.67)
c
0
exists for all t ∈ [c, x] ∩ T. Let r1 , r2 > 1 and assume that A1 (t)r1 A2 (t)r2 ∈ L∇q (c, t)
for all t ∈ [c, x] ∩ T. Let f : e
T → R be a function with integrable fractional
q
β
derivative c ∇t f (t) ∈ L∇ (c, t) for all t ∈ [c, x] ∩ T such that c ∇β−k
f (c) = 0 for all
t
0 < k ≤ n. Then
Z x
r1 r2
ω1 (t) c ∇αt 1 f (t) c ∇αt 2 f (t) ∇t
c
!(r1 +r2 )/q
Z x
β
q
1/q0
1/q
≤ Ω(x) (x − c)
ω2 (t) c ∇t f (t) ∇t
,
(4.68)
c
86
CHAPTER 4. OPIAL-TYPE INEQUALITIES
where Ω(x) =
x
R
c
ω2 (t)q B1 (t)r1 B2 (τ)r2 ∇t.
0
Rt
Proof. As before we define y(t) = c ω(τ) c ∇βτ f (τ), and so y∇ (t) = ω2 (t) c ∇βt f (t)
µ∇ -a.e. Thus c ∇βt f (t) = ω2 (t)−1/q y∇ (t)1/q .
For i = 1, 2 we have
α
Z t
i
c ∇t f (t) ≤
ĥβ−αi −1 (t, ρ(τ)) c ∇βτ f (τ) ∇τ
Zc t
≤
ĥβ−αi −1 (t, ρ(τ)) ω2 (τ)−1/q ω2 (τ)1/q
c
!1/q0
Z t
q0
q0 /q
≤
ĥβ−αi −1 (t, ρ(τ)) ω2 (t) ∇τ
c
β
c ∇τ f (τ) ∇τ
Z
t
!1/q
β
q
ω2 (τ) c ∇τ f (τ) ∇τ
c
1/q0
= Bi (t)
1/q
y(t)
,
(4.69)
where we have used the Hölder inequality with indices q0 and q. Then
r1 r2
ω1 (t) c ∇αt 1 f (t) c ∇αt 2 f (t) ≤ ω1 (t) B1 (t)r1 /q B2 (t)r2 /q y(t)(r1 +r2 )/q .
(4.70)
Integrating over (c, x] ∩ T and applying Hölder once more,
Z
x
c
ω1 (t)B1 (t)r1 /q B2 (t)r2 /q y(t)(r1 +r2 )/q ∇t
!1/q0 Z x
!1/q
Z x
q0
r1
r2
r1 +r2
≤
ω1 (t) B1 (t) B2 (t) ∇t
y(t)
∇t
0
0
c
c
1/q0
≤ Ω(x)
1/q
(x − c)
(r1 +r2 )/q
y(t)
,
(4.71)
as required.
Theorem 4.2.12. Let 0 < α1 < α2 < β and let n = bβc + 1. Let ω1 : e
T → R be
positive and essentially bounded. Let r1 , r2 > 1 and assume that there exist time
scale power functions such that
A(x) =
Z
x
ĥβ−α1 (t, c)r1 ĥβ−α2 (t, c)r2 ∇t,
(4.72)
c
is finite. Let f : e
T → R be a function with fractional derivative ∇β f ∈ L∇∞ (c, x)
4.2. OPIAL’S INEQUALITY AND GENERALIZATIONS
87
such that c ∇β−k
f (c) = 0 for all 0 < k ≤ n. Then
t
x
Z
r1
ω1 (t) c ∇αt 1 f (t)
c
r +r
r
α
c ∇t 2 f (t) 2 ∇t ≤ ∇β f ∞1 2 kω1 k∞ A(x).
(4.73)
Proof. As before c ∇αt i f (t) ≤ ∇β f ∞ ĥβ−αi (t, c) for i = 1, 2. Therefore we have
Z
x
r2
r1 ω1 (t) c ∇αt 1 f (t) c ∇αt 2 f (t) ∇t
c



Z x
r1 +r2 
β


ess sup ω1 (t)
≤ ess sup c ∇ x f (t)
ĥβ−α1 (t, c)r1 ĥβ−α2 (t, c)r2 ∇t, (4.74)
t∈[c,x]∩e
T
t∈[c,x]∩e
T
c
as required.
