# Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

Authors: Jianke Zhang; Gaofeng Wang; Xiaobin Zhi; Chang Zhou
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.


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## Body

## 1. Introduction

Fractional calculus, as an extension of ordinary calculus, has about a 300-year-old history, which [1] plays an important role in dealing with many natural dynamical processes. The fractional variational problem is an important dynamical system. In recent years, under different differentiability, several researchers [2–9] studied the fractional Euler-Lagrange equations for general fractional variational problems.It is well known that the Riemann-Liouville (RL) and Caputo derivatives have some drawbacks. The RL derivative of a constant is not zero, and it demands initial conditions of noninteger order which are not physically determined. On the other hand, the Caputo derivative requires higher conditions of regularity for differentiability, which is specified only for differentiable functions. In 2015, a new fractional derivative [10] with a nonsingular kernel was proposed. Losada and Nieto studied several properties of the Caputo-Fabrizio (CF) fractional derivative [11] and other researchers tried to study the different equations (see [12] and the references therein). Under the CF differentiability, Zhang et al. [9] proposed the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function. However, Sheikh et al. [13, 14] pointed out that the kernel of the CF fractional derivative was nonsingular but was still nonlocal. Some researchers [15, 16] also concluded that the CF fractional derivative was not a derivative with a fractional order instead of a filter with a fractional parameter.To overcome these drawbacks, Yang et al. [17] proposed a new fractional derivative involving the normalized sinc function without singular kernel. Based on the generalized Mittag-Leffler function, a new operator with fractional order [18] was introduced by Atangana and Baleanu in 2016. The kernel of the Atangana-Baleanu (AB) fractional derivative is nonlocal and nonsingular, which have all the benefits of the CF derivative. In [19], the AB and CF derivatives were used in the Allen-Cahn reaction-diffusion model, and the modified models were solved numerically and numerical simulations were presented for different values of alpha. Abdeljawad and Baleanu [20] studied some properties of the AB fractional derivative.However, the majority of real-world optimization problems are often involved in data uncertainty or imprecision owing to measurement errors or some unexpected things. Fuzziness is a kind of very common uncertainty in the problems of the real world. Farhadinia [21] was the first one who introduced the concept of fuzzy variational problem and studied the optimality conditions for this problem. Fard and Salehi [22] and Soolaki et al. [23] also established the necessary optimality conditions for fuzzy fractional variational problems using the concept of Caputo and combined Caputo differentiability based on Hukuhara difference of fuzzy functions.In this paper, we extend the concept of Atangana-Baleanu fractional derivative to fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. In our results, the fuzzy admissible curvex~(t)=[x_(t,α),x¯(t,α)] is generalized Hukuhara differentiable, and the two real-valued functions x_(t,α) and x¯(t,α) are differentiable, which generalized the results of [21–23]. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.The paper is organized as follows. Section2 presents some preliminaries needed in the sequel. The notion of the Atangana-Baleanu fuzzy fractional derivative is defined in Section 3. In Section 4, the generalized fuzzy fractional Euler-Lagrange conditions for the fuzzy fractional variational problems with natural boundary conditions are obtained. In Section 5, the weakly sufficient optimality conditions for fuzzy fractional variational problems are proposed. Finally, we give a conclusion in Section 6.


