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Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces

Year 2024, Volume: 73 Issue: 3, 724 - 748, 27.09.2024

Ömer Kişi , Rümeysa Akbıyık , Mehmet Gürdal

https://doi.org/10.31801/cfsuasmas.1474229

Abstract

The primary objective of this study is to introduce the concepts of $I_2$-deferred Cesàro summability and $I_2-$ deferred statistical convergence for double sequences in fuzzy normed spaces (FNS). Furthermore, the aim is to explore the connections between these concepts and subsequently establish several theorems pertaining to the notion of $I_2$-deferred statistical convergence in FNS for double sequences. We further define $I_2$-deferred statistical limit points and $I_2$-deferred statistical cluster points of a sequence within FNS and explore the relationships among these concepts.

Keywords

Fuzzy normed spaces deferred statistical convergence Cesàro summability I₂-statistical convergence

References

  • Agnew, R. P., On deferred Cesaro mean, Ann. Math., 33(3) (1932), 413–421. https://doi.org/10.2307/1968524
  • Bag, T., Samanta, S. K., Fuzzy bounded linear operators, Fuzzy Sets Syst., 151(3) (2005), 513–547. https://doi.org/10.1016/j.fss.2004.05.004
  • Cheng, S. C., Mordeson, J. N., Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 86 (1994), 429–436.
  • Dağadur, I., Sezgek, Ş., Deferred Cesaro mean and deferred statistical convergence of double sequences, J. Inequal. Spec. Funct., 7(4) (2016), 118–136.
  • Dağadur, I., Sezgek, Ş., Deferred statistical cluster points of double sequences, Math. Appl., 4(2) (2015), 77–90. https://doi.org/10.13164/ma.2015.06
  • Das, N. R., Das, P., Fuzzy topology generated by fuzzy norm, Fuzzy Sets Syst., 107(3) (1999), 349–354. https://doi.org/10.1016/S0165-0114(97)00302-3
  • Das, P., Kostyrko, P., Wilczynski, 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
  • Et, M., Çınar, M., Şengül, H., Deferred statistical convergence in metric spaces, Conf. Proc. Sci. Tech., 2(3) (2019), 189-193.
  • Et, M., Yılmazer, M. Ç., On deferred statistical convergence of sequences of sets, AIMS Math., 5(3) (2020), 2143–2152. https://doi.org/10.3934/math.2020142
  • Fang, J. X., A note on the completions of fuzzy metric spaces and fuzzy normed spaces, Fuzzy Sets Syst., 131(3) (2002), 399–407. https://doi.org/10.1016/S0165-0114(02)00054-4
  • Fast, H., Sur la convergence statistique, Colloq. Math., 2(3-4) (1951), 241–244.
  • Felbin, C., Finite dimensional fuzzy normed linear space, Fuzzy Sets Syst., 48(2) (1992), 239–248. https://doi.org/10.1016/0165-0114(92)90338-5
  • Fridy, J. A., Statistical limit points, Proc. Amer. Math. Soc., 118(4) (1993), 1187–1192. https://doi.org/10.2307/2160076
  • Goetschel, R., Voxman, W., Elementary fuzzy calculus, Fuzzy Sets Syst., 18(1) (1986), 31–43. https://doi.org/10.1016/0165-0114(86)90026-6
  • Gülle, E., Ulusu, U., Dündar, E., Tortop, Ş., $I_2$-deferred statistical convergence for sequences of sets, Filomat, 38(3) (2024), 891–901. https://doi.org/10.2298/FIL2403891G
  • Hazarika, B., Alotaibi, A., Mohiuddine, S. A., Statistical convergence in measure for double sequences of fuzzy-valued functions, Soft Comput., 24 (2020), 6613–6622. https://doi.org/10.1007/s00500-020-04805-y