Because the previous results are dependent on the Taylor formula, different
definitions of the fractional derivatives will result in different Opial inequalities. However, the only major difference will be in the initial conditions, and
the proofs follow with the same method as before. Consequently they are suppressed. We present an analogue to Theorem 4.2.9 and Theorem 4.2.11 for Caputo fractional derivatives.
Theorem 4.2.13. Let β > α > 0 and m := dβe. Let ω1 , ω2 : e
T → R be positive
and essentially bounded with respect to µ∇ . Suppose that the function
Z
B(t) :=
t
ĥβ−α−1 (t, ρ(τ))q ω2 (τ)−q /q ∇τ,
0
0
(4.75)
c
is defined for each t ∈ (c, x] ∩ T, and such that
x
Z
0
0
ω1 (t)q ω2 (t)−1/q B(t)1/q ∇t < ∞.
(4.76)
c
Let f ∈ AC m (e
T) have integrable fractional derivative Cc ∇βt f (t) ∈ L∇q (e
T), and
k
c ∇t f (c) = 0 for all 0 ≤ k < n. Then the inequality
x
Z
c
!2/q
Z x
α
β
β
q
1/q
ω1 (t) c ∇t f (t) c ∇t f (t) ∇t ≤ Ω(x)
ω2 (t) c ∇t f (t) ∇t
,
c
(4.77)
88
CHAPTER 4. OPIAL-TYPE INEQUALITIES
where
Ω(x) :=
Z
x
ω1 (t)q ω2 (t)−q /q B(t)∇t,
0
0
(4.78)
c
is valid.
Theorem 4.2.14. Let 0 < α1 < α2 < β and let m := dβe. Let ω1 (t) and ω2 (t) be
essentially bounded functions over e
T. Assume that there exist time scale power
functions with the property that the function
Bi (t) =
Z
t
ĥβ−αi −1 (t, ρ(τ))q ω(τ)q /q ∇τ.
0
0
(4.79)
c
0
exists for all t ∈ [c, x] ∩ T. Let r1 , r2 > 1 and assume that A1 (t)r1 A2 (t)r2 ∈ L∇q (c, t)
for all t ∈ [c, x] ∩ T. Let f ∈ AC m (e
T) have integrable fractional fractional
q
β
derivative c ∇t f (t) ∈ L∇ (c, t) for all t ∈ [c, x] ∩ T such that c ∇kt f (c) = 0 for all
0 ≤ k < n. Then
Z x
r1 r2
ω1 (t) c ∇αt 1 f (t) c ∇αt 2 f (t) ∇t
c
!(r1 +r2 )/q
Z x
β
q
1/q0
1/q
≤ Ω(x) (x − c)
ω2 (t) c ∇t f (t) ∇t
,
c
(4.80)
where Ω(x) =
4.3
R
c
x
ω2 (τ)q B1 (τ)r1 B2 (τ)r2 ∇t.
0
Applications of Opial’s Inequality
We will now use (4.59) to look into bounds for dynamic equations using a similar structure to Agarwal [3]. We shall stick to the situation where the initial
conditions are the same as the terms in the dynamical equation.
For a positive integer m, let g be a ld-continuous function defined on e
T×Rm+1 .
Let 0 < α < 1. We shall consider the dynamical equation
f ),
f )∇ = g(t, f, c ∇αt f, c ∇α+1
f, . . . , c ∇α+m
(c ∇α+m
t
t
t
(4.81)
f (c) = 0, c ∇α+i
f (c) = 0 for each 0 ≤ i ≤ m − 1,
with initial conditions c ∇−(1−α)
t
t
4.3. APPLICATIONS OF OPIAL’S INEQUALITY
α+m
f (c)
c ∇t
89
= ω, and c ∇α+m
f (t) non-decreasing. Suppose that
t
g(t, x0 , x1 , . . . , xm ) ≤
m−1
X
qk (t) |xk | ,
(4.82)
k=0
where each qk is positive and essentially bounded on (c, x] ∩ T. Thus we have
(c ∇α+m
f )∇ (t) ≤
t
m−1
X
f (t) .
