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Nuno R. O. Bastos, Rui A. C. Ferreira and Delfim F. M. Torres

THE FRACTIONAL DIFFERENCE

CALCULUS OF VARIATIONS

Nuno R. O. Bastos1, Rui A. C. Ferreira2, Delfim F. M. Torres3

1Department of Mathematics

Polytechnic Institute of Viseu

3504-510 Viseu, Portugal

e-mail: [email protected]

2Faculty of Engineering and Natural SciencesLusophone University of Humanities and Technologies

1749-024 Lisbon, Portugal


3Department of Mathematics

University of Aveiro

3810-193 Aveiro, Portugal


Abstract. We introduce a discrete-time fractional calculus of variations. First order fractional

necessary optimality conditions of Euler-Lagrange type for the time scale hZ, h > 0, are es-tablished. Examples illustrating the use of the new condition are given. They show that the

solutions to the considered fractional problems: (i) converge to the classical discrete-time solu-

tions when the fractional order of the discrete-derivatives tend to integer values, (ii) converge to

the fractional continuous-time solutions when h tends to zero.

2010 Mathematics Subject Classification: 26A33, 39A12, 49K05.

Keywords: fractional difference calculus, fractional summation by parts, calculus of variations,

Euler-Lagrange equations, time scale hZ.

1 INTRODUCTION

The Fractional Calculus is currently an important research field in several different areas:

physics (including classical and quantum mechanics as well as thermodynamics), chemistry,

biology, economics, and control theory [13, 15, 17]. It has origin more than 300 years ago when

LHopital asked Leibniz what should be the meaning of a derivative of non-integer order. After

that episode several more famous mathematicians contributed to the development of Fractional

Calculus: Abel, Fourier, Liouville, Riemann, Riesz, just to mention a few names.

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The Fractional Difference Calculus of Variations

In [12] Miller and Ross define a fractional sum of order > 0 via the solution of a lineardifference equation. Namely, they present it as (see Section 2 for the notations used here)

f(t) =1

()

ts=a

(t (s))(1)f(s). (1)

This was done in analogy with the RiemannLiouville fractional integral of order > 0,

aDx f(x) =

1

()

xa

(x s)1f(s)ds,

which can be obtained via the solution of a linear differential equation [12, 13]. Some basic

properties of the sum in (1) were obtained in [12]. More recently, F. Atici and P. Eloe [4, 5]

defined the fractional difference of order > 0, i.e., f(t) = ((1)f(t)), and developedsome of its properties that allow to obtain solutions of certain fractional difference equations.

Fractional differential calculus has been widely developed in the past few decades due mainlyto its demonstrated applications in various fields of science and engineering. The study of frac-

tional problems of the Calculus of Variations and respective Euler-Lagrange equations is a fairly

recent issue see [1, 2, 7, 9, 14] and references therein and include only the continuous case. It

is well known that discrete analogues of differential equations can be very useful in applications

[10, 11]. Therefore, we consider pertinent to start a fractional discrete-time theory of the calculus

of variations.

Our objective is two-fold. On one hand we proceed to develop the theory of fractional dif-

ference calculus, namely, we introduce the concept of left and right fractional sum/difference

(cf. Definition 2.1). On the other hand, we believe that the present work will potentiate research

not only in the fractional calculus of variations but also in solving fractional difference equa-tions, specifically, fractional equations in which left and right fractional differences appear. We

formulate our main result in the time-scale hZ, h > 0.Because the theory of fractional difference calculus is in its infancy [4, 5, 12], the paper is

self contained. We begin, in Section 2, to give the definitions and results needed throughout. In

Section 3 we present the new results of the paper. Finally, in Section 4 we show some examples.

2 PRELIMINARIES

We begin by introducing some notation used throughout. The notation is borrowed from the

calculus on time scales [6]. Let a, b Z

with a < b. We putT

= [a, b] Z

,T

= [a, b 1] Z

and T2

= [a, b 2] Z. Denote by F the set of all real valued functions defined on T. Also,we will frequently write (t) = t + 1, (t) = t 1 and f(t) = f((t)). The usual conventionsc1

t=c f(t) = 0, c T, and1

i=0 f(i) = 1 remain valid here.As usual, the forward difference is defined by f(t) = f(t) f(t). If we have a func-

tion f of two variables, f(t, s), its partial (difference) derivatives are denoted by t and s,respectively. For arbitrary x, y R define (when it makes sense)

x(y) =(x + 1)

(x + 1 y),

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where is the gamma function. The following property of the gamma function,

(x + 1) = x(x),

will be frequently used.

