Fractional Calculus of Variations in Terms of a Generalized Fractional … · 2016-08-08 · Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 871912, 24 pagesdoi:10.1155/2012/871912

Research ArticleFractional Calculus of Variations inTerms of a Generalized Fractional Integral withApplications to Physics

Tatiana Odzijewicz,1 Agnieszka B. Malinowska,2and Delfim F. M. Torres1

1 Center for Research and Development in Mathematics and Applications, Department of Mathematics,University of Aveiro, 3810-193 Aveiro, Portugal

2 Faculty of Computer Science, Białystok University of Technology, 15-351 Białystok, Poland

Correspondence should be addressed to Delfim F. M. Torres, [email protected]

Received 1 January 2012; Revised 25 February 2012; Accepted 27 February 2012

Academic Editor: Muhammad Aslam Noor

Copyright q 2012 Tatiana Odzijewicz et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study fractional variational problems in terms of a generalized fractional integral withLagrangians depending on classical derivatives, generalized fractional integrals and derivatives.We obtain necessary optimality conditions for the basic and isoperimetric problems, as wellas natural boundary conditions for free-boundary value problems. The fractional action-likevariational approach (FALVA) is extended and some applications to physics discussed.

1. Introduction

The calculus of variations is a beautiful and useful field of mathematics that deals withproblems of determining extrema (maxima or minima) of functionals [1–3]. It starts with thesimplest problem of finding a function extremizing (minimizing or maximizing) an integral

J(y) =∫b

a

F(t, y(t), y′(t)

)dt (1.1)

subject to boundary conditions y(a) = ya and y(b) = yb. In the literature, many gen-eralizations of this problem were proposed, including problems with multiple integrals,functionals containing higher-order derivatives, and functionals depending on several

2 Abstract and Applied Analysis

functions [4–6]. Of our interest is an extension proposed by Riewe in 1996-1997, wherefractional derivatives (real or complex order) are introduced in the Lagrangian [7, 8].

During the last decade, fractional problems have increasingly attracted the attentionof many researchers. As mentioned in [9], Science Watch of Thomson Reuters identifiedthe subject as an Emerging Research Front area. Fractional derivatives are nonlocal operatorsand are historically applied in the study of nonlocal or time-dependent processes [10]. Thefirst and well-established application of fractional calculus in physics was in the frameworkof anomalous diffusion, which is related to features observed in many physical systems.Here we can mention the report [11] demonstrating that fractional equations work asa complementary tool in the description of anomalous transport processes. Within thefractional approach, it is possible to include external fields in a straightforward manner. Asa consequence, in a short period of time, the list of applications expanded. Applicationsinclude chaotic dynamics [12], material sciences [13], mechanics of fractal and complexmedia [14, 15], quantum mechanics [16, 17], physical kinetics [18], long-range dissipation[19], and long-range interaction [20, 21], just to mention a few. One of the most remarkableapplications of fractional calculus appears, however, in the fractional variational calculus,in the context of classical mechanics. Riewe [7, 8] shows that a Lagrangian involvingfractional time derivatives leads to an equation of motion with nonconservative forces suchas friction. It is a remarkable result since frictional and nonconservative forces are beyondthe usual macroscopic variational treatment and, consequently, beyond the most advancedmethods of classical mechanics [22]. Riewe generalizes the usual variational calculus, byconsidering Lagrangians that are dependent on fractional derivatives, in order to dealwith nonconservative forces. Recently, several different approaches have been developed togeneralize the least action principle and the Euler-Lagrange equations to include fractionalderivatives. Results include problems depending on Caputo fractional derivatives andRiemann-Liouville fractional derivatives [23–35].

A more general unifying perspective to the subject is, however, possible, by consid-ering fractional operators depending on general kernels [36–38]. In this work, we followsuch an approach, developing a generalized fractional calculus of variations. We considervery general problems, where the classical integrals are substituted by generalized fractionalintegrals, and the Lagrangians depend not only on classical derivatives but also on gener-alized fractional operators. Problems of the type considered here, for particular kernels, areimportant in physics [39]. Here, we obtain general necessary optimality conditions, for sev-eral types of variational problems, which are valid for rather arbitrary operators and kernels.By choosing particular operators and kernels, one obtains the recent results available in theliterature of mathematical physics [39–44].

The paper is organized as follows. In Section 2, we introduce the generalized fractionaloperators and prove some of their basic properties. Section 3 is dedicated to prove integrationby parts formulas for the generalized fractional operators. Such formulas are then used inlater sections to prove necessary optimality conditions (Theorems 4.2 and 6.3). In Sections4, 5, and 6 we study three important classes of generalized variational problems: weobtain fractional Euler-Lagrange conditions for the fundamental (Section 4) and generalizedisoperimetric problems (Section 6), as well as fractional natural boundary conditions forgeneralized free-boundary value problems (Section 5). Finally, two illustrative examplesare discussed in detail in Section 7, while applications to physics are given in Section 8: inSection 8.1, we obtain the damped harmonic oscillator in quantum mechanics; in Section 8.2,we show how results from FALVA physics can be obtained. We end with Section 9 ofconclusion, pointing out an important direction of future research.

Abstract and Applied Analysis 3

2. Preliminaries

In this section, we present definitions and properties of generalized fractional operators. Asparticular cases, by choosing appropriate kernels, these operators are reduced to standardfractional integrals and fractional derivatives. Other nonstandard kernels can also beconsidered as particular cases. For more on the subject of generalized fractional calculusand applications, we refer the reader to [37]. Throughout the text, α denotes a real numberbetween zero and one. Following [45], we use round brackets for the arguments of functionsand square brackets for the arguments of operators. By definition, an operator receives andreturns a function.

Definition 2.1 (generalized fractional integral). The operator KαP is given by

KαP

[f](x) = Kα

P

[t �−→ f(t)

](x) = p

∫x

a

kα(x, t)f(t)dt + q

∫b

x

kα(t, x)f(t)dt, (2.1)

where P = 〈a, x, b, p, q〉 is the parameter set (p-set for brevity), x ∈ [a, b], p, q are realnumbers, and kα(x, t) is a kernel which may depend on α. The operator Kα

P is referred toas the operator K (K-op for simplicity) of order α and p-set P , while Kα

P [f] is called theoperation K (or K-opn) of f of order α and p-set P .

Note that if we define

G(x, t) :=

⎧⎨

⎩

pkα(x, t), if t < x,

qkα(t, x), if t ≥ x,(2.2)

then the operator KαP can be written in the form

KαP

[f](x) = Kα

P

[t �−→ f(t)

](x) =

∫b

a

G(x, t)f(t)dt. (2.3)

This is a particular case of one of the oldest and most respectable class of operators, the so-called Fredholm operators [46, 47].