qk (t) c ∇α+k
t
(4.83)
k=0
We multiply both sides of (4.96) by 2 (c ∇α+m
f )(t) to obtain
t
2(c ∇α+m
f )(t) (c ∇α+m
f )∇ (t)
t
t
≤2
m−1
X
(c ∇α+m
qk (t) c ∇α+k
f
(t)
f )(t).
t
t
(4.84)
k=0
However, unlike in [3] we cannot directly integrate the left hand side. None the
less we can establish a lower bound of (4.84) by observing that (c ∇α+m
f )(t) is
t
non-decreasing,
h
(c ∇α+m
f )(t)2
t
i∇
= (c ∇α+m
f ◦ ρ)(t) + (c ∇α+m
f )(t) (c ∇α+m
f )∇ (t)
t
t
t
≤ 2(c ∇α+m
f )(t) (c ∇α+m
f )∇ (t).
t
t
(4.85)
Thus combining (4.84) with (4.85), and integrating t over (c, x] ∩ T we obtain
(c ∇α+m
)(x)2
x
−
(c ∇α+m
f )(c)2
x
≤2
m−1 Z
X
k=0
x
qk (t) c ∇α+k
f (t) (c ∇α+m
f )(t) ∇t. (4.86)
t
t
c
After using the initial conditions, and taking absolute values we obtain the bound
m−1 Z x
X
α+m
2
α+m
2
c ∇ x f (x) ≤ |ω| + 2
∇t.
qk (t) c ∇α+k
f
(t)
∇
f
(t)
t
c t
k=0
(4.87)
c
The initial conditions allow use to use (4.59) term by term with the parameters
q = q0 = 2, ω1 (t) = qk (t) and ω2 (t) = 1. We obtain
Z x
α+m
2
∇α+m f (t)2 ∇t,
2
c ∇ x f (x) ≤ |ω| + ψ(x)
c t
c
(4.88)
90
CHAPTER 4. OPIAL-TYPE INEQUALITIES
P
1/2
where we have let ψ(x) = 2 m−1
and Ωk are the individual coefficients
k=0 Ωk (x)
for each term. We shall need the nabla analogue of Gronwall’s inequality, which
can be easily obtained by adapting the proof of [46, Theorem 6.4] with the theory
of Anderson et al [30]. In addition we shall need to assume 1 − ν(t)ψ(t) > 0. We
obtain the bound
Z x
α+m
2
2
c ∇ x f (x) ≤ |ω| +
êψ (x, ρ(τ)) |ω|2 ψ(x) ∇τ,
(4.89)
c
where we have denoted êψ (·, t0 ) to be the solution of the dynamic equation w∇ =
ψ w with initial condition w(t0 ) = 1. We now take the square root and denote the
right hand side as φ(x).
s
α+m
c ∇ x f (x) ≤ |ω|
1 + ψ(x)
x
Z
êψ (x, ρ(τ)) ∇τ = φ(x).
(4.90)
c
In order to obtain a bound for the solution of the dynamical equation, we shall
use (3.132) to observe
Z
| f (x)| ≤
x
Z x
α+m
ĥα+m−1 (x, ρ(t)) c ∇t f (t) ∇t ≤
ĥα+m−1 (x, ρ(t)) φ(t) ∇t. (4.91)
c
c
We have thus proved the following theorem.