It was mentioned in Section 1 that equality (1) was introduced in [12] as the fractional sum of

order > 0. In fact one finds the definition of left and right fractional sum (Definition 2.1) whenproving the fractional formula of summation by parts (Lemma 3.1), which is a crucial result in

order to fulfill our goal (see proof of the Euler-Lagrange equation (10) given by Theorem 3.4).

Definition 2.1. Letf F. The left fractional sum and the right fractional sum of order > 0are defined, respectively, as

at (f)(t) =

1

()

ts=a

(t (s))(1)f(s), (2)

and

tb (f)(t) =

1

()

bs=t+

(s (t))(1)f(s). (3)

Remark 2.1. The above sums (2) and (3) are defined for t = a + mod (1) and t = a mod (1), respectively, while f(t) is defined for t = a mod (1) (see [12]). Throughout we will

write (2) and (3), respectively, in the following way:

at f(t) =

1

()

ts=a

(t + (s))(1)f(s), t T,

tb f(t) =

1

()

bs=t

(s + (t))(1)f(s), t T.

Remark 2.2. The left fractional sum defined in (2) coincides with the fractional sum defined in

[12]. The analogy of (2) and (3) with the RiemannLiouville left and right fractional integralsof order > 0 is clear:

aDx f(x) =

1

()

xa

(x s)1f(s)ds,

xDb f(x) =

1

()

bx

(s x)1f(s)ds.

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It was proved in [12] that lim0 at f(t) = f(t). We do the same for the right fractional

sum using a different method. Let > 0 be arbitrary. Then,

tb f(t) =

1

()

bs=t

(s + (t))(1)f(s)

= f(t) +1

()

bs=(t)

(s + (t))(1)f(s)

= f(t) +b

s=(t)

(s + t)

()(s t + 1)f(s)

= f(t) +b

s=(t)

st1i=0 (+ i)(s t + 1)

f(s).

Therefore, lim0 tb f(t) = f(t). It is now natural to define

a0tf(t) = t

0bf(t) = f(t),

which we do here, and to write

at f(t) = f(t) +

(+ 1)

t1s=a

(t + (s))(1)f(s), t T, 0, (4)

tb f(t) = f(t) +

(+ 1)

bs=(t)

(s + (t))(1)f(s), t T, 0.

The next theorem was proved in [5].

Theorem 2.2. Letf F and > 0. Then, the equality

at f(t) = (a

t f(t))

(t + a)(1)

()f(a), t T,

holds.

Remark 2.3. It is easy to include the case = 0 in Theorem 2.2. Indeed, in view of previousdefinitions, we get

at f(t) = (a

t f(t))

(+ 1)(t + a)(1)f(a), t T, (5)

for all 0.

We now present the counterpart of Theorem 2.2 for the right fractional sum:

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Theorem 2.3. Letf F and 0. Then, the equality

t

(b)f(t) =

(+ 1) (b + (t))

(1)

f(b) + (t

b f(t)), t T

, (6)

holds.

The proof of Theorem 2.3 is very similar to the one of Theorem 2.2.

Definition 2.4. Let 0 < 1 and set = 1 . Then, the left fractional difference and theright fractional difference of order of a function f F are defined, respectively, by

at f(t) = (a

t f(t)), t T

,

and

tb f(t) = (t

b f(t))), t T

.

3 MAIN RESULTS

Our aim is to introduce the discrete-time fractional problem of the calculus of variations and

to prove corresponding necessary optimality conditions. For that we need a fractional version of

the summation by parts formula.

