# Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number

• Z. Önder Ege Univ., Izmir, Turkey
• İ. Çanak Ege Univ., Izmir, Turkey
Keywords: Sequences of fuzzy numbers, slow oscillation, slow oscillation relative to $(P_n)$, statistical convergence, two-sided conditions of Hardy type, Tauberian theorems, weighted mean summability method

### Abstract

UDC 517.5

Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and
$$P_n:=\sum_{k=0}^{n}p_k\to\infty\qquad \text{as}\qquad n\to\infty.$$
Let $(u_n)$ be a sequence of fuzzy numbers.
The weighted mean of $(u_n)$ is defined by
$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_k u_k\qquad \text{for}\qquad n =0,1,2,\ldots \,.$$
It is known that the existence of the limit $\lim u_n=\mu_{0}$ implies that of $\lim t_n=\mu_{0}.$
For the the existence of the limit $st$-$\lim t_n=\mu_{0},$ we require the boundedness of $(u_n)$ in addition to the existence of the limit $\lim u_n=\mu_{0}.$
But, in general, the converse of this implication is not true.
In this paper, we obtain Tauberian conditions, under which the existence of the limit $\lim u_n=\mu_{0}$ follows from that of $\lim t_n=\mu_{0}$ or $st$-$\lim t_n=\mu_{0}.$
These Tauberian conditions are satisfied if $(u_n)$ satisfies the two-sided condition of Hardy type relative to $(P_n).$

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Published
18.08.2021
How to Cite
Önder, Z., and İ. Çanak. “Tauberian Conditions under Which Convergence Follows from the Weighted Mean Summability and Its Statistical Extension for Sequences of Fuzzy Number”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 8, Aug. 2021, pp. 1085 -01, doi:10.37863/umzh.v73i8.584.
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Research articles