Theorem 4.3.1. Let 0 < α < 1 and m a positive integer. Let g : e
T × Rm+1 be
ld-continuous and satisfy the inequality
g(t, x0 , x1 , . . . , xm ) ≤
m
X
qk (t) |xk | ,
(4.92)
k=0
where qk : e
T is positive and essentially bounded with respect to µ∇ . Suppose that
e
f : T → R is a solution to the dynamical equation
α+m+1
f
c ∇t
f ),
= g(t, f, c ∇αt f, . . . , c ∇α+m
t
(4.93)
with initial conditions c ∇−(1−α)
f (c) = 0 and c ∇α+i
f (c) = 0 for each 0 ≤ i ≤ m − 1.
t
t
If f satisfies the conditions of (c ∇α+m
f )(c) = ω non-zero and c ∇α+m
f nont
t
4.3. APPLICATIONS OF OPIAL’S INEQUALITY
91
decreasing over (c, x] ∩ T, then f satisfies the bound
Z
| f (x)| ≤ |ω|
c
x
s

Z t


ĥα+m−1 (x, ρ(t))  1 + ψ(t)
êψ (t, ρ(τ)) ∇τ  ∇t,
c
where
ψ(x) = 2
m−1 Z
X
x
ĥm−k−1 (t, ρ(τ)) ∇τ
2
qk (t) ∇t
c
k=0
!1/2
t
Z
2
(4.94)
,
(4.95)
c
provided 1 − νψ > 0.
Comparing Theorem 4.3.1 to its analogue in Agarwal [3], we see that although we have generalized the inequality to other time scales, we see that we
have been forced to make the additional assumption of c ∇α+m
f being a nont
decreasing function. Also in order to invoke the nabla analogue of Gronwall’s
inequality, we have made the assumption that 1−ν(t) ψ(t) > 0 for all t ∈ (c, x]∩ e
T.
This is trivially true for T = R, but becomes a stronger constraint as ν gets larger.
It is a compromise we shall have to make for time scales calculus.
If instead of (4.82), suppose g satisfies
g(t, x0 , x1 , . . . , xm ) ≤ q(t) |xk |rk |x j |r j ,
(4.96)
where 0 < k < m and 0 < j < m are fixed, rk , r j > 1 and q(t) is positive and
essentially bounded over e
T. Then
α+k
rk α+ j
r j
∇
.
(c ∇α+m
f
)
(t)
≤
q(t)
∇
f
(t)
∇
f
(t)
t
c t
c t
(4.97)
We shall assume the c is right dense. Integrating over t ∈ (c, x] ∩ T we have
Z
x
(c ∇α+m
f )∇ (t) ∇t
t
x
Z
≤
c
rk α+ j
r j
∇t.
f
(t)
∇
f
(t)
q(t) c ∇α+k
t
c t
(4.98)
c
Then after taking absolute values we have the bound
Z x
α+m
c ∇ x f (x) ≤ |ω| +
q(t)
α+k
r
c ∇t f (t) k
α+ j
r j
c ∇t f (t) ∇t.
(4.99)
c
We apply the inequality (4.68) with ω1 = q(t), ω2 (t) = 1, r1 = rk , r2 = r j and
92
CHAPTER 4. OPIAL-TYPE INEQUALITIES
q = r1 + r2 . Let us write λ = r j + rk . Thus we obtain the inequality
Z x
α+m
∇α+m f (t)λ ∇t,
c ∇ x f (x) ≤ |ω| + ψ(x)
c t
(4.100)
c
where we have let ψ(x) = Ω(x)1/λ (x − c)1/λ . The analogous theorem in [3] merely
states a bound “can be obtained rather easily”. We shall establish the precise
form of this bound. Observing the map z 7→ zλ is monotonically increasing, we
wish to apply the Bihari inequality [106, Theorem 4.2] (see also [46, Theorem
6.12]), to (4.100). However because ψ(c) = 0 we see that ψ1 is not differentiable
at x = c. To fix this, we weaken (4.100) by replacing ψ with Ψ(x) = ψ(x) + for
some > 0.
Rewriting (4.100) in the form
c ∇α+m
f
(x)
x
Ψ(x)
λ 
Z x  α+m

∇
f
(t)

|ω|
 c t
 Ψ(t)λ ∇t
≤
+


λ
Ψ(x)
Ψ(t)
c
(4.101)
we obtain the bound
α+m
c ∇t f (t) < w(t) Ψ(t),
c < t < x,
(4.102)
where w is the solution to the dynamic equation
1
w = |ω|
Ψ
∇
with initial conditions
w(c) >
!∇
+ Ψλ wλ ,
|ω|
.