## 2. Preliminaries

In this section, we recall some definitions and basic concepts which will be used in this paper. LetRn be an n-dimensional Euclidean space and let R+n be its nonnegative orthant. A fuzzy set on Rn is a mapping u~:Rn→[0,1]. For each fuzzy set u~, we denote its α-level set as [u~]α=x∈Rn∣u~x≥α for any α∈(0,1] and suppu={x∈Rn:u~(x)>0} (called support of u~); the closure of suppu in Rn is written as [u~]0.Definition 1.
A compact and convex fuzzy setu~ on Rn is a fuzzy set with the following properties:(1)
u~ is normal; that is, there exists x0∈Rn such that u~(x0)=1.(2)
u~ is an upper semicontinuous function.(3)
u~(λx+(1-λ)y)≥min{u~(x),u~(y)},x,y∈Rn,λ∈[0,1].(4)
suppu~=x∈Rn∣u~≥0 is the support of u~ and its closure cl(suppu~) is compact.It is well known that theα-cut set of a fuzzy number is a closed and bounded interval [u_α(x),u¯α(x)] for any α∈[0,1].Definition 2.
The left-hand endpointu_α(x) and the right-hand endpoint u¯α(x) of a fuzzy number u~(x) satisfy the following conditions:(1)
u_α(x) is a monotonically increasing left continuous function.(2)
u¯α(x) is a monotonically decreasing left continuous function.(3)
u_α(x)≤u¯α(x),0≤α≤1.For the two fuzzy numbers[u~]α=[u_α,u¯α] and [v~]α=[v_α,v¯α], the sum [u~]α+[v~]α and the product k·[u~]α for k∈R and all 0≤α≤1 are defined as (1)u~+v~α=u_α+v_α,u¯α+v¯α;u~+v~α=u_α-v¯α,u¯α-v_α.k u ~ α = [ k u _ α , k u ¯ α ] if k≥0 and ku~α=[ku¯α,ku_α] if k≤0.The product of the two fuzzy numbersu~α=[u_α,u¯α] and v~α=[v_α,v¯α] is defined as (2)u~⊙v~α=min⁡u_αv_α,u_αv¯α,u¯αv_α,u¯αv¯α,max⁡u_αv_α,u_αv¯α,u¯αv_α,u¯αv¯α.Let us denote the class of fuzzy numbers byRf. For u~,v~∈Rf, the Hausdorff distance d:Rf×Rf→R+∪{0} is defined as (3)du~,v~=supα∈0,1⁡max⁡u_α-v_α,u¯α-v¯α.It can be seen that the distanced is a metric in Rf, and it has the following properties:(1)
du~+w~,v~+w~=du~,v~,∀u~,w~,v~∈Rf,(2)
d(ku~,kv~)=kd(u~,v~),∀u~,v~∈Rf,k∈R,(3)
du~+v~,w~+m~≤du~,v~+dw~,m~,∀u~,w~,m~,v~∈Rf,(4)
(d,Rf) is a complete metric space.Definition 3 (see [24]).
The generalized Hukuhara difference of two fuzzy numbersx~∈Rf and y~∈Rf (gH-difference for short) is defined as follows: (4)x~⊖gHy~=z~⟺x~=y~+z~ory~=x~+-1z~.Ifz~=x~⊖gHy~ exists as a fuzzy number, then its level cuts [z_α,z¯α] are given by (5)z_α=min⁡x_α-y_α,x¯α-y¯α,z¯α=max⁡x_α-y_α,x¯α-y¯α for all α∈[0,1].Definition 4 (see [21]).
Leta~∈Rf and b~∈Rf. One writes a~⪯b~ if a_α≤b_α and a¯α≤b¯α for all α∈[0,1]. One also writes a~≺b~ if a~⪯b~, and there exists an α∗∈[0,1] so that a_α∗<b_α∗ and a¯α∗<b¯α∗. Moreover, a~≈b~ if a~⪯b~ and a~⪰b~; that is, [a]α=[b]α for all α∈[0,1].We say thata~,b~∈Rf are comparable if either a~⪯b~ or a~⪰b~ and noncomparable otherwise.A functionF~:K→Rf,K⊆Rn is said to be a fuzzy function. For each α∈[0,1], we associate with F~ the family of interval-valued functions given by F~α(t)=[f_(t,α),f¯(t,α)]. Here, the endpoint functions f_(t,α),f¯(t,α):K→R are called lower and upper functions of F~, respectively.Definition 5 (see [24]).
Lett∈(a,b) and h be such that t+h∈(a,b); then, the generalized Hukuhara derivative of a fuzzy-valued function x~(t):(a,b)→Rf at t is defined as (6)DgHx~t=limh→0⁡x~t+h⊖gHx~th.IfDgHx~(t)∈Rf exists, we say that x~ is generalized Hukuhara differentiable (gH-differentiable for short) at t.Definition 6 (see [21]).
One says that the fuzzy functionx~:[a,b]→Rf with α-level set xtα=[x_(t,α),x¯(t,α)] is continuous at t∈[a,b], if crisp functions x_(t,α) and x¯(t,α) are continuous functions at t∈[a,b] for all α∈[0,1].If the fuzzy functionx~:[a,b]→Rf is continuous in the metric d(·,·), then its definite integral exists. Furthermore, (7)∫abx~t_dtα=∫abx_t,αdt,∫abx~t¯dtα=∫abx¯t,αdt.Theorem 7 (see [25]).
Letx~:[a,b]→Rf be a fuzzy function. If x~ is gH-differentiable at t0∈[a,b], then for each α∈[0,1], one of the following cases holds:(a)
x_α and x¯α are differentiable at t0 and (8)DgHx~t0α=min⁡x_′t0,α,x¯′t0,α,max⁡x_′t0,α,x¯′t0,α;(b)
(x_)-′(t0,α),(x_)+′(t0,α),(x¯)-′(t0,α) and (x¯)+′(t0,α) exist and these satisfy (x_)-′(t0,α)=(x¯)+′(t0,α) and (x_)+′(t0,α)=(x¯)-′(t0,α). Moreover, (9)DgHx~t0α=min⁡x_-′t0,α,x¯-′t0,α,max⁡x_-′t0,α,x¯-′t0,α=min⁡x_+′t0,α,x¯+′t0,α,max⁡x_+′t0,α,x¯+′t0,α.Proposition 8 (see [25]).
LetF~α(x)=[f_(x,α),f¯(x,α)]:K→Rf be a fuzzy function. If F~ is gH-differentiable at x0∈K, then for each α∈[0,1], the real-valued function f_(x,α)+f¯(x,α):K→R is differentiable at x0. Moreover, (10)∂Fα_∂xix0+∂Fα¯∂xix0=∂f_x,α+f¯x,α∂xix0.Remark 9.
Chalco-Cano et al. [25] have pointed that the concept of generalized Hukuhara differentiable fuzzy functions is more general than level-wise differentiability and Hukuhara differentiability. For a gH-differentiable fuzzy function F~α(x)=[f_(x,α),f¯(x,α)]:K→Rf, the endpoint functions f_(x,α) and f¯(x,α) are not necessarily differentiable.Definition 10 (see [23]).
LetT be an open subset of R and F~:T→Rf and x0∈T. One says that F~(x0) is a local minimum (maximum) of F~(·) if there exists some ϵ>0 such that F~(x0)⪯(⪰)F~(x) when x∈Nϵ(x0)=x∈T∣x-x0≤ϵ.Theorem 11 (see [26]).
Letf be a real-valued function differentiable on the open interval I. If f has a local extremum at x∈I, then d/dxf(x)=0.Theorem 12 (see [23]).
LetF~:[a,b]→Rf be a fuzzy function. If the local minimum of F~ is attended in the point x∗∈[a,b], then the local minimum of real-valued endpoint functions f_(x,α) and f¯(x,α) is attended in the point x∗ for all α∈[0,1]. So, one has (11)df_x,αdxx∗=df¯x,αdxx∗=0.


