Journal of Function Spaces
Volume 2018 (2018), Article ID 2740678, 15 pages
https://doi.org/10.1155/2018/2740678

Research Article

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

Jianke Zhang, Gaofeng Wang, Xiaobin Zhi, and Chang Zhou

Department of Mathematics, School of Science, Xi’an University of Posts and Telecommunications, Xi’an, China

Correspondence should be addressed to Jianke Zhang; jiankezh@163.com

Received 14 September 2017; Accepted 1 January 2018; Published 19 March 2018

Academic Editor: Talat Nazir

Copyright © 2018 Jianke Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 curve is generalized Hukuhara differentiable, and the two real-valued functions and 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. Section 2 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. Let be an -dimensional Euclidean space and let be its nonnegative orthant. A fuzzy set on is a mapping For each fuzzy set , we denote its -level set as for any and (called support of ); the closure of in is written as .

Definition 1. A compact and convex fuzzy set on is a fuzzy set with the following properties:

(1) is normal; that is, there exists such that .(2) is an upper semicontinuous function.(3).(4) is the support of and its closure is compact.

It is well known that the cut set of a fuzzy number is a closed and bounded interval for any .

Definition 2. The left-hand endpoint and the right-hand endpoint of a fuzzy number satisfy the following conditions:

(1) is a monotonically increasing left continuous function.(2) is a monotonically decreasing left continuous function.(3)

For the two fuzzy numbers and , the sum and the product for and all are defined as

if and if .

The product of the two fuzzy numbers and is defined as

Let us denote the class of fuzzy numbers by . For , the Hausdorff distance is defined as

It can be seen that the distance is a metric in , and it has the following properties:

(1),(2),(3),(4) is a complete metric space.

Definition 3 (see [24]). The generalized Hukuhara difference of two fuzzy numbers and (gH-difference for short) is defined as follows:

If exists as a fuzzy number, then its level cuts are given by

for all .

Definition 4 (see [21]). Let and . One writes if and for all . One also writes if , and there exists an so that and . Moreover, if and ; that is, for all .

We say that are comparable if either or and noncomparable otherwise.

A function is said to be a fuzzy function. For each , we associate with the family of interval-valued functions given by . Here, the endpoint functions are called lower and upper functions of , respectively.

Definition 5 (see [24]). Let and be such that ; then, the generalized Hukuhara derivative of a fuzzy-valued function at is defined as

If exists, we say that is generalized Hukuhara differentiable (gH-differentiable for short) at

Definition 6 (see [21]). One says that the fuzzy function with level set is continuous at , if crisp functions and are continuous functions at for all .

If the fuzzy function is continuous in the metric , then its definite integral exists. Furthermore,

Theorem 7 (see [25]). Let be a fuzzy function. If is differentiable at , then for each , one of the following cases holds:

(a) and are differentiable at and

(b) and exist and these satisfy and . Moreover,

Proposition 8 (see [25]). Let be a fuzzy function. If is differentiable at , then for each , the real-valued function is differentiable at . Moreover,

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 differentiable fuzzy function , the endpoint functions and are not necessarily differentiable.

Definition 10 (see [23]). Let be an open subset of and and . One says that is a local minimum (maximum) of if there exists some such that when .

Theorem 11 (see [26]). Let be a real-valued function differentiable on the open interval . If has a local extremum at , then .

Theorem 12 (see [23]). Let be a fuzzy function. If the local minimum of is attended in the point , then the local minimum of real-valued endpoint functions and is attended in the point for all . So, one has

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 , the RL fractional calculus of of order is defined as follows:

(1)The left RL fractional integral of order starting from is defined by

(2)The right RL fractional integral of order ending at is defined by

(3)The left RL fractional derivative of order starting at is defined by

(4)The right RL fractional derivative of order ending at is defined by

(5)The left Caputo fractional derivative of order starting at is defined by

If is of class , then

The new Caputo-Fabrizio (CF) fractional derivative [12, 15] can be obtained by changing the kernel by the function and with . That is,

where is a normalization function such that . It is clear that if is a constant function, then as in the usual Caputo derivative, but contrary to the usual Caputo derivative, the kernel does not have singularity for .

