# Functional sequences with fuzzy argument: convergence of level sets

## 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*f*(

_{n}*A*) with convex upper semicontinuous fuzzy number

*A*for argument; it is supposed that lim

_{n→∞}

*f*(

_{n}*x*)=

*f*(

*x*), and this convergence is uniform on each closed interval within supp

*A*. The paper proposes sufficient conditions for

*f*(

_{n}*A*) to converge in the sense that a sequence of level sets [

*f*(

_{n}*A*)]

_{α}converges with respect to Hausdorff distance

*dH*([

*f*(

_{n}*A*)]

_{α},[

*f*(

*A*)]

_{α}). It is proved that: lim

_{n→∞}

*dH*([

*f*(

_{n}*A*)]

_{α},[

*f*(

*A*)]

_{α})=0 for each 0<α≤1 assuming continuity of

*f*(

_{n}*x*) (

*n*≥1) and

*f*(

*x*), without the assumption about an existence of a derivative. Also, it is proved that a sequence

*f*(A) (

_{n}*n*≥1) converges with respect to distance ρ(

*f*(

_{n}*A*),

*f*(

*A*))=sup

_{0<α≤1}

*dH*([

*f*(

_{n}*A*)]

_{α},[

*f*(

*A*)]

_{α}) in the space of fuzzy sets, additionally assuming that

*f*(

_{n}*A*) converges uniformly on the whole supp

*A*. In this case, for the sake of finiteness of Hausdorff distance for all 0<α≤1, fuzzy set

*A*is supposed to be normal.

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