## 3. Fuzzy Atangana-Baleanu Fractional Derivative

We recall some basic concepts with regard to the RL fractional calculus, Caputo fractional derivative, and the Atangana-Baleanu fractional derivative [18]. Given a function x(t):[a,b]→R, the RL fractional calculus of x of order γ∈(0,1) is defined as follows:(1)
The left RL fractional integral of orderγ>0 starting from a is defined by (12)IRLa+γxt=1Γγ∫att-τγ-1xτdτ.(2)
The right RL fractional integral of orderγ>0 ending at b>a is defined by (13)IRLb-γxt=1Γγ∫tbτ-tγ-1xτdτ.(3)
The left RL fractional derivative of order0<γ<1 starting at a is defined by (14)DRLa+γxt=ddtIRLa+1-γxt.(4)
The right RL fractional derivative of order0<γ<1 ending at b is defined by (15)DRLb-γxt=-ddtIRLb-1-γxt.(5)
The left Caputo fractional derivative of order0<γ<1 starting at a is defined by(16)Da+γCxt=1Γ1-γddt∫at1t-τγxτ-xadτ.Ifx is of class C1, then (17)Da+γCxt=1Γ1-γ∫at1t-τγx′τdτ.The new Caputo-Fabrizio (CF) fractional derivative [12, 15] can be obtained by changing the kernel (t-τ)-γ by the function exp(-γ(t-τ)/(1-γ)) and 1/Γ(1-γ) with M(γ)/(1-γ). That is, (18)Da+γCFxt=Mγ1-γ∫atexp⁡-γt-τ1-γx′τdτ, where M(γ) is a normalization function such that M(0)=M(1)=1. It is clear that if x is a constant function, then DγCFx(t)=0 as in the usual Caputo derivative, but contrary to the usual Caputo derivative, the kernel does not have singularity for t=τ.However, Atangana and Koca [15] pointed out that the integral associate is not a fractional operator but the average of the function and its integral. Some researchers also concluded that the operator was not a derivative with a fractional order instead of a filter with a fractional parameter. The fractional parameter can then be viewed as a filter regulator. To solve the above drawbacks, Atangana and Baleanu introduced a new operator with a fractional order based upon the generalized Mittag-Leffler function [18] as follows.Definition 13 (see [18]).
Letx∈H1(a,b),b>a,γ∈(0,1); then, the definition of the Atangana-Baleanu fractional derivative in the sense of left Caputo is described as(19)Da+γABCxt=Bγ1-γ∫atEγ-γt-τγ1-γx′τdτ. And in the left RL sense, it is described by(20)Da+γABRxt=Bγ1-γddt∫atEγ-γt-τγ1-γxτdτ.The associated fractional integral is described by(21)Ia+γABxt=1-γBγxt+γBγIa+γRLxt, where B(γ) stands for normalization function such that B(0)=B(1)=1 and (22)EγZ=∑k=0∞zkΓγk+1,Reγ>0,z∈C, is the Mittag-Leffler function of one parameter.Definition 14 (see [20]).
The right Atangana-Baleanu fractional derivative with Mittag-Leffler kernel in the RL sense of orderγ∈(0,1) is defined by (23)Db-γABRxt=-Bγ1-γddt∫tbEγ-γτ-tγ1-γxτdτ.Definition 15 (see [20]).
The right Atangana-Baleanu fractional integral with Mittag-Leffler kernel in the RL sense of orderγ∈(0,1) is defined by (24)Ib-γABxt=1-γBγxt+γBγIb-γRLxt.Definition 16 (see [20]).
The right Atangana-Baleanu fractional derivative with Mittag-Leffler kernel in the Caputo sense of orderγ∈(0,1) is defined by (25)Db-γABCxt=-Bγ1-γ∫tbEγ-γτ-tγ1-γx′τdτ.Proposition 17 (see [20]).
The right Atangana-Baleanu fractional derivatives with Mittag-Leffler kernel in the RL sense and in the Caputo sense are related by the identity(26)Db-γABCxt=Db-γABRxt-Bγ1-γxbEγ-γb-tγ1-γ.Proposition 18 (see [20] (integration by parts for the Atangana-Baleanu fractional derivative in the Caputo sense)).
Letx be a continuous function and y be of class C1. Then,(27)∫0bxtDa+γABCytdt=∫0bytDb-γABRxtdt+Bγ1-γytEγ,1,-γ/1-γ,b-1xtt=0t=b,∫0bxtDb-γABCytdt=∫0bytD0+γABRxtdt-Bγ1-γytEγ,1,-γ/1-γ,0+1xtt=0t=b, where the left generalized fractional integral operator is defined by (28)Eγ,μ,ω,a+αxt=∫att-τμ-1Eγ,μαωt-τγxτdτ,t>a, and the right generalized fractional integral operator is defined by (29)Eγ,μ,ω,b-αxt=∫tbτ-tμ-1Eγ,μαωτ-tγxτdτ,t<b, where Eγ,μα(z)=∑k=0∞(α)kzk/Γ(γk+μ)k! is the generalized Mittag-Leffler function which is defined for complex γ,μ,α,(Re(γ)>0).The fuzzy gH-fractional Caputo derivative of a fuzzy-valued function was introduced in [27]. Following [27], we denote the space of all continuous fuzzy-valued functions on [a,b]⊂R by C([a,b],Rf), the class of fuzzy functions with continuous first derivatives on [a,b]⊂R by C1([a,b],Rf), and the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval [a,b] by L([a,b],Rf).We introduce the fuzzy gH-Atangana-Baleanu fractional derivative as follows.Definition 19.
LetDgHx~(t)∈C([a,b],Rf)∩L([a,b],Rf). The fuzzy gH-Atangana-Baleanu fractional differentiable fuzzy-valued function in the Caputo sense of x~(t) ([gH]γABC-differentiable for short) is defined as follows: (30)Da+γgHABCx~t=Bγ1-γ∫atEγ-γt-τγ1-γDgHx~τdτ, where 0<γ≤1,t>a.Theorem 20.
