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

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.

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 𝒥𝑦=^𝑏_𝑎𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)𝑑𝑡 (1.1) subject to boundary conditions 𝑦(𝑎)=𝑦_𝑎 and 𝑦(𝑏)=𝑦_𝑏. 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 Section 2, 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 operator  𝐾^𝛼_𝑃 is given by 𝐾^𝛼_𝑃𝑓(𝑥)=𝐾^𝛼_𝑃𝑡⟼𝑓(𝑡)(𝑥)=𝑝^𝑥_𝑎𝑘_𝛼(𝑥,𝑡)𝑓(𝑡)𝑑𝑡+𝑞^𝑏_𝑥𝑘_𝛼(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 real numbers, and 𝑘_𝛼(𝑥,𝑡) is a kernel which may depend on 𝛼. The operator 𝐾^𝛼_𝑃 is referred to as the operator  𝐾 (𝐾-op for simplicity) of order 𝛼 and 𝑝-set 𝑃, while 𝐾^𝛼_𝑃[𝑓] is called the operation  𝐾 (or 𝐾-opn) of 𝑓 of order 𝛼 and 𝑝-set 𝑃.
Note that if we define 𝐺(𝑥,𝑡)∶=𝑝𝑘_𝛼(𝑥,𝑡),if𝑡<𝑥,𝑞𝑘_𝛼(𝑡,𝑥),if𝑡≥𝑥, (2.2)
then the operator 𝐾^𝛼_𝑃 can be written in the form𝐾^𝛼_𝑃𝑓(𝑥)=𝐾^𝛼_𝑃𝑡⟼𝑓(𝑡)(𝑥)=^𝑏_𝑎𝐺(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 𝑘_𝛼 is a square integrable function on the square Δ=[𝑎,𝑏]×[𝑎,𝑏], then 𝐾^𝛼_𝑃∶𝐿₂([𝑎,𝑏])→𝐿₂([𝑎,𝑏]) is well-defined, linear, and bounded operator.

Theorem 2.3. Let 𝑘_𝛼 be a difference kernel, that is, let 𝑘_𝛼∈𝐿₁([𝑎,𝑏]) with 𝑘_𝛼(𝑥,𝑡)=𝑘_𝛼(𝑥−𝑡). Then, 𝐾^𝛼_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)∶=𝑝𝑘_𝛼(𝑡−𝜏)⋅𝑓(𝜏),if𝜏≤𝑡,𝑞𝑘_𝛼(𝜏−𝑡)⋅𝑓(𝜏),if𝜏>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_𝑎𝐹(𝜏,𝑡)𝑑𝑡𝑑𝜏=^𝑏_𝑎𝑓(𝜏)^𝑏_𝜏𝑝𝑘_𝛼(𝑡−𝜏)𝑑𝑡+^𝜏_𝑎𝑞𝑘_𝛼(≤𝜏−𝑡)𝑑𝑡𝑑𝜏_𝑏𝑎𝑓(𝜏)𝑝−𝑞𝑘_𝛼=𝑝−𝑞⋅𝑘𝑑𝜏_𝛼⋅𝑓. (2.5) It follows from Fubini’s theorem that 𝐹 is integrable on the square Δ. Moreover, 𝐾^𝛼_𝑃𝑓=^𝑏_𝑎𝑝^𝑡_𝑎𝑘_𝛼(𝑡−𝜏)𝑓(𝜏)𝑑𝜏+𝑞^𝑏_𝑡𝑘_𝛼()𝑓(𝜏)≤𝜏−𝑡𝑑𝜏𝑑𝑡_𝑏𝑎𝑝^𝑡_𝑎𝑘_𝛼(𝑡−𝜏)⋅𝑓(𝜏)𝑑𝜏+𝑞^𝑏_𝑡𝑘_𝛼(⋅𝑓(𝜏=𝜏−𝑡))𝑑𝜏𝑑𝑡_{𝑏𝑎𝑏𝑎}𝐹()≤𝑝−𝑞⋅𝑘𝜏,𝑡𝑑𝜏𝑑𝑡_𝛼⋅𝑓. (2.6) Hence, 𝐾^𝛼_𝑃∶𝐿₁([𝑎,𝑏])→𝐿₁([𝑎,𝑏]) and ‖𝐾^𝛼_𝑃‖≤||𝑝|−|𝑞||⋅‖𝑘_𝛼‖.

Remark 2.4. The  𝐾-op reduces to the left and the right Riemann-Liouville fractional integrals from a suitably chosen kernel  𝑘_𝛼(𝑥,𝑡)  and  𝑝-set  𝑃. Let  𝑘_𝛼(𝑥,𝑡)=𝑘_𝛼(𝑥−𝑡)=(1/Γ(𝛼))(𝑥−𝑡)^𝛼−1:(i)if 𝑃=⟨𝑎,𝑥,𝑏,1,0⟩, then 𝐾^𝛼_𝑃𝑓𝑥)=1(Γ(𝛼)^𝑥_𝑎(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡=∶_𝑎𝐼^𝛼_𝑥𝑓(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 𝐾^𝛼_𝑃𝑓𝑥)=1(Γ(𝛼)^𝑏_𝑥(𝑡−𝑥)^𝛼−1𝑓(𝑡)𝑑𝑡=∶_𝑥𝐼^𝛼_𝑏𝑓(𝑥) (2.8) is the standard right Riemann-Liouville fractional integral of 𝑓 of order 𝛼.

Corollary 2.5. Operators _𝑎𝐼^𝛼_𝑥,_𝑥𝐼^𝛼_𝑏∶𝐿₁([𝑎,𝑏])→𝐿₁([𝑎,𝑏]) are well defined, linear and bounded.