Theorem 2.2 (cf. Example 6 of [46]). Let α ∈ (0, 1) and P = 〈a, x, b, p, q〉. If kα is a squareintegrable function on the square Δ = [a, b] × [a, b], then Kα

P : L2([a, b]) → L2([a, b]) is well-defined, linear, and bounded operator.

Theorem 2.3. Let kα be a difference kernel, that is, let kα ∈ L1([a, b]) with kα(x, t) = kα(x − t).Then, Kα

P : L1([a, b]) → L1([a, b]) is a well defined bounded and linear operator.

Proof. Obviously, the operator is linear. Let α ∈ (0, 1), P = 〈a, t, b, p, q〉, and f ∈ L1([a, b]).Define

F(τ, t) :=

⎧⎨

⎩

∣∣pkα(t − τ)∣∣ · ∣∣f(τ)∣∣, if τ ≤ t,

∣∣qkα(τ − t)∣∣ · ∣∣f(τ)∣∣, if τ > t,

(2.4)


for all (τ, t) ∈ Δ = [a, b] × [a, b]. Since F is measurable on the square Δ, we have

∫b

a

(∫b

a

F(τ, t)dt

)

dτ =∫b

a

[∣∣f(τ)

∣∣(∫b

τ

∣∣pkα(t − τ)

∣∣dt +

∫ τ

a

∣∣qkα(τ − t)

∣∣dt

)]

dτ

≤∫b

a

∣∣f(τ)

∣∣∣∣∣∣p∣∣ − ∣∣q∣∣∣∣‖kα‖dτ

=∣∣∣∣p∣∣ − ∣∣q∣∣∣∣ · ‖kα‖ ·

∥∥f∥∥.

(2.5)

It follows from Fubini’s theorem that F is integrable on the square Δ. Moreover,

∥∥Kα

P

[f]∥∥ =

∫b

a

∣∣∣∣∣p

∫ t

a

kα(t − τ)f(τ)dτ + q

∫b

t

kα(τ − t)f(τ)dτ

∣∣∣∣∣dt

≤∫b

a

(∣∣p∣∣∫ t

a

|kα(t − τ)| · ∣∣f(τ)∣∣dτ +∣∣q∣∣∫b

t

|kα(τ − t)| · ∣∣f(τ)∣∣dτ)

dt

=∫b

a

(∫b

a

F(τ, t)dτ

)

dt

≤ ∣∣∣∣p∣∣ − ∣∣q∣∣∣∣ · ‖kα‖ ·∥∥f∥∥.

(2.6)

Hence, KαP : L1([a, b]) → L1([a, b]) and ‖Kα

P‖ ≤ ||p| − |q|| · ‖kα‖.

Remark 2.4. The K-op reduces to the left and the right Riemann-Liouville fractional integralsfrom a suitably chosen kernel kα(x, t) and p-set P . Let kα(x, t) = kα(x−t) = (1/Γ(α))(x−t)α−1:

(i) if P = 〈a, x, b, 1, 0〉, then

KαP

[f](x) =

1Γ(α)

∫x

a

(x − t)α−1f(t)dt =: aIαx

[f](x) (2.7)

is the standard left Riemann-Liouville fractional integral of f of order α;

(ii) if P = 〈a, x, b, 0, 1〉, then

KαP

[f](x) =

1Γ(α)

∫b

x

(t − x)α−1f(t)dt =: xIαb

[f](x) (2.8)

is the standard right Riemann-Liouville fractional integral of f of order α.

Corollary 2.5. Operators aIαx , xI

αb: L1([a, b]) → L1([a, b]) are well defined, linear and bounded.

The generalized fractional derivatives AαP and Bα

P are defined in terms of the gener-alized fractional integral K-op.


Definition 2.6 (generalized Riemann-Liouville fractional derivative). Let P be a givenparameter set and 0 < α < 1. The operator Aα

P is defined by AαP = D ◦K1−α

P , where D denotesthe standard derivative operator, and is referred to as the operator A (A-op) of order α andp-set P , while Aα

P [f], for a function f such that K1−αP [f] ∈ AC([a, b]), is called the operation

A (A-opn) of f of order α and p-set P .

Definition 2.7 (generalized Caputo fractional derivative). Let P be a given parameter set andα ∈ (0, 1). The operator Bα

P is defined by BαP = K1−α

P ◦ D, where D denotes the standardderivative operator, and is referred to as the operator B (B-op) of order α and p-set P , whileBαP [f], for a function f ∈ AC([a, b]), is called the operation B (B-opn) of f of order α and

p-set P .

Remark 2.8. The standard Riemann-Liouville and Caputo fractional derivatives are easilyobtained from the generalized operators Aα

P and BαP , respectively. Let k1−α(x, t) = k1−α(x−t) =

(x − t)−α/Γ(1 − α):

(i) if P = 〈a, x, b, 1, 0〉, then

AαP

[f](x) =

1Γ(1 − α)

D

[

ξ �−→∫ ξ

a

(ξ − t)−αf(t)dt

]

(x) =:aDαx

[f](x) (2.9)

is the standard left Riemann-Liouville fractional derivative of f of order α, while

BαP

[f](x) =

1Γ(1 − α)

∫x

a

(x − t)−αD[f](t)dt =:CaD

αx

[f](x) (2.10)

is the standard left Caputo fractional derivative of f of order α;

(ii) if P = 〈a, x, b, 0, 1〉, then

−AαP

[f](x) =

−1Γ(1 − α)

D

[

ξ �−→∫b

ξ

(t − ξ)−αf(t)dt

]

(x) =:xDαb

[f](x) (2.11)

is the standard right Riemann-Liouville fractional derivative of f of order α, while

−BαP

[f](x) =

−1Γ(1 − α)

∫b

x

(t − x)−αD[f](t)dt =:CxD

αb

[f](x) (2.12)

is the standard right Caputo fractional derivative of f of order α.

3. On Generalized Fractional Integration by Parts

We now prove integration by parts formulas for generalized fractional operators.


Theorem 3.1 (fractional integration by parts for the K-op). Let α ∈ (0, 1), P = 〈a, t, b, p, q〉, kαbe a square-integrable function onΔ = [a, b]×[a, b], and f, g ∈ L2([a, b]). The generalized fractionalintegral Kα

P satisfies the integration by parts formula

∫b

a

g(x)KαP

[f](x)dx =

∫b

a

f(x)KαP ∗[g](x)dx, (3.1)

where P ∗ = 〈a, t, b, q, p〉.