Ψ(c)
(4.103)
(4.104)
We have thus proved the following bound for f .
Theorem 4.3.2. Let 0 < α < 1 and m a positive integer. Let > 0, g : e
T × Rm+1
be ld-continuous and satisfy the inequality
g(t, x0 , x1 , . . . , xm ) ≤ q(t) |xk |rk |x j |r j ,
(4.105)
where q : e
T → R is positive and essentially bounded with respect to µ∇ . Suppose
4.3. APPLICATIONS OF OPIAL’S INEQUALITY
93
that f : e
T → R is a solution to the dynamical equation
α+m+1
f
c ∇t
= g(t, f, c ∇αt f, . . . , c ∇α+m
f ),
t
(4.106)
f (c) = 0 for each 0 ≤ i ≤ m − 1.
with initial conditions c ∇t−(1−α) f (c) = 0 and c ∇α+i
t
If f satisfies the conditions of (c ∇α+m
f )(c) = ω non-zero, then f satisfies the
t
bound
Z x
ĥα+m−1 (x, ρ(t)) w(t) Ψ(t) ∇t,
(4.107)
| f (x)| ≤
c
where w solves the dynamic equation
1
w = |ω|
Ψ
∇
!∇
+ Ψλ wλ ,
w(c) >
|ω|
,
Ψ(c)
(4.108)
with
Ψ(x) = + (x − c)1/(rk +r j )
!rk Z t
!r j !1/(rk +r j )
Z x Z t
2
2
ĥm−k−1 (t, ρ(τ)) ∇τ
ĥm− j−1 (t, ρ(τ)) ∇τ ∇t
×
.
c
c
c
(4.109)
This bound has not been established in the literature. Indeed even the analogue in Agarwal [3] on which it isbased does not give its explicit form. We
see that we have needed to consider the function Ψ in order to compensate for
1/ψ not being ∇-differentiable at c. It is not currently known if constraint can
be weakened. However, observe that we no longer need the conditions of nonnegativity of c ∇α+m
f or 1 − νψ > 0. Hence the two theorems are complementary
t
of each other.
In this chapter we have considered various extensions of Opial’s inequality
to fractional derivatives. We have considered both Riemann-Liouville fractional
∇-derivatives and the equivalents for Caputo fractional ∇-derivatives. We also
have considered Opial’s inequalities which feature positive powers of the factors
and also the case with bounded weights. The latter has been applied to studying
dynamical equations with fractional derivatives, extending the results of Agarwal
[3]. Observe that analogous results for Caputo fractional ∇-derivatives can be
94
CHAPTER 4. OPIAL-TYPE INEQUALITIES
obtained by replacing (4.59) with (4.77). This would require the initial conditions
to be changed to
C k
0 ≤ k < m,
(4.110)
c ∇t f (c) = 0,
and the dynamical equation (4.81) to be changed to
(Cc ∇α+m
f )∇ = g(t, f, f ∇ , f ∇ , . . . , f ∇
t
2
to reflect this.
m−1
, Cc ∇α+m
f)
t
(4.111)
Chapter 5
A Blueprint for Further Work
We have demonstrated that the important properties of fractional calculus can be
derived from a simple set of axioms for power functions. The Cauchy formula
illustrates that multiple integration can be reduced to an integral transform with a
monomial kernel. Hence, in order to extend this to fractional calculus, we need to
define what properties power functions should satisfy. The three axioms we have
chosen reflect the properties of fractional calculus we desire. First the requirement of ĥα (t, s) ≡ 0 when t < s allows us to avoid the situation of the function not
existing for non-integer α. The second property subsumes time scale monomials
as special cases of power functions. This allows us to extend integration to fractional order by treating n-th order multiple integration as a special case. Lastly,
the third property is the semi-group property for power functions, allowing us
to evaluate the convolution of two power functions. Further we have shown that
on each of the major time scales studied in the literature, there exist an example which satisfies our requirements. Thus these three properties are sufficient in
order to unify fractional calculus, fractional differences and fractional quantum
calculus.