The generalized fractional derivatives 𝐴^𝛼_𝑃 and 𝐵^𝛼_𝑃 are defined in terms of the generalized fractional integral 𝐾-op.

Definition 2.6 (generalized Riemann-Liouville fractional derivative). Let 𝑃 be a given parameter set and 0<𝛼<1. The operator 𝐴^𝛼_𝑃 is defined by 𝐴^𝛼_𝑃=𝐷∘𝐾_𝑃^1−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 𝐴^𝛼_𝑃[𝑓], for a function 𝑓 such that 𝐾_𝑃^1−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). Let 𝑃 be a given parameter set and 𝛼∈(0,1). The operator 𝐵^𝛼_𝑃 is defined by 𝐵^𝛼_𝑃=𝐾_𝑃^1−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 𝐵^𝛼_𝑃[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 operators  𝐴^𝛼_𝑃 and 𝐵^𝛼_𝑃, respectively. Let  𝑘_1−𝛼(𝑥,𝑡)=𝑘_1−𝛼(𝑥−𝑡)=(𝑥−𝑡)^−𝛼/Γ(1−𝛼):(i)if 𝑃=⟨𝑎,𝑥,𝑏,1,0⟩, then 𝐴^𝛼_𝑃𝑓𝑥)=1(Γ()𝐷1−𝛼𝜉⟼^𝜉_𝑎(𝜉−𝑡)^−𝛼𝑓(𝑡)𝑑𝑡(𝑥)=∶_𝑎𝐷^𝛼_𝑥𝑓(𝑥) (2.9)is the standard left Riemann-Liouville fractional derivative of 𝑓 of order 𝛼, while 𝐵^𝛼_𝑃𝑓𝑥)=1(Γ(1−𝛼)^𝑥_𝑎(𝑥−𝑡)^−𝛼𝐷𝑓(𝑡)𝑑𝑡=∶^𝐶_𝑎𝐷^𝛼_𝑥𝑓(𝑥) (2.10)is the standard left Caputo fractional derivative of 𝑓 of order 𝛼;(ii)if 𝑃=⟨𝑎,𝑥,𝑏,0,1⟩, then −𝐴^𝛼_𝑃𝑓(𝑥)=−1Γ()𝐷1−𝛼𝜉⟼^𝑏_𝜉(𝑡−𝜉)^−𝛼𝑓(𝑡)𝑑𝑡(𝑥)=∶_𝑥𝐷^𝛼_𝑏𝑓(𝑥) (2.11)is the standard right Riemann-Liouville fractional derivative of 𝑓 of order 𝛼, while −𝐵^𝛼_𝑃𝑓(𝑥)=−1Γ(1−𝛼)^𝑏_𝑥(𝑡−𝑥)^−𝛼𝐷𝑓(𝑡)𝑑𝑡=∶^𝐶_𝑥𝐷^𝛼_𝑏𝑓(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⟩, 𝑘_𝛼 be a square-integrable function on Δ=[𝑎,𝑏]×[𝑎,𝑏], and 𝑓,𝑔∈𝐿₂([𝑎,𝑏]). The generalized fractional integral 𝐾^𝛼_𝑃 satisfies the integration by parts formula ^𝑏_𝑎𝑔(𝑥)𝐾^𝛼_𝑃𝑓(𝑥)𝑑𝑥=^𝑏_𝑎𝑓(𝑥)𝐾^𝛼_𝑃∗𝑔(𝑥)𝑑𝑥, (3.1) where 𝑃^∗=⟨𝑎,𝑡,𝑏,𝑞,𝑝⟩.

