Short CommunicationComparison between the linearly correlated difference and the generalized Hukuhara difference of fuzzy numbers
Introduction
In the calculus theory of fuzzy number-valued functions, the definition of the difference of fuzzy numbers is still a research topic that needs to be improved due to the special structure of the space of fuzzy numbers. In the early days, in order to consider the algebraic structure of the space of fuzzy numbers, the addition and the scalar multiplication operations were introduced using Zadeh's extension principle. Naturally, the standard difference of fuzzy numbers is introduced by using the addition and scalar multiplication operations, which is similar to the definition of the difference in linear spaces. However, the standard difference suffers from many problems, such as , , etc. These shortcomings bring a lot of inconvenience to the introduction of the derivative of fuzzy number-valued functions. In 1983, a type of non-standard difference, that is, Hukuhara difference, was introduced by Puri and Ralscu [13], which was based on the corresponding difference of classical sets, because the fuzzy set can be regarded as a generalization of the classical set or the classical interval. To a certain extent, the Hukuhara difference makes up for some defects of the standard difference, but it does not always exist for any two fuzzy numbers. Essentially, the Hukuhara derivative based on the Hukuhara difference provides an important foundation for the establishment of fuzzy calculus theory. However, the problem is that the Hukuhara derivative is not a good tool for dealing with contractive fuzzy process [12]. Afterwards, Bede and Gal [4] partially improved the deficiency of the Hukuhara derivative introduced by unilaterally defining the differentiability with the help of the Hukuhara difference. In order to improve the shortcomings of the Hukuhara difference, Stefanini [17] introduced the generalized Hukuhara difference (gH-difference for short) in the space of fuzzy numbers. However, the gH-difference is only well-defined for interval numbers, and still does not always exist for fuzzy numbers. Compared with the Hukuhara difference, the gH-difference exists in a larger subset of fuzzy numbers. Correspondingly, the solution obtained by using the generalized Hukuhara derivative (gH-derivative for short) to solve the fuzzy differential equation is also better than the solution obtained by using the Hukuhara derivative. As an improvement of gH-difference, Bede and Stefanini [5] proposed the concept of the generalized difference of fuzzy numbers. Subsequently, Gomes and Barros [8] further improve the definition of the generalized difference. Although the generalized difference exists for any two fuzzy numbers, the complexity of its definition makes it difficult to calculate. Especially, the generalized derivative defined by using the generalized difference is difficult to be applied to solve fuzzy differential equations. To make up for the shortcomings of several types of differences mentioned above, Mazandarani et al. [9] introduced the granular difference in the space of fuzzy numbers.
It is well known that the arithmetic operations of fuzzy numbers are based on the Zadeh's extension principle. From the perspective of random variables, the arithmetic operations of fuzzy numbers actually imply the non-interactivity between two fuzzy numbers [19], [20], [21]. Inspired by this idea, Carlsson et al. [6] introduced the arithmetic operations of interactive fuzzy numbers by using the generalized extension principle. Generally speaking, the interactivity of a set of fuzzy numbers can be described by a joint distribution function. To avoid the use of the joint distribution function, Barros and Pedro [1] introduced the concept of linearly correlated fuzzy numbers. At the same time, they also proposed the definition of sum and difference defined by the form of the level sets for this kind of fuzzy numbers. After that, Esmi and Pedro et al. [7] constructed an operator using the coefficients of the linearly correlated fuzzy number. They found that the space of linearly correlated fuzzy numbers is linear if the basic fuzzy number is a non-symmetric fuzzy number. And in this case, the operator they introduced is a linear isomorphism from the 2-dimensional Euclidean space to the space of linearly correlated fuzzy number. Then, the addition and scalar multiplication in the space of linearly related fuzzy number can be defined by this linear isomorphism and similar operations in the Euclidean space. Therefore, the difference can be naturally derived from the addition and the scalar multiplication. However, when the basic fuzzy number is symmetric, it is impossible to directly propose a suitable difference through the addition and the scalar multiplication mentioned above, because the operator is no longer a linear isomorphism and the space of linearly correlated fuzzy number spaces is also not linear. To deal with this problem, the author [15] recently introduced an equivalence relationship in the 2-dimensional Euclidean space. With the aid of this equivalence relationship, the quotient set was defined and a bijection from the quotient set to the space of linearly correlated fuzzy numbers was constructed. Similar to the case where the basic fuzzy number is non-symmetric, a new type of difference, named the linearly correlated difference (LC-difference for short), was introduced by using the bijection. Coincidentally, the LC-difference and the gH-difference are equal for interval numbers. It is worth mentioning that LC-difference is adaptable regardless of whether the basic fuzzy number is symmetric or non-symmetric, and this difference always exists in the space of linearly correlated fuzzy numbers. Based on the LC-difference, the linearly correlated derivative (LC-derivative for short) was also proposed for the linearly correlated fuzzy number-valued function. At the same time, it should be noted that the LC-differentiability of a linearly correlated fuzzy-valued function can be conveniently characterized by the differentiability of its representation functions. As an application of the LC-derivative, the author [16] also discussed the solution of first-order linear fuzzy differential equations under the LC-differentiability. The purpose of this paper is to establish the relationship between the LC-difference and the gH-difference in the space of linearly correlated fuzzy numbers. In addition, the relationship between the LC-differentiability and the gH-differentiability will also be explored for linearly correlated fuzzy number-valued functions.
