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

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Ukrainian Mathematical Journal Aims and scope

Let (pn) be a sequence of nonnegative numbers such that p0 > 0 and

$$ {P}_n:= \sum \limits_{k=0}^n{p}_k\to \infty \kern1em \mathrm{as}\kern1em n\to \infty . $$

Let (un) be a sequence of fuzzy numbers. The weighted mean of (un) is defined by

$$ {t}_n:= \frac{1}{P_n}\sum \limits_{k=0}^n{p}_k{u}_k\kern1em \mathrm{for}\kern1em n=0,1,2,\dots $$

It is known that the existence of the limit limun = μ0 implies that limtn = μ0. For the existence of the limit st-limtn = μ0, we require the boundedness of (un) in addition to the existence of the limit limun = μ0. However, in general, the converse of this implication is not true. We establish Tauberian conditions, under which the existence of the limit limun = μ0 follows from the existence of the limit limtn = μ0 or st-limtn = μ0. These Tauberian conditions are satisfied if (un) satisfies the two-sided condition of Hardy type relative to (Pn).

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Authors and Affiliations

  1. Ege University, Izmir, Turkey

    Z. Önder & İ. Çanak

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  1. Z. Önder
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  2. İ. Çanak
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Correspondence to İ. Çanak.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 8, pp. 1085–1101, August, 2021. Ukrainian DOI: 10.37863/umzh.v73i8.584.

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Önder, Z., Çanak, İ. Tauberian Conditions Under which Convergence Follows from the Weighted Mean Summability and Its Statistical Extension for Sequences of Fuzzy Numbers. Ukr Math J 73, 1259–1277 (2022). https://doi.org/10.1007/s11253-022-01989-4

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  • DOI: https://doi.org/10.1007/s11253-022-01989-4

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