Öz
Nanda [29] 1989 yılında bütün yakınsak bulanık sayı dizilerinin tam metrik uzaylar olduğunu gösterdi. Ayrıca, Nuray [30] bulanık sayılarda lacunary istatistiksel yakınsak ve istatistiksel yakınsak diziler arasındaki ilişkileri verdi. Bununla birlikte, bulanık sayı dizilerinin çeşitli yönleri birçok yazar tarafından tartışılmıştır. Bu çalışmada, üç indisli bir bulanık sayı dizisinin lacunary istatistiksel yakınsaklığı ve üç indisli lacunary güçlü p-Cesàro toplanabilmesi kavramları incelenmiştir. Üç indisli lacunary istatistiksel Cauchy dizisi, üç indisli lacunary güçlü p-Cesàro toplanabilme ve lacunary istatistiksel olarak bulanık bir sayıya yakınsak olmayı tanımlıyoruz. Bu çeşitli kavramlar arasında bir ilişki olduğunu düşünüyoruz ve bu nedenle, makalede bu konu ile ilgili bazı temel teoremlere yer veriyoruz.
Anahtar Kelimeler
Lacunary istatistiksel yakınsaklık İstatistiksel yakınsaklık Bulanık sayı Üç indisli dizi
Teşekkür
Bu çalışmanın yazarı olarak, çalışmanın okunabilirliğinin iyileştirilmesine katkıda bulunan hakemlere teşekkür ederim.
Kaynakça
- R.P. Agnew, “On summability of multiple sequences,” Amer. J. Math., 1 (4), 62–68, 1934.
- Y. Altın, M. Et and R. Çolak, “Lacunary statistical and lacunary strongly convergence of generalized difference sequences of fuzzy numbers,” Comput. Math., 52, 1011–1020, 2006.
- J. S. Connor, “The statistical and strong p -Cesàro convergence of sequences,” Analysis, 8, 46–63, 1988.
- P. Diamond and P. Kloeden, “Metric of fuzzy space sets,” Fuzzy Sets Syst., 35, 241–249, 1990.
- A. Esi and E. Savaş, “On lacunary statistically convergent triple sequences in probabilistic normed space,” Appl. Math. Inf. Sci., 9 (5), 2529–2534, 2015.
- H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 241–244, 1951.
- X. Feng, “Ideal statistically pre-Cauchy triple sequences of fuzzy number and Orlicz functions,” Appl. Math., 12 (9), 767–774, 2021.
- A. R. Freedman and J. J. Sember, “Densities and summability,” Pac. J. Math., 95 (2), 293–305, 1981.
- J. A. Fridy, “On statistical convergence,” Analysis, 5 (4), 301–313, 1985.
- J. A. Fridy and C. Orhan, “Lacunary statistical convergence,” Pac. J. Math., 160 (1), 43–51, 1993.
- M. Gürdal, “Some types of convergence,” Diss. Doctoral Dissertation, Isparta, 2004.
- J. D. Hill, “On perfect summability of double sequences,” Bull. Am. Math. Soc., 46, 327–331, 1940.
- B. Hazarika, “Lacunary difference ideal convergent sequence spaces of fuzzy numbers,” J. Intell. Fuzzy Syst., 25, 157–166, 2013.
- Ö. Kişi, “Lacunary ideal convergence in measure for sequences of fuzzy valued functions,” J. Intell. Fuzzy Syst., 40 (3), 5517-5526, 2021.
- Ö. Kişi, “Lacunary statistical convergence in measure for double sequences of fuzzy valued functions,” Hindawi, J. Math., 12 pages, 2021.
- Ö. Kişi, V. Gürdal and M. B. Huban, “Ideal statistically limit points and ideal statistically cluster points of triple sequences of fuzzy numbers,” J. Classical Anal., 19 (2), 127–137, 2022.
- Ö. Kişi, M. B. Huban and M. Gürdal, “New results on I_2-statistically limit points and I_2-statistically cluster points of sequences of fuzzy numbers,” Hindawi, J. Funct. Spaces, 6 pages, 2021.