Proof. Define 𝐹(𝜏,𝑡)∶=𝑝𝑘_𝛼(𝑡,𝜏)⋅𝑔(𝑡)⋅𝑓(𝜏),if𝜏≤𝑡,𝑞𝑘_𝛼(𝜏,𝑡)⋅𝑔(𝑡)⋅𝑓(𝜏),if𝜏>𝑡, (3.2) for all (𝜏,𝑡)∈Δ. Applying Holder’s inequality, we obtain ^𝑏_𝑎^𝑏_𝑎𝐹(𝜏,𝑡)𝑑𝑡𝑑𝜏=^𝑏_𝑎𝑓(𝜏)^𝑏_𝜏𝑝𝑘_𝛼(𝑡,𝜏)⋅𝑔(𝑡)𝑑𝑡+^𝜏_𝑎𝑞𝑘_𝛼(⋅𝑔(𝑡≤𝜏,𝑡))𝑑𝑡𝑑𝜏_𝑏𝑎𝑓(𝜏)^𝑏_𝑎𝑝𝑘_𝛼(𝑡,𝜏)⋅𝑔(𝑡)𝑑𝑡+^𝑏_𝑎𝑞𝑘_𝛼(⋅𝑔(𝑡≤𝜏,𝑡))𝑑𝑡𝑑𝜏_𝑏𝑎𝑓(𝜏)^𝑏_𝑎𝑝𝑘_𝛼(𝑡,𝜏)²𝑑𝑡^1/2𝑏_𝑎𝑔(𝑡)²𝑑𝑡^1/2+^𝑏_𝑎𝑞𝑘_𝛼(𝜏,𝑡)²𝑑𝑡^1/2𝑏_𝑎𝑔(𝑡)²𝑑𝑡^1/2𝑑𝜏. (3.3) By Fubini’s theorem, functions 𝑘_𝛼,𝜏(𝑡)∶=𝑘_𝛼(𝑡,𝜏) and 𝑘_𝛼,𝜏(𝑡)∶=𝑘_𝛼(𝜏,𝑡) belong to 𝐿₂([𝑎,𝑏]) for almost all 𝜏∈[𝑎,𝑏]. Therefore, ^𝑏_𝑎𝑓(𝜏)^𝑏_𝑎𝑝𝑘_𝛼(𝑡,𝜏)²𝑑𝑡^1/2𝑏_𝑎𝑔(𝑡)²𝑑𝑡^1/2+^𝑏_𝑎𝑞𝑘_𝛼(𝜏,𝑡)²𝑑𝑡^1/2𝑏_𝑎𝑔(𝑡)²𝑑𝑡^1/2=𝑔𝑑𝜏_2𝑏𝑎𝑓(𝜏)𝑝𝑘_𝛼,𝜏2+𝑞𝑘_𝛼,𝜏2≤𝑔𝑑𝜏_2𝑏𝑎𝑓(𝜏)²𝑑𝜏^1/2𝑏_𝑎𝑝𝑘_𝛼,𝜏2+𝑞𝑘_𝛼,𝜏2²𝑑𝜏^1/2≤𝑔₂⋅𝑓₂𝑝𝑘_𝛼2+𝑞𝑘_𝛼2<∞. (3.4) Hence, we can use again Fubini’s theorem to change the order of integration: ^𝑏_𝑎𝑔(𝑡)𝐾^𝛼_𝑃𝑓(𝑡)𝑑𝑡=𝑝^𝑏_𝑎𝑔(𝑡)𝑑𝑡^𝑡_𝑎𝑓(𝜏)𝑘_𝛼(𝑡,𝜏)𝑑𝜏+𝑞^𝑏_𝑎𝑔(𝑡)𝑑𝑡^𝑏_𝑡𝑓(𝜏)𝑘_𝛼(𝜏,𝑡)𝑑𝜏=𝑝^𝑏_𝑎𝑓(𝜏)𝑑𝜏^𝑏_𝜏𝑔(𝑡)𝑘_𝛼(𝑡,𝜏)𝑑𝑡+𝑞^𝑏_𝑎𝑓(𝜏)𝑑𝜏^𝜏_𝑎𝑔(𝑡)𝑘_𝛼()=𝜏,𝑡𝑑𝑡_𝑏𝑎𝑓(𝜏)𝐾^𝛼_𝑃∗𝑔(𝜏)𝑑𝜏. (3.5)

Theorem 3.2. Let 0<𝛼<1 and 𝑃=⟨𝑎,𝑥,𝑏,𝑝,𝑞⟩. If 𝑘_𝛼(𝑥,𝑡)=𝑘_𝛼(𝑥−𝑡), 𝑘_𝛼,𝑓∈𝐿₁([𝑎,𝑏]), and 𝑔∈𝐶([𝑎,𝑏]), then the operator 𝐾^𝛼_𝑃 satisfies the integration by parts formula (3.1).

Proof. Define 𝐹(𝑡,𝑥)∶=𝑝𝑘_𝛼(𝑥−𝑡)⋅𝑔(𝑥)⋅𝑓(𝑡),if𝑡≤𝑥,𝑞𝑘_𝛼(𝑡−𝑥)⋅𝑔(𝑥)⋅𝑓(𝑡),if𝑡>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_𝑎𝐹(𝑡,𝑥)𝑑𝑡𝑑𝑥=^𝑏_𝑎𝑓(𝑡)^𝑏_𝑡𝑝𝑘_𝛼(𝑥−𝑡)⋅𝑔(𝑥)𝑑𝑥+^𝑡_𝑎𝑞𝑘_𝛼(⋅𝑔(𝑥≤𝑡−𝑥))𝑑𝑥𝑑𝑡_𝑏𝑎𝑓(𝑡)^𝑏_𝑎𝑝𝑘_𝛼(𝑥−𝑡)⋅𝑔(𝑥)𝑑𝑥+^𝑏_𝑎𝑞𝑘_𝛼(𝑡−𝑥)⋅𝑔(𝑥)𝑑𝑥𝑑𝑡≤𝐶^𝑏_𝑎𝑓(𝑡)^𝑏_𝑎𝑝𝑘_𝛼(𝑥−𝑡)𝑑𝑥+^𝑏_𝑎𝑞𝑘_𝛼(𝑝−𝑞𝑘𝑡−𝑥)𝑑𝑥𝑑𝑡=𝐶_𝛼𝑓<∞. (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,𝐾_𝑃^1−𝛼_∗[𝑔]∈𝐴𝐶([𝑎,𝑏]), and we are in conditions to use formula (3.1) (Theorem 3.1 or Theorem 3.2), then ^𝑏_𝑎𝑔(𝑥)𝐵^𝛼_𝑃𝑓(𝑥)𝑑𝑥=𝑓(𝑥)𝐾_𝑃^1−𝛼_∗𝑔(𝑥)^𝑏_𝑎−^𝑏_𝑎𝑓(𝑥)𝐴^𝛼_𝑃∗𝑔(𝑥)𝑑𝑥, (3.8) where 𝑃^∗=⟨𝑎,𝑡,𝑏,𝑞,𝑝⟩.

