Rough I_((λ,μ) )-Statistical Convergence of Double Sequences in Gradual Normed Linear Spaces

Year 2022, Volume: 17 Issue: 2, 405 - 428, 25.11.2022

Ömer Kişi Chiranjib Choudhury

https://doi.org/10.29233/sdufeffd.1158636

Abstract

The aim of this paper is to we examine the notion of gradually rough I_((λ,μ) )-statistical convergence of double sequences in gradual normed linear spaces (GNLS). In addition, we define the concept of gradually rough I_((λ,μ) )-statistical limit set of double sequences and obtain some algebraic and topological features of this set. Theorems are proved in the light of GNLS theory approach. Results are obtained via different perspective and new examples are established to justify the counterparts and indicate existence of introduced notions. We produce significant results that present several fundamental properties of this notion. The results established in this research work supplies an exhaustive foundation in GNLS and make a significant contribution in the theoretical development of GNLS in literature. The original aspect of this study is the first wholly up-to-date and thorough examination of the features and implementations of new introduced notions in GNLS.

Keywords

Gradual normed linear spaces, Rough convergence, I-convergence

Gradual Normlu Uzaylarda Çift Dizilerin Kaba I_((λ,μ) )-İstatistiksel Yakınsaklığı

Abstract

Bu makalenin amacı, gradual normlu lineer uzaylarda (GNLU) çift dizilerin kaba I_((λ,μ) )-istatistiksel yakınsaklığı kavramını incelemektir. Ayrıca, çift dizilerinin gradual kaba I_((λ,μ) )-istatistiksel limit kümesi kavramını tanımlayacak, bu kümenin bazı cebirsel ve topolojik özelliklerini elde edeceğiz. Teoremler, GNLU teorisi yaklaşımı ışığında ispatlanacaktır. Farklı bakış açılarıyla sonuçlar elde edilecek ve karşıtları haklı çıkarmak ve tanıtılan kavramların varlığını göstermek için yeni örnekler üretilecektir. Bu kavramların bazı temel özelliklerini sunan önemli sonuçlar elde edilecektir. Bu araştırma çalışmasında elde edilen sonuçlar, GNLU'da kapsamlı bir temel sağlayacak ve literatürde GNLS'nin teorik gelişimine önemli bir katkı sağlayacaktır. Bu çalışmanın özgün yönü, GNLS'de tanımlanan yeni kavramların özelliklerinin ve uygulamalarının tamamen güncel ve kapsamlı ilk incelemesidir.

