# Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

Authors: Tatiana Odzijewicz; Agnieszka B. Malinowska; Delfim F. M. Torres
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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


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## Body

## 1.  Introduction

The calculus of variations is a beautiful and useful field of mathematics that deals with problems of determining extrema (maxima or minima) of functionals [1–3]. It starts with the simplest problem of finding a function extremizing (minimizing or maximizing) an integral
(1.1)J(y)=∫abF(t,y(t),y′(t))dt
subject to boundary conditions y(a)=ya and y(b)=yb. In the literature, many generalizations of this problem were proposed, including problems with multiple integrals, functionals containing higher-order derivatives, and functionals depending on several functions [4–6]. Of our interest is an extension proposed by Riewe in 1996-1997, where fractional derivatives (real or complex order) are introduced in the Lagrangian [7, 8].During the last decade, fractional problems have increasingly attracted the attention of many researchers. As mentioned in [9], Science Watch of Thomson Reuters identified the subject as an Emerging Research Front area. Fractional derivatives are nonlocal operators and are historically applied in the study of nonlocal or time-dependent processes [10]. The first and well-established application of fractional calculus in physics was in the framework of anomalous diffusion, which is related to features observed in many physical systems. Here we can mention the report [11] demonstrating that fractional equations work as a complementary tool in the description of anomalous transport processes. Within the fractional approach, it is possible to include external fields in a straightforward manner. As a consequence, in a short period of time, the list of applications expanded. Applications include chaotic dynamics [12], material sciences [13], mechanics of fractal and complex media [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 remarkable applications of fractional calculus appears, however, in the fractional variational calculus, in the context of classical mechanics. Riewe [7, 8] shows that a Lagrangian involving fractional time derivatives leads to an equation of motion with nonconservative forces such as friction. It is a remarkable result since frictional and nonconservative forces are beyond the usual macroscopic variational treatment and, consequently, beyond the most advanced methods of classical mechanics [22]. Riewe generalizes the usual variational calculus, by considering Lagrangians that are dependent on fractional derivatives, in order to deal with nonconservative forces. Recently, several different approaches have been developed to generalize the least action principle and the Euler-Lagrange equations to include fractional derivatives. Results include problems depending on Caputo fractional derivatives and Riemann-Liouville fractional derivatives [23–35].A more general unifying perspective to the subject is, however, possible, by considering fractional operators depending on general kernels [36–38]. In this work, we follow such an approach, developing a generalized fractional calculus of variations. We consider very general problems, where the classical integrals are substituted by generalized fractional integrals, and the Lagrangians depend not only on classical derivatives but also on generalized fractional operators. Problems of the type considered here, for particular kernels, are important in physics [39]. Here, we obtain general necessary optimality conditions, for several 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 the literature of mathematical physics [39–44].The paper is organized as follows. In Section2, we introduce the generalized fractional operators and prove some of their basic properties. Section 3 is dedicated to prove integration by parts formulas for the generalized fractional operators. Such formulas are then used in later sections to prove necessary optimality conditions (Theorems 4.2 and 6.3). In Sections 4, 5, and 6 we study three important classes of generalized variational problems: we obtain fractional Euler-Lagrange conditions for the fundamental (Section 4) and generalized isoperimetric problems (Section 6), as well as fractional natural boundary conditions for generalized free-boundary value problems (Section 5). Finally, two illustrative examples are discussed in detail in Section 7, while applications to physics are given in Section 8: in Section 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 of conclusion, pointing out an important direction of future research.


## 2.  Preliminaries

In this section, we present definitions and properties of generalized fractional operators. As particular cases, by choosing appropriate kernels, these operators are reduced to standard fractional integrals and fractional derivatives. Other nonstandard kernels can also be considered as particular cases. For more on the subject of generalized fractional calculus and applications, we refer the reader to [37]. Throughout the text, α denotes a real number between zero and one. Following [45], we use round brackets for the arguments of functions and square brackets for the arguments of operators. By definition, an operator receives and returns a function.Definition 2.1 (generalized fractional integral).
The operatorKPα is given by
(2.1)KPα[f](x)=KPα[t⟼f(t)](x)=p∫axkα(x,t)f(t)dt+q∫xbkα(t,x)f(t)dt,
where P=〈a,x,b,p,q〉 is the parameter set (p-set for brevity), x∈[a,b], p,q are real numbers, and kα(x,t) is a kernel which may depend on α. The operator KPα is referred to as the operator  K (K-op for simplicity) of order α and p-set P, while KPα[f] is called the operation  K (or K-opn) of f of order α and p-set P.
Note that if we define(2.2)G(x,t):={pkα(x,t),ift<x,qkα(t,x),ift≥x,
then the operatorKPα can be written in the form(2.3)KPα[f](x)=KPα[t⟼f(t)](x)=∫abG(x,t)f(t)dt.
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]).
Letα∈(0,1) and  P=〈a,x,b,p,q〉. If  kα is a square integrable function on the square  Δ=[a,b]×[a,b], then  KPα:L2([a,b])→L2([a,b]) is well-defined, linear, and bounded operator.Theorem 2.3.
