Chiranjib Choudhury, Shyamal Debnath

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In this paper, we introduce the concepts of $\mathcal{I}$ and $\mathcal{I^{*}}-$convergence of sequences in gradual normed linear spaces. We study some basic properties and implication relations of the newly defined convergence concepts. Also, we introduce the notions of $\mathcal{I}$ and $\mathcal{I^{*}}-$Cauchy sequences in the gradual normed linear space and investigate the relations between them.


Gradual number; gradual normed linear space; ideal; filter; ideal convergence.

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F. Aiche and D. Dubois: Possibility and gradual number approaches to ranking methods for random fuzzy intervals. Commun. Comput. Inf. Sci. 299 (2012), 9-18.

S. Debnath and D. Rakshit: On I−statistical convergence. Iran. J. Math. Sci. Inform. 13(2) (2018), 101-109.

K. Demirci: I−limit superior and limit inferior. Math. Commun. 6 (2001), 165-172.

K. Dems: On I−Cauchy sequences. Real Anal. Exchange 30(1) (2004-2005),123-128.

D. Dubois and H. Prade: Gradual elements in a fuzzy set. Soft Comput. 12 (2007), 165-175.

M. Ettefagh, F. Y. Azari and S. Etemad: On some topological properties in gradual normed spaces. Facta Univ. Ser. Math. Inform. 35(3) (2020), 549-559.

M. Ettefagh, S. Etemad and F. Y. Azari: On some properties of sequences in gradual normed spaces. Asian-Eur. J. Math. 13(4) (2020), 2050085.

J. Fortin, D. Dubois and H. Fargier: Gradual numbers and their application to fuzzy interval analysis. IEEE Trans. Fuzzy Syst. 16(2) (2008), 388-402.

J. Gogola, M. Macaj and T. Visnyai: On Ic(q)−convergence. Ann. Math. Inform. 38 (2011), 27-36.

P. Kostyrko, M. Macaj, T. Salat and M. Sleziak: I-convergence and extremal I-limit points. Math. Slovaca 55(4) (2005), 443-464.

P. Kostyrko, T. Salat and W. Wilczynski: I-convergence. Real Anal. Exch. 26(2) (2000-2001), 669-686.

L. Lietard and D. Rocacher: Conditions with aggregates evaluated using gradual numbers. Control Cybernet. 38 (2009), 395-417.

M. Mursaleen and S. A. Mohiuddine: On ideal convergence in probabilistic normed spaces. Math. Slovaca 62(1) (2012), 49-62.

A. Nabiev, S. Pehlivan and M. Gurdal: On I−Cauchy sequences. Taiwanese J. Math. 11(2) (2007), 569-576.

I. Sadeqi and F. Y. Azari: Gradual normed linear space. Iran. J. Fuzzy Syst. 8(5) (2011), 131-139.

T. Salat, B. C. Tripathy and M. Ziman: On some properties of I−convergence. Ann. Math. Inform. 38 (2011), 27-36.

E. Savas and P. Das: A generalized statistical convergence via ideals. Appl. Math. Lett. 24(6) (2011), 826-830.

E. A. Stock: Gradual numbers and fuzzy optimization. Ph. D. Thesis, University of Colorado Denver, Denver, America, 2010.

B. C. Tripathy and B. Hazarika: Paranorm I-convergent sequence spaces. Math. Slovaca 59(4) (2009), 485-494.

L. A. Zadeh: Fuzzy sets. Inf. Control 8 (1965), 338-353.

C. Zhou: Gradual metric spaces. Appl. Math. Sci. 9(14) (2015), 689-701.

C. Zhou and J. Li: New fuzzy measure based on gradual numbers. Int. J. Math. Anal. (N.S.) 9 (2015), 101-110.

DOI: https://doi.org/10.22190/FUMI210108044C


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