3.1 Fractional summation by parts

Lemma 3.1 (Fractional summation by parts). Letf andg be real valued functions defined on TkandT, respectively. Fix 0 < 1 and put = 1 . Then,

b1t=a

f(t)at g(t) = f(b 1)g(b) f(a)g(a) +

b2t=a

t(b)f(t)g

(t)

+

( + 1)g(a)

b1

t=a

(t + a)(1)f(t) b1

t=(a)

(t + (a))(1)f(t)

. (7)

Proof. From (5) we can write

b1t=a

f(t)at g(t) =

b1t=a

f(t)(at g(t))

=b1t=a

f(t)

a

t g(t) +

( + 1)(t + a)(1)g(a)

=b1t=a

f(t)at g(t) +

b1t=a

( + 1)(t + a)(1)f(t)g(a). (8)

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Using the relation (4) we get

b1t=a

f(t)at g(t)

=b1t=a

f(t)

g(t) +

( + 1)

t1s=a

(t + (s))(1)g(s)

=b1t=a

f(t)g(t) +

( + 1)

b1t=a

f(t)t1s=a

(t + (s))(1)g(s)

=b1

t=af(t)g(t) +

( + 1)

b2

t=ag(t)

b1

s=(t)(s + (t))(1)f(s)

= f(b 1)[g(b) g(b 1)] +b2t=a

g(t)t(b)f(t),

where the third equality follows from the fact that if and are two functions defined on T

and is a function defined on T T, then the equality

b1t=a

(t)t1s=a

(t, s)(s) =b2t=a

(t)b1

s=(t)

(s, t)(s)

holds. We proceed to develop the right hand side of the equality

b1t=a

f(t)at g(t) = f(b 1)[g(b) g(b 1)] +

b2t=a

g(t)t(b)f(t)

as follows:

f(b 1)[g(b) g(b 1)] +b2t=a

g(t)t(b)f(t)

= f(b 1)[g(b) g(b 1)] +

g(t)t

(b)f(t)t=b1t=a

b2t=a g

(t)(t

(b)f(t))

= f(b 1)g(b) f(a)g(a)

( + 1)g(a)

b1s=(a)

(s + (a))(1)f(s)

+b2t=a

t

(b)f(t)

g(t) ,

where the first equality follows from the usual summation by parts formula. Putting this into (8),

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we get:

b1t=a

f(t)at g(t) = f(b 1)g(b) f(a)g(a) +

b2t=a

t

(b)f(t)

g(t)

+g(a)

( + 1)

b1t=a

(t + a)(1)

()f(t)

g(a)

( + 1)

b1s=(a)

(s + (a))(1)f(s).

The theorem is proved.

Remark 3.1. When = 1 andf andg are defined on {a, a + 1, . . . , b}, then we obtain from ourformula (7) the usual integration by parts formula in difference calculus:

b1t=a

f(t)g(t) = f(b 1)g(b) f(a)g(a) b2t=a

f(t)g(t)

= f(t)g(t)|ba b1t=a

f(t)g(t) .

3.2 Necessary optimality condition

We begin to fix two arbitrary real numbers and such that , (0, 1]. Further, we put = 1 and = 1 .

Let a function L(t,u,v,w) : TRRR R be given. We assume that the second-orderpartial derivatives Luu, Luv, Luw, Lvw , Lvv , and Lww exist and are continuous.

Consider the functional L : F R defined by

L(y()) =b1t=a

L(t, y(t), at y(t), t

b y(t)) (9)

and the problem, that we denote by (P1), of minimizing (9) subject to the boundary conditionsy(a) = A and y(b) = B (A, B R). Our aim is to derive necessary conditions of first andsecond order for problem (P1).

Definition 3.2. Forf F we define the norm

f = maxtT

|f(t)| + maxtT

|at f(t)| + max

tT|t

b f(t)|.

A function y F with y(a) = A and y(b) = B is called a local minimizer for problem (P1)provided there exists > 0 such thatL(y) L(y) for all y F with y(a) = A and y(b) = Bandy y < .

Definition 3.3. A function F is called an admissible variation for problem (P1) provided = 0 and(a) = (b) = 0.

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The next theorem presents a first order necessary condition for problem (P1).

Theorem 3.4 (The fractional discrete-time EulerLagrange equation). If y F is a local mini-

mizer for problem (P1), then

Lu[y](t) + t(b)Lv[y](t) + a

t Lw[y](t) = 0 (10)

holds for all t T2

, where the operator[] is defined by

[y](s) = (s, y(s), as y(s), s

b y(s)).

Proof. Suppose that y() is a local minimizer of L(). Let () be an arbitrary fixed admissible

variation and define the function :

() ,

()

R by

() = L(y() + ()).

