Document Type : Research Paper


Department Of Mathematics, Chandigarh University, Mohali, 140301, Punjab, India.


In this paper, we define strong deferred summability and deferred statistical summability in intuitionistic fuzzy \(n\)-normed linear spaces (briefly called IF-\(n\)-NLS) and study some of their properties. We also define deferred statistical Cauchy sequence deferred statistical completeness and characterize deferred statistical summability in such spaces. Finally, we will compare deferred statistical summability for different pairs of sequences $\alpha(\ell)$, $\gamma(\ell)$,$u(\ell)$ and $v(\ell)$ satisfying $\alpha(\ell) \leq u(\ell) < v(\ell) \leq \gamma(\ell)$.


Main Subjects

1. R.P. Agnew, On deferred Cesàro means, Ann. Math., 33 (1932), pp. 413-421.
2. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), pp. 87-96.
3. K. Atanassov, Intuitionistic fuzzy sets, In: VII ITKR’s Session, deposed in Central Sci. Technical Library of Bulg. Acad. of Sci., Sofia, 1983, 84.
4. T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11 (2003), pp. 687-706.
5. P. Baliarsingh and L. Nayak, On deferred statistical convergence of order β for fuzzy fractional difference sequence and applications, Soft Comput., 26 (2022), pp. 2625-2634.
6. L.C.D. Barros, R.C., Bassanezi and P. A.Tonelli, Fuzzy modelling in population dynamics, Ecol Modell., 128 (2000), pp. 27-33.
7. S.C. Cheng and J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, In First International Conference on Fuzzy Theory and Technology Proceedings, Abstracts and Summaries, 1992, 193-197.
8. J.S. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis, 8 (1988), pp. 47-64.
9. P. Debnath, A generalized statistical convergence in intuitionistic fuzzy n-normed linear spaces, Ann. Fuzzy Math. Inform., 12 (2016), pp. 559-572.
10. O. Duman, M.K. Khan and C. Orhan, A-statistical convergence of approximating operators, Math. Inequal. Appl., 6 (2003), pp.689-699.
11. M. Et and M.C. Yilmazer, On deferred statistical convergence of sequences of sets, AIMS Math., 5 (2020), pp. 2143-2152.
12. H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951). pp. 241-244.
13. C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets Syst., 48 (1992), pp. 239-248.
14. A.L. Fradkov and R.J. Evans, Control of chaos: Methods and applications in engineering, Annu Rev Control, 29 (2005), pp. 33-56.
15. S. G̈ahler, Lineare 2-normietre R̈aume, Math. Nachr, 28 (1965), pp. 1-43.
16. S. G̈ahler, Untersuchungen uber verallgemeinerte m-metrische R̈aume, I, II, III. Math. Nachr. 40 (1969), pp.165-189.
17. R. Giles, A computer program for fuzzy reasoning, Fuzzy Sets Syst., 4 (1980), pp. 221-234.
18. H. Gunawan and M. Mashadi, On n-normed spaces, Int J Math Math Sci., 27 (2001), pp. 631-639.
19. B. Hazarika, V. Kumar and B. Lafuerza-Guillén, Generalized ideal convergence in intuitionistic fuzzy normed linear spaces, Filomat, 27 (2013), pp. 811-820.
20. L. Hong and J.Q. Sun, Bifurcations of fuzzy nonlinear dynamical systems, Commun Nonlinear Sci Numer Simul., 11 (2006), pp. 1-12.
21. U. Kadak and S.A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the (p, q)-gamma function and related approximation theorems, Results Math., 73 (2018), pp. 1-31.
22. S. Karakus, K. Demirci and O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos Solitons Fractals., 35 (2008), pp. 763-769.
23. A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy sets syst., 12 (1984), pp.143-154.