  • Katsaras, A. K., Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12(2) (1984), 143–154. https://doi.org/10.1016/0165-0114(84)90034-4
  • Kişi, Ö., Gürdal, M., Savaş, E., On deferred statistical convergence of fuzzy variables, Appl. Appl. Math., 17(2) (2022), 1–20.
  • Kişi, Ö., Huban, M. B., Gürdal, M., New results on $I_2$-statistically limit points and $I_2$-statistically cluster points of sequences of fuzzy numbers, J. Func. Spaces, 2021 (2021), Article ID 4602823, 6 pages. https://doi.org/10.1155/2021/4602823
  • Kostyrko, P., Salat, T., Wilczyski, W., $I$-convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
  • Kumar, V., Kumar, K., On the ideal convergence of sequences of fuzzy numbers, Inf. Sci., 178(24) (2008), 4670–4678. https://doi.org/10.1016/j.ins.2008.08.013
  • Kumar, V., Sharma, A., Kumar, K., Singh, N., On $I$-limit points and I$$-cluster points of sequences of fuzzy numbers, Int. Math. Forum, 2 (2007), 2815–2822.
  • Küçükaslan, M., Yilmaztürk, M., On deferred statistical convergence of sequences, Kyungpook Math. J., 56(2) (2016), 357–366. http://dx.doi.org/10.5666/KMJ.2016.56.2.357
  • Matloka, M., Sequences of fuzzy numbers, Busefal, 28 (1986), 28–37.
  • 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
  • Mursaleen, M., Edely, O.H.H., Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1) (2003), 223–231. https://doi.org/10.1016/j.jmaa.2003.08.004
  • Nanda, S., On sequences of fuzzy numbers, Fuzzy Sets Syst., 33(1) (1989), 123–126. https://doi.org/10.1016/0165-0114(89)90222-4
  • Nuray, F., Savaş, E., Statistical convergence of sequences of fuzzy numbers, Math. Slovaca, 45(3) (1995), 269-273.
  • Raj, K., Mohiuddine, S. A., Jasrotia, S., Characterization of summing operators in multiplier spaces of deferred Norlund summability, Positivity, 27 (2023), Article 9. https://doi.org/10.1007/s11117-022-00961-7
  • Savaş, E., Das, P., A generalized statistical convergence via ideals, Appl. Math. Lett., 24(6) (2011), 826–830. https://doi.org/10.1016/j.aml.2010.12.022
  • 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.
  • Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(5) (1951), 361–375. https://doi.org/10.2307/2308747
  • Şencimen, C., Pehlivan, S., Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets Syst., 159(3) (2008), 361–370. https://doi.org/10.1016/j.fss.2007.06.008
  • Şengül, H., Et, M., Işık, M., On $I$-deferred statistical convergence of order $\alpha$, Filomat, 33(9) (2019), 2833–2840. https://doi.org/10.2298/FIL1909833S
  • Sezgek, Ş., Dağadur, I., On strongly deferred Cesaro mean of double sequences, J. Math. Anal., 8(3) (2017), 43–53.
  • Tripathy, B. C., Debnath, S., Rakshit, D., On I$$-statistically limit points and $I$-statistically cluster poins of sequences of fuzzy numbers, Mathematica, 63(86)(1) (2021), 140–147. https://doi.org/10.24193/mathcluj.2021.1.13
  • Ulusu, U., Gülle, E., Deferred Cesaro summability and statistical convergence for double sequences of sets, J. Intell. Fuzzy Syst., 42(4) (2022), 4095–4103. https://doi.org/10.3233/JIFS-212486
  • Xiao, J., Zhu, X., On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst., 125(2) (2002), 153-161. https://doi.org/10.1016/S0165-0114(00)00136-6
  • Zadeh, L.A., Fuzzy sets, Infor. Control, 8(3) (1965), 338–353.
  • Zygmund, A., Trigonometric Series, Cambridge University Press, New York, 1959.

