Functional sequences with fuzzy argument: convergence of level sets

Authors

  • Igor Ya. Spectorsky Educational and Scientific Complex "Institute for Applied System Analysis" of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine https://orcid.org/0000-0003-4863-7986

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2019.3.12

Keywords:

fuzzy number, level set, functional sequence, convergence

Abstract

The main consideration subject is functional sequences fn(A) with convex upper semicontinuous fuzzy number A for argument; it is supposed that limn→∞fn(x)=f(x), and this convergence is uniform on each closed interval within suppA. The paper proposes sufficient conditions for fn(A) to converge in the sense that a sequence of level sets [fn(A)]α converges with respect to Hausdorff distance dH([fn(A)]α,[f(A)]α). It is proved that: limn→∞dH([fn(A)]α,[f(A)]α)=0 for each 0<α≤1 assuming continuity of fn(x) (n≥1) and f(x), without the assumption about an existence of a derivative. Also, it is proved that a sequence fn(A) (n≥1) converges with respect to distance ρ(fn(A),f(A))=sup0<α≤1dH([fn(A)]α,[f(A)]α) in the space of fuzzy sets, additionally assuming that fn(A) converges uniformly on the whole suppA. In this case, for the sake of finiteness of Hausdorff distance for all 0<α≤1, fuzzy set A is supposed to be normal.

Author Biography

Igor Ya. Spectorsky, Educational and Scientific Complex "Institute for Applied System Analysis" of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv

Igor Ya. Spectorsky,

Ph.D, an associate professor at the department of “Mathematical Methods of System Analysis” of Educational and Scientific Complex "Institute for Applied System Analysis" of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine.

Current scientific areas: convergence of functional sequences with fuzzy argument, polynomial representation of boolean functions.

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Published

2019-10-07

Issue

Section

Mathematical methods, models, problems and technologies for complex systems research