LetDgHx~(t)∈C([a,b],Rf)∩L([a,b],Rf) be such that xtα=[x_(t,α),x¯(t,α)] for t∈[a,b] and 0≤α≤1. If the real-valued functions x_(t,α) and x¯(t,α) are Atangana-Baleanu-differentiable in the Caputo sense at t∈[a,b], then the fuzzy function x~(t) is [gH]γABC-differentiable at t∈[a,b] and (31)Da+γgHABCx~tα=min⁡Da+γABCx_t,α,Da+γABCx¯t,α,max⁡Da+γABCx_t,α,Da+γABCx¯t,α, where Da+γABCx_(t,α) and Da+γABCx¯(t,α) defined in (19).Proof.
Let the real-valued functionsx_(t,α) and x¯(t,α) be Atangana-Baleanu-differentiable in the Caputo sense at t∈[a,b], which means that x_(t,α) and x¯(t,α) are differentiable at t∈[a,b]. From case (a) of Theorem 7, we have that x~ is gH-differentiable; then, by Definition 19, we have that x~ is gHγABC-differentiable. Using case (a) of Theorem 7, for a<τ<t,1-γ>0, we have(32)Da+γgHABCx~tα=Bγ1-γ∫atEγ-γt-τγ1-γmin⁡x_′τ,α,x¯′τ,α,max⁡x_′τ,α,x¯′τ,αdτ=Bγ1-γmin⁡∫atEγ-γt-τγ1-γx_′τ,αdτ,∫atEγ-γt-τγ1-γx¯′τ,αdτ,Bγ1-γmax⁡∫atEγ-γt-τγ1-γx_′τ,αdτ,∫atEγ-γt-τγ1-γx¯′τ,αdτ=min⁡Da+γABCx_t,α,Da+γABCx¯t,α,max⁡Da+γABCx_t,α,Da+γABCx¯t,α.The proof is complete.Remark 21.
According to Remark9, a gH-differentiable fuzzy function x~α(t)=[x_(t,α),x¯(t,α)]:K→Rf, the endpoint functions x_(t,α) and x¯(t,α) are not necessarily differentiable. So, the endpoint functions x_(t,α) and x¯(t,α) are also not necessarily Atangana-Baleanu-differentiable in the Caputo sense. For example [25], we consider the fuzzy mapping x~:K→Rf defined by x(t,α)=C·(t-1), where C is a fuzzy interval defined via its α-levels by Cα=[1+α,2(3-α)]. It is well known that, in this case, x~ is a generalization of a linear function. Then,(33)xt,α=1+αt-1,23-αt-1ift≥1,23-αt-1,1+αt-1if0<t<1. We can see that the endpoint functions x_(t,α) and x¯(t,α) are not necessarily differentiable at t=1. So, x_(t,α) and x¯(t,α) are not necessarily Atangana-Baleanu-differentiable in the Caputo sense at t=1. However, x(t,α) is gH-differentiable and x′(t,α)=C for all t>0.Theorem 22.
LetDgHx~(t)∈C([a,b],Rf)∩L([a,b],Rf) be such that xtα=[x_(t,α),x¯(t,α)] for t∈[a,b] and 0≤α≤1. If x_-′(t,α),x_+′(t,α), x¯-′(t,α), and x¯+′(t,α) exist and these satisfied x_-′(t,α)=x¯+′(t,α) and x_+′(t,α)=x¯-′(t,α) at t∈a,b, then the fuzzy function x~(t) is gHγABC-differentiable at t∈[a,b] and (34)Da+γgHABCx~tα=min⁡Da+γ+ABCx_t,α,Da+γ+ABCx¯t,α,max⁡Da+γ+ABCx_t,α,Da+γ+ABCx¯t,α, where (35)Da+γ+ABCx_t,α=Bγ1-γ∫atEγ-γt-τγ1-γx_+′τ,αdτ,Da+γ+ABCx¯t,α=Bγ1-γ∫atEγ-γt-τγ1-γx¯+′τ,αdτ.Proof.
Letx_-′(t,α),x_+′(t,α), x¯-′(t,α), and x¯+′(t,α) exist, and these satisfy x_-′(t,α)=x¯+′(t,α) and x_+′(t,α)=x¯-′(t,α) at t∈[a,b]. From case (b) of Theorem 7, we have that x~ is gH-differentiable; then, by Definition 19, we have that x~ is gHγABC-differentiable. Using case (b) of Theorem 7, for a<τ<t,1-γ>0, we have (36)DgHx~tα=min⁡x_-′t,α,x¯-′t,α,max⁡x_-′t,α,x¯-′t,α=min⁡x_+′t,α,x¯+′t,α,max⁡x_+′t,α,x¯+′t,α,Da+γgHABCx~tα=Bγ1-γ∫atEγ-γt-τγ1-γDgHxταdτ=Bγ1-γ∫atEγ-γt-τγ1-γmin⁡x_+′τ,α,x¯+′τ,α,max⁡x_+′τ,α,x¯+′τ,αdτ=Bγ1-γmin⁡∫atEγ-γt-τγ1-γx_+′τ,αdτ,∫atEγ-γt-τγ1-γx¯+′τ,αdτ,Bγ1-γmax⁡∫atEγ-γt-τγ1-γx_+′τ,αdτ,∫atEγ-γt-τγ1-γx¯+′τ,αdτ=min⁡Da+γ+ABCx_t,α,Da+γ+ABCx¯t,α,max⁡Da+γ+ABCx_t,α,Da+γ+ABCx¯t,α.The proof is complete.Definition 23.
Letx~:[a,b]→Rf be gHγABC-differentiable at t∈(a,b). If x_(t,α) and x¯(t,α) are differentiable at t∈(a,b), one says that x~ is 1-gHγABC-differentiable at t if(37)1Da+γgHABCx~tα=Da+γABCx_t,α,Da+γABCx¯t,α,0≤α≤1,and that x~ is 2-gHγABC-differentiable at t∈(a,b) if(38)2Da+γgHABCx~tα=Da+γABCx¯t,α,Da+γABCx_t,α,0≤α≤1, where(39)Da+γABCx_t,α=Bγ1-γ∫atEγ-γt-τγ1-γddτx_τ,αdτ,Da+γABCx¯t,α=Bγ1-γ∫atEγ-γt-τγ1-γddτx¯τ,αdτ.Ifx_(t,α) and x¯(t,α) are not differentiable at t∈(a,b), but x_-′(t,α),x_+′(t, α), x¯-′(t,α), and x¯+′(t,α) exist and these satisfy x_-′(t,α)=x¯+′(t,α) and x_+′(t,α)=x¯-′(t,α) at t, we say that x~ is 3-gHγABC-differentiable at t if(40)3Da+γgHABCx~tα=Da+γ+ABCx_t,α,Da+γ+ABCx¯t,α,0≤α≤1, and that x~ is 4-gHγABC-differentiable at t∈(a,b) if(41)4Da+γgHABCx~tα=Da+γ+ABCx¯t,α,Da+γ+ABCx_t,α,0≤α≤1, where Da+γ+ABCx¯(t,α) and Da+γ+ABCx_(t,α) as in Theorem 22.Definition 24 (see [27]).
Letx~:[a,b]→Rf be a fuzzy function. A point t∈(a,b) is said to be a switching point for the gHγABC-differentiability of x, if in any neighborhood V of t there exist points t1<t<t2 such thattype (I) at t1 (37) holds while (38) does not hold and at t2 (38) holds and (37) does not hold, ortype (II) at t1 (38) holds while (37) does not hold and at t2 (37) holds and (38) does not hold.Theorem 25.
Letx~:[a,b]→Rf be a fuzzy function on [a,b]:(i)
Ifx~ is [(1)-gH]-differentiable at t∈[a,b], then Da+γgHABCx~tα=[Da+γABCx_t,α,Da+γABCx¯(t,α)].(ii)
Ifx~ is [(2)-gH]-differentiable at t∈[a,b], then Da+γgHABCx~tα=Da+γABCx¯t,α,Da+γABCx_t,α.Proof.
The proof is similar to [27].


