Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number

  • Z. Önder Ege Univ., Izmir, Turkey
  • İ. Çanak Ege Univ., Izmir, Turkey
Keywords: Sequences of fuzzy numbers, slow oscillation, slow oscillation relative to $(P_n)$, statistical convergence, two-sided conditions of Hardy type, Tauberian theorems, weighted mean summability method


UDC 517.5

Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and
P_n:=\sum_{k=0}^{n}p_k\to\infty\qquad \text{as}\qquad n\to\infty.
Let $(u_n)$ be a sequence of fuzzy numbers.
The weighted mean of $(u_n)$ is defined by
u_k\qquad \text{for}\qquad n =0,1,2,\ldots \,.
It is known that the existence of the limit $\lim u_n=\mu_{0}$ implies that of $\lim t_n=\mu_{0}.$
For the 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 $\lim u_n=\mu_{0}.$
But, in general, the converse of this implication is not true.
In this paper, we obtain Tauberian conditions, under which the existence of the limit $\lim u_n=\mu_{0}$ follows from that of $\lim t_n=\mu_{0}$ 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 $(P_n).$


L. A. Zadeh, Fuzzy sets, Inform. and Control, 8, 338 – 353 (1965). DOI: https://doi.org/10.1016/S0019-9958(65)90241-X

D. Dubois, H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci., 9, № 6, 613 – 626 (1978), https://doi.org/10.1080/00207727808941724 DOI: https://doi.org/10.1080/00207727808941724

R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18, № 1, 31 – 43 (1986), https://doi.org/10.1016/0165-0114(86)90026-6 DOI: https://doi.org/10.1016/0165-0114(86)90026-6

M. Matloka, Sequences of fuzzy numbers, Busefal, 28, 28 – 37 (1986).

S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets and Systems, 33, № 1, 123 – 126 (1989), https://doi.org/10.1016/0165-0114(89)90222-4 DOI: https://doi.org/10.1016/0165-0114(89)90222-4

B. C. Tripathy, A. Baruah, M. Et, M. Gungor, On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers, Iran. J. Sci. and Technol. Trans. A Sci., 36, № 2, 147 – 155 (2012).

I. ¸Canak, On the Riesz mean of sequences of fuzzy real numbers, J. Intell. Fuzzy Systems, 26, № 6, 2685 – 2688 (2014), https://doi.org/10.3233/IFS-130938 DOI: https://doi.org/10.3233/IFS-130938

Z. Önder, S. A. Sezer, I. ¸Canak, A Tauberian theorem for the weighted mean method of summability of sequences of fuzzy numbers, J. Intell. Fuzzy Systems, 28, № 3, 1403 – 1409 (2015). DOI: https://doi.org/10.3233/IFS-141424

H. Fast, Sur la convergence statistique, Colloq. Math., 2, 241 – 244 (1951), https://doi.org/10.4064/cm-2-3-4-241-244 DOI: https://doi.org/10.4064/cm-2-3-4-241-244

I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66, 361 – 375 (1959), https://doi.org/10.2307/2308747 DOI: https://doi.org/10.2307/2308747

A. Zygmund, Trigonometric series, Cambridge Univ. Press (1959).

F. Nuray, E. Sava¸s, Statistical convergence of sequences of fuzzy numbers, Math. Slovaca, 45, № 3, 269 – 273 (1995).

E. Sava¸s, On statistically convergent sequences of fuzzy numbers, Inform. Sci., 137, 277 – 282 (2001), https://doi.org/10.1016/S0020-0255(01)00110-4 DOI: https://doi.org/10.1016/S0020-0255(01)00110-4

S. Aytar, S. Pehlivan, Statistical convergence of sequences of fuzzy numbers and sequences of $α$ -cuts, Int. J. Gen. Syst., 37, № 2, 231 – 237 (2008), https://doi.org/10.1080/03081070701251075 DOI: https://doi.org/10.1080/03081070701251075

F. Ba¸sar, Summability theory and its applications, Bentham Sci. Publ., (2012). DOI: https://doi.org/10.2174/97816080545231120101

J. S. Kwon, On statistical and $p$-Cesaro convergence of fuzzy numbers ` , Korean J. Comput. and Appl. Math., 7, № 1, 195 – 203 (2000). DOI: https://doi.org/10.1007/BF03009937

Ö. Talo, F. Ba¸sar, On the slowly decreasing sequences of fuzzy numbers, Abstr. and Appl. Anal., 2013, Article ID 891986 (2013), 7 p., https://doi.org/10.1155/2013/891986 DOI: https://doi.org/10.1155/2013/891986

Ö. Talo, C. Bal, On statistical summability $(N,P)$ of sequences of fuzzy numbers, Filomat, 30, № 3, 873 – 884 (2016), https://doi.org/10.2298/FIL1603873T DOI: https://doi.org/10.2298/FIL1603873T

M. Et, B. C. Tripathy, A. J. Dutta, On pointwise statistical convergence of order $alpha$ of sequences of fuzzy mappings, Kuwait J. Sci., 41, № 3, 17 – 30 (2014).

F. Moricz, Ordinary convergence follows from statistical summability $(C, 1)$ in the case of slowly decreasing or oscillating sequences, Colloq. Math., 99, № 2, 207 – 219 (2004), https://doi.org/10.4064/cm99-2-6 DOI: https://doi.org/10.4064/cm99-2-6

F. Moricz, Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences, Analysis, 24, № 2, 127 – 145 (2004), https://doi.org/10.1524/anly.2004.24.14.127 DOI: https://doi.org/10.1524/anly.2004.24.14.127

D. Dubois, H. Prade, Fuzzy sets and systems, Acad. Press (1980).

B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer (2013), https://doi.org/10.1007/978-3-642-35221-8 DOI: https://doi.org/10.1007/978-3-642-35221-8

P. V. Subrahmanyam, Cesàro summability for fuzzy real numbers. $p$-adic analysis, summability theory, fuzzy analysis and applications (INCOPASFA) (Chennai, 1998) ` , J. Anal., 7, 159 – 168 (1999).

G. A. Mikhalin, Theorems of Tauberian type for $(J,,p n)$ summation methods, Ukr. Math. J., 29, № 6, 564 – 569 (1977). DOI: https://doi.org/10.1007/BF01085962

J. Boos, Classical and modern methods in summability, Oxford Univ. Press (2000).

How to Cite
Önder, Z., and İ. Çanak. “Tauberian Conditions under Which Convergence Follows from the Weighted Mean Summability and Its Statistical Extension for Sequences of Fuzzy Number”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 8, Aug. 2021, pp. 1085 -01, doi:10.37863/umzh.v73i8.584.
Research articles