Letkα be a difference kernel, that is, let  kα∈L1([a,b]) with  kα(x,t)=kα(x-t). Then,  KPα:L1([a,b])→L1([a,b]) is a well defined bounded and linear operator.Proof.
Obviously, the operator is linear. Letα∈(0,1), P=〈a,t,b,p,q〉, and f∈L1([a,b]). Define
(2.4)F(τ,t):={|pkα(t-τ)|⋅|f(τ)|,ifτ≤t,|qkα(τ-t)|⋅|f(τ)|,ifτ>t,
for all (τ,t)∈Δ=[a,b]×[a,b]. Since F is measurable on the square Δ, we have
(2.5)∫ab(∫abF(τ,t)dt)dτ=∫ab[|f(τ)|(∫τb|pkα(t-τ)|dt+∫aτ|qkα(τ-t)|dt)]dτ≤∫ab|f(τ)|||p|-|q||‖kα‖dτ=||p|-|q||⋅‖kα‖⋅‖f‖.
It follows from Fubini’s theorem that F is integrable on the square Δ. Moreover,
(2.6)‖KPα[f]‖=∫ab|p∫atkα(t-τ)f(τ)dτ+q∫tbkα(τ-t)f(τ)dτ|dt≤∫ab(|p|∫at|kα(t-τ)|⋅|f(τ)|dτ+|q|∫tb|kα(τ-t)|⋅|f(τ)|dτ)dt=∫ab(∫abF(τ,t)dτ)dt≤||p|-|q||⋅‖kα‖⋅‖f‖.
Hence, KPα:L1([a,b])→L1([a,b]) and ∥KPα∥≤||p|-|q||·∥kα∥.Remark 2.4.
TheK-op reduces to the left and the right Riemann-Liouville fractional integrals from 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)
ifP=〈a,x,b,1,0〉, then
(2.7)KPα[f](x)=1Γ(α)∫ax(x-t)α-1f(t)dt=:Iaxα[f](x)
is the standard left Riemann-Liouville fractional integral of f of order α;(ii)
ifP=〈a,x,b,0,1〉, then
(2.8)KPα[f](x)=1Γ(α)∫xb(t-x)α-1f(t)dt=:Ixbα[f](x)
is the standard right Riemann-Liouville fractional integral of f of order α.Corollary 2.5.
OperatorsaIxα,xIbα:L1([a,b])→L1([a,b]) are well defined, linear and bounded.The generalized fractional derivativesAPα and BPα are defined in terms of the generalized fractional integral K-op.Definition 2.6 (generalized Riemann-Liouville fractional derivative).
LetP be a given parameter set and 0<α<1. The operator APα is defined by APα=D∘KP1-α, where D denotes the standard derivative operator, and is referred to as the operator A (A-op) of order α and p-set P, while APα[f], for a function f such that KP1-α[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).
LetP be a given parameter set and α∈(0,1).The operator BPα is defined by BPα=KP1-α∘D,where D denotes the standard derivative operator, and is referred to as the operator B (B-op) of order α and p-set P, while BPα[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 easily obtained from the generalized operatorsAPα and BPα, respectively. Let  k1-α(x,t)=k1-α(x-t)=(x-t)-α/Γ(1-α):(i)
ifP=〈a,x,b,1,0〉, then
(2.9)APα[f](x)=1Γ(1-α)D[ξ⟼∫aξ(ξ-t)-αf(t)dt](x)=∶aDxα[f](x)is the standard left Riemann-Liouville fractional derivative off of order α, while
(2.10)BPα[f](x)=1Γ(1-α)∫ax(x-t)-αD[f](t)dt=∶aCDxα[f](x)is the standard left Caputo fractional derivative off of order α;(ii)
ifP=〈a,x,b,0,1〉, then
(2.11)-APα[f](x)=-1Γ(1-α)D[ξ⟼∫ξb(t-ξ)-αf(t)dt](x)=∶xDbα[f](x)is the standard right Riemann-Liouville fractional derivative off of order α, while
(2.12)-BPα[f](x)=-1Γ(1-α)∫xb(t-x)-αD[f](t)dt=∶xCDbα[f](x)is the standard right Caputo fractional derivative off 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 theK-op).
Letα∈(0,1),  P=〈a,t,b,p,q〉,  kα be a square-integrable function on  Δ=[a,b]×[a,b], and  f,g∈L2([a,b]). The generalized fractional integral  KPα satisfies the integration by parts formula
(3.1)∫abg(x)KPα[f](x)dx=∫abf(x)KP*α[g](x)dx,
where P*=〈a,t,b,q,p〉.Proof.
Define(3.2)F(τ,t):={|pkα(t,τ)|⋅|g(t)|⋅|f(τ)|,ifτ≤t,|qkα(τ,t)|⋅|g(t)|⋅|f(τ)|,ifτ>t,
for all (τ,t)∈Δ. Applying Holder’s inequality, we obtain
(3.3)∫ab(∫abF(τ,t)dt)dτ=∫ab[|f(τ)|(∫τb|pkα(t,τ)|⋅|g(t)|dt+∫aτ|qkα(τ,t)|⋅|g(t)|dt)]dτ≤∫ab[|f(τ)|(∫ab|pkα(t,τ)|⋅|g(t)|dt+∫ab|qkα(τ,t)|⋅|g(t)|dt)]dτ≤∫ab{|f(τ)|[(∫ab|pkα(t,τ)|2dt)1/2(∫ab|g(t)|2dt)1/2+(∫ab|qkα(τ,t)|2dt)1/2(∫ab|g(t)|2dt)1/2]}dτ.