Year 2024, Volume: 73 Issue: 3, 724 - 748, 27.09.2024

Ömer Kişi , Rümeysa Akbıyık , Mehmet Gürdal

https://doi.org/10.31801/cfsuasmas.1474229

Abstract

References

  • Agnew, R. P., On deferred Cesaro mean, Ann. Math., 33(3) (1932), 413–421. https://doi.org/10.2307/1968524
  • Bag, T., Samanta, S. K., Fuzzy bounded linear operators, Fuzzy Sets Syst., 151(3) (2005), 513–547. https://doi.org/10.1016/j.fss.2004.05.004
  • Cheng, S. C., Mordeson, J. N., Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 86 (1994), 429–436.
  • Dağadur, I., Sezgek, Ş., Deferred Cesaro mean and deferred statistical convergence of double sequences, J. Inequal. Spec. Funct., 7(4) (2016), 118–136.
  • Dağadur, I., Sezgek, Ş., Deferred statistical cluster points of double sequences, Math. Appl., 4(2) (2015), 77–90. https://doi.org/10.13164/ma.2015.06
  • Das, N. R., Das, P., Fuzzy topology generated by fuzzy norm, Fuzzy Sets Syst., 107(3) (1999), 349–354. https://doi.org/10.1016/S0165-0114(97)00302-3
  • Das, P., Kostyrko, P., Wilczynski, 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
  • Et, M., Çınar, M., Şengül, H., Deferred statistical convergence in metric spaces, Conf. Proc. Sci. Tech., 2(3) (2019), 189-193.
  • Et, M., Yılmazer, M. Ç., On deferred statistical convergence of sequences of sets, AIMS Math., 5(3) (2020), 2143–2152. https://doi.org/10.3934/math.2020142
  • Fang, J. X., A note on the completions of fuzzy metric spaces and fuzzy normed spaces, Fuzzy Sets Syst., 131(3) (2002), 399–407. https://doi.org/10.1016/S0165-0114(02)00054-4
  • Fast, H., Sur la convergence statistique, Colloq. Math., 2(3-4) (1951), 241–244.
  • Felbin, C., Finite dimensional fuzzy normed linear space, Fuzzy Sets Syst., 48(2) (1992), 239–248. https://doi.org/10.1016/0165-0114(92)90338-5
  • Fridy, J. A., Statistical limit points, Proc. Amer. Math. Soc., 118(4) (1993), 1187–1192. https://doi.org/10.2307/2160076
  • Goetschel, R., Voxman, W., Elementary fuzzy calculus, Fuzzy Sets Syst., 18(1) (1986), 31–43. https://doi.org/10.1016/0165-0114(86)90026-6
  • Gülle, E., Ulusu, U., Dündar, E., Tortop, Ş., $I_2$-deferred statistical convergence for sequences of sets, Filomat, 38(3) (2024), 891–901. https://doi.org/10.2298/FIL2403891G
  • Hazarika, B., Alotaibi, A., Mohiuddine, S. A., Statistical convergence in measure for double sequences of fuzzy-valued functions, Soft Comput., 24 (2020), 6613–6622. https://doi.org/10.1007/s00500-020-04805-y
  • Katsaras, A. K., Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12(2) (1984), 143–154. https://doi.org/10.1016/0165-0114(84)90034-4
  • Kişi, Ö., Gürdal, M., Savaş, E., On deferred statistical convergence of fuzzy variables, Appl. Appl. Math., 17(2) (2022), 1–20.
  • Kişi, Ö., Huban, M. B., Gürdal, M., New results on $I_2$-statistically limit points and $I_2$-statistically cluster points of sequences of fuzzy numbers, J. Func. Spaces, 2021 (2021), Article ID 4602823, 6 pages. https://doi.org/10.1155/2021/4602823
  • Kostyrko, P., Salat, T., Wilczyski, W., $I$-convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
  • Kumar, V., Kumar, K., On the ideal convergence of sequences of fuzzy numbers, Inf. Sci., 178(24) (2008), 4670–4678. https://doi.org/10.1016/j.ins.2008.08.013
  • Kumar, V., Sharma, A., Kumar, K., Singh, N., On $I$-limit points and I$$-cluster points of sequences of fuzzy numbers, Int. Math. Forum, 2 (2007), 2815–2822.
  • Küçükaslan, M., Yilmaztürk, M., On deferred statistical convergence of sequences, Kyungpook Math. J., 56(2) (2016), 357–366. http://dx.doi.org/10.5666/KMJ.2016.56.2.357
  • Matloka, M., Sequences of fuzzy numbers, Busefal, 28 (1986), 28–37.
  • 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
  • Mursaleen, M., Edely, O.H.H., Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1) (2003), 223–231. https://doi.org/10.1016/j.jmaa.2003.08.004
  • Nanda, S., On sequences of fuzzy numbers, Fuzzy Sets Syst., 33(1) (1989), 123–126. https://doi.org/10.1016/0165-0114(89)90222-4
  • Nuray, F., Savaş, E., Statistical convergence of sequences of fuzzy numbers, Math. Slovaca, 45(3) (1995), 269-273.
  • Raj, K., Mohiuddine, S. A., Jasrotia, S., Characterization of summing operators in multiplier spaces of deferred Norlund summability, Positivity, 27 (2023), Article 9. https://doi.org/10.1007/s11117-022-00961-7
  • Savaş, E., Das, P., A generalized statistical convergence via ideals, Appl. Math. Lett., 24(6) (2011), 826–830. https://doi.org/10.1016/j.aml.2010.12.022
  • 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.
  • Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(5) (1951), 361–375. https://doi.org/10.2307/2308747
  • Şencimen, C., Pehlivan, S., Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets Syst., 159(3) (2008), 361–370. https://doi.org/10.1016/j.fss.2007.06.008
  • Şengül, H., Et, M., Işık, M., On $I$-deferred statistical convergence of order $\alpha$, Filomat, 33(9) (2019), 2833–2840. https://doi.org/10.2298/FIL1909833S
  • Sezgek, Ş., Dağadur, I., On strongly deferred Cesaro mean of double sequences, J. Math. Anal., 8(3) (2017), 43–53.
  • Tripathy, B. C., Debnath, S., Rakshit, D., On I$$-statistically limit points and $I$-statistically cluster poins of sequences of fuzzy numbers, Mathematica, 63(86)(1) (2021), 140–147. https://doi.org/10.24193/mathcluj.2021.1.13
  • Ulusu, U., Gülle, E., Deferred Cesaro summability and statistical convergence for double sequences of sets, J. Intell. Fuzzy Syst., 42(4) (2022), 4095–4103. https://doi.org/10.3233/JIFS-212486
  • Xiao, J., Zhu, X., On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst., 125(2) (2002), 153-161. https://doi.org/10.1016/S0165-0114(00)00136-6
  • Zadeh, L.A., Fuzzy sets, Infor. Control, 8(3) (1965), 338–353.
  • Zygmund, A., Trigonometric Series, Cambridge University Press, New York, 1959.
There are 40 citations in total.

