Research Article
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Year 2022, Volume: 71 Issue: 1, 13 - 24, 30.03.2022

Mehmet Gürdal , Ekrem Savaş

https://doi.org/10.31801/cfsuasmas.890982

Abstract

References

  • Das, P., Kostyrko, P., Wilczyski, W., Malik, P., $I$ and $I^*$-convergence of double sequences, Math. Slovaca, 58(5) (2008), 605–620. https://doi.org/10.2478/s12175-008-0096-x
  • Engelking, R., General Topology, PWN-Polish Science Publishers, Warsaw, 1977.
  • Erceg, M. A., Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69(1) (1979), 205–230. https://doi.org/10.1016/0022-247X(79)90189-6
  • Fast, H., Sur la convergence statistique, Colloq. Math., 2(3-4) (1951), 241–244.
  • George, A., Veeramani, P., On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(3) (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7
  • George, A., Veeramani, P., On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90(3) (1997), 365–368. https://doi.org/10.1016/S0165-0114(96)00207-2
  • Gregori, V., Lopez-Crevillen, A., Morillas, S., Sapena, A., On convergence in fuzzy metric spaces, Topology Appl., 156(18) (2009), 3002–3006. https://doi.org/10.1016/j.topol.2008.12.043
  • Gregori, V., Minana, J. J., Std-convergence in fuzzy metric spaces, Fuzzy Sets and Systems, 267 (2015), 140–143. https://doi.org/10.1016/j.fss.2014.05.007
  • Gregori, V., Mi˜nana, J. J., Morillas, S., A note on convergence in fuzzy metric spaces, Iran J. Fuzzy Syst., 11(4) (2014), 75–85. https://doi.org/10.22111/IJFS.2014.1625
  • Gregori, V., Mi˜nana, J. J., Morillas, S., Sapena, A., Cauchyness and convergence in fuzzy metric spaces, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 111(1) (2017), 25–37. https://doi.org/10.1007/s13398-015-0272-0
  • Gürdal, M., Some types of convergence, Doctoral Dissertation, Suleyman Demirel University, Isparta, 2004.
  • Gürdal, M., Açık, I., On I-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl., 11(2) (2008), 349–354. https://doi.org/10.7153/mia-11-26
  • Gürdal, M., Huban, M. B., On I-convergence of double sequences in the topology induced by random 2-norms, Mat. Vesnik, 66(1) (2014), 73–83.
  • Gürdal, M., Sarı, N., Savaş, E., A-statistically localized sequences in n-normed spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(2) (2020), 1484 1497. https://doi.org/10.31801/cfsuasmas.704446
  • Gürdal, M., Şahiner, A., Extremal I-limit points of double sequences, Appl. Math. E-Notes, 8 (2008), 131–137.
  • Hazarika, B., Alotaibi, A., Mohiuddine, S. A., Statistical convergence inmeasure for double sequences of fuzzy-valued functions, Soft Comput., 24 (2020), 6613–6622. https://doi.org/10.1007/s00500-020-04805-y
  • Kaleva, O., Seikkala, S., On fuzzy metric spaces, Fuzzy Sets and Systems, 12(3) (1984), 215–229. https://doi.org/10.1016/0165-0114(84)90069-1
  • Kramosil, I., Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetika, 11(5) (1975), 336–344.
  • Li, C., Zhang, Y., Zhang, J., On statistical convergence in fuzzy metric spaces, J. Intell. Fuzzy Systems, 39(3) (2020), 3987–3993. https://doi.org/10.3233/JIFS-200148
  • Kostyrko, S., Salat, T., Wilczynski, W., I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
  • Mohiuddine, S. A., Asiri, A., Hazarika, B., Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst., 48(5) (2019), 492–506. https://doi.org/10.1080/03081079.2019.1608985
  • Mohiuddine, S. A., Hazarika, B., Some classes of ideal convergent sequences and generalized difference matrix operator, Filomat, 31(6) (2017), 1827–1834. https://doi.org/10.2298/FIL1706827M
  • Mohiuddine, S. A., Hazarika, B., Alotaibi, A., On statistical convergence of double sequences of fuzzy valued functions, J. Intell. Fuzzy Systems, 32 (2017), 4331–4342. https://doi.org/10.3233/JIFS-16974
  • Mohiuddine, S. A., Hazarika, B., Alghamdi, M. A., Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33(14) (2019), 4549–4560. https://doi.org/10.2298/FIL1914549M
  • Mursaleen, M., Mohiuddine, S. A., On ideal convergence in probabilistic normed spaces, Math. Slovaca, 62(1) (2012), 49–62. https://doi.org/10.2478/s12175-011-0071-9
  • Mursaleen, M., Mohiuddine, S. A., Osama Edely, H. H., On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl., 59(2) (2010), 603–611. https://doi.org/10.1016/j.camwa.2009.11.002
  • Nabiev, A., Pehlivan, S., Gürdal, M., On I-Cauchy sequence, Taiwanese J. Math., 11(2) (2007), 569–576.
  • Nabiev, A. A., Sava¸s, E., G¨urdal, M., I-localized sequences in metric spaces, Facta Univ. Ser. Math. Inform., 35(2) (2020), 459–469. https://doi.org/10.22190/FUMI2002459N
  • Rath, D., Tripathy, B. C., On statistically convergence and statistically Cauchy sequences, Indian J. Pure Appl. Math., 25(4) (1994), 381–386.
  • Şahiner, A. Gürdal, M., Düden, F. K., Triple sequences and their statistical convergence, Selcuk J. Appl. Math., 8(2) (2007), 49–55.
  • Şahiner, A., Tripathy, B. C., Some I-related properties of triple sequences, Selcuk J. Appl. Math., 9(2) (2008), 9–18.
  • Savaş, E., Gürdal, M., Certain summability methods in intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Systems, 27(4) (2014), 1621–1629. https://doi.org/10.3233/IFS-141128
  • Savaş, E., Gürdal, M., Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces, J. Intell. Fuzzy Systems, 27 (2014), 2067–2075. https://doi.org/10.3233/IFS- 141172
  • Savaş, E., Gürdal, M., I-statistical convergence in probabilistic normed spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77(4) (2015), 195–204.
  • Savaş, E., Gürdal, M., Ideal convergent function sequences in random 2-normed spaces, Filomat, 30(3) (2016), 557–567. https://doi.org/10.2298/FIL1603557S
  • Savaş, E., Mursaleen, M., On statistically convergent double sequences of fuzzy numbers, Inform. Sci., 162(3-4) (2004), 183–192. https://doi.org/10.1016/j.ins.2003.09.005
  • Savaş, E., Yamancı, U., Gürdal, M., I-lacunary statistical convergence of weighted g via modulus functions in 2-normed spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2) (2019), 2324–2332. https://doi.org/10.31801/cfsuasmas.573396
  • Schweizer, B., Sklar, A., Statistical metric spaces, Pacific J. Math., 10(1) (1960), 314–334.
  • Yamancı, U., Gürdal, M., Std-statistical convergence in intuitionistic fuzzy normed spaces, Notes on Intuitionistic Fuzzy Sets, 22(2) (2016), 52–58.
  • Yamancı, U., Sava¸s, E., Gürdal, M., I-localized sequence in two normed spaces, Malaysian J. Math. Sci., 14(3) (2020), 491–503.
  • Zadeh, L.A., Fuzzy sets, Inform. Control, 8(3) (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