By Fubini’s theorem, functions kα,τ(t):=kα(t,τ) and k̂α,τ(t):=kα(τ,t) belong to L2([a,b]) for almost all τ∈[a,b]. Therefore,
(3.4)∫ab{|f(τ)|[(∫ab|pkα(t,τ)|2dt)1/2(∫ab|g(t)|2dt)1/2+(∫ab|qkα(τ,t)|2dt)1/2(∫ab|g(t)|2dt)1/2]}dτ=‖g‖2∫ab[|f(τ)|(‖pkα,τ‖2+‖qk̂α,τ‖2)]dτ≤‖g‖2(∫ab|f(τ)|2dτ)1/2(∫ab|‖pkα,τ‖2+‖qk̂α,τ‖2|2dτ)1/2≤‖g‖2⋅‖f‖2(‖pkα‖2+‖qkα‖2)<∞.
Hence, we can use again Fubini’s theorem to change the order of integration:
(3.5)∫abg(t)KPα[f](t)dt=p∫abg(t)dt∫atf(τ)kα(t,τ)dτ+q∫abg(t)dt∫tbf(τ)kα(τ,t)dτ=p∫abf(τ)dτ∫τbg(t)kα(t,τ)dt+q∫abf(τ)dτ∫aτg(t)kα(τ,t)dt=∫abf(τ)KP*α[g](τ)dτ.Theorem 3.2.
Let0<α<1 and  P=〈a,x,b,p,q〉. If  kα(x,t)=kα(x-t),  kα,f∈L1([a,b]), and  g∈C([a,b]), then the operator  KPα satisfies the integration by parts formula (3.1).Proof.
Define(3.6)F(t,x):={|pkα(x-t)|⋅|g(x)|⋅|f(t)|,ift≤x,|qkα(t-x)|⋅|g(x)|⋅|f(t)|,ift>x,
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,
(3.7)∫ab(∫abF(t,x)dt)dx=∫ab[|f(t)|(∫tb|pkα(x-t)|⋅|g(x)|dx+∫at|qkα(t-x)|⋅|g(x)|dx)]dt≤∫ab[|f(t)|(∫ab|pkα(x-t)|⋅|g(x)|dx+∫ab|qkα(t-x)|⋅|g(x)|dx)]dt≤C∫ab[|f(t)|(∫ab|pkα(x-t)|dx+∫ab|qkα(t-x)|dx)]dt=C||p|-|q||‖kα‖‖f‖<∞.
Hence, we can use Fubini’s theorem to change the order of integration in iterated integrals.Theorem 3.3 (generalized fractional integration by parts).
Letα∈(0,1) and  P=〈a,t,b,p,q〉. If functions  f,KP*1-α[g]∈AC([a,b]), and we are in conditions to use formula (3.1) (Theorem 3.1 or Theorem 3.2), then
(3.8)∫abg(x)BPα[f](x)dx=f(x)KP*1-α[g](x)|ab-∫abf(x)AP*α[g](x)dx,
where P*=〈a,t,b,q,p〉.Proof.
From Definition2.7, we know that BPα[f](x)=KP1-α[D[f]](x). It follows that
(3.9)∫abg(x)BPα[f](x)dx=∫abg(x)KP1-α[D[f]](x)dx.
By relation (3.1),
(3.10)∫abg(x)BPα[f](x)dx=∫abD[f](x)KP*1-α[g](x)dx,
and the standard integration by parts formula implies (3.8):
(3.11)∫abg(x)BPα[f](x)dx=f(x)KP*1-α[g](x)|ab-∫abf(x)D[KP*1-α[g]](x)dx.Corollary 3.4 (cf. [48]).
Let0<α<1. If  f,xIb1-α[g]∈AC([a,b]), then
(3.12)∫abg(x)aCDxα[f](x)dx=f(x)xIb1-α[g](x)|x=ax=b+∫abf(x)xDbα[g](x)dx.


## 4.  The Generalized Fundamental Variational Problem

By∂iF, we denote the partial derivative of a function F with respect to its ith argument. We consider the problem of finding a function y=t↦y(t), t∈[a,b], that gives an extremum (minimum or maximum) to the functional
(4.1)J(y)=KP1α[t⟼F(t,y(t),y′(t),BP2β[y](t),KP3γ[y](t))](b)
subject to the boundary conditions
(4.2)y(a)=ya,y(b)=yb,
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 {·}P2,P3β,γ defined by
(4.3){y}P2,P3β,γ(t)=(t,y(t),y′(t),BP2β[τ⟼y(τ)](t),KP3γ[τ⟼y(τ)](t)).
