On sequences of fuzzy sets and fuzzy set-valued mappings
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RESEARCH
Open Access
On sequences of fuzzy sets and fuzzy set-valued mappings Masamichi Kon1* and Hiroaki Kuwano2 * Correspondence: [email protected] 1 Graduate School of Science and Technology, Hirosaki University, 3, Bunkyo, Hirosaki, Aomori 036-8561, Japan Full list of author information is available at the end of the article
Abstract Based on level sets of fuzzy sets, we propose definitions of limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings. Then, their properties are derived. Limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings are fuzzified ones of them for crisp sets, where ‘crisp’ means ‘non-fuzzy’. MSC: Primary 03E72; secondary 90C70 Keywords: sequence of fuzzy sets; fuzzy set-valued mapping; fuzzy set-valued analysis
1 Introduction and preliminaries The usefulness and importance of limits of sequences of crisp sets, and limits (continuity) and derivatives of crisp set-valued mappings have been recognized in many areas, for example, variational analysis, set-valued optimization, stability theory, sensitivity analysis, etc. For details, see, for example, [–]. The concept of limits of sequences of crisp sets is interesting and important for itself, and it is necessary to introduce the concepts of limits and derivatives of crisp set-valued mappings. Typical and important applications of them are (i) set-valued optimization and (ii) stability theory and sensitivity analysis for mathematical models. For the case (ii), consider the following system. Some mathematical model outputs the set of optimal values W ∗ (u) ⊂ R and the set of optimal solutions S∗ (u) ⊂ Rn for a given input parameter u ∈ Rm . Then W ∗ and S∗ are crisp set-valued mappings. Stability theory deals with the continuity of W ∗ and S∗ . Sensitivity analysis deals with the derivative of W ∗ . In this article, limits of sequences of fuzzy sets, and limits and derivatives of fuzzy setvalued mappings are considered. They are generalizations of them for crisp sets. The aim of this article is to propose those concepts and to investigate their properties systematically. Some research works deal with limits of sequences of fuzzy numbers or fuzzy sets with bounded supports [–], while few research works deal with limits of sequences of fuzzy sets. In addition, some research works deal with limits (continuity) and derivatives of fuzzy number or fuzzy set with bounded support-valued mappings [, , ], while few research works deal with limits (continuity) and derivatives of fuzzy set-valued mappings. Furthermore, their approaches need some assumptions that level sets of fuzzy sets are nonempty and compact. Our new approach in this article, however, does not need those assumptions. Limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings ©2013 Kon and Kuwano; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), wh
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