Concerning the semi-group property (3.16), we have shown it is a sufficient
condition for an index law of fractional integration. In order to show this, we have
proved a ∇-analogue of the Dirichlet theorem. The index law in turn has allowed
us to derive various extensions to Taylor’s theorem. Taylor’s original theorem
allows one to approximate a function using polynomials whose coefficients are
95
96
CHAPTER 5. A BLUEPRINT FOR FURTHER WORK
related to the function’s derivatives evaluated at one point. The extension via
fractional derivatives allows one to approximate the function in the more general class of power functions. The Taylor theorems we have derived allow us to
consider Opial type inequalities. This class of integral inequalities allows one
to bound expressions involving a function and its derivative by an easier expression which only consists of the derivative. These inequalities can be applied to
dynamical equations to give estimations on the growth of the solution functions.
Our work is related to that of Anastassiou [21]. However, there are errors
in the work which we highlight. Firstly, Anastassiou defines power functions
without properly considering their domain. He defines a function T × T → R
(t − s)α
,
ĥα (t, s) =
Γ(α + 1)
(5.1)
when T = R. This is consistent with our definition when t > s. However, as
noted above, we have chosen to define ĥα (t, s) = 0 for t < s. Anastassiou has
not, so (5.1) is undefined whenever t < s and α is a non-integer. Our definition
however, is properly defined in this case.
Secondly, Anastassiou has not properly proved a Dirichlet theorem, and this
causes errors in the index law. Anastassiou claims without proof that
Z tZ
a
a
τ
f (τ, u) ∇u ∇τ =
Z tZ
t
f (τ, u) ∇τ ∇u.
a
(5.2)
u
However, the integers are a counterexample, as the left hand side includes the
diagonal but the right hand side does not. This is due to the notation of the nabla
Rt
integral. Because we have denoted u to be integrating over (u, t] ∩ T, the points
τ = u are not included. In this thesis we have taken care to prove the Dirichlet
formula, which will be useful to not just fractional calculus but to the theory of
time scales calculus as a whole.
Due to the incorrect Dirichlet theorem, Anastassiou has proved an incorrect
index law and an incorrect Taylor type theorem. Anastassiou claims that the
index law for fractional calculus is given by
Jaα Jaβ f (t) + D( f, α, β, T, t) = Jaα+β f (t).
(5.3)
97
However, the presence of D( f, α, β, T, t) has not appeared in the literature of fractional differences or the literature of fractional q-integrals. This error propagates
through the work and Anastassiou obtains an erroneous Taylor type formula
f (t) =
m−1
X
∇k
ĥk (t, a) f (a) + D( f
∇m
)+
Z
t
ĥµ−2 (t, ρ(τ)) ∇µ−1
a∗ f (τ) ∇τ,
(5.4)
a
k=0
where D( f ∇ ) = D( f ∇ , µ−1, m−µ+1, T, t). Again this expression is inconsistent
with the established literature. However, our work is consistent with the literature
and more general.
m
m
The errors flow into studying inequalities as well. Anastassiou presents a
number of integral inequalities, including Opial’s inequality, but due to the problems with the Taylor formula they are incorrect. In order to compensate for the
m
presence of the D( f ∇ , µ − 1, m − µ + 1, T, t), Anastassiou frames the inequalities
in terms of an auxiliary function
m
A(t) = f (t) − D( f ∇ , µ − 1, m − µ + 1, T, t).
(5.5)
We have shown this is superfluous. Our work shows that the functional D is
not required, so our Opial inequalities use only the function f and its fractional
derivatives. Further we have proved Opial inequalities which contain factors
raised to positive powers, and inequalities with bounded weights. This makes
our results easier to apply when studying fractional dynamic equations.
One criticism of Anastassiou’s work is that while the theory allows one to
describe the continuum case and discrete together, it not obvious that the theory
extends to arbitrary time scales. This is true of our theory as well. We do not have
results which give the existence of the power functions on time scales. Indeed
the method of extending monomials to power functions appears to be different in
the different time scales we have considered, thus a unifying mechanic for this
procedure appears to not exist. Worse, we do not have concrete results concerning properties of a time scale under which we can assert that power functions
exist for a time scale. In this sense the theory is very much a work in progress.