An investigation on the triple ideal convergent sequences in fuzzy metric spaces

Year 2022, Volume: 71 Issue: 1, 13 - 24, 30.03.2022

Mehmet Gürdal , Ekrem Savaş

https://doi.org/10.31801/cfsuasmas.890982

Abstract

The notion of ideal convergence is a process of generalizing of statistical convergence which is dependent on the idea of the ideal $I$ of subsets of the set positive integer numbers. In this study we also present the concept of ideal convergence for triple sequences in fuzzy metric spaces (FMS) in the manner of George and Veeramani and the terms of ideal Cauchy sequence and $I^{∗}$-Cauchy sequence in FMS and study their certain properties.

Keywords

Ideal Cauchy sequence fuzzy metric triple sequence ideal convergence

References

  • Das, P., Kostyrko, P., Wilczyski, W., Malik, P., $I$ and $I^*$-convergence of double sequences, Math. Slovaca, 58(5) (2008), 605–620. https://doi.org/10.2478/s12175-008-0096-x
  • Engelking, R., General Topology, PWN-Polish Science Publishers, Warsaw, 1977.
  • Erceg, M. A., Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69(1) (1979), 205–230. https://doi.org/10.1016/0022-247X(79)90189-6
  • Fast, H., Sur la convergence statistique, Colloq. Math., 2(3-4) (1951), 241–244.
  • George, A., Veeramani, P., On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(3) (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7
  • George, A., Veeramani, P., On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90(3) (1997), 365–368. https://doi.org/10.1016/S0165-0114(96)00207-2
  • Gregori, V., Lopez-Crevillen, A., Morillas, S., Sapena, A., On convergence in fuzzy metric spaces, Topology Appl., 156(18) (2009), 3002–3006. https://doi.org/10.1016/j.topol.2008.12.043
  • Gregori, V., Minana, J. J., Std-convergence in fuzzy metric spaces, Fuzzy Sets and Systems, 267 (2015), 140–143. https://doi.org/10.1016/j.fss.2014.05.007
  • Gregori, V., Mi˜nana, J. J., Morillas, S., A note on convergence in fuzzy metric spaces, Iran J. Fuzzy Syst., 11(4) (2014), 75–85. https://doi.org/10.22111/IJFS.2014.1625
  • Gregori, V., Mi˜nana, J. J., Morillas, S., Sapena, A., Cauchyness and convergence in fuzzy metric spaces, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 111(1) (2017), 25–37. https://doi.org/10.1007/s13398-015-0272-0
  • Gürdal, M., Some types of convergence, Doctoral Dissertation, Suleyman Demirel University, Isparta, 2004.
  • Gürdal, M., Açık, I., On I-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl., 11(2) (2008), 349–354. https://doi.org/10.7153/mia-11-26
  • Gürdal, M., Huban, M. B., On I-convergence of double sequences in the topology induced by random 2-norms, Mat. Vesnik, 66(1) (2014), 73–83.
  • Gürdal, M., Sarı, N., Savaş, E., A-statistically localized sequences in n-normed spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(2) (2020), 1484 1497. https://doi.org/10.31801/cfsuasmas.704446
  • Gürdal, M., Şahiner, A., Extremal I-limit points of double sequences, Appl. Math. E-Notes, 8 (2008), 131–137.
  • Hazarika, B., Alotaibi, A., Mohiuddine, S. A., Statistical convergence inmeasure for double sequences of fuzzy-valued functions, Soft Comput., 24 (2020), 6613–6622. https://doi.org/10.1007/s00500-020-04805-y
  • Kaleva, O., Seikkala, S., On fuzzy metric spaces, Fuzzy Sets and Systems, 12(3) (1984), 215–229. https://doi.org/10.1016/0165-0114(84)90069-1
  • Kramosil, I., Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetika, 11(5) (1975), 336–344.
  • Li, C., Zhang, Y., Zhang, J., On statistical convergence in fuzzy metric spaces, J. Intell. Fuzzy Systems, 39(3) (2020), 3987–3993. https://doi.org/10.3233/JIFS-200148
