Deferred statistical convergence of sequences in octonion-valued metric spaces

Year 2025, Volume: 74 Issue: 3, 375 - 394, 23.09.2025

Selim Çetin , Ömer Kişi , Mehmet Gürdal

https://doi.org/10.31801/cfsuasmas.1611730

Abstract

By employing octonions, which offer a higher-dimensional and non-associative algebraic structure, octonion-valued metric spaces generalize conventional metric spaces. Every ring forms a module over itself, and every field forms a vector space over itself, as is commonly known. It should be noted, nevertheless, that octonions do not form a module over themselves and so cannot even be regarded as a ring because they lack the multiplicative union condition. The metric spaces we have defined and the findings produced in these spaces are very intriguing because of this aspect. Consequently, various conclusions pertaining to summability theory are examined utilizing some essential concepts associated with these mathematical structures. In particular, we present the concepts of deferred statistical convergence and deferred strong Cesàro summability in octonion-valued metric spaces and explore the connections between them. Additionally, we introduce and discuss the concepts of strong deferred invariant convergence, deferred invariant convergence in octonion-valued metric spaces, and deferred invariant statistical convergence.

Keywords

Deferred statistical convergence , invariant , prime numbers , octanion metric space

References

• Agnew, R. P., On deferred Cesàro means, Ann. Math., 33(3) (1932), 413–421. https://doi.org/10.2307/1968524.
• Albert, A. A., On a certain algebra of quantum mechanics, Ann. Math., 35(1) (1934), 65–73. https://doi.org/10.2307/1968118.
• Azam, A., Fisher, B., Khan, M., Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32(3) (2011), 243–253. https://doi.org/10.1080/01630563.2011.533046.
• Baez, J. C., The octonions, Bull. Amer. Math. Soc., 39 (2002), 145–205. https://doi.org/10.1090/S0273-0979-01-00934-X.
• Belen, C., Mohiuddine, S. A., Generalized weighted statistical convergence and application, Appl. Math. Comput., 219(18) (2013), 9821–9826. https://doi.org/10.1016/j.amc.2013.03.115.
• Connor, J., The statistical and strongly $p$ Cesàro convergence of sequences, Analysis, 8 (1988), 47–63. https://doi.org/10.1524/anly.1988.8.12.47.
• Conway, J. H., Smith, D. A., On Quaternions and Octonions: Their geometry, arithmetic, and symmetry, Bull. Amer. Math. Soc., 42 (2005), 229–243.
• Çetin, S., Mursaleen, M., Kişi, Ö., Gürdal, M., Octonion-valued metric spaces and the concept of convergence in these spaces. Preprint. 2024.
• Dray, T., Manogue, C., The Geometry of the Octonions, World Scientific, 2015. https://doi.org/10.1142/8456.
• El-Sayed, A. A., Omran, S., Asad, A. J., Fixed point theorems in quaternion valued metric spaces, Abstr. Appl. Anal., Article ID 258958, (2014), 1–9. https://doi.org/10.1155/2014/258985.
• Et, M., Cinar, M., Kandemir, H. S., Deferred statistical convergence of order $\alpha$ in metric spaces, AIMS Mathematics 5(4) (2020), 3731–3740. https://doi.org/10.3934/math.2020241.
• Fast, H., Sur la convergence statistique, Colloq. Math., 2(3-4) (1951), 241–244.
• Fiorenza, D., Sati, H., Schreiber, U., Super-exceptional embedding construction of the heterotic M5: Emergence of SU(2)-flavor sector, J. Geom. Phys., 170 (2021), 104349. https://doi.org/10.1016/j.geomphys.2021.104349.
• Freedman, A. R., Sember, J., Raphael, M., Some Cesàro-type summability spaces, Proc. London Math. Soc., 37(3) (1978), 508–520. https://doi.org/10.1112/PLMS2FS3-37.3.508.
• Gürdal, M., Şahiner, A., Açık, I., Approximation theory in 2-Banach spaces, Nonlinear Anal., 71(5–6) (2009), 1654– 1661. https://doi.org/10.1016/j.na.2009.01.030.
• Gürdal, M., Şahiner, A., Statistical approximation with a sequence of 2-Banach spaces, Math. Comput. Model., 55(3-4) (2012), 471–479. https://doi.org/10.1016/j.mcm.2011.08.026.
