On Henstock-Stieltjes Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions

In this paper we introduce the notion of the Henstock-Stieltjes (HS) integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.


Introduction
As it is well known, the Henstock (H) integral for a real function was first defined by Henstock [1] in 1963.The Henstock (H) integral is a lot powerful and easier than the Lebesgue, Wiener and Richard Phillips Feynman integrals.Furthermore, it is also equal to the Denjoy and the Perron integrals [1] [2].In 2000, Congxin Wu and Zengtai Gong [3] introduced the notion of the Henstock (H) integrals of interval-valued functions and fuzzynumber-valued functions and obtained a number of their properties.In 2016, Yoon [4] introduced the intervalvalued Henstock-Stieltjes integral on time scales and investigated some properties of these integrals.In 1998, Lim et al. [5] introduced the notion of the Henstock-Stieltjes (HS) integral of real-valued function which was a generalization of the Henstock (H) integral and obtained its properties.
In this paper, we tend to introduce the notion of the Henstock-Stieltjes (HS) integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.
The paper is organized as follows.In Section two, we tend to give the preliminary terminology used in the present paper.Section three is dedicated to discussing the Henstock-Stieltjes (HS) integral of interval-valued functions.In Section four, we tend to introduce the Henstock-Stieltjes (HS) integral of fuzzy-number-valued functions.The last section provides conclusions., ;
For brevity, we write where the sum ∑ is understood to be over P, we write ( ) ( ) We write ( ) ( )d as the distance between intervals A and B.
We write ( ) ( ) , then there exists a unique integral value.
Proof Let the integral value is not unique and let Since for all 0, ε > there exists a ( ) → R be an increasing function.Then an interval-valued function , by Definition 3.3 there exists a unique interval number 0 0 0  with the property, for any 0 ε > there exists a ( ) Therefore, by Definition 2.3 we can obtain , then there exists a unique 1 H R ∈ with the property, given 0 ε > there exists a ( ) .
It is similar to find ( ) .
ii) Let ( ) ( ) Similarly, for four cases above we have ii) The proof is similar to Theorem 2.8 in [5].
Proof By definition of distance, then A  is called a fuzzy number.If A  satisfy the following conditions: 1) convex, 2) normal, 3) upper semi-continuous, 4) has the compact support, then A  is called a compact fuzzy number.
Let  R denote the set of all fuzzy numbers and C  R denote the set of all compact fuzzy numbers.Definition 4.2 [6] is called the distance between A  and .B  Lemma 4.1 [9] , where ( ) where ( ) IHS integral, we are able to get the properties of ( ) FHS integral, for example, 1) the linear, 2) monotone, 3) interval additive properties of ( ) FHS integral.

Conclusion
In this paper, we proposed the definition of the Henstock-Stieltjes (HS) integrals of interval-valued functions and fuzzy-number-valued functions and investigated some properties of those integrals.
be an increasing function and let f, g are Henstock-Stieltjes (HS) integrable with respect to α on [ ]

,
a b and the integral value is defined by


Using Theorem 4.1 and the properties of ( )