  • Kostyrko, S., Salat, T., Wilczynski, W., I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
  • Mohiuddine, S. A., Asiri, A., Hazarika, B., Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst., 48(5) (2019), 492–506. https://doi.org/10.1080/03081079.2019.1608985
  • Mohiuddine, S. A., Hazarika, B., Some classes of ideal convergent sequences and generalized difference matrix operator, Filomat, 31(6) (2017), 1827–1834. https://doi.org/10.2298/FIL1706827M
  • Mohiuddine, S. A., Hazarika, B., Alotaibi, A., On statistical convergence of double sequences of fuzzy valued functions, J. Intell. Fuzzy Systems, 32 (2017), 4331–4342. https://doi.org/10.3233/JIFS-16974
  • Mohiuddine, S. A., Hazarika, B., Alghamdi, M. A., Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33(14) (2019), 4549–4560. https://doi.org/10.2298/FIL1914549M
  • Mursaleen, M., Mohiuddine, S. A., On ideal convergence in probabilistic normed spaces, Math. Slovaca, 62(1) (2012), 49–62. https://doi.org/10.2478/s12175-011-0071-9
  • Mursaleen, M., Mohiuddine, S. A., Osama Edely, H. H., On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl., 59(2) (2010), 603–611. https://doi.org/10.1016/j.camwa.2009.11.002
  • Nabiev, A., Pehlivan, S., Gürdal, M., On I-Cauchy sequence, Taiwanese J. Math., 11(2) (2007), 569–576.
  • Nabiev, A. A., Sava¸s, E., G¨urdal, M., I-localized sequences in metric spaces, Facta Univ. Ser. Math. Inform., 35(2) (2020), 459–469. https://doi.org/10.22190/FUMI2002459N
  • Rath, D., Tripathy, B. C., On statistically convergence and statistically Cauchy sequences, Indian J. Pure Appl. Math., 25(4) (1994), 381–386.
  • Şahiner, A. Gürdal, M., Düden, F. K., Triple sequences and their statistical convergence, Selcuk J. Appl. Math., 8(2) (2007), 49–55.
  • Şahiner, A., Tripathy, B. C., Some I-related properties of triple sequences, Selcuk J. Appl. Math., 9(2) (2008), 9–18.
  • Savaş, E., Gürdal, M., Certain summability methods in intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Systems, 27(4) (2014), 1621–1629. https://doi.org/10.3233/IFS-141128
  • Savaş, E., Gürdal, M., Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces, J. Intell. Fuzzy Systems, 27 (2014), 2067–2075. https://doi.org/10.3233/IFS- 141172
  • Savaş, E., Gürdal, M., I-statistical convergence in probabilistic normed spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77(4) (2015), 195–204.
  • Savaş, E., Gürdal, M., Ideal convergent function sequences in random 2-normed spaces, Filomat, 30(3) (2016), 557–567. https://doi.org/10.2298/FIL1603557S
  • Savaş, E., Mursaleen, M., On statistically convergent double sequences of fuzzy numbers, Inform. Sci., 162(3-4) (2004), 183–192. https://doi.org/10.1016/j.ins.2003.09.005
  • Savaş, E., Yamancı, U., Gürdal, M., I-lacunary statistical convergence of weighted g via modulus functions in 2-normed spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2) (2019), 2324–2332. https://doi.org/10.31801/cfsuasmas.573396
  • Schweizer, B., Sklar, A., Statistical metric spaces, Pacific J. Math., 10(1) (1960), 314–334.
  • Yamancı, U., Gürdal, M., Std-statistical convergence in intuitionistic fuzzy normed spaces, Notes on Intuitionistic Fuzzy Sets, 22(2) (2016), 52–58.
  • Yamancı, U., Sava¸s, E., Gürdal, M., I-localized sequence in two normed spaces, Malaysian J. Math. Sci., 14(3) (2020), 491–503.
  • Zadeh, L.A., Fuzzy sets, Inform. Control, 8(3) (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
There are 41 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Mehmet Gürdal 0000-0003-0866-1869

