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Ideal convergence in 2-fuzzy 2-normed spaces

Year 2017, Volume: 46 Issue: 1, 149 - 162, 01.02.2017

Abstract

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.

References

  • 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. Ş“imşek , 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. Ş“imşek , 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. S’alát, 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. Ş“ençimen, 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.
Year 2017, Volume: 46 Issue: 1, 149 - 162, 01.02.2017

Abstract

References

  • 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. Ş“imşek , 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. Ş“imşek , 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. S’alát, 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. Ş“ençimen, 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.
There are 46 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Mohammad H.m. Rashid

Ljubi\v{s}a D.r. Ko\v{c}inac

Publication Date February 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 1

Cite

APA Rashid, M. H., & Ko\v{c}inac, L. D. (2017). Ideal convergence in 2-fuzzy 2-normed spaces. Hacettepe Journal of Mathematics and Statistics, 46(1), 149-162.
AMA Rashid MH, Ko\v{c}inac LD. Ideal convergence in 2-fuzzy 2-normed spaces. Hacettepe Journal of Mathematics and Statistics. February 2017;46(1):149-162.
Chicago Rashid, Mohammad H.m., and Ljubi\v{s}a D.r. Ko\v{c}inac. “Ideal Convergence in 2-Fuzzy 2-Normed Spaces”. Hacettepe Journal of Mathematics and Statistics 46, no. 1 (February 2017): 149-62.
EndNote Rashid MH, Ko\v{c}inac LD (February 1, 2017) Ideal convergence in 2-fuzzy 2-normed spaces. Hacettepe Journal of Mathematics and Statistics 46 1 149–162.
IEEE M. H. Rashid and L. D. Ko\v{c}inac, “Ideal convergence in 2-fuzzy 2-normed spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, pp. 149–162, 2017.
ISNAD Rashid, Mohammad H.m. - Ko\v{c}inac, Ljubi\v{s}a D.r. “Ideal Convergence in 2-Fuzzy 2-Normed Spaces”. Hacettepe Journal of Mathematics and Statistics 46/1 (February 2017), 149-162.
JAMA Rashid MH, Ko\v{c}inac LD. Ideal convergence in 2-fuzzy 2-normed spaces. Hacettepe Journal of Mathematics and Statistics. 2017;46:149–162.
MLA Rashid, Mohammad H.m. and Ljubi\v{s}a D.r. Ko\v{c}inac. “Ideal Convergence in 2-Fuzzy 2-Normed Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, 2017, pp. 149-62.
Vancouver Rashid MH, Ko\v{c}inac LD. Ideal convergence in 2-fuzzy 2-normed spaces. Hacettepe Journal of Mathematics and Statistics. 2017;46(1):149-62.