A Tauberian condition under which convergence follows from the weighted mean summability of sequences of fuzzy number

Abstract

Let $(u_n)$ be a sequence of fuzzy numbers and $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and
$$P_n:=\sum_{k=0}^{n}p_k\to\infty\,\,\,\,\text{as}\,\,\,\,n\to\infty.$$
The weighted mean of $(u_n)$ is defined by
$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_k u_k\,\,\,\,\text{for}\,\,\,\,n =0,1,2,...$$
It is well known that convergence of $(u_n)$ implies that of the sequence $(t_n)$ of its weighted means. However, the converse of this implication is not true in general. In this paper, we investigate under which conditions convergence of $(u_n)$ follows from its weighted mean summability. We prove a Tauberian theorem including condition of slow decrease with respect to the weighted mean summability method for sequences of fuzzy numbers.