The equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets☆
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 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 such that the level function 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 metric is regarded in general as a kind of important metric in fuzzy number space . However, few work has been done so far about the relationship between the convergence with respect to the 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 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.
Section snippets
Preliminaries
In this section, we recall some basic concepts and definitions of fuzzy numbers. Throughout this paper the pair will be a metric space. The Hausdorff metric is defined on the collection of all nonempty compact subsets of X denoted by :where .
Especially, if and are closed intervals on real line , then we have .
For the results about fuzzy number space , we recall that
Convergences of sequences of fuzzy numbers
In the present section, we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric, the sendograph metric and the metric. First of all, we compare the convergence with respect to the metric with the convergence with respect to the Lebesgue measure . Theorem 3.1 For each , let , then Proof For every and , we haveBy , we
Applications to the characterization of compact sets
In this section, we apply the main result of Section 3 to give a new characterization of compact sets in fuzzy number space with the sendograph metric. To this end, the following several definitions and lemmas will be used in the sequel. Definition 4.1 For , if for arbitrary , there exists such thatthen we say that U is equi-right-continuous at . Definition 4.2 Let . If for each , there exists such that for all then we[12]
[4]
Conclusion
With the help of the convergence of sequences of fuzzy numbers with respect to the Lebesgue measure, we discuss the relationship among convergences of sequences of fuzzy numbers with respect to the endograph metric, the sendograph metric and the metric. We show that all the previous convergences are equivalent under proper conditions. As an application, we give a new characterization of compact sets in fuzzy number space with the sendograph metric. We hope that our results would provide some
Acknowledgements
The authors are grateful to the editors and the referees for their valuable comments and suggestions that improved the presentation of this paper.
References (28)
Statistical limit points of sequences of fuzzy numbers
Information Sciences
(2004)- et al.
Statistically monotonic and statistically bounded sequences of fuzzy numbers
Information Sciences
(2006) Neoclassical analysis: fuzzy continuity and convergence
Fuzzy Sets and Systems
(1995)- et al.
On topological properties of the Choquet weak convergence of capacity functionals of random sets
Information Sciences
(2007) On the compactness of fuzzy numbers with sendograph
Fuzzy Sets and Systems
(2004)- et al.
Endographic approach on supremum and infimum of fuzzy numbers
Information Sciences
(2004) Sendograph metric and relatively compact sets of fuzzy sets
Fuzzy Sets and Systems
(2006)- et al.
Topologocal properties on the space of fuzzy sets
Journal of Mathematical Analysis and Applications
(2000) On the convergence of fuzzy sets
Fuzzy Sets and Systems
(1985)- et al.
On fuzzy metric spaces
Fuzzy Sets and Systems
(1984)
On the ideal convergence of sequences of fuzzy numbers
Information Sciences
On sequences of fuzzy numbers
Fuzzy Sets and Systems
On the continuity of the concave integral
Fuzzy Sets and Systems
Fuzzy metric and convergence based on the symmetric difference
Fuzzy Sets and Systems
Cited by (4)
The core of a double sequence of fuzzy numbers
2021, Fuzzy Sets and SystemsCitation Excerpt :Moreover concepts of limit inferior and limit superior are vital ingredients for various mathematical concepts such as semicontinuity of real-valued functions. Since the notion of sequences of fuzzy numbers was first introduced by Matloka [24], several types of convergence have been studied for a sequence of fuzzy numbers and the relationships among them have been investigated [32,42,49]. Fuzzy number space is partially ordered and supremum and infimum of any bounded set of fuzzy numbers always exist (see [46]).
Random metric space of fuzzy numbers
2015, International Journal of General SystemsOn interrelationships between fuzzy metric structures
2013, Iranian Journal of Fuzzy Systems
- ☆
Project supported by National Natural Science Foundation of China (No: 70773029).