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 (l, 2) if lim(n ->infinity) 1/l(n)((2)) Sigma(n)(k=1) x(k)/kl(k) = L where l(n)((2)) = Sigma(n)(k=1) 1/kl(k) similar to log(log n). We prove that the ordinary convergence of (x(n)) implies its (l, 2) summability. The converse implication is not necessarily true. Namely, the (l, 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 (l, 2) summable sequences of fuzzy numbers into convergent sequences of fuzzy numbers with preserving the limit.