Some Results on f-Statistical Convergence of Double Sequences

Year 2025, Volume: 7 Issue: 1, 62 - 75, 24.07.2025

Ömer Kişi , Mehmet Gürdal

https://doi.org/10.54286/ikjm.1531097

Abstract

This study focuses on developing and examining new definitions related to statistical convergence, modulus functions, and σ-density in 2-normed spaces. We explore various relationships between statistical f_σ-convergence, f_σ-statistical convergence, f_σ-statistically Cauchy sequences, f_σ-statistically bounded double sequences, and strongly f_(σ_s )-convergent double sequences in 2 normed spaces. Additionally, we analyze some topological properties associated with these concepts.

Keywords

Double sequences , modulus function , statistical convergence

References

• A. Aizpuru, M.C. Listan-Garcia, F. Rambla-Barreno, Density by moduli and statistical convergence, Quaestiones Mathematicae, 37(4), (2014) 525–530.
• S. Banach, Théorie des Operations Lineaires, Warszawa, 1932.
• V.Baláz, T. Salát, Uniform density u and corresponding I_u-convergence, Mathematical Communications, 11(1), (2006) 1-7.
• V.K. Bhardwaj, S. Dhawan, S. Gupta, Density by moduli and statistical boundedness, Abstract and Applied Analysis. 2016, (2016) Article ID 2143018, 1-6.
• J.S. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis, 8, (1988) 47-63.
• C. Çakan, B. Altay, M. Mursaleen, σ-convergence and σ-core of double sequences, Applied Mathematics Letters, 19, (2006) 1122-1128.
• R. Çolak, Some Relations Between the Sets of f-Statistically Convergent Sequences, In 4th International Conference on Computational Mathematics and Engineering Sciences (CMES-2019), Springer International Publishing, 4, (2010) 307-314.
• C. Diminnie, S. Gähler, A. White, 2-inner product spaces, Demonstratio Mathematica, 6, (1973) 525-536. H. Fast, Sur la convergence statistique, Colloquim Mathematicum, 2, (1951) 241-244.
• S. Gähler, 2-metrische Räume und ihre topologische Struktur, Mathematische Nachrichten, 26, (1963) 115-148.
• A.T. Gülcan, On the statistical convergence and density by moduli, 2024, preprint.
• M. Gürdal, S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bulletin Mathematics, 33, (2009) 257-264.
• G.G. Lorentz, A contribution to theory of divergent sequences, Acta Mathematica, 80, (1948) 167-190.
• I.J. Maddox, Inclusions between FK-spaces and Kuttner’s theorem, Mathematical Proceedings of the Cambridge Philosophical Society, 101(3), (1987) 523-527.
• M. Mursaleen, Matrix transformations between some new sequence spaces, Houston Journal of Mathematics, 9(4), (1983) 505-509.
• M. Mursaleen, On some new invariant matrix methods of summability, Quarterly Journal of Mathematics, 34, (1983) 77-86.
• M. Mursaleen, O.H.H. Edely, Statistical convergence of double sequences, Journal of Mathematical Analysis and Applications, 288,(2003) 223-231.
• M. Mursaleen, O.H.H. Edely, On the invariant mean and statistical convergence, Applied Mathematics Letters, 22(11), (2009) 1700-1704.
• H. Nakano, Concave modulars, Journal of the Mathematical Society of Japan, 5, (1953) 29-49.
• A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Matheatische Annalen, 53, (1900) 289-321.
• R.A. Raimi, Invariant means and invariant matrix methods of summability, Duke Mathematical Journal, 30 (1963) 81-94.
• D.Rath, B.C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian Journal of Pure and Applied Mathematics, 25, (1994) 381-381.
• E. Savaş, F. Nuray, On σ-statistically convergence and lacunary σ-statistically convergence, Mathematica Slovaca, 43(3), (1993) 309-315.