Proof. From Definition 2.7, we know that 𝐵^𝛼_𝑃[𝑓](𝑥)=𝐾_𝑃^1−𝛼[𝐷[𝑓]](𝑥). It follows that ^𝑏_𝑎𝑔(𝑥)𝐵^𝛼_𝑃𝑓(𝑥)𝑑𝑥=^𝑏_𝑎𝑔(𝑥)𝐾_𝑃^1−𝛼𝐷𝑓(𝑥)𝑑𝑥. (3.9) By relation (3.1), ^𝑏_𝑎𝑔(𝑥)𝐵^𝛼_𝑃𝑓(𝑥)𝑑𝑥=^𝑏_𝑎𝐷𝑓(𝑥)𝐾_𝑃^1−𝛼_∗𝑔(𝑥)𝑑𝑥, (3.10) and the standard integration by parts formula implies (3.8): ^𝑏_𝑎𝑔(𝑥)𝐵^𝛼_𝑃𝑓(𝑥)𝑑𝑥=𝑓(𝑥)𝐾_𝑃^1−𝛼_∗[𝑔](𝑥)^𝑏_𝑎−^𝑏_𝑎𝑓(𝑥)𝐷𝐾_𝑃^1−𝛼_∗𝑔(𝑥)𝑑𝑥. (3.11)

Corollary 3.4 (cf. [48]). Let 0<𝛼<1. If 𝑓,_𝑥𝐼_𝑏^1−𝛼[𝑔]∈𝐴𝐶([𝑎,𝑏]), then ^𝑏_𝑎𝑔(𝑥)^𝐶_𝑎𝐷^𝛼_𝑥𝑓(𝑥)𝑑𝑥=𝑓(𝑥)_𝑥𝐼_𝑏^1−𝛼𝑔(𝑥)^𝑥=𝑏_𝑥=𝑎+^𝑏_𝑎𝑓(𝑥)_𝑥𝐷^𝛼_𝑏𝑔(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. 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 𝒥𝑦=𝐾^𝛼_𝑃1𝑡⟼𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),𝐵^𝛽_𝑃2𝑦(𝑡),𝐾^𝛾_𝑃3𝑦(𝑡)(𝑏) (4.1) subject to the boundary conditions 𝑦(𝑎)=𝑦_𝑎,𝑦(𝑏)=𝑦_𝑏, (4.2) where 𝛼,𝛽,𝛾∈(0,1), 𝑃₁=⟨𝑎,𝑏,𝑏,1,0⟩, and 𝑃_𝑗=⟨𝑎,𝑡,𝑏,𝑝_𝑗,𝑞_𝑗⟩, 𝑗=2,3. For simplicity of notation, we introduce the operator {⋅}_𝑃^𝛽,𝛾_2,𝑃3 defined by 𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)=𝑡,𝑦(𝑡),𝑦^′(𝑡),𝐵^𝛽_𝑃2𝜏⟼𝑦(𝜏)(𝑡),𝐾^𝛾_𝑃3𝜏⟼𝑦(𝜏)(𝑡). (4.3) With the new notation, one can write (4.1) simply as 𝒥(𝑦)=𝐾^𝛼_𝑃1[𝐹{𝑦}_𝑃^𝛽,𝛾_2,𝑃3](𝑏). The operator 𝐾^𝛼_𝑃1 has kernel 𝑘_𝛼(𝑥,𝑡), and operators 𝐵^𝛽_𝑃2 and 𝐾^𝛾_𝑃3 have kernels ℎ_1−𝛽(𝑡,𝜏) and ℎ_𝛾(𝑡,𝜏), respectively. In the sequel, we assume the following:(H1)Lagrangian 𝐹∈𝐶¹([𝑎,𝑏]×ℝ⁴;ℝ),(H2)functions 𝐴^𝛽_𝑃∗2[𝜏↦𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹{𝑦}_𝑃^𝛽,𝛾_2,𝑃3(𝜏)], 𝐾^𝛾_𝑃∗3[𝜏↦𝑘_𝛼(𝑏,𝜏)𝜕₅𝐹{𝑦}_𝑃^𝛽,𝛾_2,𝑃3(𝜏)], 𝐷[𝑡↦𝜕₃𝐹{𝑦}_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)], and 𝑡↦𝑘_𝛼(𝑏,𝑡)𝜕₂𝐹{𝑦}_𝑃^𝛽,𝛾_2,𝑃3(𝑡) are continuous on (𝑎,𝑏),(H3)functions 𝑡↦𝜕₃𝐹{𝑦}_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡), 𝐾_𝑃^1−𝛽_∗2[𝜏↦𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹{𝑦}_𝑃^𝛽,𝛾_2,𝑃3(𝜏)]∈𝐴𝐶([𝑎,𝑏]);(H4)kernels 𝑘_𝛼(𝑥,𝑡), ℎ_1−𝛽(𝑡,𝜏), and ℎ_𝛾(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 function 𝑦∈𝐶¹([𝑎,𝑏];ℝ) is said to be admissible for the fractional variational problem (4.1)-(4.2), if functions 𝐵^𝛽_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-Lagrange equation 𝑘_𝛼(𝑏,𝑡)𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,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 function 𝑦=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 𝒥𝑦=𝐽(𝜀)=𝐾^𝛼_𝑃1𝑡⟼𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),𝐵^𝛽_𝑃2𝑦(𝑡),𝐾^𝛾_𝑃3𝑡)𝑏)=𝑦((_𝑏𝑎𝑘_𝛼()𝐹(𝑡)(𝑡),𝑑𝑏,𝑡𝑡,𝑦+𝜀𝜂𝑦(𝑡)(𝑡)𝑑𝑡+𝜀𝜂,𝐵^𝛽_𝑃2𝑦+𝜀𝜂(𝑡),𝐾^𝛾_𝑃3𝑦+𝜀𝜂(𝑡)𝑑𝑡. (4.5) A necessary condition for 𝑦 to be an extremizer is given by 𝑑𝐽𝑑𝜀_𝜀=0=0⟺𝐾^𝛼_𝑃1𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3𝜂+𝜕₃𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3𝐷𝜂+𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3𝐵^𝛽_𝑃2𝜂+𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3𝐾^𝛾_𝑃3𝜂𝑏)⟺(=0_𝑏𝑎𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝜂(𝑡)+𝜕₃𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑑𝜂(𝑡)𝑑𝑡+𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝐵^𝛽_𝑃2𝜂(𝑡)+𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝐾^𝛾_𝑃3𝜂(𝑡)𝑘_𝛼(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𝜕₃𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼()𝑑𝑏,𝑡𝜂(𝑡)𝑑𝑡𝑑𝑡=𝜕₃𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)𝜂(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑑𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)𝜂(𝑡)𝑑𝑡,_𝑏𝑎𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)𝐵^𝛽_𝑃2𝜂(𝑡)𝑑𝑡=𝐾_𝑃^1−𝛽_∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)𝜂(𝑡)^𝑏_𝑎−^𝑏_𝑎𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)𝜂(𝑡)𝑑𝑡,_𝑏𝑎𝑘_𝛼(𝑏,𝑡)𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝐾^𝛾_𝑃3𝜂(𝑡)𝑑𝑡=^𝑏_𝑎𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)𝜂(𝑡)𝑑𝑡, (4.7) where 𝑃^∗_𝑗=⟨𝑎,𝑡,𝑏,𝑞_𝑗,𝑝_𝑗⟩,𝑗=2,3. Because 𝜂(𝑎)=𝜂(𝑏)=0, (4.6) simplifies to ^𝑏_𝑎𝑘_𝛼(𝑏,𝑡)𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,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 𝒥𝑦=_𝑎𝐼^𝛼_𝑏𝑡⟼𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦(𝑡)(𝑏) (4.9) in the class 𝑦∈𝐶¹([𝑎,𝑏];ℝ) subject to the boundary conditions 𝑦(𝑎)=𝑦_𝑎,𝑦(𝑏)=𝑦_𝑏, (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_𝑎𝐷^𝛽_𝜏[𝑦](𝜏)) has continuous Riemann-Liouville fractional derivative _𝑡𝐷^𝛽_𝑏, then 𝜕₂𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦(𝑡)⋅(𝑏−𝑡)^𝛼−1−𝑑𝜕𝑑𝑡³𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦(𝑡)⋅(𝑏−𝑡)^𝛼−1+_𝑡𝐷^𝛽_𝑏𝜏⟼(𝑏−𝜏)^𝛼−1𝜕₄𝐹𝜏,𝑦(𝜏),𝑦^′(𝜏),^𝐶_𝑎𝐷^𝛽_𝜏𝑦(𝜏)(𝑡)=0 (4.11) for all 𝑡∈(𝑎,𝑏).