Proof. Define

F(τ, t) :=

⎧⎪⎨

⎪⎩

∣∣pkα(t, τ)

∣∣ · ∣∣g(t)∣∣ · ∣∣f(τ)∣∣, if τ ≤ t,

∣∣qkα(τ, t)∣∣ · ∣∣g(t)∣∣ · ∣∣f(τ)∣∣, if τ > t,

(3.2)

for all (τ, t) ∈ Δ. Applying Holder’s inequality, we obtain

∫b

a

(∫b

a

F(τ, t)dt

)

dτ =∫b

a

[∣∣f(τ)

∣∣(∫b

τ

∣∣pkα(t, τ)∣∣ · ∣∣g(t)∣∣dt +

∫ τ

a

∣∣qkα(τ, t)∣∣ · ∣∣g(t)∣∣dt

)]

dτ

≤∫b

a

[∣∣f(τ)

∣∣(∫b

a

∣∣pkα(t, τ)∣∣ · ∣∣g(t)∣∣dt +

∫b

a

∣∣qkα(τ, t)∣∣ · ∣∣g(t)∣∣dt

)]

dτ

≤∫b

a

⎧⎨

⎩

∣∣f(τ)∣∣

⎡

⎣

(∫b

a

∣∣pkα(t, τ)∣∣2dt

)1/2(∫b

a

∣∣g(t)∣∣2dt

)1/2

+

(∫b

a

∣∣qkα(τ, t)∣∣2dt

)1/2(∫b

a

∣∣g(t)∣∣2dt

)1/2⎤

⎦

⎫⎬

⎭dτ.

(3.3)

By Fubini’s theorem, functions kα,τ(t) := kα(t, τ) and kα,τ(t) := kα(τ, t) belong to L2([a, b]) foralmost all τ ∈ [a, b]. Therefore,

∫b

a

⎧⎨

⎩

∣∣f(τ)∣∣

⎡

⎣

(∫b

a

∣∣pkα(t, τ)∣∣2dt

)1/2(∫b

a

∣∣g(t)∣∣2dt

)1/2

+

(∫b

a

∣∣qkα(τ, t)∣∣2dt

)1/2(∫b

a

∣∣g(t)∣∣2dt

)1/2⎤

⎦

⎫⎬

⎭dτ

=∥∥g∥∥2

∫b

a

[∣∣f(τ)∣∣(∥∥pkα,τ

∥∥2 +∥∥∥qkα,τ

∥∥∥2

)]dτ


≤ ∥∥g∥∥2(∫b

a

∣∣f(τ)

∣∣2dτ

)1/2(∫b

a

∣∣∣∥∥pkα,τ

∥∥2 +∥∥∥qkα,τ

∥∥∥2

∣∣∣2dτ

)1/2

≤ ∥∥g∥∥2 ·∥∥f∥∥2

(∥∥pkα∥∥2 +∥∥qkα

∥∥2

)< ∞.

(3.4)

Hence, we can use again Fubini’s theorem to change the order of integration:

∫b

a

g(t)KαP

[f](t)dt = p

∫b

a

g(t)dt∫ t

a

f(τ)kα(t, τ)dτ + q

∫b

a

g(t)dt∫b

t

f(τ)kα(τ, t)dτ

= p

∫b

a

f(τ)dτ∫b

τ

g(t)kα(t, τ)dt + q

∫b

a

f(τ)dτ∫ τ

a

g(t)kα(τ, t)dt

=∫b

a

f(τ)KαP ∗[g](τ)dτ.

(3.5)

Theorem 3.2. Let 0 < α < 1 and P = 〈a, x, b, p, q〉. If kα(x, t) = kα(x − t), kα, f ∈ L1([a, b]), andg ∈ C([a, b]), then the operator Kα

P satisfies the integration by parts formula (3.1).

Proof. Define

F(t, x) :=

⎧⎪⎨

⎪⎩

∣∣pkα(x − t)∣∣ · ∣∣g(x)∣∣ · ∣∣f(t)∣∣, if t ≤ x,

∣∣qkα(t − x)∣∣ · ∣∣g(x)∣∣ · ∣∣f(t)∣∣, if t > x,

(3.6)

for all (t, x) ∈ Δ = [a, b] × [a, b]. Since g is a continuous function on [a, b], it is bounded on[a, b], that is, there exists a real number C > 0 such that |g(x)| ≤ C for all x ∈ [a, b]. Therefore,

∫b

a

(∫b

a

F(t, x)dt

)

dx =∫b

a

[∣∣f(t)

∣∣(∫b

t

∣∣pkα(x − t)∣∣ · ∣∣g(x)∣∣dx +

∫ t

a

∣∣qkα(t − x)∣∣ · ∣∣g(x)∣∣dx

)]

dt

≤∫b

a

[∣∣f(t)

∣∣(∫b

a

∣∣pkα(x − t)∣∣ · ∣∣g(x)∣∣dx +

∫b

a

∣∣qkα(t − x)∣∣ · ∣∣g(x)∣∣dx

)]

dt

≤ C

∫b

a

[∣∣f(t)

∣∣(∫b

a

∣∣pkα(x − t)∣∣dx +

∫b

a

∣∣qkα(t − x)∣∣dx

)]

dt

= C∣∣∣∣p∣∣ − ∣∣q∣∣∣∣‖kα‖

∥∥f∥∥ < ∞.

(3.7)

Hence, we can use Fubini’s theorem to change the order of integration in iterated integrals.


Theorem 3.3 (generalized fractional integration by parts). Let α ∈ (0, 1) and P = 〈a, t, b, p, q〉.If functions f, K1−α

P ∗ [g] ∈ AC([a, b]), and we are in conditions to use formula (3.1) (Theorem 3.1 orTheorem 3.2), then

∫b

a

g(x)BαP

[f](x)dx = f(x)K1−α

P ∗[g](x)∣∣∣b

a−∫b

a

f(x)AαP ∗[g](x)dx, (3.8)

where P ∗ = 〈a, t, b, q, p〉.