Keywords

Gradual normed linear uzaylar, Kaba yakınsaklık, I-yakınsaklık

References

• F. Aiche and D. Dubois, “Possibility and gradual number approaches to ranking methods for random fuzzy intervals,” Commun. Comput. Inf. Sci.,, 299, 9–18, 2012.
• S. Aytar, “Rough statistical covergence,” Numer. Funct. Anal. Optimiz., 29 (3), 291–303, 2008.
• C. Choudhury and S. Debnath, “On I-convergence of sequences in gradual normed linear spaces,” Facta Univ. Ser. Math. Inform., 36 (3), 595–604, 2021.
• P. Das, P. Kostyrko, W. Wilczynski and P. Malik, “I and I^*-convergence of double sequences,” Math. Slovaca, 58, 605–620, 2008.
• P. Das and P. Malik, “On the statistical and I-variation of double sequences,” Tatra Mt. Math. Publ., 40, 91–112, 2008.
• P. Das, E. Savaş and S. K. Ghosal, “On generalizations of certain summability methods using ideals,” Appl. Math. Letters, 24, 1509–1514, 2011.
• D. Dubois and H. Prade, “Gradual elements in a fuzzy set,” Soft Comput., 12, 165–175, 2007.
• E. Dündar, “On Rough I_2-convergence,” Numer. Funct. Anal. Optimiz., 37 (4), 480–491, 2016.
• E. Dündar and C. Çakan, “Rough convergence of double sequences,” Demonstr. Math., 47 (3), 638–651, 2014.
• M. Ettefagh, F. Y. Azari and S. “Etemad, “On some topological properties in gradual normed spaces,” Facta Univ. Ser. Math. Inform., 35 (3), 549–559, 2020.
• M. Ettefagh, S. Etemad and F. Y. Azari, “On some properties of sequences in gradual normed spaces,” Asian-Eur. J. Math., 13 (4), 2050085, 2020.
• A. Ghosh and P. Malik, “Rough I^λ-statistical convergence of sequences,” Bull. Allahabad Math. Soc., in press.
• M. Gürdal, “On ideal convergent sequences in 2-normed spaces,” Thai J. Math., 4 (1), 85–91, 2012.
• H. Fast, “Surla convergence statistique,” Colloq. Math., 2, 241–244, 1951.
• J. Fortin, D. Dubois and H. Fargier, “Gradual numbers and their application to fuzzy interval analysis,” IEEE Trans. Fuzzy Syst., 16, 388–402, 2008.
• A.R. Freedman and J.J. Sember, “Densities and summability,” Pacific J. Math., 95, 293–305, 1981.
• J. A. Fridy, “On statistical convergence,” Analysis, 5, 301–313, 1985.
• Ö. Kişi and E. Dündar, “Rough I_2-lacunary statistical convergence of double sequences,” J. Inequal. Appl., 230, 16 pages, 2018.
• P. Kostyrko, T. Salát and W. Wilczyński, “I-convergence,” Real Anal. Exchange, 26 (2), 669–686, 2000.
• L. Lietard and D. Rocacher, “Conditions with aggregates evaluated using gradual numbers,” Control Cybernet., 38, 395–417, 2009.
• P. Malik and M. Maity, “On rough statistical convergence of double sequences in normed linear spaces,” Afr. Math., 27, 141–148, 2016.
• P. Malik and M. Maity, “On rough convergence of double sequences in normed linear spaces,” Bull. Allahabad Math. Soc., 28 (1), 89–99, 2013.
• P. Malik, M. Maity and A. Ghosh, “A note on rough I-convergence of double sequences,” [Online]. Available: https://arxiv.org/abs/1603.01363
• P. Malik, M. Maity and A. Ghosh, “Rough I-statistical convergence of sequences,” [Online]. Available: https://arxiv.org/abs/1611.03224v2
• M. Mursaleen, “λ-statistical convergence,” Math. Slovaca, 50 (1), 111–115, 2000.
• S. K. Pal, D. Chandra and S. Dutta, “Rough ideal convergence,” Hacettepe J. Math. Stat., 42 (6), 633–640, 2013.
• H. X. Phu, “Rough convergence in normed linear spaces,” Numer. Funct. Anal. Optimiz., 22, 201–224, 2001.
• I. Sadeqi and F. Y. Azari, “Gradual normed linear space,” Iran. J. Fuzzy Syst., 8 (5), 131–139, 2011.
• T. Salát, “On statistically convergent sequences of real numbers,” Math. Slovaca, 30, 139–150, 1980.
• E. Savaş and P. Das, “A generalized statistical convergence via ideals,” App. Math. Letters, 24, 826–830, 2011.
• E. Savaş and M. Gürdal, “A generalized statistical convergence in intuitionistic fuzzy normed spaces,” Science Asia, 41, 289–294, 2015.
• E. Savaş and R. F. Patterson, “(λ,μ)-double sequence spaces via Orlicz function,” J. Comput. Anal. Appl., 10 (1), 101–111, 2008.
• H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloq. Math., 2, 73–74, 1951.
• E. A. Stock, “Gradual numbers and fuzzy optimization,” PhD. Thesis, University of Colorado Denver, Denver, America, 2010.
• L. A. Zadeh, “Fuzzy set,” Inf. Control. 8 (3), 338–353, 1965.