This function has a minimum at = 0, so we must have (0) = 0, i.e.,

b1t=a

Lu[y](t)

(t) + Lv[y](t)at (t) + Lw[y](t)t

b (t)

= 0,

which we may write, equivalently, as

Lu[y](t)(t)|t=(b) +

b2t=a

Lu[y](t)(t)

+

b1t=a

Lv[y](t)at (t) +

b1t=a

Lw[y](t)tb (t) = 0. (11)

Using Lemma 3.1, and the fact that (a) = (b) = 0, we get for the third term in (11) that

b1t=a

Lv[y](t)at (t) =

b2t=a

t

(b)Lv[y](t)

(t). (12)

Using (6) we can write

b1

t=a

Lw[y](t)tb (t)

= b1t=a

Lw[y](t)(tb (t))

= b1t=a

Lw[y](t)

t

(b)(t)

(+ 1)(b + (t))(1)(b)

=

b1t=a

Lw[y](t)t(b)(t)

(b)

(+ 1)

b1t=a

(b + (t))(1)Lw[y](t)

.

(13)

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It follows that

b1t=a

Lw[y](t)t

(b)(t)

=b1t=a

Lw[y](t)

(t) +

(+ 1)

b1s=(t)

(s + (t))(1)(s)

=b1t=a

Lw[y](t)(t) +

(+ 1)

b2t=a

Lw[y](t)b1

s=(t)

(s + (t))(1)(s)

=b1

t=aLw[y](t)(t) +

(+ 1)

b1

t=a(t)

t1

s=a(t + (s))(1)Lw[y](s)

=b1t=a

(t)at Lw[y](t).

(14)

We apply again the usual summation by parts formula, this time to (14), to obtain:

b1t=a

(t)at Lw[y](t)

=b2t=a

(t)at Lw[y](t) + ((b) ((b)))a

t Lw[y](t)|t=(b)

=

(t)at Lw[y](t)

t=b1

t=a

b2t=a

(t)(at Lw[y](t))

+ (b)at Lw[y](t)|t=(b) (b 1)a

t Lw[y](t)|t=(b)

= (b)at Lw[y](t)|t=(b) (a)a

t Lw[y](t)|t=a

b2t=a

(t)at Lw[y](t).

(15)

Since (a) = (b) = 0 it follows, from (14) and (15), that

b1

t=a Lw[y](t)t(b)(t) =

b2

t=a (t)a

t Lw[y](t)

and, after inserting in (13), that

b1t=a

Lw[y](t)tb (t) =

b2t=a

(t)at Lw[y](t). (16)

By (12) and (16) we may write (11) as

b2t=a

Lu[y](t) + t

(b)Lv[y](t) + a

t Lw[y](t)

(t) = 0.

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Since the values of(t) are arbitrary for t T2

, the EulerLagrange equation (10) holds along

y.

Remark 3.2. If the initial condition y(a) = A is not present (i.e., y(a) is free), we can usestandard techniques to show that the following supplementary condition must be fulfilled:

Lv(a) +

( + 1)

b1t=a

(t + a)(1)Lv[y](t)

b1

t=(a)

(t + (a))(1)Lv[y](t)

+ Lw(a) = 0. (17)

Similarly, ify(b) = B is not present (i.e., y(b) is free), the equality

Lu((b)) + Lv((b)) Lw((b))

+

(+ 1)

b1t=a

(b + (t))(1)Lw[y](t)

b2t=a

((b) + (t))(1)Lw[y](t)

= 0 (18)

holds. We just note that the first term in (18) arises from the first term on the left hand side of

(11). Equalities (17) and (18) are the fractional discrete-time natural boundary conditions.

3.3 Generalization

Theorem 3.4 can be generalized from the time scale Z to hZ, h > 0. The proof is tech-nical, and we present the result here without proof. Illustrative examples of the application of

Theorem 3.6 are given in Section 4.

We begin with some standard notation of the calculus of variations on time scales [ 8]. Let

hZ = {hk : h > 0, k Z} and a, b hZ with a < b. The previous considered notation, validfor h = 1, is now generalized for t hZ by denoting (t) = t + h, (t) = t h, and (t) = h.We put T = [a, b] hZ, T = [a, (b)] hZ and T

2

= [a, 2(b)] hZ.