- I. G. Kull, “Multiplication of summable double series,” Uch. zap. Tartusskogo un-ta, 62, 3–59, 1958, (in Russian).
- P. Kumar, V. Kumar and S. S. Bhatia, “Multiple sequences of fuzzy numbers and their statistical convergence,” Math. Sci., 6 (2), 1–6, 2012.
- J. S. Kwon and H. T. Shim, “Remark on lacunary statistical convergence of fuzzy numbers,” Fuzzy Sets Syst., 123, 85–88, 2001.
- J. S. Kwon and S. H. Sung, “On lacunary statistical and p-Cesàro summability of fuzzy numbers,” J. Fuzzy Math., 9 (3), 603–610, 2001.
- B. V. Limaye and M. Zeltser. “On the Pringsheim convergence of double series,” Proc. Est. Acad. Sci., 58 (2), 108–121, 2009.
- M. Matloka, “Sequences of fuzzy numbers,” Busefal, 28, 28–37, 1986.
- F. Móricz, “Some remarks on the notion of regular convergence of multiple series,” Acta Math. Hungarica, 41 (1–2), 161–168, 1983.
- F. Móricz, “Statistical convergence of multiple sequences,” Arch. Math., 81, 82–89, 2003.
- S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,” Appl. Math. Inf. Sci., 6 (3), 581–585, 2012.
- M. Mursaleen and M. Başarır, “On some new sequence spaces of fuzzy numbers, Indian J. Pure Appl. Math., 34 (9), 1351–1357, 2003.
- M. Mursaleen, H. M. Srivastava and S. K. Sharma, “Generalized statistically convergent sequences of fuzzy numbers,” J. Intell. Fuzzy Syst., 30, 1511–1518, 2016.
- S. Nanda, “On sequence of fuzzy numbers,” Fuzzy Sets Syst., 33, 123–126, 1989.
- F. Nuray, “Lacunary statistical convergence of sequences of fuzzy numbers Fuzzy Sets Syst., 99 (3), 353–355, 1998.
- F. Nuray and E. Savaş, “Statistical convergence of sequences of fuzzy numbers,” Math. Slovaca, 45 (3), 269–273, 1995.
- A. Pringsheim, “Zur theorie der zweifach unendlichen Zahlenfolgen,” Math. Ann., 53, 289–321, 1900.
- R. T. Rockafellar and R. T.-B. Wets, Variational Analysis. Grundlehren der Mathematischen Wissenschaften 317, Springer-Verlag, 1997, 733 pages, third printing New York, 2009.
- T. Šalát, “On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139–150, 1980.
- E. Savaş, “A note on double sequence of fuzzy numbers,” Turk. J. Math., 20, 175–178, 1996.
- E. Savaş, “On statistically convergent sequences of fuzzy numbers,” Inf. Sci., 137, 272–282, 2001.
- E. Savaş, “On lacunary statistically convergent double sequences of fuzzy numbers,” Appl. Math. Lett., 21, 134–141, 2008.
- E. Savaş, “A note on double lacunary statistical sigma-convergence of fuzzy numbers,” Soft Comput., 16 (4), 591–595, 2012.
- E. Savaş and M. Gürdal, “Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces,” J. Intell. Fuzzy Syst., 27 (4), 2067-2075, 2014.
- E. Savaş and M. Mursaleen, “On statistically convergent double sequences of fuzzy numbers,” Inf. Sci., 162 (3–4), 183-192, 2004.
- E. Savaş and R. F. Patterson, “Lacunary statistical convergence of multiple sequences,” Appl. Math. Lett., 19 (6), 527–534, 2006.
- I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Am. Math. Mon., 66, 361–375, 1959.
- A. Şahiner, M. Gürdal and F. K. Düden, “Triple sequences and their statistical convergence,” Selcuk J. Appl. Math., 8 (2), 49–55, 2007.
- B. C. Tripathy and A. Baruah, “Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers,” Kyungpook Math. J., 50, 565–574, 2010.
- U. Ulusu and F. Nuray, “On strongly lacunary summability of sequences of sets,” J. Appl. Math. Bioinform., 3 (3), 75–88, 2013.