With the new notation, one can write (4.1) simply as 𝒥(y)=KP1α[F{y}P2,P3β,γ](b). The operator KP1α has kernel kα(x,t), and operators BP2β and KP3γ have kernels h1-β(t,τ) and hγ(t,τ), respectively. In the sequel, we assume the following:(H1)
LagrangianF∈C1([a,b]×ℝ4;ℝ),(H2)
functionsAP2*β[τ↦kα(b,τ)∂4F{y}P2,P3β,γ(τ)], KP3*γ[τ↦kα(b,τ)∂5F{y}P2,P3β,γ(τ)], D[t↦∂3F{y}P2,P3β,γ(t)kα(b,t)], and t↦kα(b,t)∂2F{y}P2,P3β,γ(t) are continuous on (a,b),(H3)
functionst↦∂3F{y}P2,P3β,γ(t)kα(b,t), KP2*1-β[τ↦kα(b,τ)∂4F{y}P2,P3β,γ(τ)]∈AC([a,b]);(H4)
kernelskα(x,t), h1-β(t,τ), and hγ(t,τ) are such that we are in conditions to use Theorems 3.1 or 3.2, and Theorem 3.3.Definition 4.1.
A functiony∈C1([a,b];ℝ)is said to be admissible for the fractional variational problem (4.1)-(4.2), if functionsBP2β[y]andKP3γ[y]exist and are continuous on the interval[a,b], andysatisfies the given boundary conditions (4.2).Theorem 4.2.
Ify is a solution to problem (4.1)-(4.2), then  y satisfies the generalized Euler-Lagrange equation
(4.4)kα(b,t)∂2F{y}P2,P3β,γ(t)-ddt(∂3F{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4F{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5F{y}P2,P3β,γ(τ)](t)=0
for all t∈(a,b).Proof.
Suppose thaty is an extremizer of 𝒥. Consider the value of 𝒥 at a nearby function ŷ=y+εη, where ε∈ℝ is a small parameter, and η∈C1([a,b];ℝ) is an arbitrary function with continuous B-op and K-op. We require that η(a)=η(b)=0. Let
(4.5)J(ŷ)=J(ε)=KP1α[t⟼F(t,ŷ(t),ŷ′(t),BP2β[ŷ](t),KP3γ[ŷ](t))](b)=∫abkα(b,t)F(t,y(t)+εη(t),ddt(y(t)+εη(t)),BP2β[y+εη](t),KP3γ[y+εη](t))dt.
A necessary condition for y to be an extremizer is given by
(4.6)dJdε|ε=0=0⟺KP1α[∂2F{y}P2,P3β,γη+∂3F{y}P2,P3β,γD[η]+∂4F{y}P2,P3β,γBP2β[η]+∂5F{y}P2,P3β,γKP3γ[η]](b)=0⟺∫ab(∂2F{y}P2,P3β,γ(t)η(t)+∂3F{y}P2,P3β,γ(t)ddtη(t)+∂4F{y}P2,P3β,γ(t)BP2β[η](t)+∂5F{y}P2,P3β,γ(t)KP3γ[η](t)ddt)kα(b,t)dt=0.
Using classical and generalized fractional integration by parts formulas (Theorems 3.1 or 3.2, and Theorem 3.3),
(4.7)∫ab∂3F{y}P2,P3β,γ(t)kα(b,t)ddtη(t)dt=∂3F{y}P2,P3β,γ(t)kα(b,t)η(t)|ab-∫abddt(∂3F{y}P2,P3β,γ(t)kα(b,t))η(t)dt,∫ab∂4F{y}P2,P3β,γ(t)kα(b,t)BP2β[η](t)dt=KP2*1-β[τ⟼kα(b,τ)∂4F{y}P2,P3β,γ(τ)](t)η(t)|ab-∫abAP2*β[τ⟼kα(b,τ)∂4F{y}P2,P3β,γ(τ)](t)η(t)dt,∫abkα(b,t)∂5F{y}P2,P3β,γ(t)KP3γ[η](t)dt=∫abKP3*γ[τ⟼kα(b,τ)∂5F{y}P2,P3β,γ(τ)](t)η(t)dt,
where Pj*=〈a,t,b,qj,pj〉,j=2,3. Because η(a)=η(b)=0, (4.6) simplifies to
(4.8)∫ab{kα(b,t)∂2F{y}P2,P3β,γ(t)-ddt(∂3F{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4F{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5F{y}P2,P3β,γ(τ)](t)ddt(∂3F{y}P2,P3β,γ(t)kα(b,t))}η(t)dt=0.
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.
Ify is a solution to the problem of minimizing or maximizing
(4.9)J(y)=aIbα[t⟼F(t,y(t),y′(t),aCDtβ[y](t))](b)
in the class y∈C1([a,b];ℝ) subject to the boundary conditions
(4.10)y(a)=ya,y(b)=yb,
where α,β∈(0,1), F∈C1([a,b]×ℝ3;ℝ), and τ↦(b-τ)α-1∂4F(τ,y(τ),y′(τ),aCDτβ[y](τ)) has continuous Riemann-Liouville fractional derivative tDbβ, then
(4.11)∂2F(t,y(t),y′(t),aCDtβ[y](t))⋅(b-t)α-1-ddt{∂3F(t,y(t),y′(t),aCDtβ[y](t))⋅(b-t)α-1}+tDbβ[τ⟼(b-τ)α-1∂4F(τ,y(τ),y′(τ),aCDτβ[y](τ))](t)=0
for all t∈(a,b).Proof.
Choosekα(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 the Riemann-Liouville fractional derivative.Corollary 4.4.