• Gürdal, M., Sarı, N., Savaş, E., $A$-statistically localized sequences in n-normed spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(2) (2020), 1484–1497. https://doi.org/10.31801/cfsuasmas.704446.
• Gürdal, M., Yamancı, U., Statistical convergence and some questions of operator theory, Dynam. Systems Appl., 24(7) (2015), 305–311.
• Kadak, U., Mohiuddine, S. A., Generalized statistically almost convergence based on the difference opera- tor which includes the (p,q)-gamma function and related approximation theorems, Results Math., 73(9) (2018), https://doi.org/10.1007/s00025-018-0789-6.
• Kansu, M., Tanışlı, E. M., Demir, S., Octonion form of duality-invariant field equations for dyons, Turk. J. Phys., 44(1) (2020), 10–23. https://doi.org/10.3906/fiz-1910-7.
• Koşar, C., Küçükaslan, M., Et, M., On asymptotically deferred statistical equivalence of sequences, Filomat, 31(16) (2017), 5139–5150. https://doi.org/10.2298/FIL1716139K.
• Küçükaslan, M., Yılmaztürk, M., On deferred statistical convergence of sequences, Kyungpook Math. J., 56(2) (2016), 357–366. https://doi.org/10.5666/KMJ.2016.56.2.357.
• Maddox, I. J., Elements of Functional Analysis, Cambridge at the University Press, 1970.
• Mohiuddine, S. A., Alamri, B. A. S., Generalization of equi-statistical 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(3) (2019), 1955–1973. https://doi.org/10.1007/s13398-018-0591-z.
• Mohiuddine, S. A., Statistical weighted $A$-summability with application to Korovkin’s type approximation theorem, J. Inequal. Appl., 2016(101) (2016). https://doi.org/10.1186/s13660-016-1040-1.
• Mursaleen, M., Çetin, S., Kişi, Ö., Gürdal, M., An analysis of ideal convergence of sequences in Octonion-valued metric spaces. Preprint. 2024.
• Mursaleen, M., Edely, O. H. H., On the invariant mean and statistical convergence, Appl. Math. Lett., 22(11) (2009), 1700–1704. https://doi.org/10.1016/j.aml.2009.06.005.
• Mursaleen, M., Invariant mean and some matrix transformations, Thamkang J. Math., 10 (1979), 183–188.
• Mursaleen, M., On some new invariant matrix methods of summability, Quart. J. Math. Oxford, 34(2) (1983), 77–86. http://dx.doi.org/10.1093/qmath/34.1.77.
• Nabiev, A. A., Sava¸s, E., Gürdal, M., Statistically localized sequences in metric spaces, J. Appl. Anal. Comput., 9(2) (2019), 739–746. https://doi.org/10.11948/2156-907X.20180157.
• Nuray, F., Strongly deferred invariant convergence and deferred invariant statistical convergence, J. Comput. Sci. Comput. Math., 10(1) (2020), 1–6. http://dx.doi.org/10.20967/jcscm.2020.01.001.
• Okubo, S., Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge, 1995.
• Quan, J. J., Çetin, S., Kişi, Ö., Gürdal, M., On certain results of statistical convergence in octonion-valued metric spaces, 2020, Preprint.
• Raimi, R. A., Invariant means and invariant matrix methods of summability, Duke Math. J., 30(1) (1963), 81–94. http://dx.doi.org/10.1215/S0012-7094-63-03009-6.
• Savaş, E., Kişi, Ö., Gürdal, M., On statistical convergence in credibility space, Numer. Funct. Anal. Optim., 43 (2022), 987–1008. https://doi.org/10.1080/01630563.2022.2070205.
• Savaş, E., Nuray, F., On $\sigma$-statistically convergence and lacunary $\sigma$-statistically convergence, Math. Slovaca, 43(3) (1993), 309–315.
• Savaş, E., On lacunary strong $\sigma$-convergence, Indian J. Pure Appl. Math., 21 (1990), 359–365.
• Savaş, E., Some sequence spaces involving invariant means, Indian J. Math., 31 (1989), 1–8.
• Schaefer, P., Infinite matrices and invariant means, Proc. Amer. Math. Soc., 36(1) (1972), 104–110. https://doi.org/10.2307/2039044.
• Takahashi, K., Fujita, M., Hashimoto, M., Remarks on Octonion-valued neural networks with application to robot manipulator control, (2021), 1–6, IEEE International Conference on Mechatronics (ICM), Kashiwa, Japan, 2021.
• Wu, J., Xu, L., Wu, F., Kong, Y., Senhadji, L., Shu, H., Deep octonion networks, Neurocomputing, 397 (2020), 179–191. https://doi.org/10.1016/j.neucom.2020.02.053.
• Yamancı, U., Gürdal, M., Statistical convergence and operators on Fock space, New York J. Math., 22 (2016), 199–207.