 # TAUBERIAN THEOREMS FOR THE EULER-NORLUND MEAN-CONVERGENT SEQUENCES OF FUZZY NUMBERS

Document Type: Research Paper

Authors

1 Department of Mathematics and Computer Sciences, University of Prishtina, Avenue Mother Teresa, No-4, Prishtine, 10000, Kosova

2 Department of Mathematics, Frat University, Elazig, 23119, Turkey

Abstract

Fuzzy set theory has entered into a large variety of disciplines of sciences,
technology and humanities having established itself as an extremely versatile
interdisciplinary research area. Accordingly different notions of fuzzy
structure have been developed such as fuzzy normed linear space, fuzzy
topological vector space, fuzzy sequence space etc. While reviewing the
literature in fuzzy sequence space, we have seen that the notion of Tauberian
theorems for the Euler-N\"{o}rlund mean-convergent sequences of fuzzy numbers
has not been developed. In the present paper, we introduce some new concepts
about statistical convergence of sequences of fuzzy numbers. The main purpose
of this paper is to study Tauberian theorems for the Euler-N\"{o}rlund
mean-convergent sequences of fuzzy numbers and investigate some other kind of
convergences named Euler-N\"{o}rlund mean-level convergence so as to fill up
the existing gaps in the literature. The results which we obtained in this
study are much more general than those obtained by others.

Keywords

### References

 Y. Altin, M. Mursaleen and H. Altinok, Statistical summability (C; 1) for sequences of fuzzy
real numbers and a Tauberian theorem, Journal of Intelligent and Fuzzy Systems, 21 (2010),
379{384.
 S. Aytar, M. A. Mammadov and S. Pehlivan, Statistical limit inferior and limit superior for
sequences of fuzzy numbers, Fuzzy Sets and Systems, 157(7) (2006), 976{985.
 B. Bede and S. G. Gal, Almost periodic fuzzy number valued functions, Fuzzy Sets and
Systems, 147 (2004), 385{403.
 N. L. Braha, Tauberian conditions under which 􀀀statistical convergence follows from statistical
summability (V; ), Miskolc Math. Notes, 16(2) (2015), 695{703.
 M. Et, H. Altinok and R. Colak, On -statistical convergence of di erence sequences of fuzzy
numbers, Inform. Sci., 176(15) (2006), 2268{2278.
 H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241{244.
 J. X. Fang and H. Huang, On the level convergence of a sequence of fuzzy numbers, Fuzzy
Sets and Systems, 147(3) (2004), 417{435.
 A. Gokhan, M. Et and M. Mursaleen, Almost lacunary statistical and strongly almost lacunary
convergence of sequences of fuzzy numbers, Math. Comput. Modelling, 49(3-4) (2009),
548{555.
 J. S. Kwon, On statistical and p-Cesaro convergence of fuzzy numbers, The Korean Journal
of Computational and Applied Mathematics, 7 (2000), 195{203.
 L. Leindler,  Uber die de la Vallee-Pousinsche summierbarkeit allgemeiner orthogonalreihen,
Acta Math. Acad. Sci. Hungar., 16 (1965), 375{387.
 M. Matloka, Sequence of fuzzy numbers, BUSEFAL, 28 (1986), 28{37.
 F. Moricz, Tauberian conditions under which statistical convergence follows from statistical
summability (C, 1), J. Math. Anal. Appl., 275 (2002), 277{287.
 S. Nanda, On sequence of fuzzy numbers, Fuzzy Sets and Systems, 33 (1989), 123{126.
 I. J. Schoenberg, The integrability of certain functions and related summability methods,
Amer. Math. Monthly, 66 (1959), 361{375.
92 N. L. Braha and M. Et
 H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2
(1951), 73{74.
 C. Wu and G. Wang, Convergence of sequences of fuzzy numbers and xed point theorems for
increasing fuzzy mappings and application, Theme: Fuzzy intervals. Fuzzy Sets and Systems,
130(3) (2002), 383{390.
 L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.
 A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, London and
New York, 1979.