Proof. Choose 𝑘_𝛼(𝑥,𝑡)=(1/Γ(𝛼))(𝑥−𝑡)^𝛼−1, ℎ_1−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. Let 𝛽,𝛾∈(0,1). If 𝑦 is a solution to the problem ^𝑏_𝑎𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦(𝑡),_𝑎𝐼^𝛾_𝑡𝑦(𝑡)𝑑𝑡⟶extr,𝑦∈𝐶¹]),𝑦(𝑎)([𝑎,𝑏;ℝ=𝑦_𝑎,𝑦(𝑏)=𝑦_𝑏, (4.12) then 𝜕₂𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦(𝑡),_𝑎𝐼^𝛾_𝑡𝑦𝑡)−𝑑(𝜕𝑑𝑡³𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦(𝑡),_𝑎𝐼^𝛾_𝑡𝑦𝑡)+_𝑡(𝐷^𝛽_𝑏𝜏⟼𝜕₄𝐹𝜏,𝑦(𝜏),𝑦^′(𝜏),^𝐶_𝑎𝐷^𝛽_𝜏𝑦(𝜏),_𝑎𝐼^𝛾_𝜏𝑦𝜏)𝑡)+((_𝑡𝐼^𝛽_𝑏𝜏⟼𝜕₅𝐹𝜏,𝑦(𝜏),𝑦^′(𝜏),^𝐶_𝑎𝐷^𝛽_𝜏𝑦(𝜏),_𝑎𝐼^𝛾_𝜏𝑦(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 𝑘_𝛼(𝑥,𝑡)=1, ℎ_1−𝛽(𝑡,𝜏)=(1/Γ(1−𝛽))(𝑡−𝜏)^−𝛽, and ℎ_𝛾(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 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 𝑦(𝑎)isfreeand𝑦(𝑏)=𝑦_𝑏. (5.1)

Theorem 5.1. If 𝑦 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 𝜕₃𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑎)𝑘_𝛼(𝑏,𝑎)+𝐾_𝑃^1−𝛽_∗2𝜏⟼𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)𝑘_𝛼(𝑏,𝜏)(𝑎)=0 (5.2) holds.