## 4. The Necessary Optimality Conditions for Fuzzy Fractional Variational Problems

In this section, we consider the following fuzzy fractional variational problemFFVP with the Atangana-Baleanu fractional derivative in the Caputo sense. Let the fuzzy function x~(t)=[x_(t,α),x¯(t,α)]∈C1([0,b],Rf)∩L([0,b],Rf):FFVPmin⁡J~x~=∫0bL~t,x~t,Di0+γgHABCx~tdt, with x~(0)=x~0 and x~(b)=x~b and i=1,2, where x~0,x~b∈Rf. The assumptions are as follows:( 1 )  L~:[0,b]×Rf2→Rf is continuously differentiable with respect to all its arguments, which means L_ and L¯∈C1([0,b],Rf), and the first variation of the functional L~ must be vanished.( 2 ) Given any x~, the map t↦Db-βgHABC(∂3L~) is continuous.Hereafter, we denote by∂iL the partial derivative of the function L with respect to its ith argument. The fuzzy function x~(t) is i-gHγABC-differentiable with respect to the independent variable t for i∈{1,2}, which is denoted as Di0+γgHABCx~(t). We consider γ∈(0,1); Di0+γgHABCx~(t) denotes the Atangana-Baleanu fuzzy fractional derivative as follows: (42)Di0+γgHABCx~t=Bγ1-γ∫0tEγ-γt-τγ1-γDi,gHx~τdτ, for i=1,2.Theorem 26.
Assume thatx~∈C1([0,b],Rf). Then, its fuzzy gH-Atangana-Baleanu fractional derivative in the Caputo sense is continuous on the closed interval [0,b].Proof.
From Definition19, the fuzzy gH-Atangana-Baleanu fractional derivative in the Caputo sense of x~ is defined as follows: (43)Da+γgHABCx~t=Bγ1-γ∫atEγ-γt-τγ1-γDgHx~τdτ, where 0<γ≤1,t>a, and a=0.
Sincex~∈C1([0,b],Rf), DgHx~ is continuous on the closed interval [0,b]. It can be shown that Da+γgHABCx~(t) is also continuous on the closed interval [0,b].
Similar to [22], we define the concepts ofadmissible curve andfuzzy weak neighborhood under the assumption of fuzzy gH-Atangana-Baleanu fractional differentiability.Definition 27.
One says that the fuzzy curvex~(t)=[x_(t,α),x¯(t,α)] is admissible if it satisfies the end conditions and the two functions x_(t,α) and x¯(t,α) have a continuous Atangana-Baleanu fractional derivative in the Caputo sense of order γ,(0<γ,α<1) in [0,b]. One denotes the set of all admissible curves by X~0d.Definition 28.
A fuzzy weak neighborhoodN~ε(x~) of a fuzzy curve x^~=x^~(t) is the set of all admissible curves x~∈X~0d satisfying, for all t∈[0,b],(44)dx~t,x^~t≤ε,∀t∈0,b,dDi0+γgHABCx~t,Di0+γgHABCx^~t≤ε,∀t∈0,b,where ε is a small positive real number and i=1,2.
For the problem ofFFVP, it is to find the admissible curves x~∈X~0d in a fuzzy weak neighborhood, if any exists, such that J~ is minimized. Let x~∗=x~∗(t) be a minimized fuzzy curve of FFVP, which means(45)J~x~⪰J~x~∗ for all admissible curves x~∈X~0d in the fuzzy weak neighborhood N~ε(x~∗). According to Definition 4, (46)J_αx~≥J_αx~∗,J¯αx~≥J¯αx~∗ for all α∈[0,1] and all admissible curves x~∈X~0d in N~ε(x~∗).Proposition 29.
Letx~(t) and h~(t) be all admissible curves in X~0d. Then, for k∈R+ and γ∈[0,1], (47)D0+γgHABCx~t+kh~t=min⁡D0+γABCx_t,α+kh_t,α,D0+γABCx¯t,α+kh¯t,α,max⁡D0+γABCx_t,α+kh_t,α,D0+γABCx¯t,α+kh¯t,α=min⁡D0+γABCx_t,α+kD0+γABCh_t,α,D0+γABCx¯t,α+kD0+γABCh¯t,α,max⁡D0+γABCx_t,α+kD0+γABCh_t,α,D0+γABCx¯t,α+kD0+γABCh¯t,α.Proof.
According to the assumptions,x~(t) and h~(t) are all admissible curves in X~0d and the fuzzy functions x~(t) and h~(t) are i-gHγABC-differentiable with respect to the independent variable t for i∈{1,2}.
From Definitions23 and 27, x_(t,α),h_(t,α),x¯(t,α), and h¯(t,α) are differentiable at t∈(0,b). So, by Definition 19, we have (48)D0+γgHABCx~t+kh~t=Bγ1-γ∫0tEγ-γt-τγ1-γDgHx~+kh~τdτ.
Sincek∈R+, we get (49)x~t+kh~tα=x_t,α+kh_t,α,x¯t,α+kh¯t,α.
Using Theorem7(a), we have (50)DgHx~+kh~τα=min⁡x_′τ,α+kh_′τ,α,x¯′τ,α+kh¯′t,α,max⁡x_′τ,α+kh_′τ,α,x¯′τ,α+kh¯′t,α.
By the above equations, we obtain(51)D0+γgHABCx~t+kh~t=min⁡D0+γABCx_t,α+kh_t,α,D0+γABCx¯t,α+kh¯t,α,max⁡D0+γABCx_t,α+kh_t,α,D0+γABCx¯t,α+kh¯t,α=min⁡D0+γABCx_t,α+kD0+γABCh_t,α,D0+γABCx¯t,α+kD0+γABCh¯t,α,max⁡D0+γABCx_t,α+kD0+γABCh_t,α,D0+γABCx¯t,α+kD0+γABCh¯t,α.To develop the necessary conditions for the problem ofFFVP, we assume that x~∗(t) is the desired function; let ϵ∈R+ and define a family of curves x~(t)=x~∗(t)+ϵh~(t), where h~(t)=[h_(t,α),h¯(t,α)] is an arbitrary admissible variation and the two real-valued functions h_(t,α) and h¯(t,α) are differentiable for all α∈[0,1]. We present the necessary condition for the problem of FFVP as follows.Theorem 30.
Letx~∗ be a solution of FFVP. Then, x~∗ is a solution of the fractional Euler-Lagrange equations(52)∂2L_Z+Db-γABR∂4L_Z=0,∂3L_Z+Db-γABR∂5L_Z=0,∂2L¯Z+Db-γABR∂4L¯Z=0,∂3L¯Z+Db-γABR∂5L¯Z=0,for all t∈[0,b], where (53)Z=t,x_∗t,α,x¯∗t,α,Di0+γABCx_∗t,α,Di0+γABCx¯∗t,α,i∈1,2.Proof.
