The equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets

Abstract

In this paper, with the help of the convergence of sequences of fuzzy numbers with respect to the Lebesgue measure, we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric, the sendograph metric and the $L_p$ metric. We prove that these convergences are equivalent under proper conditions. In addition, by applying our result, we give a new characterization of compact sets in fuzzy number space with the sendograph metric.

Introduction

Since the concept of fuzzy numbers was firstly introduced in 1970s, it has been studied extensively from many different aspects of the theory and applications such as fuzzy topology, fuzzy analysis, fuzzy logic and fuzzy decision making, etc. (see [1], [2], [3], [4], [6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]).

The theory of fuzzy numbers is a powerful tool for modelling uncertainty and for processing vague or subjective information in mathematical models. In particular, the convergence of sequences of fuzzy numbers plays an important role in studying the continuity of fuzzy integrals, fuzzy control problems and fuzzy optimization problems which are widely used in fuzzy information theory and fuzzy signal systems (see [5], [8], [9], [10], [11], [23]). Recently, many authors have discussed the convergence of sequences of fuzzy numbers and obtained many important results (see [4], [13], [14], [15], [16], [17], [20], [24], [25]). Kaleva [16] firstly defined three kinds of convergences in fuzzy number space, and studied the relationship between these convergences. Diamond and Kloeden [4] investigated the relationship among convergences of sequences of fuzzy numbers with respect to the supremum metric, the endograph metric, the sendograph metric and the $L_p$ metric. Medar and Flores [20] compared the level convergence with convergences of sequences of fuzzy numbers with respect to several types of metrics, including the endograph metric, the sendograph metric and the supremum metric. It has been showed that all the previous convergences are equivalent in the subspace of elements $u\in E^n$ such that the level function $r\in [0\text{,}1]\to [u{]}^r\in \mathcal{K}({\mathbb{R}}^n)$ is continuous with respect to the Hausdorff metric. Aytar [1], Aytar and Pehlivan [2], Kumar and Kumar [18], Nuray [22], Wu and Li [24], Wu and Wang [25] and many others, also discussed sequences of fuzzy numbers later on.

On the other hand, the concept of compactness is one of most important concepts in fuzzy analysis. So far some authors have studied the characterization of compact sets in fuzzy number space with the sendograph metric. Diamond and Kloeden [4] firstly gave a characterization of compact sets in fuzzy number space with the sendograph metric without the proof. Fan [6] pointed out that the characterization in [4] was incorrect, and got a correct characterization of compact sets in fuzzy number space with the sendograph metric. Greco [12] studied the characterization of relatively compact sets in the space of fuzzy sets which are upper semi-continuous and normal with nonempty compact support, and then obtained a general characterization.

The $L_p$ metric is regarded in general as a kind of important metric in fuzzy number space $E^n$. However, few work has been done so far about the relationship between the convergence with respect to the $L_p$ metric and those convergences with respect to other metrics. Therefore, we discuss the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric, the sendograph metric and the $L_p$ metric. We verify that these convergences are equivalent under proper conditions. Finally, by using the above-mentioned result, we obtain a new characterization of compact sets in fuzzy number space with the sendograph metric.