Nonlinear Operators in Fixed Point Theory with Applications to Fractional Differential and Integral EquationsView this Special Issue
Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability
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.
Fractional calculus, as an extension of ordinary calculus, has about a 300-year-old history, which  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  with a nonsingular kernel was proposed. Losada and Nieto studied several properties of the Caputo-Fabrizio (CF) fractional derivative  and other researchers tried to study the different equations (see  and the references therein). Under the CF differentiability, Zhang et al.  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.  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  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 , 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  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  was the first one who introduced the concept of fuzzy variational problem and studied the optimality conditions for this problem. Fard and Salehi  and Soolaki et al.  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.
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 ). 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 ). 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 ). 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 ). 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 ). 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 ). 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.  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 ). 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 ). Let be a real-valued function differentiable on the open interval . If has a local extremum at , then .
Theorem 12 (see ). 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 . 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  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  as follows.
Definition 13 (see ). 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 byThe 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 ). The right Atangana-Baleanu fractional derivative with Mittag-Leffler kernel in the RL sense of order is defined by
Definition 15 (see ). The right Atangana-Baleanu fractional integral with Mittag-Leffler kernel in the RL sense of order is defined by
Definition 16 (see ). The right Atangana-Baleanu fractional derivative with Mittag-Leffler kernel in the Caputo sense of order is defined by
Proposition 17 (see ). 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  (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 . Following , 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 haveThe 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 , 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 ifand 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 ). 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 .
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 , 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