However, recently there has been research which points us in the direction of
obtaining answers.
98
CHAPTER 5. A BLUEPRINT FOR FURTHER WORK
Independent of us, a recent paper by Bastos et al [38] has considered the
problem of defining fractional calculus on time scales through the generalized
Laplace transform. To see this first consider the Laplace transform of the Caputo
fractional derivative on the real numbers.
Theorem 5.0.3 ([112]). Let α > 0, and n = dαe. Let f ∈ C n (0, ∞). Suppose there
exist t1 > 0 and constants B, q > 0 such that f (n) ∈ L1 (0, t1 ) and f (n) (t) ≤ Beqt
for all t > t1 . If the Laplace transforms of f and f (n) exist and limt→∞ f k (t) = 0
for each 0 ≤ k ≤ n − 1, then
L
C
Dα0+ f (z)
α
= z L f (z) −
n−1
X
f (k) (0) zα−k−1 ,
(5.6)
k=0
where L is the Laplace transform and C Dα0+ is the Caputo fractional derivative
with base point 0.
The authors observe that the Laplace transform has an analogue in time scales
calculus and use this to define a Caputo fractional delta derivative on an arbitrary
time scale.
Definition 5.0.4. Let α > 0 and f : T → R. The Bastos fractional integral of f
of order α is defined as
"
#
−1 F(z)
α
IT f (t) = LT
(t),
(5.7)
zα
where F(z) is the generalized Laplace transform of f .
Definition 5.0.5. Let α > 0 and n = dαe. The Bastos fractional derivative of a
function f : T → R of order α is given by
f
(α)
(t) =
L−1
T


n−1
X
 α

k
∆
α−k−1
z F(z) −
 (t).
f (0+) z
(5.8)
k=0
Importantly for our theory, the authors consider the properties of fractional
derivatives of time scale monomials.
Theorem 5.0.6 ([38]). Let α > 0 and n = dαe. If k ≥ n, then
"
h(α)
k (t, 0)
=
L−1
T
1
zk+1−α
#
(t).
(5.9)
99
Thus it seems reasonable that the power functions we have defined also satisfy a similar identity under the nabla Laplace transform.
Conjecture 1. Let T be a time scale with min T = t0 , and let L∇ be the nabla
analogue of the Laplace transform on T. For α > −1, define a collection of
functions fα (t, s) : T × T → R as fα = 0 whenever t ≤ s, and for t > s,
"
fα (t, s) :=
L−1
∇
#
êz (s, t0 )
(t),
zα+1
s ∈ T,
(5.10)
where êz (·, t0 ) is the solution to the initial value problem
w∇ =
−zw
,
1 − νz
w(t0 ) = 1.
(5.11)
Then the collection fα satisfy the requirement of power functions.
The equation (5.10) is simply an extension of the result of the Laplace transform of monomials. Indeed
L∇ ĥk (t, t0 ) =
1
sk+1
,
(5.12)
where the generalized Laplace transform is with respect to nabla calculus. The
exponential function in the enumerator is a shifting of the Laplace transform (see
Bohner and Guseinov [44]), allowing us to replace t0 with s > t0 .
Observe that the conjecture does not necessarily make calculating power
functions any easier. In real analysis, contour integration is the main method
of inverting the Laplace transform. However, there is no analogue for arbitrary
time scales calculus. Indeed, as far as we know, while there are results concerning the existence of the Laplace transform of a function f , there are no results
concerning the existence of the inverse Laplace transform of a function F. Thus
the conjecture even if true does not guarantee that power functions exist on a
given time scale.
These criticisms also apply to the theory in Bastos et al [38]. The paper does
not discuss conditions under which the inversion is possible, and so the authors
cannot give conditions when the fractional integral or the fractional derivative
exists. Just as we have done, the authors consider properties of these operators
100
CHAPTER 5. A BLUEPRINT FOR FURTHER WORK
under the assumption of existence, and therefore their work is not too dissimilar
to our theory. It is hoped that the thesis will spur research in this area, as we have
shown which properties of power functions are relevant to the study of fractional
calculus.
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