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

Z. Önder; İ. Çanak

doi:10.37863/umzh.v73i8.584

Z. Önder Ege Univ., Izmir, Turkey
İ. Çanak Ege Univ., Izmir, Turkey

DOI: https://doi.org/10.37863/umzh.v73i8.584

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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