Tauberian Conditions Under which Convergence Follows from the Weighted Mean Summability and Its Statistical Extension for Sequences of Fuzzy Numbers

Let (p(n)) be a sequence of nonnegative numbers such that p(0) > 0 and P-n:= Sigma(n)(k=0)p(k)->infinity as n -> infinity. Let (u(n)) be a sequence of fuzzy numbers. The weighted mean of (u(n)) is defined by t(n) := 1/P-n Sigma(n)(k=0)pkuk for n = 0,1,1,2,... It is known that the existence of the limit lim u(n) = mu(0) implies that lim t(n) = mu(0). For 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). However, in general, the converse of this implication is not true. We establish Tauberian conditions, under which the existence of the limit lim u(n) = mu(0) follows from the existence of the limit 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).