Details

Primary Language English
Subjects Operations Research İn Mathematics
Journal Section Research Articles
Authors

Ömer Kişi 0000-0001-6844-3092

Rümeysa Akbıyık 0009-0004-8751-0325

Mehmet Gürdal 0000-0003-0866-1869

Publication Date September 27, 2024
Submission Date April 26, 2024
Acceptance Date June 3, 2024
Published in Issue Year 2024 Volume: 73 Issue: 3

Cite

APA Kişi, Ö., Akbıyık, R., & Gürdal, M. (2024). Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(3), 724-748. https://doi.org/10.31801/cfsuasmas.1474229
AMA Kişi Ö, Akbıyık R, Gürdal M. Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2024;73(3):724-748. doi:10.31801/cfsuasmas.1474229
Chicago Kişi, Ömer, Rümeysa Akbıyık, and Mehmet Gürdal. “Some Results on $I_2$-Deferred Statistically Convergent Double Sequences in Fuzzy Normed Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, no. 3 (September 2024): 724-48. https://doi.org/10.31801/cfsuasmas.1474229.
EndNote Kişi Ö, Akbıyık R, Gürdal M (September 1, 2024) Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 3 724–748.
IEEE Ö. Kişi, R. Akbıyık, and M. Gürdal, “Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 73, no. 3, pp. 724–748, 2024, doi: 10.31801/cfsuasmas.1474229.
ISNAD Kişi, Ömer et al. “Some Results on $I_2$-Deferred Statistically Convergent Double Sequences in Fuzzy Normed Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/3 (September 2024), 724-748. https://doi.org/10.31801/cfsuasmas.1474229.
JAMA Kişi Ö, Akbıyık R, Gürdal M. Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:724–748.
MLA Kişi, Ömer et al. “Some Results on $I_2$-Deferred Statistically Convergent Double Sequences in Fuzzy Normed Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 73, no. 3, 2024, pp. 724-48, doi:10.31801/cfsuasmas.1474229.
Vancouver Kişi Ö, Akbıyık R, Gürdal M. Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(3):724-48.

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

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