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

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 $(P_n).$