Proof. From Definition 2.7, we know that BαP [f](x) = K1−α

P [D[f]](x). It follows that

∫b

a

g(x)BαP

[f](x)dx =

∫b

a

g(x)K1−αP

[D[f]](x)dx. (3.9)

By relation (3.1),

∫b

a

g(x)BαP

[f](x)dx =

∫b

a

D[f](x)K1−α

P ∗[g](x)dx, (3.10)

and the standard integration by parts formula implies (3.8):

∫b

a

g(x)BαP

[f](x)dx = f(x)K1−α

P ∗ [g](x)∣∣∣b

a−∫b

a

f(x)D[K1−α

P ∗[g]](x)dx. (3.11)

Corollary 3.4 (cf. [48]). Let 0 < α < 1. If f,xI1−αb [g] ∈ AC([a, b]), then

∫b

a

g(x)CaDαx

[f](x)dx = f(x)xI

1−αb

[g](x)∣∣∣x=b

x=a+∫b

a

f(x)xDαb

[g](x)dx. (3.12)

4. The Generalized Fundamental Variational Problem

By ∂iF, we denote the partial derivative of a function F with respect to its ith argument. Weconsider the problem of finding a function y = t �→ y(t), t ∈ [a, b], that gives an extremum(minimum or maximum) to the functional

J(y) = KαP1

[t �−→ F

(t, y(t), y′(t), Bβ

P2

[y](t), Kγ

P3

[y](t))]

(b) (4.1)

subject to the boundary conditions

y(a) = ya, y(b) = yb, (4.2)


where α, β, γ ∈ (0, 1), P1 = 〈a, b, b, 1, 0〉, and Pj = 〈a, t, b, pj , qj〉, j = 2, 3. For simplicity of

notation, we introduce the operator {·}β,γP2,P3defined by

{y}β,γP2,P3

(t) =(t, y(t), y′(t), Bβ

P2

[τ �−→ y(τ)

](t), Kγ

P3

[τ �−→ y(τ)

](t)). (4.3)

With the new notation, one can write (4.1) simply as J(y) = KαP1[F{y}β,γP2,P3

](b). The operator

KαP1

has kernel kα(x, t), and operators Bβ

P2and K

γ

P3have kernels h1−β(t, τ) and hγ(t, τ),

respectively. In the sequel, we assume the following:

(H1) Lagrangian F ∈ C1([a, b] × R4;R),

(H2) functions Aβ

P ∗2[τ �→ kα(b, τ)∂4F{y}β,γP2,P3

(τ)], Kγ

P ∗3[τ �→ kα(b, τ)∂5F{y}β,γP2,P3

(τ)], D[t �→∂3F{y}β,γP2,P3

(t)kα(b, t)], and t �→ kα(b, t)∂2F{y}β,γP2,P3(t) are continuous on (a, b),

(H3) functions t �→ ∂3F{y}β,γP2,P3(t)kα(b, t), K

1−βP ∗2[τ �→ kα(b, τ)∂4F{y}β,γP2,P3

(τ)] ∈ AC([a, b]);

(H4) kernels kα(x, t), h1−β(t, τ), and hγ(t, τ) are such that we are in conditions to useTheorems 3.1 or 3.2, and Theorem 3.3.

Definition 4.1. A function y ∈ C1([a, b];R) is said to be admissible for the fractional variationalproblem (4.1)-(4.2), if functions Bβ

P2[y] and K

γ

P3[y] exist and are continuous on the interval [a, b],

and y satisfies the given boundary conditions (4.2).

Theorem 4.2. If y is a solution to problem (4.1)-(4.2), then y satisfies the generalized Euler-Lagrangeequation

kα(b, t)∂2F{y}β,γP2,P3

(t) − d

dt

(∂3F

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4F

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5F

{y}β,γP2,P3

(τ)](t) = 0

(4.4)

for all t ∈ (a, b).

Proof. Suppose that y is an extremizer of J. Consider the value of J at a nearby functiony = y + εη, where ε ∈ R is a small parameter, and η ∈ C1([a, b];R) is an arbitrary functionwith continuous B-op and K-op. We require that η(a) = η(b) = 0. Let

J(y) = J(ε) = KαP1

[t �−→ F

(t, y(t), y′(t), Bβ

P2

[y](t), Kγ

P3

[y](t))]

(b)

=∫b

a

kα(b, t)F(t, y(t) + εη(t),

d

dt

(y(t) + εη(t)

), B

β

P2

[y + εη

](t), Kγ

P3

[y + εη

](t))dt.

(4.5)


A necessary condition for y to be an extremizer is given by

dJ

dε

∣∣∣∣ε=0

= 0 ⇐⇒ KαP1

[∂2F

{y}β,γP2,P3

η + ∂3F{y}β,γP2,P3

D[η]

+ ∂4F{y}β,γP2,P3

Bβ

P2

[η]+ ∂5F

{y}β,γP2,P3

Kγ

P3

[η]](b) = 0

⇐⇒∫b

a

(∂2F

{y}β,γP2,P3

(t)η(t) + ∂3F{y}β,γP2,P3

(t)d

dtη(t)

+ ∂4F{y}β,γP2,P3

(t)Bβ

P2

[η](t) + ∂5F

{y}β,γP2,P3

(t)Kγ

P3

[η](t))kα(b, t)dt = 0.

(4.6)

Using classical and generalized fractional integration by parts formulas (Theorems 3.1 or 3.2,and Theorem 3.3),

∫b

a

∂3F{y}β,γP2,P3

(t)kα(b, t)d

dtη(t)dt = ∂3F

{y}β,γP2,P3

(t)kα(b, t)η(t)∣∣∣b

a

−∫b

a

d

dt

(∂3F

{y}β,γP2,P3

(t)kα(b, t))η(t)dt,

∫b

a

∂4F{y}β,γP2,P3

(t)kα(b, t)Bβ

P2

[η](t)dt = K

1−βP ∗2

[τ �−→ kα(b, τ)∂4F

{y}β,γP2,P3

(τ)](t)η(t)

∣∣∣b

a

−∫b

a

Aβ

P ∗2

[τ �−→ kα(b, τ)∂4F

{y}β,γP2,P3

(τ)](t)η(t)dt,

∫b

a


(t)Kγ

P3

[η](t)dt =

∫b

a

Kγ

P ∗3

[τ �−→ kα(b, τ)∂5F

{y}β,γP2,P3

(τ)](t)η(t)dt,

(4.7)

where P ∗j = 〈a, t, b, qj , pj〉, j = 2, 3. Because η(a) = η(b) = 0, (4.6) simplifies to

∫b

a

{kα(b, t)∂2F

{y}β,γP2,P3

(t) − d

dt

(∂3F

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4F

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5F

{y}β,γP2,P3

(τ)](t)}η(t)dt = 0.

(4.8)

We obtain (4.4) by application of the fundamental lemma of the calculus of variations (see,e.g., [49, Section 2.2]).

The following corollary gives an extension of the main result of [28].

Corollary 4.3. If y is a solution to the problem of minimizing or maximizing

J(y)=aIαb

[t �−→ F

(t, y(t), y′(t), C

aDβt

[y](t))]

(b) (4.9)


in the class y ∈ C1([a, b];R) subject to the boundary conditions

y(a) = ya, y(b) = yb, (4.10)

where α, β ∈ (0, 1), F ∈ C1([a, b] × R3;R), and τ �→ (b − τ)α−1∂4F(τ, y(τ), y′(τ), C

aDβτ [y](τ)) has

continuous Riemann-Liouville fractional derivative tDβ

b, then

∂2F(t, y(t), y′(t), C

aDβt

[y](t))· (b − t)α−1 − d

dt

{∂3F

(t, y(t), y′(t), C

aDβt

[y](t))· (b − t)α−1

}

+ tDβ

b

[τ �−→ (b − τ)α−1∂4F

(τ, y(τ), y′(τ), C

aDβτ

[y](τ))]

(t) = 0

(4.11)

for all t ∈ (a, b).

