Mohammad H.M. RASHİD, Ljubisa D.R. KOCİNAC

Ideal convergence in 2-fuzzy 2-normed spaces

In this paper we introduce the notion of $\mathcal{I}$-convergence and $\mathcal{I}$-Cauchyness of sequences in 2-fuzzy 2-normed spaces and established some basic results related to these notions. Further, we define $\mathcal{I}$-limit and $\mathcal{I}$-cluster points of a sequence in a 2-fuzzy 2-normed linear space and investigate the relations between these concepts.

Keywords:

2-fuzzy 2-norm ideal convergence, ideal Cauchy sequence,

PDF

___

S. Aytar, Statistical limit points of sequences of fuzzy numbers, Inform. Sci. 165 (2004), 129138.
S. Aytar, M. Mammadov, S. Pehlivan, Statistical limit inferior and limit superior for sequences of fuzzy numbers, Fuzzy Sets Syst. 157:7 (2006), 976985.
S. Aytar, S. Pehlivan, Statistical cluster and extreme limit points of sequences of fuzzy numbers, Inform. Sci. 177 (2007), 32903296.
T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11:3 (2003), 687705.
T. Bag, S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst. 151:3 (2005), 513547.
T. Bag, S.K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy Sets Syst. 159 (2008), 670684.
M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007), 715729.
A.I. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185193.
H. Çakalli, S. Ersan, Strongly lacunary ward continuity in 2-normed spaces, Sci. World J. 2014 (2014), Art. ID 479679, 5 pp.
H. Çakalli, S. Ersan, New types of continuity in 2-normed spaces, Filomat 30:3 (2016), 525532.
A. Caserta, G. Di Maio, Lj.D.R. Kocinac, Statistical convergence in function spaces, Abstr. Appl. Anal. 2011 (2011), Article ID 420419, 11 pages.
S.C. Cheng, J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc 86 (1994), 429436.
P. Das, S.K. Ghosal, Some further results on I-Cauchy sequences and condition (AP), Comput. Math. Appl. 59 (2010), 25972600.
P. Das, S. Pal, S.K. Ghosal, Further investigations of ideal summability in 2-normed spaces, Appl. Math. Letters 24 (2011), 3943.
P. Das, E. Savaş, On I-statistically pre-Cauchy sequences, Taiwanese J. Math. 18:1 (2014), 115126.
K. Dems, On I-Cauchy sequences, Real Anal. Exch. 30:1 (2004-2005), 123128.
G. Di Maio, Lj.D.R. Kocinac, Statistical convergence in topology, Topology Appl. 156:1 (2008), 2845.
S. Ersan, H. Çakalli, Ward continuity in 2-normed spaces, Filomat 29:7 (2015), 15071513.
S. Gähler, 2-metrishe Räume und ihr topologishe struktur, Math. Nachr. 26 (1963), 115148.
H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241244.
C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48 (1992), 239 248.
J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301313.
B. Hazarika, On ideal convergent sequences in fuzzy normed linear spaces, Afrika Matematika 25:4 (2014), 987999.
B. Hazarika, E. Savaş, Some I-convergent lamda-summable difference sequence spaces of fuzzy real numbers dened by a sequence of Orlicz, Math. Comp. Modelling 54 (2011), 29862998.
O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst. 12 (1984), 215229.
S. Karakuş, K. Demirci, O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals 35 (2008), 763769.
V. Karakaya, N.N. Şimşek , M. Ertürk, F. Gürsoy, Statistical A-convergence of sequences of functions in intuitionistic fuzzy normed spaces, Abstr. Appl. Anal. 2012 (2012).
V. Karakaya, N.N. Şimşek , M. Ertürk, F. Gürsoy, On ideal convergence of sequences of functions in intuitionistic fuzzy normed spaces, Appl. Math. Inf. Sci. 8:5 (2014), 23072313.
A.K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets Syst. 12 (1984), 143154.
P. Kostyrko, T. Salát, W. Wilczynski, I-convergence, Real Anal. Exchange, 26:2 (2000-2001), 669686.
P. Kostyrko, M. Macaj, T. Salát, M. Sleziak, I-convergence and extremal I-limit points, Math. Slovaca 55 (2005), 443464.
A.K. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326334.
V. Kumar, K Kumar, On the ideal convergence of sequences of fuzzy numbers, Inform. Sci. 178 (2008), 46704678.
M. Mursaleen, S.A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons and Fractals 41 (2009), 24142421.
A. Nabin, S. Pehlivan, M. Gürdal, On I-Cauchy sequences, Taiwanese J. Math. 11:2 (2007), 569576.
F. Nuray, E. Savaş, Statistical convergence of sequences of fuzzy numbers, Math. Slovaca 45 (1995), 269273.
W. Raymond, Y. Freese, J. Cho, Geometry of linear 2-normed spaces, N.Y. Nova Science Publishers, Huntington, 2001.
E. Savaş, On statistically convergent sequence of fuzzy numbers, Inform. Sci. 137 (2001), 272282.
E. Savaş, On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function, J. Inequal. Appl. 2010 (2010), Art. ID 482392, 8 pp.
E. Savaş, Some I-convergent sequence spaces of fuzzy numbers dened by innite matrix, Math. Comp. Appl. 18:2 (2013), 8493.
C. Şençimen, S. Pehlivan, Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets Syst. 159 (2008), 361370.
H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 7334.
R.M. Somasundaram, T. Beaula, Some aspects of 2-fuzzy 2-normed linear spaces, Bull. Malays. Math. Sci. Soc. (2) 32:2 (2009), 211221.
L.A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965), 338353.
J. Zhang, The continuity and boundedness of fuzzy linear operators in fuzzy normed space, J. Fuzzy Math. 13:3 (2005), 519536.
A. Zygmund, Trigonometric Series, 2nd edition, Cambridge University Press, Cambridge, 1979.