- U. Yamancı and M. Gürdal, “On lacunary ideal convergence in random normed space,” J. Math., 8 pages, 2013.
Öz
It was Nanda [29] in 1989 who demonstrate that all convergent fuzzy number sequences are complete metric spaces. Nuray [30] gave the relations between lacunary statistical convergent and statistical convergent sequences in fuzzy numbers. Various aspects of fuzzy number sequences have been discussed by many different authors. In this work, we peruse the notions of lacunary statistical convergence of a triple sequence of fuzzy numbers and triple lacunary strongly p-Cesàro summability. We define triple lacunary statistically Cauchy sequence, triple lacunary strongly p-Cesàro summable and lacunary statistically convergent to a fuzzy number. We think that there is a relationship between these various concepts. Thus in this article, we include some basic theorems on this subject.
Anahtar Kelimeler
Lacunary statistical convergence Statistical convergence Fuzzy number Triple sequence
Kaynakça
- R.P. Agnew, “On summability of multiple sequences,” Amer. J. Math., 1 (4), 62–68, 1934.
- Y. Altın, M. Et and R. Çolak, “Lacunary statistical and lacunary strongly convergence of generalized difference sequences of fuzzy numbers,” Comput. Math., 52, 1011–1020, 2006.
- J. S. Connor, “The statistical and strong p -Cesàro convergence of sequences,” Analysis, 8, 46–63, 1988.
- P. Diamond and P. Kloeden, “Metric of fuzzy space sets,” Fuzzy Sets Syst., 35, 241–249, 1990.
- A. Esi and E. Savaş, “On lacunary statistically convergent triple sequences in probabilistic normed space,” Appl. Math. Inf. Sci., 9 (5), 2529–2534, 2015.
- H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 241–244, 1951.
- X. Feng, “Ideal statistically pre-Cauchy triple sequences of fuzzy number and Orlicz functions,” Appl. Math., 12 (9), 767–774, 2021.
- A. R. Freedman and J. J. Sember, “Densities and summability,” Pac. J. Math., 95 (2), 293–305, 1981.
- J. A. Fridy, “On statistical convergence,” Analysis, 5 (4), 301–313, 1985.
- J. A. Fridy and C. Orhan, “Lacunary statistical convergence,” Pac. J. Math., 160 (1), 43–51, 1993.
- M. Gürdal, “Some types of convergence,” Diss. Doctoral Dissertation, Isparta, 2004.
- J. D. Hill, “On perfect summability of double sequences,” Bull. Am. Math. Soc., 46, 327–331, 1940.
- B. Hazarika, “Lacunary difference ideal convergent sequence spaces of fuzzy numbers,” J. Intell. Fuzzy Syst., 25, 157–166, 2013.
- Ö. Kişi, “Lacunary ideal convergence in measure for sequences of fuzzy valued functions,” J. Intell. Fuzzy Syst., 40 (3), 5517-5526, 2021.
- Ö. Kişi, “Lacunary statistical convergence in measure for double sequences of fuzzy valued functions,” Hindawi, J. Math., 12 pages, 2021.
- Ö. Kişi, V. Gürdal and M. B. Huban, “Ideal statistically limit points and ideal statistically cluster points of triple sequences of fuzzy numbers,” J. Classical Anal., 19 (2), 127–137, 2022.
- Ö. Kişi, M. B. Huban and M. Gürdal, “New results on I_2-statistically limit points and I_2-statistically cluster points of sequences of fuzzy numbers,” Hindawi, J. Funct. Spaces, 6 pages, 2021.
- I. G. Kull, “Multiplication of summable double series,” Uch. zap. Tartusskogo un-ta, 62, 3–59, 1958, (in Russian).
- P. Kumar, V. Kumar and S. S. Bhatia, “Multiple sequences of fuzzy numbers and their statistical convergence,” Math. Sci., 6 (2), 1–6, 2012.
- J. S. Kwon and H. T. Shim, “Remark on lacunary statistical convergence of fuzzy numbers,” Fuzzy Sets Syst., 123, 85–88, 2001.