Proof. Under the boundary conditions (5.1), we do not require 𝜂 in the proof of Theorem 4.2 to vanish at 𝑡=𝑎. Therefore, following the proof of Theorem 4.2, we obtain 𝜕₃𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑎)𝑘_𝛼(𝑏,𝑎)𝜂(𝑎)+𝜂(𝑎)𝐾_𝑃^1−𝛽_∗2𝜏⟼𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)𝑘_𝛼(𝑎)+𝑏,𝜏)(_𝑏𝑎𝜂(𝑡)𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼()−𝑑𝑏,𝑡𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)𝑘_𝛼(𝑏,𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)𝑘_𝛼(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 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. Let 𝒥 be the functional given by 𝒥𝑦=_𝑎𝐼^𝛼_𝑏𝑡⟼𝐹𝑡,𝑦(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦(𝑡)(𝑏). (5.4) Let 𝑦 be a minimizer of 𝒥 satisfying the boundary condition 𝑦(𝑏)=𝑦_𝑏. Then, 𝑦 satisfies the Euler-Lagrange equation (𝑏−𝑡)^𝛼−1𝜕₂𝐹𝑡,𝑦(𝑡),^𝐶_𝑎𝐷^𝛼_𝑡𝑦(𝑡)+_𝑡𝐷^𝛼_𝑏𝜏⟼(𝑏−𝜏)^𝛼−1𝜕₃𝐹𝜏,𝑦(𝜏),^𝐶_𝑎𝐷^𝛽_𝜏𝑦(𝜏)(𝑡)=0 (5.5) and the natural boundary condition _𝑎𝐼_𝑏^1−𝛽𝜏⟼(𝑏−𝜏)^𝛼−1𝜕₃𝐹𝜏,𝑦(𝜏),^𝐶_𝑎𝐷^𝛽_𝜏𝑦(𝜏)(𝑎)=0. (5.6)

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⟩, ℎ_1−𝛽(𝑡−𝜏)=(1/Γ(1−𝛽))(𝑡−𝜏)^−𝛽, and 𝑘_𝛼(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]). Let 𝒥 be the functional given by 𝒥𝑦=^𝑏_𝑎𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),𝐵^𝛽_𝑃2𝑦(𝑡),𝐾^𝛾_𝑃3𝑦(𝑡)𝑑𝑡. (5.7) If 𝑦 is a minimizer to 𝒥 satisfying the boundary condition 𝑦(𝑏)=𝑦_𝑏, then 𝑦 satisfies the Euler-Lagrange equation 𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝐴^𝛽_𝑃∗2𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)+𝐾^𝛾_𝑃∗3𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)=0 (5.8) and the natural boundary condition 𝜕₃𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑎)+𝐾_𝑃^1−𝛽_∗2𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑎)=0. (5.9)

Proof. Choose, in the problem defined by (4.1) and (5.1), 𝑘_𝛼(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)=𝑦_𝑎,𝑦(𝑏)=𝑦_𝑏, (6.1) and an isoperimetric constraint of the form ℐ𝑦=𝐾^𝛼_𝑃1𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑏)=𝜉, (6.2) we look for the one that extremizes (i.e., minimizes or maximizes) a functional 𝒥𝑦=𝐾^𝛼_𝑃1𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑏). (6.3) Operators 𝐾^𝛼_𝑃1, 𝐵^𝛽_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 it satisfies the Euler-Lagrange equation (4.4) associated with functional in (6.2), that is, 𝑘_𝛼(𝑏,𝑡)𝜕₂𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)=0, (6.4) where 𝑃^∗_𝑗=⟨𝑎,𝑡,𝑏,𝑞_𝑗,𝑝_𝑗⟩, 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 𝑘_𝛼(𝑏,𝑡)𝜕₂𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)=0 (6.5) 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 𝑃^∗_𝑗=⟨𝑎,𝑡,𝑏,𝑞_𝑗,𝑝_𝑗⟩, 𝑗=2,3.

Proof. Consider a two-parameter family of the form 𝑦=𝑦+𝜀₁𝜂₁+𝜀₂𝜂₂, where, for each 𝑖∈{1,2}, we have 𝜂_𝑖(𝑎)=𝜂_𝑖(𝑏)=0. First, we show that we can select 𝜀₂𝜂₂ such that 𝑦 satisfies (6.2). Consider the quantity ℐ(𝑦)=𝐾^𝛼_𝑃1[𝐺{𝑦}_𝑃^𝛽,𝛾_2,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 by parts formulas (Theorems 3.1 or 3.2, and Theorem 3.3), we obtain that 𝜕𝐼̂𝜕𝜀_2(0,0)=^𝑏_𝑎𝜂₂(𝑡)𝑘_𝛼(𝑏,𝑡)𝜕₂𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)𝑑𝑡, (6.6) where 𝑃^∗_𝑗=⟨𝑎,𝑡,𝑏,𝑞_𝑗,𝑝_𝑗⟩, 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 𝜕𝐼/̂𝜕𝜀₂|_(0,0)≠0. According to the implicit function theorem, there exists a function 𝜀₂(⋅) defined in a neighborhood of 0 such that 𝐼(̂𝜀₁,𝜀₂(𝜀₁))=0. Let 𝐽(̂𝜀₁,𝜀₂)=𝒥(𝑦). Function 𝐽̂ 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 𝜕𝐽̂𝜕𝜀_1(0,0)=^𝑏_𝑎𝑘_𝛼(𝑏,𝑡)𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)𝜂₁(𝑡)𝜕𝐼̂𝑑𝑡,𝜕𝜀_1(0,0)=^𝑏_𝑎𝑘_𝛼(𝑏,𝑡)𝜕₂𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)𝜂₁(𝑡)𝑑𝑡, (6.7) one has ^𝑏_𝑎𝑘_𝛼(𝑏,𝑡)𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏𝑡)𝑘)(−𝜆_𝛼(𝑏,𝑡)𝜕₂𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐺𝑦_𝑃^𝛽,𝛾_2,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), it follows 𝑘_𝛼(𝑏,𝑡)𝜕₂𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐹𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏𝑡)𝑘)(−𝜆_𝛼(𝑏,𝑡)𝜕₂𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐺𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)=0, (6.9) that is, 𝑘_𝛼(𝑏,𝑡)𝜕₂𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)𝑘_𝛼(𝑏,𝑡)−𝐴^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₄𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)+𝐾^𝛾_𝑃∗3𝜏⟼𝑘_𝛼(𝑏,𝜏)𝜕₅𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝜏)(𝑡)=0 (6.10) with 𝐻=𝐹−𝜆𝐺.