Considerx~∗+ϵh~ as an admissible variation for ϵ∈R+, and h~(t)=[h_(t,α),h¯(t,α)]:[0,b]→Rf is an arbitrary admissible variation. The two real-valued functions h_(t,α) and h¯(t,α) are differentiable for all α∈[0,1]. Let j~(ϵ)=J~(x~∗+ϵh~): (54)j~ϵ=∫0bL~t,x~∗t+ϵh~t,Di0+γgHABCx~∗t+ϵh~tdt for i=1,2. From Proposition 29, we have (55)D0+γgHABCx~∗t+ϵh~t=min⁡D0+γABCx_∗t,α+ϵh_t,α,D0+γABCx¯∗t,α+ϵh¯t,α,max⁡D0+γABCx_∗t,α+ϵh_t,α,D0+γABCx¯∗t,α+ϵh¯t,α=min⁡D0+γABCx_∗t,α+ϵD0+γABCh_t,α,D0+γABCx¯∗t,α+ϵD0+γABCh¯t,α,max⁡D0+γABCx_∗t,α+ϵD0+γABCh_t,α,D0+γABCx¯∗t,α+ϵD0+γABCh¯t,α.
Then, the lower and upper bounds ofj~ are (56)j_ϵ=∫0bL_⋯αdt,j¯ϵ=∫0bL¯⋯αdtrespectively, where (57)⋯α=t,x_∗t,α+ϵh_t,α,x¯∗t,α+ϵh¯t,α,D0+γABCx_∗t,α+ϵD0+γABCh_t,α,D0+γABCx¯∗t,α+ϵD0+γABCh¯t,α.
Sincex~ is a solution of FFVP, then (58)dj_ϵdϵϵ=0=dj¯ϵdϵϵ=0=0.
Computingj_′(ϵ)ϵ=0, we get(59)∫0b∂2L_Zh_+∂3L_Zh¯+∂4L_ZD0+γABCh_+∂5L_ZD0+γABCh¯dt=0,where(60)Z=t,x_∗t,α,x¯∗t,α,Di0+γgHABCx_∗t,α,Di0+γgHABCx¯∗t,α,i∈1,2.Using Proposition 18, we obtain(61)∫0b∂4L_ZD0+γABCh_dt=∫0bh_Db-γABR∂4L_Zdt+Bγ1-γh_Eγ,1,-γ/1-γ,b-1∂4L_Zt=0t=b,∫0b∂5L_ZD0+γABCh¯dt=∫0bh¯Db-γABR∂5L_Zdt+Bγ1-γh¯Eγ,1,-γ/1-γ,b-1∂5L_Zt=0t=b.
Considering the end conditions ofh_(0,α)=h¯(0,α)=h_(b,α)=h¯(b,α)=0 and substituting (61) into (59), then (62)∫0b∂2L_Z+Db-γABR∂4L_Zh_+∂3L_Z+Db-γABR∂5L_Zh¯dt=0.
Sinceh_(t,α) and h¯(t,α) are arbitrary elsewhere, we get(63)∂2L_Z+Db-γABR∂4L_Z=0,∂3L_Z+Db-γABR∂5L_Z=0 for all t∈[0,b]. Following the scheme of obtaining (59), (61), and (63) and computing j¯′(ϵ)|ϵ=0=0, we can also obtain (64)∂2L¯Z+Db-γABR∂4L¯Z=0,∂3L¯Z+Db-γABR∂5L¯Z=0 for all t∈[0,b], where(65)Z=t,x_∗t,α,x¯∗t,α,Di0+γgHABCx_∗t,α,Di0+γgHABCx¯∗t,α,i∈1,2.
Next, we consider a more general class of fractional variational problems, forA∈(0,b) and the functionalGFFVPmin⁡J~x~=∫AbL~t,x~t,Di0+γgHABCx~tdt, with x~(A)=x~A and x~(b)=x~b and i=1,2, where x~A,x~b∈Rf. The assumptions are the same as previously stated for L~(t,x~(t),Di0+γgHABCx~(t)).Theorem 31.
Ifx~ is a solution of GFFVP, then x~ satisfies(66)Db-γABR∂4L_Z-DA-γABR∂4L_Z=0,Db-γABR∂5L_Z-DA-γABR∂5L_Z=0,Db-γABR∂4L-Z-DA-γABR∂4L-Z=0,Db-γABR∂5L-Z-DA-γABR∂5L-Z=0,on [0,A],(67)∂2L_Z+Db-γABR∂4L_Z=0,∂3L_Z+Db-γABR∂5L_Z=0,∂2L¯Z+Db-γABR∂4L¯Z=0,∂3L¯Z+Db-γABR∂5L¯Z=0,on [A,b], and(68)Eγ,1,-γ/1-γ,A-1∂4L_Z-Eγ,1,-γ/1-γ,b-1∂4L_Z=0,Eγ,1,-γ/1-γ,A-1∂5L_Z-Eγ,1,-γ/1-γ,b-1∂5L_Z=0,Eγ,1,-γ/1-γ,A-1∂4L¯Z-Eγ,1,-γ/1-γ,b-1∂4L¯Z=0,Eγ,1,-γ/1-γ,A-1∂5L¯Z-Eγ,1,-γ/1-γ,b-1∂5L¯Z=0, at t=0.Proof.
Letx~∗ be a solution of GFFVP. Consider x~∗+ϵh~ as an admissible variation for ϵ∈R+ with ϵ≪1, and h~(t)=[h_(t,α),h¯(t,α)]:[0,b]→Rf is an arbitrary admissible variation. The two real-valued functions h_(t,α) and h¯(t,α) are differentiable for all α∈[0,1]. Let j~(ϵ)=J~(x~∗+ϵh~): (69)j~ϵ=∫AbL~t,x~∗t+ϵh~t,Di0+γgHABCx~∗t+ϵh~tdt for i=1,2. From Proposition 29, we have(70)D0+γgHABCx~∗t+ϵh~t=min⁡D0+γABCx_∗t,α+ϵD0+γABCh_t,α,D0+γABCx¯∗t,α+ϵD0+γABCh¯t,α,max⁡D0+γABCx_∗t,α+ϵD0+γABCh_t,α,D0+γABCx¯∗t,α+ϵD0+γABCh¯t,α.
The lower and upper bounds ofj~ are (71)j_ϵ=∫AbL_⋯αdt,j¯ϵ=∫AbL¯⋯αdt, respectively, where (72)⋯α=t,x_∗t,α+ϵh_t,α,x¯∗t,α+ϵh¯t,α,D0+γABCx_∗t,α+ϵD0+γABCh_t,α,D0+γABCx¯∗t,α+ϵD0+γABCh¯t,α.
Sincex~ is a solution of GFFVP, then (73)dj_ϵdϵϵ=0=dj¯ϵdϵϵ=0=0.
Computingj_′(ϵ)|ϵ=0, we get (74)∫Ab∂2L_Zh_+∂3L_Zh¯+∂4L_ZD0+γABCh_+∂5L_ZD0+γABCh¯dt=∫Ab∂2L_Zh_+∂3L_Zh¯dt+∫0b∂4L_ZD0+γABCh_+∂5L_ZD0+γABCh¯dt-∫0A∂4L_ZD0+γABCh_+∂5L_ZD0+γABCh¯dt=∫Ab∂2L_Zh_+∂3L_Zh¯dt+∫0bh_Db-γABR∂4L_Zdt+Bγ1-γh_Eγ,1,-γ/1-γ,b-1∂4L_Zt=0t=b+∫0bh¯Db-γABR∂5L_Zdt+Bγ1-γh¯Eγ,1,-γ/1-γ,b-1∂5L_Zt=0t=b-∫0Ah_DA-γABR∂4L_Zdt-Bγ1-γh_Eγ,1,-γ/1-γ,A-1∂4L_Zt=0t=A-∫0Ah¯DA-γABR∂5L_Zdt-Bγ1-γh¯Eγ,1,-γ/1-γ,A-1∂5L_Zt=0t=A=0, where (75)Z=t,x_∗t,α,x¯∗t,α,Di0+γgHABCx_∗t,α,Di0+γgHABCx¯∗t,α,i∈1,2.According to the end conditions ofh_(A,α)=h¯(A,α)=h_(b,α)=h¯(b,α)=0, the above equation can be deduced as follows:(76)∫0Ah_Db-γABR∂4L_Z-DA-γABR∂4L_Z+h¯Db-γABR∂5L_Z-DA-γABR∂5L_Zdt+∫Abh_∂2L_Z+Db-γABR∂4L_Z+h¯∂3L_Z+Db-γABR∂5L_Zdt+Bγ1-γh_0,αEγ,1,-γ/1-γ,A-1∂4L_Z-Eγ,1,-γ/1-γ,b-1∂4L_Z+h¯0,αEγ,1,-γ/1-γ,A-1∂5L_Z-Eγ,1,-γ/1-γ,b-1∂5L_Z=0.Sinceh_(t,α),h¯(t,α),h_(0,α), and h¯(0,α) are arbitrary elsewhere, we get the three necessary conditions(77)Db-γABR∂4L_Z-DA-γABR∂4L_Z=0,Db-γABR∂5L_Z-DA-γABR∂5L_Z=0, on [0,A], (78)∂2L_Z+Db-γABR∂4L_Z=0,∂3L_Z+Db-γABR∂5L_Z=0, on [A,b], and (79)Eγ,1,-γ/1-γ,A-1∂4L_Z-Eγ,1,-γ/1-γ,b-1∂4L_Z=0,Eγ,1,-γ/1-γ,A-1∂5L_Z-Eγ,1,-γ/1-γ,b-1∂5L_Z=0, at t=0.Following the scheme above and computingj¯′(ϵ)ϵ=0=0, we can also obtain (80)Db-γABR∂4L¯Z-DA-γABR∂4L¯Z=0,Db-γABR∂5L¯Z-DA-γABR∂5L¯Z=0, on [0,A], (81)∂2L¯Z+Db-γABR∂4L¯Z=0,∂3L¯Z+Db-γABR∂5L¯Z=0, on [A,b], and (82)Eγ,1,-γ/1-γ,A-1∂4L¯Z-Eγ,1,-γ/1-γ,b-1∂4L¯Z=0,Eγ,1,-γ/1-γ,A-1∂5L¯Z-Eγ,1,-γ/1-γ,b-1∂5L¯Z=0, at t=0.Remark 32.
The Euler-Lagrange equations (52) are easily extended to the case of several variations.


