Functional sequences with fuzzy argument: convergence of level sets
I. Ya. Spectorsky · System research and information technologies · 2019
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.