The Henstock–Stieltjes integral for fuzzy-number-valued functions

Abstract

In this paper, we firstly define and discuss the Henstock–Stieltjes integral for fuzzy-number-valued functions which is an extension of the usual fuzzy Riemann–Stieltjes integral. In addition, several necessary and sufficient conditions of the integrability for fuzzy-number-valued functions are given by means of the Henstock–Stieltjes integral of real-valued functions and Henstock integral of fuzzy-number-valued functions. Secondly, the continuity and the differentiability of the primitive for the fuzzy Henstock–Stieltjes integral are discussed. We find that there exists a fuzzy-number-valued function which is fuzzy Henstock–Stieltjes integrable, but whose primitive is not α-differentiable almost everywhere. Thirdly, we introduce some quadrature rules for the fuzzy Henstock–Stieltjes integral by giving error bounds for the mappings of bounded variation and of Lipschitz type. We also consider the generalization of classical quadrature rules, such as midpoint-type, trapezoidal and Simpson’s quadrature. Finally, we propose the concept of weak equi-integrability for sequences of fuzzy Henstock–Stieltjes integrable functions. Under this concept, we prove two convergence theorems for sequences of the fuzzy Henstock–Stieltjes integrable functions. At the same time, the formula of integration by parts is also studied.

Introduction

Since the concept of fuzzy sets [31] was first introduced by Zadeh in 1965, it has been studied extensively from many different aspects of the theory and applications, especially in information science, such as linguistic information system and approximate reasoning [32], [33], [34], [35], fuzzy topology [1], [2], fuzzy analysis [20], fuzzy decision making, fuzzy logic [36], [37] and so on. Recently, in order to complete the theory of fuzzy integrals and to meet the solving need of the fuzzy differential equations [3], [9], [13], fuzzy integrals of fuzzy-number-valued functions have been studied by many authors from different points of views, including Nanda [19], Wu and Gong [26] and other authors [4], [8], [10], [15], [21], [29]. On the other hand, these integrals have also been successfully applied to various application fields, such as, in decision-making models [17], image processing, pattern recognition [15], and others.

In the classical real analysis, as a natural extension for Riemann integral and Lebesgue integral, the Stieltjes integral plays an important role in probability theory, stochastic processes, physics, econometrics, biometrics and numerical analysis [5], [6], [7], [24]. In fact, the establishment and development of the Stieltjes integral was related to the moment of inertia in physics [18]. Until 1909, Riesz presented a general expression for the linear functional of the space of the continuous functions in a finite interval by Stieltjes integral [23]. After Riesz’ work, people find that the Stieltjes integral is a powerful tool in several branches of mathematics. In a fuzzy environment, in 1968, Zadeh studied and defined the probability measure of a fuzzy event by using the Lebesgue-Stieltjes integral of the membership function [38].

It is well known that the notion of the Stieltjes integral for fuzzy-number-valued functions was originally proposed by Nanda [19] in 1989. Nevertheless, as Wu et al. [27] pointed out that the existence of supremum and infimum for a bounded set of fuzzy numbers was not simple as initially thought. That is, Nanda’s concept of fuzzy Riemann–Stieltjes integral in [19] was incorrect. To overcome these limitation, generalizations of the fuzzy Riemann–Stieltjes integral were considered by many scholars [8], [21], [22], [28]. In 1998, Wu [28] proposed the concept of fuzzy Riemann–Stieltjes integral by means of the representation theorem of fuzzy-number-valued functions, whose membership function could be obtained by solving a nonlinear programming problem, but it is difficult to calculate and extend to the higher-dimensional space. In 2006, Ren et al. introduced the concept of two kinds of fuzzy Riemann–Stieltjes integral for fuzzy-number-valued functions [21], [22] and showed that a continuous fuzzy-number-valued function was fuzzy Riemann–Stieltjes integrable with respect to a real-valued increasing function. However, we note that if a fuzzy-number-valued function has some kind of discontinuity or non-integrability, the existing methods have been restricted. In real analysis, the Henstock integral is designed to integrate highly oscillatory functions which the Lebesgue integral fails to do. It is well-known that the Henstock integral includes the Riemann, improper Riemann, Lebesgue and Newton integrals [11], [16]. Though such an integral was defined by Denjoy in 1912 and also by Perron in 1914, it was difficult to handle using their definitions. But with the Riemann-type definition introduced more recently by Henstock [11] in 1963 and also independently by Kurzweil [14], the definition is now simple. Furthermore, the proof involving the integral also turns out to be easy. For more detailed results about the Henstock integral, we refer to [16]. In 2001, Wu et al. introduced and discussed the fuzzy Henstock integral of the fuzzy-number-valued functions which was an extension of Kaleva integral.

To overcome the limitations of the existing studies and to characterize continuous linear functionals on the space of Henstock integrable fuzzy-number-valued functions, in this paper, the concept of the Henstock–Stieltjes integral for fuzzy-number-valued functions is defined and discussed, and some useful results for this integral are shown. We shall prove that the fuzzy Henstock–Stieltjes integral is an extension of the usual fuzzy Riemann–Stieltjes integral. Meanwhile, the integrability, the continuity and the differentiability of the primitive, numerical calculus of the integration, and the convergence theorems of the fuzzy Henstock–Stieltjes integral are also obtained. Since the fuzzy Henstock–Stieltjes integral is a generalization of the fuzzy Henstock integral, the paper can also be viewed as an extension of our earlier work in [26].

The rest of this paper is organized as follows. To make our analysis possible, in Section 2 we shall review the relevant concepts and properties of fuzzy sets and the definition of α-differentiability for fuzzy-number-valued functions. In addition, the monotone convergence theorem of the Henstock–Stieltjes integral for real-valued functions is proposed in this paper. Section 3 is devoted to discussing the properties of the fuzzy Henstock–Stieltjes integral. In Section 4 we shall focus our attention on the class of the fuzzy Henstock–Stieltjes integrable functions. In Section 5 the continuity and the differentiability of the primitive for the fuzzy Henstock–Stieltjes integral are discussed. We find that there exists a fuzzy-number-valued function which is fuzzy Henstock–Stieltjes integrable, but whose primitive is not α-differentiable almost everywhere. In Section 6 we introduce some quadrature rules for the fuzzy Henstock–Stieltjes integral of mappings by giving error bounds for mappings of bounded variation and of Lipschitz type. We also consider the generalization of classical quadrature rules, such as midpoint-type, trapezoidal and Simpson’s quadrature. In Section 7 we obtain two convergence theorems for sequences of the fuzzy Henstock–Stieltjes integrals. Finally, the formula of integration by parts is presented.