Proof. Choose kα(x, t) = (1/Γ(α))(x − t)α−1, h1−β(t, τ) = (1/Γ(1 − β))(t − τ)−β, and P2 =〈a, t, b, 1, 0〉. Then, the K-op, the A-op, and the B-op reduce to the left fractional integral,the left Riemann-Liouville, and the left Caputo fractional derivatives, respectively. Therefore,problem (4.9)-(4.10) is a particular case of problem (4.1)-(4.2) and (4.11) follows from (4.4)with ∂5F = 0.

The following result is the Caputo analogous to the main result of [50] done for theRiemann-Liouville fractional derivative.

Corollary 4.4. Let β, γ ∈ (0, 1). If y is a solution to the problem

∫b

a

F(t, y(t), y′(t),CaD

βt

[y](t),aI

γt

[y](t))dt −→ extr,

y ∈ C1([a, b];R),

y(a) = ya, y(b) = yb,

(4.12)

then

∂2F(t, y(t), y′(t),CaD

βt

[y](t),aI

γt

[y](t))− d

dt∂3F

(t, y(t), y′(t),CaD

βt

[y](t),aI

γt

[y](t))

+tD

β

b

[τ �−→ ∂4F

(τ, y(τ), y′(τ),CaD

βτ

[y](τ),aI

γτ

[y](τ))]

(t)

+tIβ

b

[τ �−→ ∂5F

(τ, y(τ), y′(τ),CaD

βτ

[y](τ),aI

γτ

[y](τ))]

(t) = 0

(4.13)

holds for all t ∈ (a, b).

Proof. The Euler-Lagrange equation (4.13) follows from (4.4) by choosing p-sets P1 =〈a, b, b, 1, 0〉, P2 = P3 = 〈a, t, b, 1, 0〉, and kernels kα(x, t) = 1, h1−β(t, τ) = (1/Γ(1 − β))(t − τ)−β,and hγ(t, τ) = (1/Γ(γ))(t − τ)γ−1.


Remark 4.5. In the particular case when the Lagrangian F of Corollary 4.4 does not depend onthe fractional integral and the classical derivative, one obtains from (4.13) the Euler-Lagrangeequation of [51].

5. Generalized Free-Boundary Variational Problems

Assume now that, in problem (4.1)-(4.2), the boundary conditions (4.2) are substituted by

y(a) is free and y(b) = yb. (5.1)

Theorem 5.1. If y is a solution to the problem of extremizing functional (4.1) with (5.1) as boundaryconditions, then y satisfies the Euler-Lagrange equation (4.4). Moreover, the extra natural boundarycondition

∂3F{y}β,γP2,P3

(a)kα(b, a) +K1−βP ∗2

[τ �−→ ∂4F

{y}β,γP2,P3

(τ)kα(b, τ)](a) = 0 (5.2)

holds.

Proof. Under the boundary conditions (5.1), we do not require η in the proof of Theorem 4.2to vanish at t = a. Therefore, following the proof of Theorem 4.2, we obtain

∂3F{y}β,γP2,P3

(a)kα(b, a)η(a) + η(a)K1−βP ∗2

[τ �−→ ∂4F

{y}β,γP2,P3

(τ)kα(b, τ)](a)

+∫b

a

η(t)(∂2F

{y}β,γP2,P3

(t)kα(b, t) − d

dt

(∂3F

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ ∂4F

{y}β,γP2,P3

(τ)kα(b, τ)](t) +K

γ

P ∗3

[τ �−→ ∂5F

{y}β,γP2,P3

(τ)kα(b, τ)](t))dt = 0

(5.3)

for every admissible η ∈ C1([a, b];R) with η(b) = 0. In particular, condition (5.3) holds forthose η that fulfill η(a) = 0. Hence, by the fundamental lemma of the calculus of variations,(4.4) is satisfied. Now, let us return to (5.3) and let η again be arbitrary at point t = a. Inserting(4.4), we obtain the natural boundary condition (5.2).

Corollary 5.2. Let J be the functional given by

J(y)=aIαb

[t �−→ F

(t, y(t),CaD

βt

[y](t))]

(b). (5.4)

Let y be a minimizer of J satisfying the boundary condition y(b) = yb. Then, y satisfies the Euler-Lagrange equation

(b − t)α−1∂2F(t, y(t),CaD

αt

[y](t))+

tDα

b

[τ �−→ (b − τ)α−1∂3F

(τ, y(τ),CaD

βτy(τ)

)](t) = 0

(5.5)


and the natural boundary condition

aI1−βb

[τ �−→ (b − τ)α−1∂3F

(τ, y(τ),CaD

βτy(τ)

)](a) = 0. (5.6)

Proof. Let functional (4.1) be such that it does not depend on the classical (integer) derivativey′(t) and on the K-op. If P2 = 〈a, t, b, 1, 0〉, h1−β(t − τ) = (1/Γ(1 − β))(t − τ)−β, and kα(x −t) = (1/Γ(α))(x − t)α−1, then the B-op reduces to the left fractional Caputo derivative and wededuce (5.5) and (5.6) from (4.4) and (5.2), respectively.

Corollary 5.3 (cf. Theorem 3.17 of [38]). Let J be the functional given by

J(y) =∫b

a

F(t, y(t), y′(t), Bβ

P2

[y](t), Kγ

P3

[y](t))dt. (5.7)

If y is a minimizer to J satisfying the boundary condition y(b) = yb, then y satisfies the Euler-Lagrange equation

∂2F{y}β,γP2,P3

(t) − d

dt∂3F

{y}β,γP2,P3

(t) −Aβ

P ∗2

[∂4F

{y}β,γP2,P3

](t) +K

γ

P ∗3

[∂5F

{y}β,γP2,P3

](t) = 0 (5.8)

and the natural boundary condition

∂3F{y}β,γP2,P3

(a) +K1−βP ∗2

[∂4F

{y}β,γP2,P3

](a) = 0. (5.9)

Proof. Choose, in the problem defined by (4.1) and (5.1), kα(x, t) ≡ 1. Then, (5.8) and (5.9)follow from (4.4) and (5.2), respectively.