Letβ,γ∈(0,1). If  y is a solution to the problem
(4.12)∫abF(t,y(t),y′(t),aCDtβ[y](t),aItγ[y](t))dt⟶extr,y∈C1([a,b];R),y(a)=ya,y(b)=yb,
then
(4.13)∂2F(t,y(t),y′(t),aCDtβ[y](t),aItγ[y](t))-ddt∂3F(t,y(t),y′(t),aCDtβ[y](t),aItγ[y](t))+Dbβt[τ⟼∂4F(τ,y(τ),y′(τ),aCDτβ[y](τ),aIτγ[y](τ))](t)+Ibβt[τ⟼∂5F(τ,y(τ),y′(τ),aCDτβ[y](τ),aIτγ[y](τ))](t)=0
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 LagrangianF of Corollary 4.4 does not depend on the fractional integral and the classical derivative, one obtains from (4.13) the Euler-Lagrange equation 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
(5.1)y(a)isfreeandy(b)=yb.Theorem 5.1.
Ify is a solution to the problem of extremizing functional (4.1) with (5.1) as boundary conditions, then  y satisfies the Euler-Lagrange equation (4.4). Moreover, the extra natural boundary condition
(5.2)∂3F{y}P2,P3β,γ(a)kα(b,a)+KP2*1-β[τ⟼∂4F{y}P2,P3β,γ(τ)kα(b,τ)](a)=0
holds.Proof.
Under the boundary conditions (5.1), we do not require η in the proof of Theorem 4.2 to vanish at t=a. Therefore, following the proof of Theorem 4.2, we obtain
(5.3)∂3F{y}P2,P3β,γ(a)kα(b,a)η(a)+η(a)KP2*1-β[τ⟼∂4F{y}P2,P3β,γ(τ)kα(b,τ)](a)+∫abη(t)(∂2F{y}P2,P3β,γ(t)kα(b,t)-ddt(∂3F{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼∂4F{y}P2,P3β,γ(τ)kα(b,τ)](t)+KP3*γ[τ⟼∂5F{y}P2,P3β,γ(τ)kα(b,τ)](t)ddt)dt=0
for every admissible η∈C1([a,b];ℝ) with η(b)=0. In particular, condition (5.3) holds for those 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.
Let𝒥 be the functional given by
(5.4)J(y)=aIbα[t⟼F(t,y(t),aCDtβ[y](t))](b).
Let y be a minimizer of 𝒥 satisfying the boundary condition y(b)=yb. Then, y satisfies the Euler-Lagrange equation
(5.5)(b-t)α-1∂2F(t,y(t),aCDtα[y](t))+Dbαt[τ⟼(b-τ)α-1∂3F(τ,y(τ),aCDτβy(τ))](t)=0
and the natural boundary condition
(5.6)aIb1-β[τ⟼(b-τ)α-1∂3F(τ,y(τ),aCDτβy(τ))](a)=0.Proof.
Let functional (4.1) be such that it does not depend on the classical (integer) derivative y′(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 we deduce (5.5) and (5.6) from (4.4) and (5.2), respectively.Corollary 5.3 (cf. Theorem  3.17 of [38]).
Let𝒥 be the functional given by
(5.7)J(y)=∫abF(t,y(t),y′(t),BP2β[y](t),KP3γ[y](t))dt.
If y is a minimizer to 𝒥 satisfying the boundary condition y(b)=yb, then y satisfies the Euler-Lagrange equation
(5.8)∂2F{y}P2,P3β,γ(t)-ddt∂3F{y}P2,P3β,γ(t)-AP2*β[∂4F{y}P2,P3β,γ](t)+KP3*γ[∂5F{y}P2,P3β,γ](t)=0
and the natural boundary condition
(5.9)∂3F{y}P2,P3β,γ(a)+KP2*1-β[∂4F{y}P2,P3β,γ](a)=0.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

Letξ∈ℝ. Among all functions y:[a,b]→ℝ satisfying boundary conditions
(6.1)y(a)=ya,y(b)=yb,
and an isoperimetric constraint of the form
(6.2)I(y)=KP1α[G{y}P2,P3β,γ](b)=ξ,
we look for the one that extremizes (i.e., minimizes or maximizes) a functional
(6.3)J(y)=KP1α[F{y}P2,P3β,γ](b).
Operators KP1α, BP2β, and KP3γ, 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 functiony:[a,b]→ℝ is said to be admissible for problem (6.1)–(6.3) if functions BP2β[y] and KP3γ[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 functiony∈C1([a,b],ℝ)  is said to be an extremal for  ℐ  if it satisfies the Euler-Lagrange equation (4.4) associated with functional in (6.2), that is,
(6.4)kα(b,t)∂2G{y}P2,P3β,γ(t)-ddt(∂3G{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4G{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5G{y}P2,P3β,γ(τ)](t)=0,
where Pj*=〈a,t,b,qj,pj〉, j=2,3, and t∈(a,b).Theorem 6.3.
Ify 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
(6.5)kα(b,t)∂2H{y}P2,P3β,γ(t)-ddt(∂3H{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4H{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5H{y}P2,P3β,γ(τ)](t)=0
for all t∈(a,b), where H(t,y,u,v,w)=F(t,y,u,v,w)-λG(t,y,u,v,w) and Pj*=〈a,t,b,qj,pj〉, j=2,3.Proof.