Ekrem Savaş 0000-0003-2135-3094

Publication Date March 30, 2022
Submission Date March 4, 2021
Acceptance Date July 9, 2021
Published in Issue Year 2022 Volume: 71 Issue: 1

Cite

APA Gürdal, M., & Savaş, E. (2022). An investigation on the triple ideal convergent sequences in fuzzy metric spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(1), 13-24. https://doi.org/10.31801/cfsuasmas.890982
AMA Gürdal M, Savaş E. An investigation on the triple ideal convergent sequences in fuzzy metric spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. March 2022;71(1):13-24. doi:10.31801/cfsuasmas.890982
Chicago Gürdal, Mehmet, and Ekrem Savaş. “An Investigation on the Triple Ideal Convergent Sequences in Fuzzy Metric Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 1 (March 2022): 13-24. https://doi.org/10.31801/cfsuasmas.890982.
EndNote Gürdal M, Savaş E (March 1, 2022) An investigation on the triple ideal convergent sequences in fuzzy metric spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 1 13–24.
IEEE M. Gürdal and E. Savaş, “An investigation on the triple ideal convergent sequences in fuzzy metric spaces”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 1, pp. 13–24, 2022, doi: 10.31801/cfsuasmas.890982.
ISNAD Gürdal, Mehmet - Savaş, Ekrem. “An Investigation on the Triple Ideal Convergent Sequences in Fuzzy Metric Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/1 (March 2022), 13-24. https://doi.org/10.31801/cfsuasmas.890982.
JAMA Gürdal M, Savaş E. An investigation on the triple ideal convergent sequences in fuzzy metric spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:13–24.
MLA Gürdal, Mehmet and Ekrem Savaş. “An Investigation on the Triple Ideal Convergent Sequences in Fuzzy Metric Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 1, 2022, pp. 13-24, doi:10.31801/cfsuasmas.890982.
Vancouver Gürdal M, Savaş E. An investigation on the triple ideal convergent sequences in fuzzy metric spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(1):13-24.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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