6. Generalized Isoperimetric Problems

Let ξ ∈ R. Among all functions y : [a, b] → R satisfying boundary conditions

y(a) = ya, y(b) = yb, (6.1)

and an isoperimetric constraint of the form

I(y) = KαP1

[G{y}β,γP2,P3

](b) = ξ, (6.2)

we look for the one that extremizes (i.e., minimizes or maximizes) a functional

J(y) = KαP1

[F{y}β,γP2,P3

](b). (6.3)

Operators KαP1, Bβ

P2, and K

γ

P3, as well as function F, are the same as in problem (4.1)-(4.2).

Moreover, we assume that functional (6.2) satisfies hypotheses (H1)–(H4).


Definition 6.1. A function y : [a, b] → R is said to be admissible for problem (6.1)–(6.3)if functions B

β

P2[y] and K

γ

P3[y] exist and are continuous on [a, b], and y satisfies the given

boundary conditions (6.1) and the given isoperimetric constraint (6.2).

Definition 6.2. An admissible function y ∈ C1([a, b],R) is said to be an extremal for I if itsatisfies the Euler-Lagrange equation (4.4) associated with functional in (6.2), that is,

kα(b, t)∂2G{y}β,γP2,P3

(t) − d

dt

(∂3G

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4G

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5G

{y}β,γP2,P3

(τ)](t) = 0,

(6.4)

where P ∗j = 〈a, t, b, qj , pj〉, j = 2, 3, and t ∈ (a, b).

Theorem 6.3. If y is a solution to the isoperimetric problem (6.1)–(6.3) and is not an extremal for I,then there exists a real constant λ such that

kα(b, t)∂2H{y}β,γP2,P3

(t) − d

dt

(∂3H

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4H

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5H

{y}β,γP2,P3

(τ)](t) = 0

(6.5)

for all t ∈ (a, b), whereH(t, y, u, v,w) = F(t, y, u, v,w)−λG(t, y, u, v,w) and P ∗j = 〈a, t, b, qj , pj〉,

j = 2, 3.

Proof. Consider a two-parameter family of the form y = y + ε1η1 + ε2η2, where, for eachi ∈ {1, 2}, we have ηi(a) = ηi(b) = 0. First, we show that we can select ε2η2 such that y satisfies(6.2). Consider the quantity I(y) = Kα

P1[G{y}β,γP2,P3

](b). Looking to I(y) as a function of ε1, ε2,

we define I(ε1, ε2) = I(y) − ξ. Thus, I(0, 0) = 0. On the other hand, applying integration byparts formulas (Theorems 3.1 or 3.2, and Theorem 3.3), we obtain that

∂I

∂ε2

∣∣∣∣∣(0,0)

=∫b

a

η2(t)(kα(b, t)∂2G

{y}β,γP2,P3

(t) − d

dt

(∂3G

{y}β,γP2,P3

(t)kα(b, t))

− Aβ

P ∗2

[τ �−→ kα(b, τ)∂4G

{y}β,γP2,P3

(τ)](t)

+Kγ

P ∗3

[τ �−→ kα(b, τ)∂5G

{y}β,γP2,P3

(τ)](t))dt,

(6.6)

where P ∗j = 〈a, t, b, qj , pj〉, j = 2, 3. We assume that y is not an extremal for I. Hence,

the fundamental lemma of the calculus of variations implies that there exists a function η2such that ∂I/∂ε2|(0,0) /= 0. According to the implicit function theorem, there exists a functionε2(·) defined in a neighborhood of 0 such that I(ε1, ε2(ε1)) = 0. Let J(ε1, ε2) = J(y).Function J has an extremum at (0, 0) subject to I(0, 0) = 0, and we have proved that


∇I(0, 0)/= 0. The Lagrange multiplier rule asserts that there exists a real number λ such that∇(J(0, 0) − λI(0, 0)) = 0. Because

∂J

∂ε1

∣∣∣∣∣(0,0)

=∫b

a

(kα(b, t)∂2F

{y}β,γP2,P3

(t) − d

dt

(∂3F

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4F

{y}β,γP2,P3

(τ)](t)

+Kγ

P ∗3

[τ �−→ kα(b, τ)∂5F

{y}β,γP2,P3

(τ)](t))η1(t)dt,

∂I

∂ε1

∣∣∣∣∣(0,0)

=∫b

a

(kα(b, t)∂2G

{y}β,γP2,P3

(t) − d

dt

(∂3G

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4G

{y}β,γP2,P3

(τ)](t)

+Kγ

P ∗3

[τ �−→ kα(b, τ)∂5G

{y}β,γP2,P3

(τ)](t))η1(t)dt,

(6.7)

one has

∫b

a

{kα(b, t)∂2F

{y}β,γP2,P3

(t) − d

dt

(∂3F

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4F

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5F

{y}β,γP2,P3

(τ)](t)

− λ

(kα(b, t)∂2G

{y}β,γP2,P3

(t) − d

dt

(∂3G

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4G

{y}β,γP2,P3

(τ)](t)

+Kγ

P ∗3

[τ �−→ kα(b, τ)∂5G

{y}β,γP2,P3

(τ)](t))}

η1(t)dt = 0.

(6.8)

From the fundamental lemma of the calculus of variations (see, e.g., [49], Section 2.2), itfollows


(t) − d

dt

(∂3F

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4F

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5F

{y}β,γP2,P3

(τ)](t)

− λ

(kα(b, t)∂2G

{y}β,γP2,P3

(t) − d

dt

(∂3G

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4G

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5G

{y}β,γP2,P3

(τ)](t))

= 0,

(6.9)


that is,

kα(b, t)∂2H{y}β,γP2,P3

(t) − d

dt

(∂3H

{y}β,γP2,P3

(t)kα(b, t))

−Aβ

P ∗2

[τ �−→ kα(b, τ)∂4H

{y}β,γP2,P3

(τ)](t) +K

γ

P ∗3

[τ �−→ kα(b, τ)∂5H

{y}β,γP2,P3

(τ)](t) = 0

(6.10)

withH = F − λG.

Corollary 6.4. Let y be a minimizer to the isoperimetric problem

J(y)=aIαb

[t �−→ F

(t, y(t),CaD

βt

[y](t))]

(b) −→ min,

I(y)=aIαb

[t �−→ G

(t, y(t),CaD

βt

[y](t))]

(b) = ξ,

y(a) = ya, y(b) = yb.

(6.11)

If y is not an extremal of I, then there exists a constant λ such that y satisfies

(b − t)α−1∂2H(t, y(t),CaD

αt

[y](t))+ tD

β

b

[τ �−→ (b − τ)α−1∂3H

(τ, y(τ),CaD

βτ

[y](τ))]

(t) = 0

(6.12)

for all t ∈ (a, b), whereH(t, y, v) = F(t, y, v) − λG(t, y, v).