Section snippets
Preliminaries
In this section, we will review some basic concepts and operations related to fuzzy numbers and linearly correlated fuzzy numbers which are derived from [1], [3], [7], [10], [11], [13]. Throughout this paper, let denote the set of all real numbers and let denote the class of fuzzy sets with the following properties:
- (i)
A is normal, i.e., there exists such that ;
- (ii)
A is fuzzy convex, that is, for all and all ;
- (iii)
A is upper semicontinuous;
The relationship between the LC-difference and the gH-difference
In this section, we will examine the relationship between the LC-difference and the gH-difference in the space of linearly correlated fuzzy numbers by distinguishing whether the basic fuzzy number is symmetric. Theorem 3.1 Let be a non-symmetric fuzzy number and let . Assume that there exist two tuples and in such that , . Then If , then the gH-difference exists and ; If and the gH-difference exists, then ;
Proof
(i) Without loss
The relationship between the LC-derivative and the gH-derivative
Using the relationship between the LC-difference and the gH-difference, in this section, we will discuss the relationship between the LC-differentiability and the gH-differentiability for linearly correlated fuzzy number-valued functions. Definition 4.1 [5] Let be a fuzzy number-valued function. For , f is said to be gH-differentiable at provided that the limit exists in the sense of the metric d. For any , the metric d is defined by
Conclusions
In this paper, we discussed the relationship of the LC-difference and the gH-difference in the space of linearly correlated fuzzy numbers. The difference between the two types of differences lies mainly in the situation where the basic fuzzy number is non-symmetric. If the basic fuzzy number is symmetric, the LC-difference and the gH-difference are completely coincident. As we all know, the LC-difference always exists no matter whether the basic fuzzy number is symmetric or not, but the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11701425), the Science and Technology Planning Project of Gansu Province (No. 21JR1RE287) and the Fuxi Scientific Research Innovation Team of Tianshui Normal University (No. FXD2020-03). The author will express his deep gratitude to the editor-in-chief, the area editor and the anonymous reviewers for their constructive suggestions.
References (21)
- et al.
Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations
Fuzzy Sets Syst.
(2005) - et al.
Generalized differentiability of fuzzy-valued functions
Fuzzy Sets Syst.
(2013) - et al.
Fréchet derivative for linearly correlated fuzzy function
Inf. Sci.
(2018) - et al.
A note on the generalized difference and the generalized differentiability
Fuzzy Sets Syst.
(2015) A note on the extension principle for fuzzy sets
J. Math. Anal. Appl.
(1978)- et al.
Population growth model via interactive fuzzy differential equation
Inf. Sci.
(2019) - et al.
Differentials of fuzzy functions
J. Math. Anal. Appl.
(1983) Calculus for linearly correlated fuzzy number-valued functions
Fuzzy Sets Syst.
(2022)First-order linear fuzzy differential equations on the space of linearly correlated fuzzy numbers
Fuzzy Sets Syst.
(2022)A generalization of Hukuhara difference and division for interval and fuzzy arithmetic
Fuzzy Sets Syst.
(2010)