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Abstract

In this paper, following a very recent and new approach of [1], we discuss a relation between two kinds of $I$ - convergence of sequences, namely, $I$ -statistical convergence of weight g and $I$ -lacunary statistical convergence of weight g, where $g : ℕ \to [0, \infty)$ is a function satisfying $lim_{n \to \infty} g (n) = \infty$ and $\frac{n}{g (n)} ↛ 0$ as n→ ∞ for fuzzy numbers. These methods are based on the methods of statistical convergence and lacunary statistical summability for sequences. Analogs of inclusion and consistency theorems for the sequential methods are established.

Keywords

Ideal filter fuzzy numbers

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References

Balcerzak

, Das

, Filipczak

and Swaczyna

, Generalized kinds of density and the associated ideals, Acta Mathematica Hungarica 147(1) (2015), 97–115.

Bhunia

, Das

and Pal

, Restricting statistical convergenge, Acta Math Hungar 134(1-2) (2012), 153–161.

Colak

, Statistical convergence of order α, Modern methods in Analysis and its Applications, New Delhi, India, Anamaya Pub., 2010, pp. 121–129.

Colak

, Altin

and Mursaleen

, On some sets difference sequence of fuzzy numbers, Soft Computing 15 (2011), 787–793.

Colak

and Bektas

C.A.

, λ-statistical convergence of order α, Acta Math Scientia 31B(3) (2011), 953–959.

Das

, Savas

and Ghosal

S.K.

, On generalizations of certain summability methods using ideals, Appl Math Letters 24 (2011), 1509–1514.

Das

and Savas

, On

I

-statistical and

I

-lacunary statistical convergence of order α, Bull of The Iranian Math Soc 40(2) (2014), 459–472.

Diomand

and Kloeden

, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 33 (1989), 123–126.

Fast

, Sur la convergence statistique, Colloq Math 2 (1951), 241–244.

10.

Freedman

A.R.

, Sember

J.J.

and Rapnael

, Some Cesaro type summability spaces, Proc London Math Soc 37 (1978), 508–520.

11.

Fridy

J.A.

, On statistical convergence, Analysis 5 (1985), 301–313.

12.

Fridy

J.A.

and Orhan

, Lacunary statistical convergence, Pacific J Math 160 (1993), 43–51.

13.

Karakas

, Altin

and Altinok

, On generalized statistical convergence of order Ã§ of sequences of fuzzy numbers, J Intell Fuzzy Systems 26(4) (2014), 1909–1917.

14.

Kostyrko

, Šalát

and Wilczynki

I

-convergence, Real Anal Exchange 26(2) (2000/2001), 669–685.

15.

Kwon

J.S.

, On statistical and-Cearo convergence of fuzzy numbers, Korean J of Comput and Appl Math 7(1) (2000), 195–203.

16.

Lahiri

B.K.

and Das

I

and

I^{*}

-convergence in topological spaces, Math Bohem 130 (2005), 153–160.

17.

Lahiri

B.K.

and Das

I

and

I^{*}

-convergence of nets,-, Real Anal Exchange 33 (2008), 431–442.

18.

, Lacunary statistical convergence and inclusion properties between lacunary methods, Int J Math Math Sc 23(3) (2000), 175–180.

19.

Matloka

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

20.

Mursaleen and Basarir

, On some new sequence spaces of fuzzy numbers, Indian Jour Pure Appl Math 34(9) (2003), 1351–1357.

21.

Nanda

, On sequences of fuzzy numbers, Fuzzy Sets and Systems 33 (1989), 123–126.

22.

Nuray

, Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets and Systems 45(3) (1998), 269–273.

23.

Nuray

and Savaş

, Statistical convergence of sequences of fuzzy numbers, Mathematica Slovaca 99(3) (1994), 353–355.

24.

Qiu

, Lu

, Zhang

and Lan

, Algebraic properties and topological properties of the quotient space of fuzzy numbers based on Mares equivalence relation, Fuzzy Sets and Systems 245 (2014), 63–82.

25.

Qiu

and Zhang

, Symmetric fuzzy numbers and additive equivalence of fuzzy numbers, Soft Computing 17 (2013), 1471–1477.

26.

Šalát

, On Statistically convergent sequences of real numbers, Math Slovaca 30 (1980), 139–150.

27.

Savaş

, A note on double sequence of Fuzzy numbers, Turk J Math 20 (1996), 175–178.

28.

Savas

, A note on sequence of Fuzzy numbers, Information Sciences 124 (2000), 297–300.

29.

Savaş

, On statistically convergent sequence of Fuzzy numbers, Information Sciences 137 (2001), 272–282.

30.

Savaş

, On strongly λ – summable sequences of fuzzy numbers, Inform Sci 125 (2000), 181–186.

31.

Savaş

, On

I

-asymptotically lacunary statistical equivalent sequences, Advan Differ Equa 2013 (2013), 111.

32.

Savaş

, Das

and Dutta

, A note on strong matrix summability via ideals, Appl Math Letters 25(4) (2012), 733–738.

33.

Savaş

and Das

, A generalized statistical convergence via ideals, Appl Math Letters 24 (2011), 826–830.

34.

Savaş

, (A)(Delta) - Double sequence spaces of fuzzy numbers via orlicz function, Iranian Journal of Fuzzy Systems 8(2) (2011), 91–103. Published: JUN.

35.

Savaş

, On fuzzy real-valued double A-sequence spaces defined by Orlicz function, Mathematical Communications 16(2) (2011), 609–619.

36.

Savaş

, On some double lacunary sequence spaces of fuzzy numbers, Math Comput Appl 15(3) (2010), 439–448.

37.

Savaş

, New double sequence spaces of fuzzy numbers, Quaest Math 33(4) (2010), 449–456.

38.

Savaş

, On lacunary strong σ -convergence, Indian J Pure Appl Math 21(4) (1990), 359–365.

39.

Savaş

, On some summability methods using ideals and fuzzy numbers, J Intell Fuzzy Systems 28(4) (2015), 1931–1936.

40.

Savaş

I_{θ}

- statistical and-Cesàro summability of fuzzy numbers, J Intell Fuzzy Systems(Preprint).

41.

Savas

and Karakaya

, Some new sequence spaces defined by lacunary sequences, Math Slovaca 57(4) (2007), 393–399.

42.

Srivastava

P.D.

and Ojha

, λ-Statistical convergence of fuzzy numbers and fuzzy functions of order α , Soft Computing 18 (2014), 1027–1032.

43.

Schoenberg

I.J.

, The integrability of certain functions and related summability methods, Amer Math Monthly 66 (1959), 361–375.

44.

Tripathy

B.C.

, Sen

and Nath

I

-convergence in probabilistic n-normed space, Soft Comput 16 (2012), 1021–1027.

45.

Zadeh

, Fuzzy sets, Infor Control 8 (1965), 338–353.

On I -lacunary statistical convergence of weight g of fuzzy numbers

Abstract

Keywords

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