Consider a two-parameter family of the formŷ=y+ε1η1+ε2η2, where, for each i∈{1,2}, we have ηi(a)=ηi(b)=0. First, we show that we can select ε2η2 such that ŷ satisfies (6.2). Consider the quantity ℐ(ŷ)=KP1α[G{ŷ}P2,P3β,γ](b). Looking to ℐ(ŷ) as a function of ε1,ε2, we define Î(ε1,ε2)=ℐ(ŷ)-ξ. Thus, Î(0,0)=0. On the other hand, applying integration by parts formulas (Theorems 3.1 or 3.2, and Theorem 3.3), we obtain that
(6.6)∂Î∂ε2|(0,0)=∫abη2(t)(kα(b,t)∂2G{y}P2,P3β,γ(t)-ddt(∂3G{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4G{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5G{y}P2,P3β,γ(τ)](t)ddt)dt,
where Pj*=〈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 η2 such that ∂Î/∂ε2|(0,0)≠0. According to the implicit function theorem, there exists a function ε2(·) defined in a neighborhood of 0 such that Î(ε1,ε2(ε1))=0. Let Ĵ(ε1,ε2)=𝒥(ŷ). Function Ĵ has an extremum at (0,0) subject to Î(0,0)=0, and we have proved that ∇Î(0,0)≠0. The Lagrange multiplier rule asserts that there exists a real number λ such that ∇(Ĵ(0,0)-λÎ(0,0))=0. Because
(6.7)∂Ĵ∂ε1|(0,0)=∫ab(kα(b,t)∂2F{y}P2,P3β,γ(t)-ddt(∂3F{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4F{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5F{y}P2,P3β,γ(τ)](t))η1(t)dt,ddt∂Î∂ε1|(0,0)=∫ab(kα(b,t)∂2G{y}P2,P3β,γ(t)-ddt(∂3G{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4G{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5G{y}P2,P3β,γ(τ)](t)ddt)η1(t)dt,
one has
(6.8)∫ab{kα(b,t)∂2F{y}P2,P3β,γ(t)-ddt(∂3F{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4F{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5F{y}P2,P3β,γ(τ)](t)-λ(kα(b,t)∂2G{y}P2,P3β,γ(t)-ddt(∂3G{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4G{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5G{y}P2,P3β,γ(τ)](t))ddt}η1(t)dt=0.
From the fundamental lemma of the calculus of variations (see, e.g., [49], Section 2.2), it follows
(6.9)kα(b,t)∂2F{y}P2,P3β,γ(t)-ddt(∂3F{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4F{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5F{y}P2,P3β,γ(τ)](t)-λ(kα(b,t)∂2G{y}P2,P3β,γ(t)-ddt(∂3G{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4G{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5G{y}P2,P3β,γ(τ)](t)ddt)=0,
that is,
(6.10)kα(b,t)∂2H{y}P2,P3β,γ(t)-ddt(∂3H{y}P2,P3β,γ(t)kα(b,t))-AP2*β[τ⟼kα(b,τ)∂4H{y}P2,P3β,γ(τ)](t)+KP3*γ[τ⟼kα(b,τ)∂5H{y}P2,P3β,γ(τ)](t)=0
with H=F-λG.Corollary 6.4.
Lety be a minimizer to the isoperimetric problem
(6.11)J(y)=aIbα[t⟼F(t,y(t),aCDtβ[y](t))](b)⟶min⁡,I(y)=aIbα[t⟼G(t,y(t),aCDtβ[y](t))](b)=ξ,y(a)=ya,y(b)=yb.
If y is not an extremal of ℐ, then there exists a constant λ such that y satisfies
(6.12)(b-t)α-1∂2H(t,y(t),aCDtα[y](t))+tDbβ[τ⟼(b-τ)α-1∂3H(τ,y(τ),aCDτβ[y](τ))](t)=0
for all t∈(a,b), where H(t,y,v)=F(t,y,v)-λG(t,y,v).Proof.
Letkα(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〉, and P2=〈a,t,b,1,0〉. Then, the K-op and the B-op reduce to the left fractional integral and the left fractional Caputo derivative, respectively. Therefore, problem (6.11) is a particular case of 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]).
Lety be a minimizer to
(6.13)J(y)=∫abF(t,y(t),y′(t),BP2β[y](t),KP3γ[y](t))dt⟶min⁡,I(y)=∫abG(t,y(t),y′(t),BP2β[y](t),KP3γ[y](t))dt=ξ,y(a)=ya,y(b)=yb.
If y is not an extremal of ℐ, then there exists a constant λ such that y satisfies
(6.14)∂2H{y}P2,P3β,γ(t)-ddt∂3H{y}P2,P3β,γ(t)-AP2*β[∂4H{y}P2,P3β,γ](t)+KP3*γ[∂5H{y}P2,P3β,γ](t)=0
for all t∈(a,b), where H(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, the generalized fractional integral KP1α becomes 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 fundamental problem (4.1)–(4.2) (Example 7.1), the other an isoperimetric problem (6.1)–(6.3) (Example 7.2).Example 7.1.