Corollary 6.4. Let 𝑦 be a minimizer to the isoperimetric problem 𝒥𝑦=_𝑎𝐼^𝛼_𝑏𝑡⟼𝐹𝑡,𝑦(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦𝑡)𝑏)ℐ𝑦=((⟶min,_𝑎𝐼^𝛼_𝑏𝑡⟼𝐺𝑡,𝑦(𝑡),^𝐶_𝑎𝐷^𝛽_𝑡𝑦𝑡)𝑏)𝑦(𝑎)((=𝜉,=𝑦_𝑎,𝑦(𝑏)=𝑦_𝑏. (6.11) If 𝑦 is not an extremal of ℐ, then there exists a constant 𝜆 such that 𝑦 satisfies (𝑏−𝑡)^𝛼−1𝜕₂𝐻𝑡,𝑦(𝑡),^𝐶_𝑎𝐷^𝛼_𝑡𝑦(𝑡)+_𝑡𝐷^𝛽_𝑏𝜏⟼(𝑏−𝜏)^𝛼−1𝜕₃𝐻𝜏,𝑦(𝜏),^𝐶_𝑎𝐷^𝛽_𝜏𝑦(𝜏)(𝑡)=0 (6.12) for all 𝑡∈(𝑎,𝑏), where 𝐻(𝑡,𝑦,𝑣)=𝐹(𝑡,𝑦,𝑣)−𝜆𝐺(𝑡,𝑦,𝑣).

Proof. Let 𝑘_𝛼(𝑥,𝑡)=(1/Γ(𝛼))(𝑥−𝑡)^𝛼−1, ℎ_1−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]). Let 𝑦 be a minimizer to 𝒥𝑦=^𝑏_𝑎𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),𝐵^𝛽_𝑃2𝑦(𝑡),𝐾^𝛾_𝑃3𝑦𝑡)ℐ𝑦=(𝑑𝑡⟶min,_𝑏𝑎𝐺𝑡,𝑦(𝑡),𝑦^′(𝑡),𝐵^𝛽_𝑃2𝑦(𝑡),𝐾^𝛾_𝑃3𝑦𝑡)𝑦(𝑎)(𝑑𝑡=𝜉,=𝑦_𝑎,𝑦(𝑏)=𝑦_𝑏. (6.13) If 𝑦 is not an extremal of ℐ, then there exists a constant 𝜆 such that 𝑦 satisfies 𝜕₂𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝑑𝜕𝑑𝑡³𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)−𝐴^𝛽_𝑃∗2𝜕₄𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)+𝐾^𝛾_𝑃∗3𝜕₅𝐻𝑦_𝑃^𝛽,𝛾_2,𝑃3(𝑡)=0 (6.14) 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 𝑘_𝛼(𝑥,𝑡)≡1. Then, the generalized fractional integral 𝐾^𝛼_𝑃1 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. Let  𝛼,𝛽∈(0,1), 𝜉∈ℝ, 𝑃₁=⟨0,1,1,1,0⟩, and  𝑃₂=⟨0,𝑡,1,1,0⟩. Consider the following problem: 𝒥𝑦=𝐾^𝛼_𝑃1𝑡⟼𝑡𝐾^𝛽_𝑃2𝑦(𝑡)+1−𝐾^𝛽_𝑃2𝑦(𝑡)²1)𝑦(0)(1)=√(⟶min,=1,𝑦24+¹₀𝑟_𝛽()11−𝜏1+𝜏^23/2𝑑𝜏, (7.1) with kernel ℎ_𝛽 such that ℎ_𝛽(𝑡,𝜏)=ℎ_𝛽(𝑡−𝜏) and ℎ_𝛽(0)=1. Here, the resolvent 𝑟_𝛽(𝑡) is related to the kernel ℎ_𝛽(𝑡) by 𝑟_𝛽(𝑡)=ℒ⁻¹[𝑠↦(1/𝑠ℎ_𝛽(𝑠))−1](𝑡), ℎ_𝛽(𝑠)=ℒ[𝑡↦ℎ_𝛽(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 𝑦(𝑡)=11+𝑡^23/2+^𝑡₀𝑟_𝛽()1𝑡−𝜏1+𝜏^23/2𝑑𝜏 (7.2) is the solution to the Volterra integral equation of first kind (see, e.g., Equation 16, page 114 of [47]) 𝐾^𝛽_𝑃2𝑦𝑡)=𝑡√(1+𝑡²1+𝑡², (7.3) it satisfies our generalized Euler-Lagrange equation (4.4), that is, 𝐾^𝛽_𝑃∗2𝜏⟼𝑘_𝛼(𝑏,𝜏)−𝐾^𝛽_𝑃∗2𝑦(𝜏)1−𝐾^𝛽_𝑃∗2𝑦(𝜏)²+𝜏(𝑡)=0. (7.4) In particular, for the kernel ℎ_𝛽(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: if 𝛼,𝛽>0, then the Mittag-Leffler function is defined by𝐸_𝛼,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ𝛼𝑘+𝛽. (7.5) This function appears naturally in the solution of fractional differential equations, as a generalization of the exponential function [52].

