Logarithmic means of sequences of fuzzy numbers and a Tauberian theorem

Abstract

A sequence $$(x_n)$$ of fuzzy numbers is said to be summable to a fuzzy number L by the logarithmic mean method $$(\ell ,2)$$ if $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\ell _n^{(2)}}\sum _{k=1}^{n}\frac{x_k}{k\ell _k}=L \end{aligned}$$where $$\begin{aligned} \ell _n^{(2)}=\sum _{k=1}^{n}\frac{1}{k\ell _k}\sim \log (\log n). \end{aligned}$$We prove that the ordinary convergence of $$(x_n)$$ implies its $$(\ell ,2)$$ summability. The converse implication is not necessarily true. Namely, the $$(\ell ,2)$$ summability of $$(x_n)$$ may not imply the convergence of $$(x_n)$$. However, under certain additional conditions the converse may hold. Such conditions are called Tauberian conditions, and the resulting theorem is said to be a Tauberian theorem. In this paper, we provide necessary and sufficient Tauberian conditions to transform $$(\ell ,2)$$ summable sequences of fuzzy numbers into convergent sequences of fuzzy numbers with preserving the limit.