- J. S. Kwon and S. H. Sung, “On lacunary statistical and p-Cesàro summability of fuzzy numbers,” J. Fuzzy Math., 9 (3), 603–610, 2001.
- B. V. Limaye and M. Zeltser. “On the Pringsheim convergence of double series,” Proc. Est. Acad. Sci., 58 (2), 108–121, 2009.
- M. Matloka, “Sequences of fuzzy numbers,” Busefal, 28, 28–37, 1986.
- F. Móricz, “Some remarks on the notion of regular convergence of multiple series,” Acta Math. Hungarica, 41 (1–2), 161–168, 1983.
- F. Móricz, “Statistical convergence of multiple sequences,” Arch. Math., 81, 82–89, 2003.
- S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,” Appl. Math. Inf. Sci., 6 (3), 581–585, 2012.
- M. Mursaleen and M. Başarır, “On some new sequence spaces of fuzzy numbers, Indian J. Pure Appl. Math., 34 (9), 1351–1357, 2003.
- M. Mursaleen, H. M. Srivastava and S. K. Sharma, “Generalized statistically convergent sequences of fuzzy numbers,” J. Intell. Fuzzy Syst., 30, 1511–1518, 2016.
- S. Nanda, “On sequence of fuzzy numbers,” Fuzzy Sets Syst., 33, 123–126, 1989.
- F. Nuray, “Lacunary statistical convergence of sequences of fuzzy numbers Fuzzy Sets Syst., 99 (3), 353–355, 1998.
- F. Nuray and E. Savaş, “Statistical convergence of sequences of fuzzy numbers,” Math. Slovaca, 45 (3), 269–273, 1995.
- A. Pringsheim, “Zur theorie der zweifach unendlichen Zahlenfolgen,” Math. Ann., 53, 289–321, 1900.
- R. T. Rockafellar and R. T.-B. Wets, Variational Analysis. Grundlehren der Mathematischen Wissenschaften 317, Springer-Verlag, 1997, 733 pages, third printing New York, 2009.
- T. Šalát, “On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139–150, 1980.
- E. Savaş, “A note on double sequence of fuzzy numbers,” Turk. J. Math., 20, 175–178, 1996.
- E. Savaş, “On statistically convergent sequences of fuzzy numbers,” Inf. Sci., 137, 272–282, 2001.
- E. Savaş, “On lacunary statistically convergent double sequences of fuzzy numbers,” Appl. Math. Lett., 21, 134–141, 2008.
- E. Savaş, “A note on double lacunary statistical sigma-convergence of fuzzy numbers,” Soft Comput., 16 (4), 591–595, 2012.
- E. Savaş and M. Gürdal, “Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces,” J. Intell. Fuzzy Syst., 27 (4), 2067-2075, 2014.
- E. Savaş and M. Mursaleen, “On statistically convergent double sequences of fuzzy numbers,” Inf. Sci., 162 (3–4), 183-192, 2004.
- E. Savaş and R. F. Patterson, “Lacunary statistical convergence of multiple sequences,” Appl. Math. Lett., 19 (6), 527–534, 2006.
- I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Am. Math. Mon., 66, 361–375, 1959.
- A. Şahiner, M. Gürdal and F. K. Düden, “Triple sequences and their statistical convergence,” Selcuk J. Appl. Math., 8 (2), 49–55, 2007.
- B. C. Tripathy and A. Baruah, “Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers,” Kyungpook Math. J., 50, 565–574, 2010.
- U. Ulusu and F. Nuray, “On strongly lacunary summability of sequences of sets,” J. Appl. Math. Bioinform., 3 (3), 75–88, 2013.
- U. Yamancı and M. Gürdal, “On lacunary ideal convergence in random normed space,” J. Math., 8 pages, 2013.
Ayrıntılar
| Birincil Dil | Türkçe |
|---|---|
| Konular | Matematik |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 25 Kasım 2022 |
| Yayımlandığı Sayı | Yıl 2022 Cilt: 17 Sayı: 2 |