Example 7.2. Let  𝛼,𝛽∈(0,1), 𝜉∈ℝ, and  𝜉∉{±1/4}. Consider the following problem: 𝒥𝑦=₀𝐼^𝛼₁1+𝑦^′+^𝐶₀𝐷^𝛽_𝑡𝑦²1)ℐ𝑦=(⟶min,₀𝐼^𝛼₁𝑦^′+^𝐶₀𝐷^𝛽_𝑡𝑦²1)𝑦(0)(1)=(=𝜉,=0,𝑦¹₀𝐸_1−𝛽,1−(1−𝜏)^1−𝛽1−16𝜉²4𝜉𝑑𝜏, (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⟩ and kernels 𝑘_𝛼(𝑥−𝑡)=(1/Γ(𝛼))(𝑥−𝑡)^𝛼−1 and ℎ_1−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 𝑦(𝑡)=^𝑡₀𝐸_1−𝛽,1−(𝑡−𝜏)^1−𝛽1−16𝜉²4𝜉𝑑𝜏 (7.7)(i)is not an extremal for ℐ,(ii)satisfies 𝑦′+^𝐶₀𝐷^𝛽_𝑡[𝑦]=1−16𝜉²/4𝜉.Moreover, (7.7) satisfies (6.5) for 𝜆=2𝜉, that is, −𝑑)𝑑𝑡(1−𝑡^𝛼−1𝑦^′(𝑡)+^𝐶₀𝐷^𝛽_𝑡𝑦𝑡)1(1+𝑦^′(𝑡)+^𝐶₀𝐷^𝛽_𝑡𝑦(𝑡)²+−4𝜉_𝑡𝐷^𝛽₁𝜏⟼(1−𝜏)^𝛼−1𝑦^′(𝜏)+^𝐶₀𝐷^𝛽_𝜏𝑦𝜏)1(1+𝑦^′(𝜏)+^𝐶₀𝐷^𝛽_𝜏𝑦(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 the following result: if 𝑦 is a solution to the problem of extremizing 𝒥𝑦=^𝑏_𝑎𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)𝑘_𝛼(𝑏,𝑡)𝑑𝑡 (8.1) subject to 𝑦(𝑎)=𝑦_𝑎 and 𝑦(𝑏)=𝑦_𝑏, where 𝛼∈(0,1), then 𝜕₂𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)=1𝑘_𝛼()⋅𝑑𝑏,𝑡𝑘𝑑𝑡^𝛼(𝑏,𝑡)𝜕₃𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡). (8.2) We recognize on the right hand side of (8.2) the generalized weak dissipative parameter 𝛿(𝑡)=1𝑘_𝛼()⋅𝑑𝑏,𝑡𝑘𝑑𝑡^𝛼(𝑏,𝑡). (8.3)

8.1. Quantum Mechanics of the Damped Harmonic Oscillator

As a first application, let us consider kernel 𝑘_𝛼(𝑏,𝑡)=e^{𝛼(𝑏−𝑡)} and the Lagrangian 𝐿=1𝑦,𝑦̇2𝑚𝑦²̇−𝑉𝑦, (8.4) 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: (𝑡)(𝑡)1𝑦̈−𝛼𝑦̇=−𝑚𝑉^′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. 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] 𝐿𝑡,𝑦,𝑦̇=𝑚(𝑡)𝑦²̇2−𝜔²𝑦²2, (8.6) which describes a dynamical oscillatory system with exponentially increasing time-dependent mass, where 𝜔 is the frequency and 𝑚(𝑡)=𝑚₀e^−𝛾𝑏e^𝛾𝑡=𝑚₀e^𝛾𝑡, 𝑚₀=𝑚₀e^−𝛾𝑏. Using our generalized FALVA Euler-Lagrange equation (8.2) with kernel 𝑘_𝛼(𝑏,𝑡) to Lagrangian (8.6), we obtain 𝑦̈(𝑡)+𝛿(𝑡)+𝛾𝑦̇(𝑡)+𝜔²𝑦(𝑡)=0. (8.7) We study two particular kernels.(1)If we choose kernel 𝑘_𝛼(𝑏,𝑡)=𝜌+1^1−𝛼Γ(𝛼)𝑏^𝜌+1−𝑡^𝜌+1𝑡^𝜌, (8.8) defined in [53], then the Euler-Lagrange equation is 𝜕₂𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)=(1−𝛼)𝜌+1𝑡^𝜌𝑏^𝜌+1−𝑡^𝜌+1𝜕₃𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡). (8.9) In particular, when 𝜌→0, (8.8) becomes the kernel of the Riemann-Liouville fractional integral, and equation (8.9) gives 𝜕₂𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)=1−𝛼𝜕𝑏−𝑡³𝐹𝑡,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+1𝑡^𝜌𝑏^𝜌+1−𝑡^𝜌+1⟶0,if𝑡⟶∞or𝑡⟶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−𝑡0,if𝑡⟶∞,1−𝛼𝑏,if𝑡⟶0. (8.12) 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 kernel 𝑘_𝛼(𝑏,𝑡)=(cosh𝑏−cosh𝑡)^𝛼−1, then 𝜕₂𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)−𝑑𝜕𝑑𝑡³𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡)=−(𝛼−1)sinh𝑡𝜕cosh𝑏−cosh𝑡³𝐹𝑡,𝑦(𝑡),𝑦^′(𝑡),𝛿(𝑡)=−(𝛼−1)sinh𝑡⟶cosh𝑏−cosh𝑡𝛼−1,if𝑡⟶∞,0,if𝑡⟶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 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.

Acknowledgments

This work was supported by FEDER funds through COMPETE—Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia), within project PEst-C/MAT/UI4106/2011 with COMPETE no. FCOMP-01-0124-FEDER-022690. T. Odzijewicz was also supported by FCT through the Ph.D. fellowship SFRH/BD/33865/2009; A. B. Malinowska by Białystok University of Technology Grant S/WI/02/2011; D. F. M. Torres by FCT through the Portugal-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 Doctoral Programme Mathematics and Applications of Universities of Aveiro and Minho.