Proof. Let kα(x, t) = (1/Γ(α))(x − t)α−1, h1−β(t, τ) = (1/Γ(1−β))(t−τ)−β, P1 = 〈a, b, b, 1, 0〉, andP2 = 〈a, t, b, 1, 0〉. Then, the K-op and the B-op reduce to the left fractional integral and theleft fractional Caputo derivative, respectively. Therefore, problem (6.11) is a particular caseof problem (6.1)–(6.3), and (6.12) follows from (6.5)with ∂3H = ∂5H = 0.

Corollary 6.5 (cf. Theorem 3.22 of [38]). Let y be a minimizer to

J(y) =∫b

a

F(t, y(t), y′(t), Bβ

P2

[y](t), Kγ

P3

[y](t))dt −→ min,

I(y) =∫b

a

G(t, y(t), y′(t), Bβ

P2

[y](t), Kγ

P3

[y](t))dt = ξ,

y(a) = ya, y(b) = yb.

(6.13)

If y is not an extremal of I, then there exists a constant λ such that y satisfies

∂2H{y}β,γP2,P3

(t) − d

dt∂3H

{y}β,γP2,P3

(t) −Aβ

P ∗2

[∂4H

{y}β,γP2,P3

](t) +K

γ

P ∗3

[∂5H

{y}β,γP2,P3

](t) = 0

(6.14)

for all t ∈ (a, b), whereH(t, y, u, v,w) = F(t, y, u, v,w) − λG(t, y, u, v,w).


Proof. Let, in problem (6.1)–(6.3), P1 = 〈a, b, b, 1, 0〉 and kernel kα(x, t) ≡ 1. Then, thegeneralized fractional integral Kα

P1becomes the classical integral and (6.14) follows from

(6.5).

7. Illustrative Examples

We illustrate our results through two examples with different kernels: one of a funda-mental problem (4.1)–(4.2) (Example 7.1), the other an isoperimetric problem (6.1)–(6.3)(Example 7.2).

Example 7.1. Let α, β ∈ (0, 1), ξ ∈ R, P1 = 〈0, 1, 1, 1, 0〉, and P2 = 〈0, t, 1, 1, 0〉. Consider thefollowing problem:

J(y) = KαP1

[

t �−→ tKβ

P2

[y](t) +

√

1 −(K

β

P2

[y](t))2]

(1) −→ min,

y(0) = 1, y(1) =√24

+∫1

0rβ(1 − τ)

1

(1 + τ2)3/2dτ,

(7.1)

with kernel hβ such that hβ(t, τ) = hβ(t − τ) and hβ(0) = 1. Here, the resolvent rβ(t) is relatedto the kernel hβ(t) by rβ(t) = L−1[s �→ (1/shβ(s)) − 1](t), hβ(s) = L[t �→ hβ(t)](s), where Land L−1 are the direct and the inverse Laplace operators, respectively. We apply Theorem 4.2with Lagrangian F given by F(t, y, u, v,w) = tw +

√1 −w2. Because

y(t) =1

(1 + t2)3/2+∫ t

0rβ(t − τ)

1

(1 + τ2)3/2dτ (7.2)

is the solution to the Volterra integral equation of first kind (see, e.g., Equation 16, page 114of [47])

Kβ

P2

[y](t) =

t√1 + t2

1 + t2, (7.3)

it satisfies our generalized Euler-Lagrange equation (4.4), that is,

Kβ

P ∗2

⎡

⎢⎢⎣τ �−→ kα(b, τ)

⎛

⎜⎜⎝

−Kβ

P ∗2

[y](τ)

√

1 −(K

β

P ∗2

[y](τ))2

+ τ

⎞

⎟⎟⎠

⎤

⎥⎥⎦(t) = 0. (7.4)

In particular, for the kernel hβ(t − τ) = cosh(β(t − τ)), the boundary conditions are y(0) = 1

and y(1) = 1 + β2(1 − √2), and the solution is y(t) = (1/(1 + t2)3/2) + β2(1 −

√1 + t2) (cf. [47],

page 22).


In the next example, we make use of the Mittag-Leffler function of two parameters: ifα, β > 0, then the Mittag-Leffler function is defined by

Eα,β(z) =∞∑

k=0

zk

Γ(αk + β

) . (7.5)

This function appears naturally in the solution of fractional differential equations, as ageneralization of the exponential function [52].

Example 7.2. Let α, β ∈ (0, 1), ξ ∈ R, and ξ /∈ {±1/4}. Consider the following problem:

J(y) = 0Iα1

[√

1 +(y′ + C

0Dβt

[y])2]

(1) −→ min,

I(y)=0Iα1

[(y′ + C

0Dβt

[y])2](1) = ξ,

y(0) = 0, y(1) =∫1

0E1−β,1

(−(1 − τ)1−β

)√1 − 16ξ2

4ξdτ,

(7.6)

which is an example of (6.1)–(6.3) with p-sets P1 = 〈0, 1, 1, 1, 0〉 and P2 = 〈0, t, 1, 1, 0〉 andkernels kα(x − t) = (1/Γ(α))(x − t)α−1 and h1−β(t − τ) = (1/Γ(1 − β))(t − τ)−β. Function H of

Theorem 6.3 is given byH(t, y, u, v,w) =√1 + (u + v)2 − λ(u + v)2. One can easily check (see

[52], page 324) that

y(t) =∫ t

0E1−β,1

(−(t − τ)1−β

)√1 − 16ξ2

4ξdτ (7.7)

(i) is not an extremal for I,(ii) satisfies y′ + C

0Dβt [y] =

√1 − 16ξ2/4ξ.

Moreover, (7.7) satisfies (6.5) for λ = 2ξ, that is,

− d

dt

⎛

⎜⎜⎝(1 − t)α−1

(y′(t) + C

0Dβt

[y](t))

⎛

⎜⎜⎝

1√

1 +(y′(t) + C

0Dβt

[y](t))2

− 4ξ

⎞

⎟⎟⎠

⎞

⎟⎟⎠

+tD

β

1

⎡

⎢⎢⎣τ �−→ (1 − τ)α−1

(y′(τ) + C

0Dβτ

[y](τ))

⎛

⎜⎜⎝

1√

1 +(y′(τ) + C

0Dβτ

[y](τ))2

− 4ξ

⎞

⎟⎟⎠

⎤

⎥⎥⎦(t) = 0

(7.8)

for all t ∈ (0, 1). We conclude that (7.7) is an extremal for problem (7.6).