Letα,β∈(0,1),ξ∈ℝ,P1=〈0,1,1,1,0〉,and  P2=〈0,t,1,1,0〉.Consider the following problem:
(7.1)J(y)=KP1α[t⟼tKP2β[y](t)+1-(KP2β[y](t))2](1)⟶min⁡,y(0)=1,y(1)=24+∫01rβ(1-τ)1(1+τ2)3/2dτ,
with kernel hβ such that hβ(t,τ)=hβ(t-τ) and hβ(0)=1. Here, the resolvent rβ(t) is related to the kernel hβ(t) by rβ(t)=ℒ-1[s↦(1/sh̃β(s))-1](t), h̃β(s)=ℒ[t↦hβ(t)](s), where ℒ and ℒ-1 are the direct and the inverse Laplace operators, respectively. We apply Theorem 4.2 with Lagrangian F given by F(t,y,u,v,w)=tw+1-w2. Because
(7.2)y(t)=1(1+t2)3/2+∫0trβ(t-τ)1(1+τ2)3/2dτ
is the solution to the Volterra integral equation of first kind (see, e.g., Equation 16, page 114 of [47])
(7.3)KP2β[y](t)=t1+t21+t2,
it satisfies our generalized Euler-Lagrange equation (4.4), that is,
(7.4)KP2*β[τ⟼kα(b,τ)(-KP2*β[y](τ)1-(KP2*β[y](τ))2+τ)](t)=0.
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(7.5)Eα,β(z)=∑k=0∞zkΓ(αk+β).
This function appears naturally in the solution of fractional differential equations, as a generalization of the exponential function [52].Example 7.2.
Letα,β∈(0,1),ξ∈ℝ,and  ξ∉{±1/4}.Consider the following problem:
(7.6)J(y)=I1α0[1+(y′+0CDtβ[y])2](1)⟶min⁡,I(y)=0I1α[(y′+0CDtβ[y])2](1)=ξ,y(0)=0,y(1)=∫01E1-β,1(-(1-τ)1-β)1-16ξ24ξdτ,
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〉 and kernels 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 by H(t,y,u,v,w)=1+(u+v)2-λ(u+v)2. One can easily check (see [52], page 324) that
(7.7)y(t)=∫0tE1-β,1(-(t-τ)1-β)1-16ξ24ξdτ(i)
is not an extremal forℐ,(ii)
satisfiesy′+0CDtβ[y]=1-16ξ2/4ξ.Moreover, (7.7) satisfies (6.5) for λ=2ξ, that is,
(7.8)-ddt((1-t)α-1(y′(t)+0CDtβ[y](t))(11+(y′(t)+0CDtβ[y](t))2-4ξ))+D1βt[τ⟼(1-τ)α-1(y′(τ)+0CDτβ[y](τ))(11+(y′(τ)+0CDτβ[y](τ))2-4ξ)](t)=0
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 the following result: if y is a solution to the problem of extremizing
(8.1)J(y)=∫abF(t,y(t),y′(t))kα(b,t)dt
subject to y(a)=ya and y(b)=yb, where α∈(0,1), then
(8.2)∂2F(t,y(t),y′(t))-ddt∂3F(t,y(t),y′(t))=1kα(b,t)⋅ddtkα(b,t)∂3F(t,y(t),y′(t)).
We recognize on the right hand side of (8.2) the generalized weak dissipative parameter
(8.3)δ(t)=1kα(b,t)⋅ddtkα(b,t).


### 8.1.  Quantum Mechanics of the Damped Harmonic Oscillator

As a first application, let us consider kernelkα(b,t)=eα(b-t) and the Lagrangian
(8.4)L(y,ẏ)=12mẏ2-V(y),
where V(y) is the potential energy and m stands for mass. The Euler-Lagrange equation (8.2) gives the following second-order ordinary differential equation:
(8.5)ÿ(t)-αẏ(t)=-1mV′(y(t)).
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. FALVA functionals are particular cases of (8.1), where the fractional time integral introduces only one parameter α. Let us consider the Caldirola-Kanai Lagrangian [39, 40, 42]
(8.6)L(t,y,ẏ)=m(t)(ẏ22-ω2y22),
which describes a dynamical oscillatory system with exponentially increasing time-dependent mass, where ω is the frequency and m(t)=m0e-γbeγt=m̅0eγt, m̅0=m0e-γb. Using our generalized FALVA Euler-Lagrange equation (8.2) with kernel kα(b,t) to Lagrangian (8.6), we obtain
(8.7)ÿ(t)+(δ(t)+γ)ẏ(t)+ω2y(t)=0.
We study two particular kernels.(1)
If we choose kernel(8.8)kα(b,t)=(ρ+1)1-αΓ(α)(bρ+1-tρ+1)tρ,
defined in [53], then the Euler-Lagrange equation is
(8.9)∂2F(t,y(t),y′(t))-ddt∂3F(t,y(t),y′(t))=((1-α)(ρ+1)tρbρ+1-tρ+1)∂3F(t,y(t),y′(t)).