The forward difference is defined by

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

(t).

The delta h-integral or h-sum off is

ba

f(t)t =

b

h1

k= ah

f(kh)h .

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If we have a function f of two variables, f(t, s), its partial (difference) derivatives are denotedby t,h and s,h, respectively. For arbitrary x, y R define (when it makes sense)

x(y) = hy(xh + 1)

(xh

+ 1 y),

where is the gamma function.Often, the left fractional delta integration of order > 0 is denoted as a

t f(t) and the

right fractional delta integration of order > 0 by tb f(t). Here, like Ross et. al. in [16], we

omit the subscript t on the operator and we will write ah f(t) and h

b f(t).

Definition 3.5. Letf F. The left fractional h-sum is defined as

ah f(t) = h

f(t) +

(+ 1) t

a

(t + h (s))(1)f(s)s, t T, 0,

while the right fractional h-sum of order 0 is given by

hb f(t) = h

f(t) +

(+ 1)

(b)(t)

(s + h (t))(1)f(s)s, t T .

As before, we begin to fix two arbitrary real numbers and such that , (0, 1], andwe put := 1 and := 1 . Given a function L(t,u,v,w) : T R R R R weconsider the problem

L(y()) = b

a

L(t, y(t), ahy(t), h

b y(t))t min

y(a) = A , y(b) = B ,

(Ph)

where A, B R are given.

Theorem 3.6 (The h-fractional EulerLagrange equation for problem (Ph)). If y F is a localminimizer for problem (Ph), then the following equality must be satisfied:

Lu[y](t) + h(b)Lv[y](t) + a

hLw[y](t) = 0, t T

2 , (19)

where the operator[] is defined by

[y](s) = (s, y(s), ahy(s), h

b y(s)) .

Remark 3.3. The Euler-Lagrange equation (19) coincides with (10) in the particular case h = 1.If the initial condition y(a) = A is not present (y(a) is free), one can prove that the followingsupplementary condition must be fulfilled:

hLv(a) +

(+ 1)

ba

(t + h a)(1)Lv [y](t)t

b(a)

(t + h (a))(1)Lv[y](t)t

+ Lw(a) = 0. (20)

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Similarly, ify(b) = B is not present (y(b) is free), the following equality

hLu((b)) + h

Lv((b)) h

Lw((b))

+

(+ 1)

ba

(b + h (t))(1)Lw[y](t)t

(b)a

((b) + h (t))(1)Lw[y](t)t

= 0, (21)

holds. The natural boundary conditions (20) and (21) reduce to (17) and (18) when h = 1.

4 EXAMPLES

In this section we present two illustrative examples. The results were obtained using the open

source Computer Algebra System Maxima (cf. http://maxima.sourceforge.net).

All computations were done running Maxima on an Intel CoreTM2 Duo, CPU of 2.27GHz

with 3Gb of RAM.

Example 4.1. Let us consider the following problem:

J(y) =

10

1

2(0

hy(t))

2 y(t)

t min , y(0) = 0 , y(1) = 0 . (22)

We begin by considering problem (22) with = 1 and different values of h. Our numericalresults show that when h tends to zero the extremal of the problem tends to the extremal of thecorresponding continuous classical problem of the calculus of variations. More precisely, when

h 0 problem (22) tends to the fractional continuous problem [3, Example 2] with solutiongiven by

y(t) =1

2t(1 t)

when = 1. This is illustrated in Figure 1. The computation times ranged from 0.02 seconds forh = 0.5 to 0.44 seconds forh = 0.0625.

Consider now problem (22) with h = 0.05 and different values of . Our numerical resultsshow that when tends to one our extremal tends to the classical (integer order) discrete-time

case. This is illustrated in Figure 2. The computation times ranged from 2.5 minutes forh = 0.7to 6.5 minutes forh = 0.99.

