# Some conditions under which slow oscillation of a sequence of fuzzy numbers follows from Ces\{a}ro summability of its generator sequence

Document Type : Research Paper

Author

Department of Mathematics, Ege University, 35100, Izmir, Turkey

Abstract

Let $(u_n)$ be a sequence of fuzzy numbers.  We recover the slow oscillation of $(u_n)$ of fuzzy numbers from the Ces\{a}ro summability of its generator sequence and some additional conditions imposed on $(u_n)$. Further, fuzzy analogues of some well known classical Tauberian theorems for Ces\{a}ro summability method are established as particular cases.

Keywords

#### References

\bibitem{alt1} Y. Alt{\i}n, M. Mursaleen and H. Alt{\i}nok, {\it Statistical summability (C,1) for sequences of fuzzy real numbers and a Tauberian theorem}, J. Intell. Fuzzy Syst., {\bf 21}\textbf{(6)} (2010), 379--384.

\bibitem{alt2} H. Alt{\i}nok and M. Mursaleen, {\it $\Delta$-Statistically boundedness for sequences of fuzzy numbers}, Taiwanese J. Math., {\bf 15}\textbf{(5)} (2011), 2081--2093.

\bibitem{col} R. \d{C}olak, Y. Alt{\i}n and M. Mursaleen, {\it On some sets of difference sequences of fuzzy numbers}, Soft Comput., {\bf 15}\textbf{(4)} (2011), 787--793

\bibitem{dub} D. Dubois and  H. Prade, {\it Fuzzy numbers: An overview, Analysis of Fuzzy Information, vol. 1}, Mathematical Logic, CRC Press, Boca, FL, 1987, 3--39.

\bibitem{dut} A. J. Dutta and B. C. Tripathy, {\it On $I$-acceleration convergence of sequences of fuzzy
real numbers}, Math. Model. Anal., {\bf 17}\textbf{(4)} (2012), 549--557.

\bibitem{fas} H. Fast, {\it Sur la convergence statistique}, Colloq. Math, {\bf 2} (1951), 241--244.

\bibitem{har} G. H. Hardy, {\it Divergent Series}, 2nd ed.,  New York, NY, 1991.

\bibitem{mat} M. Matloka,  {\it Sequences of fuzzy numbers}, BUSEFAL, {\bf 28} (1986), 28--37.

\bibitem{mor} F. M\'{o}ricz, {\it Necessary and sufficient Tauberian conditions, under which convergence follows from summability $(C,1)$}, Bull. London Math. Soc., {\bf 26} (1994), 288--294.

\bibitem{mor2} F. M\'{o}ricz, {\it Ordinary convergence follows from statistical summability $(C,1)$ in the case of slowly decreasing or oscillating sequences}, Colloq. Math., {\bf 99}\textbf{(2)} (2004), 207--219.

\bibitem{nan} S. Nanda, {\it On sequences of fuzzy numbers}, Fuzzy Sets and Systems, {\bf 33}\textbf{(1)} (1989), 123--126.

\bibitem{sch} I. J. Schoenberg, {\it The integrability of certain functions and related summability methods}, Amer. Math. Monthly, {\bf 66} (1959) 361--375.

\bibitem{sub} P. V. Subrahmanyam, {\it Ces\{a}ro summability of fuzzy real numbers}, J. Anal., {\bf 7} (1999), 159--168.

\bibitem{tal} \"{O}. Talo and  C. \d{C}akan, {\it On the Ces\`{a}ro convergence of sequences of fuzzy numbers}, Appl. Math. Lett., {\bf 25} (2012), 676--681.

\bibitem{tri} B. C. Tripathy and  A. Baruah, {\it N\"{o}rlund and Riesz mean of sequences of fuzzy real number}, Appl. Math. Lett., {\bf 23}\textbf{(5)} (2010), 651--655.

\bibitem{tri1} B. C. Tripathy and A. Baruah, {\it Lacunary statistically convergent and lacunary strongly
convergent generalized difference sequences of fuzzy real numbers}, Kyungpook Math. J., {\bf 50}\textbf{(4)} (2010), 565--574.

\bibitem{tri3} B. C. Tripathy, A. Baruah, M. Et and M. G\"{u}ng\"{o}r, {\it On almost statistical convergence of
new type of generalized difference sequence of fuzzy numbers}, Iran. J.
Sci. Technol. Trans. A Sci., {\bf 36}\textbf{(2)} (2012), 147--155.

\bibitem{tri2} B. C. Tripathy and S. Borgohain,  {\it Some classes of difference sequence spaces of fuzzy
real numbers defined by Orlicz function},  Adv. Fuzzy Syst., Art. ID 216414, (2011), 6.

\bibitem{tri5} B. C. Tripathy and S. Debnath, {\it On generalized difference sequence spaces of fuzzy
numbers}, Acta Scientiarum Technology, {\bf 35}\textbf{(1)} (2013),  117--121.

\bibitem{tri4} B. C. Tripathy and A. J. Dutta,  {\it Lacunary bounded variation sequence of fuzzy real
numbers}, J. Intell. Fuzzy Systems, {\bf 24}\textbf{(1)} (2013), 185--189.

\bibitem{zad} L. A. Zadeh, Fuzzy sets, {\it Information and Control}, {\bf 8} (1965), 338--353.