8. Applications to Physics

If the functional (4.1) does not depend on B-op and K-op, then Theorem 4.2 gives thefollowing result: if y is a solution to the problem of extremizing

J(y) =∫b

a

F(t, y(t), y′(t)

)kα(b, t)dt (8.1)

subject to y(a) = ya and y(b) = yb, where α ∈ (0, 1), then

∂2F(t, y(t), y′(t)

) − d

dt∂3F

(t, y(t), y′(t)

)=

1kα(b, t)

· d

dtkα(b, t)∂3F

(t, y(t), y′(t)

). (8.2)

We recognize on the right hand side of (8.2) the generalized weak dissipative parameter

δ(t) =1

kα(b, t)· d

dtkα(b, t). (8.3)

8.1. Quantum Mechanics of the Damped Harmonic Oscillator

As a first application, let us consider kernel kα(b, t) = eα(b−t) and the Lagrangian

L(y, y

)=

12my2 − V

(y), (8.4)

where V (y) is the potential energy andm stands for mass. The Euler-Lagrange equation (8.2)gives the following second-order ordinary differential equation:

y(t) − αy(t) = − 1mV ′(y(t)

). (8.5)

Equation (8.5) coincides with (14) of [44], obtained by modification of Hamilton’s principle.

8.2. Fractional Action-Like Variational Approach (FALVA)

We now extend some of the recent results of [39, 41–43], where the fractional action-like variational approach (FALVA) was proposed to model dynamical systems. FALVAfunctionals are particular cases of (8.1), where the fractional time integral introduces onlyone parameter α. Let us consider the Caldirola-Kanai Lagrangian [39, 40, 42]

L(t, y, y

)= m(t)

(y2

2−ω2y

2

2

)

, (8.6)

which describes a dynamical oscillatory system with exponentially increasing time-depend-ent mass, where ω is the frequency and m(t) = m0e−γbeγt = m0eγt, m0 = m0e−γb. Using our


generalized FALVA Euler-Lagrange equation (8.2) with kernel kα(b, t) to Lagrangian (8.6),we obtain

y(t) +(δ(t) + γ

)y(t) +ω2y(t) = 0. (8.7)

We study two particular kernels.

(1) If we choose kernel

kα(b, t) =

(ρ + 1

)1−α

Γ(α)

(bρ+1 − tρ+1

)tρ, (8.8)

defined in [53], then the Euler-Lagrange equation is

∂2F(t, y(t), y′(t)

) − d

dt∂3F

(t, y(t), y′(t)

)=

((1 − α)

(ρ + 1

)tρ

bρ+1 − tρ+1

)

∂3F(t, y(t), y′(t)

). (8.9)

In particular, when ρ → 0, (8.8) becomes the kernel of the Riemann-Liouvillefractional integral, and equation (8.9) gives

∂2F(t, y(t), y′(t)

) − d

dt∂3F

(t, y(t), y′(t)

)=

1 − α

b − t∂3F

(t, y(t), y′(t)

), (8.10)

which is the Euler-Lagrange equation proved in [42]. For ρ /= 0, we have

δ(t) =(1 − α)

(ρ + 1

)tρ

bρ+1 − tρ+1−→ 0, if t −→ ∞ or t −→ 0. (8.11)

Therefore, both at the very early time and at very large time, dissipation disappears.Moreover, if ρ → 0, then

δ(t) =1 − α

b − t−→

⎧⎪⎨

⎪⎩

0, if t −→ ∞,

1 − α

b, if t −→ 0.

(8.12)

This shows that, at the origin of time, the time-dependent dissipation becomesstationary and that, at very large time, no dissipation, of any kind, exists.

(2) If we choose kernel kα(b, t) = (cosh b − cosh t)α−1, then

∂2F(t, y(t), y′(t)

) − d

dt∂3F

(t, y(t), y′(t)

)= −(α − 1)

sinh tcosh b − cosh t

∂3F(t, y(t), y′(t)

),

δ(t) = −(α − 1)sinh t

cosh b − cosh t−→

⎧⎨

⎩

α − 1, if t −→ ∞,

0, if t −→ 0.

(8.13)


In contrast with previous case, item 1, here dissipation does not disappear at late-time dynamics.

We note that there is a small inconsistence in [42], regarding the coefficient of y(t) in(8.7), and a small inconsistence in [39], regarding a sign of (8.13).

9. Conclusion

In this paper, we unify, subsume, and significantly extend the necessary optimality conditionsavailable in the literature of the fractional calculus of variations. It should be mentioned,however, that since fractional operators are nonlocal, it can be extremely challenging to findanalytical solutions to fractional problems of the calculus of variations, and, in many cases,solutions may not exist. In our paper, we give two examples with analytic solutions, andmany more can be found borrowing different kernels from [47]. On the other hand, onecan easily choose examples for which the fractional Euler-Lagrange differential equationsare hard to solve, and, in that case, one needs to use numerical methods [54–57]. Thequestion of existence of solutions to fractional variational problems is a complete open areaof research. This needs attention. Indeed, in the absence of existence, the necessary conditionsfor extremality are vacuous: one cannot characterize an entity that does not exist in the firstplace. For solving a problem of the fractional calculus of variations, one should proceedalong the following three steps: (i) first, prove that a solution to the problem exists; (ii)second, verify the applicability of necessary optimality conditions; (iii) finally, apply thenecessary conditions which identify the extremals (the candidates). Further elimination, ifnecessary, identifies the minimizer(s) of the problem. All three steps in the above procedureare crucial. As mentioned by Young in [58], the calculus of variations has born from thestudy of necessary optimality conditions, but any such theory is “naive” until the existenceof minimizers is verified. The process leading to the existence theorems was introduced byTonelli in 1915 by the so-called direct method [59]. During two centuries, mathematicianswere developing “the naive approach to the calculus of variations”. There were, of course,good reasons why the existence problem was only solved in the beginning of XX century,two hundred years after necessary optimality conditions began to be studied: see [60, 61] andreferences therein. Similar situation happens now with the fractional calculus of variations:the subject is only fifteen years old and is still in the “naive period”. We believe time has cometo address the existence question, and this will be considered in a forthcoming paper.

Acknowledgments

This work was supported by FEDER funds through COMPETE—Operational ProgrammeFactors of Competitiveness (Programa Operacional Factores de Competitividade) and byPortuguese funds through the Center for Research and Development in Mathematics and Appli-cations (University of Aveiro) and the Portuguese Foundation for Science and Technology(FCT—Fundacao para a Ciencia e a Tecnologia), within project PEst-C/MAT/UI4106/2011with COMPETE no. FCOMP-01-0124-FEDER-022690. T. Odzijewicz was also supported byFCT through the Ph.D. fellowship SFRH/BD/33865/2009; A. B. Malinowska by BiałystokUniversity of Technology Grant S/WI/02/2011; D. F. M. Torres by FCT through thePortugal-Austin (USA) cooperation project UTAustin/MAT/0057/2008. This paper is part of


the first author’s Ph.D., which is carried out at the University of Aveiro under the DoctoralProgramme Mathematics and Applications of Universities of Aveiro and Minho.

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