24. S.S. Kim and Y.J. Cho, Strict convexity in linear n-normed spaces, Demonst. Math, 29 (1996), pp. 739-744.
25. M. Küçükaslan and M. Yılmaztürk, On deferred statistical convergence of sequences, Kyungpook Math. J, 56 (2016), pp. 357-366.
26. V. Kumar and K. Kumar, On ideal convergence of sequences in intuitionistic fuzzy normed spaces, Selcuk J. Appl. Math., 10 (2009),
pp. 27-41.
27. V. Kumar and M. Mursaleen, On (λ, μ)-statistical convergence of double sequence on intuitionstic fuzzy normed spaces, Filomat, 25 (2011), pp. 109-120.
28. R. Malceski, Strong n-convex n-normed spaces, Mat. Bilt., 21 (1997), pp. 81-102.
29. S. Melliani, M. Kuçukaslan, H. Sadiki and L.S. Chadli, Deferred statistical convergence of sequences in intuitionistic fuzzy normed spaces, Notes IFS, 24 (2018), pp. 64-78.
30. H.I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans Am Math Soc., 347 (1995), pp. 1811-1819.
31. A. Misiak, n-inner product spaces, Math. Nachr. 140 (1989), pp. 299-319.
32. S.A. Mohiuddine and Q.D. Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos, Solitons Fractals, 42 (2009), pp. 1731-1737.
33. S.A. Mohiuddine, A. Asiri and B. Hazarika, Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst., 48 (2019), pp. 492-506.
34. S.A. Mohiuddine and B.A.S. Alamri, Generalization of equistatistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM, 113 (2019), pp. 1955-1973.
35. S.A. Mohiuddine, B. Hazarika and M.A. Alghamdi, Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33 (2019), pp. 4549-4560.
36. M. Mursaleen and Q.D. Lohani, Intuitionistic fuzzy 2-normed space and some related concepts, Chaos, Solitons Fractals, 42 (2009), pp. 224-234.
37. M. Mursaleen and S.A. Mohiuddine, On lacunary statistical convergence w.r.t the intuitionistic fuzzy normed space, J. Comput. Appl. Math., 233 (2009), pp. 142-149.
38. A. Narayanan and S. Vijayabalaji, Fuzzy n-normed linear space, Int J Math Math Sci., 24 (2005), pp. 3963-3977.
39. L. Nayak, M. Mursaleen and P. Baliarsingh, On deferred statistical A-convergence of fuzzy sequence and applications, Iran. J. Fuzzy Syst., 19 (2022), pp. 119-131.
40. K. Raj, S.A. Mohiuddine and S. Jasrotia, Characterization of summing operators in multiplier spaces of deferred Nörlund summability, Positivity, 27 (2023), Article 9.
41. R. Saadati and J.H. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals, 27 (2006), pp. 331-344.
42. T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980), pp. 139-150.
43. I.J. Schoenberg, The integrability of certain functions and related summability methods, Am. Math. Mon., 66 (1959), 361-775.
44. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math, 10 (1960), pp. 313-334.
45. M. Sen and P. Debnath, Lacunary statistical convergence in intuitionistic fuzzy n-normed linear spaces, Math. Comput. Model. Dyn. Syst., 54 (2011), pp. 2978-2985.
46. M. Sen and P. Debnath, Statistical convergence in intuitionistic fuzzy n-normed linear spaces, Fuzzy Inf. Eng., 3 (2011), pp. 259-273.
47. H. Şengül, M. Et and M. Işık, On I-deferred statistical convergence of order α, Filomat, 33 (2019), pp. 2833-2840.
48. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, In Colloq. Math, 2 (1951), pp. 73-74.
49. S. Vijayabalaji, N. Thillaigovindan and Y.B. Jun, Intuitionistic fuzzy n-normed linear space, Bull. Korean Math. Soc., 44 (2007), pp. 291.
50. L.A. Zadeh, Fuzzy sets, Inform. Cont., 8 (1965), pp. 338–353.
51. A. Zygmund, Trigonometric series, Cambridge university press, United Kingdom, 2002.