In particular, when ρ→0, (8.8) becomes the kernel of the Riemann-Liouville fractional integral, and equation (8.9) gives
(8.10)∂2F(t,y(t),y′(t))-ddt∂3F(t,y(t),y′(t))=1-αb-t∂3F(t,y(t),y′(t)),
which is the Euler-Lagrange equation proved in [42]. For ρ≠0, we have
(8.11)δ(t)=(1-α)(ρ+1)tρbρ+1-tρ+1⟶0,ift⟶∞ort⟶0.
Therefore, both at the very early time and at very large time, dissipation disappears. Moreover, if ρ→0, then
(8.12)δ(t)=1-αb-t⟶{0,ift⟶∞,1-αb,ift⟶0.
This shows that, at the origin of time, the time-dependent dissipation becomes stationary and that, at very large time, no dissipation, of any kind, exists.(2)
If we choose kernelkα(b,t)=(coshb-cosht)α-1, then
(8.13)∂2F(t,y(t),y′(t))-ddt∂3F(t,y(t),y′(t))=-(α-1)sinh⁡tcosh⁡b-cosh⁡t∂3F(t,y(t),y′(t)),δ(t)=-(α-1)sinh⁡tcosh⁡b-cosh⁡t⟶{α-1,ift⟶∞,0,ift⟶0.
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 ẏ(t) in (8.7), and a small inconsistence in [39], regarding a sign of (8.13).


## 8.1.  Quantum Mechanics of the Damped Harmonic Oscillator

As a first application, let us consider kernelkα(b,t)=eα(b-t) and the Lagrangian
(8.4)L(y,ẏ)=12mẏ2-V(y),
where V(y) is the potential energy and m stands for mass. The Euler-Lagrange equation (8.2) gives the following second-order ordinary differential equation:
(8.5)ÿ(t)-αẏ(t)=-1mV′(y(t)).
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. FALVA functionals are particular cases of (8.1), where the fractional time integral introduces only one parameter α. Let us consider the Caldirola-Kanai Lagrangian [39, 40, 42]
(8.6)L(t,y,ẏ)=m(t)(ẏ22-ω2y22),
which describes a dynamical oscillatory system with exponentially increasing time-dependent mass, where ω is the frequency and m(t)=m0e-γbeγt=m̅0eγt, m̅0=m0e-γb. Using our generalized FALVA Euler-Lagrange equation (8.2) with kernel kα(b,t) to Lagrangian (8.6), we obtain
(8.7)ÿ(t)+(δ(t)+γ)ẏ(t)+ω2y(t)=0.
We study two particular kernels.(1)
If we choose kernel(8.8)kα(b,t)=(ρ+1)1-αΓ(α)(bρ+1-tρ+1)tρ,
defined in [53], then the Euler-Lagrange equation is
(8.9)∂2F(t,y(t),y′(t))-ddt∂3F(t,y(t),y′(t))=((1-α)(ρ+1)tρbρ+1-tρ+1)∂3F(t,y(t),y′(t)).
In particular, when ρ→0, (8.8) becomes the kernel of the Riemann-Liouville fractional integral, and equation (8.9) gives
(8.10)∂2F(t,y(t),y′(t))-ddt∂3F(t,y(t),y′(t))=1-αb-t∂3F(t,y(t),y′(t)),
which is the Euler-Lagrange equation proved in [42]. For ρ≠0, we have
(8.11)δ(t)=(1-α)(ρ+1)tρbρ+1-tρ+1⟶0,ift⟶∞ort⟶0.
Therefore, both at the very early time and at very large time, dissipation disappears. Moreover, if ρ→0, then
(8.12)δ(t)=1-αb-t⟶{0,ift⟶∞,1-αb,ift⟶0.
This shows that, at the origin of time, the time-dependent dissipation becomes stationary and that, at very large time, no dissipation, of any kind, exists.(2)
If we choose kernelkα(b,t)=(coshb-cosht)α-1, then
(8.13)∂2F(t,y(t),y′(t))-ddt∂3F(t,y(t),y′(t))=-(α-1)sinh⁡tcosh⁡b-cosh⁡t∂3F(t,y(t),y′(t)),δ(t)=-(α-1)sinh⁡tcosh⁡b-cosh⁡t⟶{α-1,ift⟶∞,0,ift⟶0.
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 ẏ(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 conditions available 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 find analytical 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, and many more can be found borrowing different kernels from [47]. On the other hand, one can easily choose examples for which the fractional Euler-Lagrange differential equations are hard to solve, and, in that case, one needs to use numerical methods [54–57]. The question of existence of solutions to fractional variational problems is a complete open area of research. This needs attention. Indeed, in the absence of existence, the necessary conditions for extremality are vacuous: one cannot characterize an entity that does not exist in the first place. For solving a problem of the fractional calculus of variations, one should proceed along 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 the necessary conditions which identify the extremals (the candidates). Further elimination, if necessary, identifies the minimizer(s) of the problem. All three steps in the above procedure are crucial. As mentioned by Young in [58], the calculus of variations has born from the study of necessary optimality conditions, but any such theory is “naive” until the existence of minimizers is verified. The process leading to the existence theorems was introduced by Tonelli in 1915 by the so-called direct method [59]. During two centuries, mathematicians were 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] and references 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 come to address the existence question, and this will be considered in a forthcoming paper.



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*Source: 871912-2012-05-13.xml*