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]). Let ; then, the definition of the Atangana-Baleanu fractional derivative in the sense of left Caputo is described as

And in the left RL sense, it is described by

The associated fractional integral is described by

where stands for normalization function such that and

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 order is defined by

Definition 15 (see [20]). The right Atangana-Baleanu fractional integral with Mittag-Leffler kernel in the RL sense of order is defined by

Definition 16 (see [20]). The right Atangana-Baleanu fractional derivative with Mittag-Leffler kernel in the Caputo sense of order is defined by

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

Proposition 18 (see [20] (integration by parts for the Atangana-Baleanu fractional derivative in the Caputo sense)). Let be a continuous function and be of class . Then,

where the left generalized fractional integral operator is defined by

and the right generalized fractional integral operator is defined by

where is the generalized Mittag-Leffler function which is defined for complex .

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 by , the class of fuzzy functions with continuous first derivatives on by , and the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval by .

We introduce the fuzzy gH-Atangana-Baleanu fractional derivative as follows.

Definition 19. Let . The fuzzy gH-Atangana-Baleanu fractional differentiable fuzzy-valued function in the Caputo sense of (-differentiable for short) is defined as follows:

where

Theorem 20. Let be such that for and . If the real-valued functions and are Atangana-Baleanu-differentiable in the Caputo sense at , then the fuzzy function is -differentiable at and

where and defined in (19).

Proof. Let the real-valued functions and be Atangana-Baleanu-differentiable in the Caputo sense at , which means that and are differentiable at . From case (a) of Theorem 7, we have that is differentiable; then, by Definition 19, we have that is -differentiable. Using case (a) of Theorem 7, for , we have

The proof is complete.

Remark 21. According to Remark 9, a -differentiable fuzzy function , the endpoint functions and are not necessarily differentiable. So, the endpoint functions and are also not necessarily Atangana-Baleanu-differentiable in the Caputo sense. For example [25], we consider the fuzzy mapping defined by , where is a fuzzy interval defined via its levels by . It is well known that, in this case, is a generalization of a linear function. Then,

We can see that the endpoint functions and are not necessarily differentiable at . So, and are not necessarily Atangana-Baleanu-differentiable in the Caputo sense at . However, is differentiable and for all .

Theorem 22. Let be such that for and . If , , and exist and these satisfied and at , then the fuzzy function is -differentiable at and

where

Proof. Let , , and exist, and these satisfy and at . From case (b) of Theorem 7, we have that is differentiable; then, by Definition 19, we have that is -differentiable. Using case (b) of Theorem 7, for , we have

The proof is complete.

Definition 23. Let be -differentiable at . If and are differentiable at , one says that is differentiable at if

and that is differentiable at if

where

If and are not differentiable at , but , , , and exist and these satisfy and at , we say that is differentiable at if

and that is differentiable at if

where and as in Theorem 22.

Definition 24 (see [27]). Let be a fuzzy function. A point is said to be a switching point for the differentiability of , if in any neighborhood of there exist points such that

 type (I) at (37) holds while (38) does not hold and at (38) holds and (37) does not hold, or type (II) at (38) holds while (37) does not hold and at (37) holds and (38) does not hold.

Theorem 25. Let be a fuzzy function on :

(i)If is [(1)-gH]-differentiable at , then .(ii)If is [(2)-gH]-differentiable at , then .

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 problem with the Atangana-Baleanu fractional derivative in the Caputo sense. Let the fuzzy function :

with and and , where . The assumptions are as follows:

   is continuously differentiable with respect to all its arguments, which means and , and the first variation of the functional must be vanished.

Given any , the map is continuous.

Hereafter, we denote by the partial derivative of the function with respect to its th argument. The fuzzy function is -differentiable with respect to the independent variable for , which is denoted as . We consider ; denotes the Atangana-Baleanu fuzzy fractional derivative as follows:

for

Theorem 26. Assume that . Then, its fuzzy gH-Atangana-Baleanu fractional derivative in the Caputo sense is continuous on the closed interval

Proof. From Definition 19, the fuzzy gH-Atangana-Baleanu fractional derivative in the Caputo sense of is defined as follows:

where , and .
Since , is continuous on the closed interval It can be shown that is also continuous on the closed interval
Similar to [22], we define the concepts of admissible curve and fuzzy weak neighborhood under the assumption of fuzzy gH-Atangana-Baleanu fractional differentiability.

Definition 27. One says that the fuzzy curve is admissible if it satisfies the end conditions and the two functions and have a continuous Atangana-Baleanu fractional derivative in the Caputo sense of order in . One denotes the set of all admissible curves by .

Definition 28. A fuzzy weak neighborhood of a fuzzy curve is the set of all admissible curves satisfying, for all ,

where is a small positive real number and
For the problem of , it is to find the admissible curves in a fuzzy weak neighborhood, if any exists, such that is minimized. Let be a minimized fuzzy curve of , which means

for all admissible curves in the fuzzy weak neighborhood . According to Definition 4,

for all and all admissible curves in .