Example 4.2. Let us consider the following problem:

J(y) =

10

1

2(0

hy(t))

2 t min , y(0) = 0 , y(1) = 1 . (23)

We consider problem (23) with = 0.75 and different values of h. Our numerical resultsshow that when h tends to zero our extremal tends to the fractional continuous extremal. Indeed,

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0

0.05

0.10

0.15

0 1

+

+

+

++

+

+

+

+*

*

*

*

*

**

* * **

*

*

*

*

*

*

y(t)

t

Figure 1: Extremal y(t) for problem of Example 4.1 with = 1 and different hs (: h = 0.50; :

h = 0.25; +: h = 0.125; : h = 0.0625).

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 1

++

++

++

+ ++ + + + + + +

++

++

++*

*

**

**

** *

* * * * **

**

**

**

y(t)

t

Figure 2: Extremal y(t) for problem of Example 4.1 with h = 0.05 and differents (: = 0.70; :

= 0.75; +: = 0.95; : = 0.99).

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when h 0 problem (23) tends to the fractional continuous problem of the calculus of variations[1, Example 1], with solution given by

y(x) = 2

x

0

dt

[(1 t)(x t)]0.25 . (24)

This is illustrated in Figure 3. The computation times ranged from 10.46 seconds forh = 0.5 to25 minutes forh = 1/30.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1

+

+

+

+

+

+

+

+

+

*

*

*

*

*

*

*

*

*

*

*

*

**

*

*

*

y(t)

t

Figure 3: Extremal y(t) for problem of Example 4.2 with = 0.75 and differenths (: h = 0.50; +:h = 0.125; : h = 0.0625; : h = 1/30). The solid line represent (24).

ACKNOWLEDGEMENTS

This work is part of the first authors PhD project which is carried out at the University of

Aveiro under the Doctoral Programme Mathematics and Applications of Universities of Aveiro

and Minho. The second author was supported by The Portuguese Foundation for Science and

Technology (FCT) through the PhD fellowship SFRH/BD/39816/2007; the third author was sup-ported by FCT through the R&D unit Centre for Research on Optimization and Control (CEOC),

cofinanced by FEDER/POCI 2010.

REFERENCES

[1] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational prob-

lems, J. Math. Anal. Appl. 272 (1) (2002) 368379.

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15/15


[2] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control

problems, Nonlinear Dynam. 38 (1-4) (2004) 323337.

[3] O. P. Agrawal, A general finite element formulation for fractional variational problems, J.

Math. Anal. Appl. 337 (2008), no. 1, 112.

[4] F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J.

Difference Equ. 2 (2) (2007) 165176.

[5] F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc.

Amer. Math. Soc. 137 (3) (2009) 981989.

[6] M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl. 13 (3-4) (2004)

339349.

[7] R. A. El-Nabulsi and D. F. M. Torres, Necessary optimality conditions for fractionalaction-like integrals of variational calculus with Riemann-Liouville derivatives of order

(, ), Math. Meth. Appl. Sci. 30 (15) (2007), 19311939.

[8] R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales, in

Mathematical control theory and finance, Springer, Berlin, 2008, 149159.

[9] G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control

theory, Nonlinear Dynam. 53 (3) (2008) 215222.

[10] R. Hilscher and V. Zeidan, Nonnegativity and positivity of quadratic functionals in discrete

calculus of variations: survey, J. Difference Equ. Appl. 11 (2005), no. 9, 857875.

[11] W. G. Kelley and A. C. Peterson, Difference equations Academic Press, Boston, MA,

1991.

[12] K. S. Miller and B. Ross, Fractional difference calculus, in Univalent functions, fractional

calculus, and their applications (K oriyama, 1988), 139152, Horwood, Chichester, 1989.

[13] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional

Differential Equations John Wiley and Sons, Inc., New York, 1993.

[14] S. I. Muslih and D. Baleanu, Fractional Euler-Lagrange equations of motion in fractional

space, J. Vib. Control 13 (2007), no. 9-10, 12091216.

[15] M. D. Ortigueira, Fractional central differences and derivatives, J. Vib. Control 14 (2008),

no. 9-10, 12551266.

[16] B. Ross, S. G. Samko and E. R. Love, Functions that have no first order derivative might

have fractional derivatives of all orders less than one, Real Anal. Exchange 20 (1994/95),

no. 1, 140157.

[17] M. F. Silva, J. A. Tenreiro Machado and R. S. Barbosa, Using fractional derivatives in

joint control of hexapod robots, J. Vib. Control 14 (2008), no. 9-10, 14731485.

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