## 5. The Weakly Sufficient Optimality Conditions for Fuzzy Fractional Variational Problems

In this section, we introduce the weak optimality conditions for fuzzy fractional variational problems. Let us consider the following multiobjective fractional variational problems defined for eachα∈[0,1]: MFVPαminJ_αx~,J¯αx~whereJ_αx~=∫0bL_t,x~t,Di0+γgHABCx~tdt,J¯αx~=∫0bL¯t,x~t,Di0+γgHABCx~tdt with x~(t)∈C1([0,b],Rf)∩L([0,b],Rf), x~(0)=x~0, x~(b)=x~b, and i=1,2, where x~0,x~b∈Rf. The assumptions are the same as the problem of FFVP.Lemma 33.
Ifx~∗ is a Pareto efficient solution for MFVPα, for each α∈[0,1], then x~∗ is a solution for FFVP.Proof.
Suppose thatx~∗ is not a solution of FFVP; then, there exists x^~∈N~ϵ(x~∗). From Definition 28, we get x^~∈C1([0,b],Rf)∩L([0,b],Rf) such that J~x^~⪯J~(x~∗). In particular, there exists α∗ such that (83)J_α∗x^~≤J_α∗x~∗,J¯α∗x^~≤J¯α∗x~∗ with a strict inequality, which contradicts the fact that x~∗ is a Pareto efficient solution for MFVPα.
Consider a weighting scalar problem forMFVPα as follows: WSPαminJ_αx~+J¯αx~whereJ_αx~=∫0bL_t,x~t,Di0+γgHABCx~tdt,J¯αx~=∫0bL¯t,x~t,Di0+γgHABCx~tdt with x~(t)∈C1([0,b],Rf)∩L([0,b],Rf), x~(0)=x~0, x~(b)=x~b, and i=1,2, where x~0,x~b∈Rf. The assumptions are the same as the problem of FFVP.Lemma 34.
Ifx~∗ is a local optimal solution of WSPα, for each α∈[0,1], then x~∗ is a solution of FFVP.Proof.
SinceWSPα is a weighting scalar problem for MFVPα and the weights are strictly positive, then x~∗ is a Pareto efficient solution for MFVPα for all α∈[0,1]. From Lemma 33, x~∗ is a solution for FFVP.For the real-valued fractional variational problemWSPα, we have the following necessary optimality condition.Theorem 35.
Letx~∗ be a local optimal solution of WSPα. Then, x~∗ is a solution of the fractional Euler-Lagrange equations(84)∂2L_+L¯Z+Db-γABR∂4L_+L¯Z=0,∂3L_+L¯Z+Db-γABR∂5L_+L¯Z=0, for all t∈[0,b], where (85)Z=t,x_∗t,α,x¯∗t,α,Di0+γgHABCx_∗t,α,Di0+γgHABCx¯∗t,α,i∈1,2.Proof.
Letx~∗(t)=[x_∗(t,α),x¯∗(t,α)]∈X~0d be a solution of WSPα and h~(t)=[h_(t,α),h¯(t,α)]:[0,b]→Rf be [(i)-gH]γABC-differentiable fuzzy function with the end condition of h_(0,α)=h¯(0,α)=h_(b,α)=h¯(b,α)=0 for i=1,2 and α,γ∈[0,1]. For ϵ≪1, we have x~∗+ϵh~∈N~ϵ(x~∗).
Letj be the function defined in a neighborhood of zero as (86)jϵ=J_x~∗+ϵh~+J¯x~∗+ϵh~.
Since the functionx~∗(t) is a local optimal solution of WSPα, then ϵ=0 is a minimizer of j(ϵ). Therefore, we get (87)djϵdϵϵ=0=0.
Computingdj(ϵ)/dϵϵ=0, we obtain (88)djϵdϵϵ=0=dJ_αx~∗+ϵh~+J¯αx~∗+ϵh~dϵϵ=0=∫0bddϵL_+L¯t,x~∗t+ϵh~t,Di0+γgHABCx~∗t+ϵh~tϵ=0dt=∫0bddϵL_+L¯⋄ϵ=0dt=∫0b∂2L_+L¯Zh_t,α+∂3L_+L¯Zh¯t,α+∂4L_+L¯ZD0+γABCh_t,α+∂5L_+L¯ZD0+γABCh¯t,αdt,where(89)⋄=t,x_∗t,α+ϵh_t,α,x¯∗t,α+ϵh¯t,α,Di0+γgHABCx_∗t,α+ϵD0+γABCh_t,α,Di0+γgHABCx¯∗t,α+ϵD0+γABCh¯t,α,Z=t,x_∗t,α,x¯∗t,α,Di0+γgHABCx_∗t,α,Di0+γgHABCx¯∗t,α,i=1,2.Using Proposition 18, we have(90)∫0b∂4L_+L¯ZD0+γABCh_dt=∫0bh_Db-γABR∂4L_+L¯Zdt+Bγ1-γh_Eγ,1,-γ/1-γ,b-1∂4L_+L¯Zt=0t=b,∫0b∂5L_+L¯ZD0+γABCh¯dt=∫0bh¯Db-γABR∂5L_+L¯Zdt+Bγ1-γh¯Eγ,1,-γ/1-γ,b-1∂5L_+L¯Zt=0t=b.
So, we get(91)djϵdϵϵ=0=∫0b∂2L_+L¯Z+Db-γABR∂4L_+L¯Zh_t,α+∂3L_+L¯Z+Db-γABR∂5L_+L¯Zh¯t,αdt=0.
Sinceh~(t)=[h_(t,α),h¯(t,α)]∈X~0d is arbitrary elsewhere, we have the fractional Euler-Lagrange equations (84).Definition 36.
One says that the real-valued functionL(t,x,y,z,q) is convex in K⊆R5 if ∂jL exist and are continuous for j=1,2,3,4,5, and the condition (92)Lt,x+x1,y+y1,z+z1,q+q1-Lt,x,y,z,q≥∂2Lt,x,y,z,qx1+∂3Lt,x,y,z,qy1+∂4Lt,x,y,z,qz1+∂5Lt,x,y,z,qq1holds for every (t,x,y,z,q),(t,x+x1,y+y1,z+z1,q+q1)∈K.Theorem 37.
If functionL_+L¯ as in WSPα is convex in [a,b]×R5, then each solution of the fractional Euler-Lagrange equations (84) minimizes FFVP, when restricted to the boundary conditions of x~(0)=x~0 and x~(b)=x~b.Proof.
Letx~(t)=[x_(t,α),x¯(t,α)]∈X~0d be a solution of (84); h~(t)=[h_(t,α),h¯(t,α)]:[0,b]→Rf is [(i)-gH]γABC-differentiable fuzzy function with the end condition of h_(0,α)=h¯(0,α)=h_(b,α)=h¯(b,α)=0 for i=1,2 and α,γ∈[0,1]. For ϵ≪1, we have x~+ϵh~∈N~ϵ(x~).
Since the functionL_+L¯ as in WSPα is convex, we get (93)J_αx~+ϵh~+J¯αx~+ϵh~-J_αx~+J¯αx~=∫0bL_+L¯t,x~t+ϵh~t,Di0+γgHABCx~t+ϵh~t-L_+L¯t,x~t,Di0+γgHABCx~tdt=∫0bL_+L¯⋄-L_+L¯Zdt≥∫0b∂2L_+L¯Zϵh_t,α+∂3L_+L¯Zϵh¯t,α+∂4L_+L¯ZϵD0+γABCh_t,α+∂5L_+L¯ZϵD0+γABCh¯t,αdt, where (94)⋄=t,x_∗t,α+ϵh_t,α,x¯∗t,α+ϵh¯t,α,Di0+γgHABCx_∗t,α+ϵD0+γABCh_t,α,Di0+γgHABCx¯∗t,α+ϵD0+γABCh¯t,α,Z=t,x_∗t,α,x¯∗t,α,Di0+γgHABCx_∗t,α,Di0+γgHABCx¯∗t,α,i=1,2.
Using Proposition18, we can obtain (90). According to the end condition of h_(0,α)=h¯(0,α)=h_(b,α)=h¯(b,α)=0 and (90), we have (95)J_αx~+ϵh~+J¯αx~+ϵh~-J_αx~+J¯αx~≥∫0b∂2L_+L¯Z+Db-γABR∂4L_+L¯Zϵh_t,α+∂3L_+L¯Z+Db-γABR∂5L_+L¯Zϵh¯t,αdt.
Sincex~(t) is a solution of (84), then (96)J_αx~+ϵh~+J¯αx~+ϵh~≥J_αx~+J¯αx~, which means that x~(t) is a local optimal solution of WSPα. From Lemma 34, x~(t) is also a solution of FFVP.