Proposition 29. Let and be all admissible curves in . Then, for and ,

Proof. According to the assumptions, and are all admissible curves in and the fuzzy functions and are -differentiable with respect to the independent variable for .
From Definitions 23 and 27, , and are differentiable at . So, by Definition 19, we have

Since , we get

Using Theorem 7(a), we have

By the above equations, we obtain

To develop the necessary conditions for the problem of , we assume that is the desired function; let and define a family of curves , where is an arbitrary admissible variation and the two real-valued functions and are differentiable for all . We present the necessary condition for the problem of as follows.

Theorem 30. Let be a solution of . Then, is a solution of the fractional Euler-Lagrange equations

for all , where

Proof. Consider as an admissible variation for , and is an arbitrary admissible variation. The two real-valued functions and are differentiable for all . Let :

for From Proposition 29, we have

Then, the lower and upper bounds of are

respectively, where

Since is a solution of , then

Computing , we get

where

Using Proposition 18, we obtain

Considering the end conditions of and substituting (61) into (59), then

Since and are arbitrary elsewhere, we get

for all Following the scheme of obtaining (59), (61), and (63) and computing , we can also obtain

for all , where

Next, we consider a more general class of fractional variational problems, for and the functional

with and and , where . The assumptions are the same as previously stated for .

Theorem 31. If is a solution of , then satisfies

on ,

on , and

at

Proof. Let be a solution of . Consider as an admissible variation for with , and is an arbitrary admissible variation. The two real-valued functions and are differentiable for all . Let :

for From Proposition 29, we have

The lower and upper bounds of are

respectively, where

Since is a solution of , then

Computing , we get

where

According to the end conditions of , the above equation can be deduced as follows:

Since , and are arbitrary elsewhere, we get the three necessary conditions

on ,

on , and

at

Following the scheme above and computing , we can also obtain

on ,

on , and

at

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 each :

with , , , and , where . The assumptions are the same as the problem of .

Lemma 33. If is a Pareto efficient solution for , for each , then is a solution for .

Proof. Suppose that is not a solution of ; then, there exists . From Definition 28, we get such that . In particular, there exists such that

with a strict inequality, which contradicts the fact that is a Pareto efficient solution for .
Consider a weighting scalar problem for as follows:

with , , , and , where . The assumptions are the same as the problem of .

Lemma 34. If is a local optimal solution of , for each , then is a solution of .

Proof. Since is a weighting scalar problem for and the weights are strictly positive, then is a Pareto efficient solution for for all . From Lemma 33, is a solution for .

For the real-valued fractional variational problem , we have the following necessary optimality condition.

Theorem 35. Let be a local optimal solution of . Then, is a solution of the fractional Euler-Lagrange equations

for all , where

Proof. Let be a solution of and be -differentiable fuzzy function with the end condition of for and . For , we have .
Let be the function defined in a neighborhood of zero as

Since the function is a local optimal solution of , then is a minimizer of . Therefore, we get

Computing , we obtain

where

Using Proposition 18, we have

So, we get

Since is arbitrary elsewhere, we have the fractional Euler-Lagrange equations (84).

Definition 36. One says that the real-valued function is convex in if exist and are continuous for , and the condition

holds for every .

Theorem 37. If function as in is convex in , then each solution of the fractional Euler-Lagrange equations (84) minimizes , when restricted to the boundary conditions of and

Proof. Let be a solution of (84); is -differentiable fuzzy function with the end condition of for and . For , we have .
Since the function as in is convex, we get

where

Using Proposition 18, we can obtain (90). According to the end condition of and (90), we have

Since is a solution of (84), then

which means that is a local optimal solution of . From Lemma 34, is also a solution of .

6. Example

Example 1. We consider the following to find the minimum of

We first derive the level set of as follows:

According to Definition 23, if , then

and if , then

From Theorem 30, the Euler-Lagrange equations are given as

with for .

According to Theorem 2.1 and Proposition   in [20], we have

For (101), by applying to both sides, we obtain

By applying to both sides of (103), we get

since, according to Remark   in [20], the nonzero function whose fractional ABC derivative is zero.

From , we have

So, the solution is

for and .

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 of 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 11701446, 11601420, 11401469, and 60974082) and the Science Plan Foundation of the Education Bureau of Shaanxi Province (no. 2013JK 1130).

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