## 6. Example

Example 1.
We consider the followingFFVP to find the minimum of(97)minJ~x~=∫01Di0+γgHABCx~t2dt,i=1,2,0<α,γ<1subjecttox~0=0,x~1=2+α,4-α·γBγΓγ.
We first derive theα-level set of J~ as follows: (98)J~x~α=∫01Di0+γgHABCx~t2_,Di0+γgHABCx~t2¯dt,i=1,2.
According to Definition23, if i=1, then (99)Di0+γgHABCx~tα=D0+γABCx_t,α,D0+γABCx¯t,α, and if i=2, then (100)Di0+γgHABCx~tα=D0+γABCx¯t,α,D0+γABCx_t,α.From Theorem30, the Euler-Lagrange equations are given as(101)D1-γABRD0+γABCx_t,α=0,D1-γABRD0+γABCγx¯t,α=0, with x~(0)=0,x~(1)=[2+α,4-α]·γ/B(γ)Γ(γ) for 0<α,γ<1.According to Theorem 2.1 and Proposition3.4 in [20], we have (102)I1-γABD1-γABRft=ft,I0+γABD0+γABCft=ft-f0.For (101), by applying I1-γAB to both sides, we obtain(103)D0+γABCx_t,α=0,D0+γABCx¯t,α=0.By applyingI0+γAB to both sides of (103), we get(104)x_t,α=x_0,α+k1·γtγ-1BγΓγ,x¯t,α=x¯0,α+k2·γtγ-1BγΓγ, since, according to Remark  3.1 in [20], the nonzero function g(t)=γtγ-1/B(γ)Γ(γ) whose fractional ABC derivative is zero.Fromx~(0)=0,x~(1)=[2+α,4-α]·γ/B(γ)Γ(γ), we have(105)x_t,α=2+α·γtγ-1BγΓγ,x¯t,α=4-α·γtγ-1BγΓγ.So, the solution is(106)x~tα=2+α,4-α·γtγ-1BγΓγ for 0<α,γ<1 and t∈[0,1].


## 7. Conclusions

In this paper, We studied the AB fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-AB fractional differentiability, we proposed the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on the AB fractional derivative. The advantage of the new fractional derivative has no singularity and no locality. However, the Euler-Lagrange equations ofFFVP are very difficult to solve; we should develop numerical methods to solve this problem. As future works, we should also study which type of the fractional order derivative is more appropriate in the actual variational problem.



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*Source: 2740678-2018-03-19.xml*