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Year 2025, Volume: 20 Issue: 1, 103 - 125, 25.05.2025

Ömer Kişi , Selim Çetin , Mehmet Gürdal

https://doi.org/10.29233/sdufeffd.1643711

Abstract

References

  • A. El-Sayed Ahmed, S. Omran and A. J. Asad, "Fixed point theorems in quaternion valued metric spaces", Abstract Applied Analysis, Article ID 258958, 1-9, 2014.
  • A. Aizpuru, M. C. Listan-Garcia and F. Rambla-Barreno, "Density by moduli and statistical convergence", Quaestiones Mathematicae, 1-6, 2014.
  • A. Azam, B. Fisher and M. Khan, "Common fixed point theorems in complex valued metric spaces", Numerical Functional Analysis and Optimization, 32, 243-253, 2011.
  • J. C. Baez, "The octonions", Bulletin of the American Mathematical Society, 39, 145-205, 2002.
  • Q. B. Cai, S. Çetin, Ö. Kişi and M. Gürdal, "Octonion-valued b-metric spaces and results on its application", preprint, 2025.
  • J. H. Conway and D. A. Smith, "On quaternions and octonions: their geometry, arithmetic, and symmetry", Bulletin of the American Mathematical Society, 42, 229-243, 2005.
  • S. Çetin, M. Mursaleen, Ö. Kişi and M. Gürdal, "Octonion-valued metric spaces and the concept of convergence in these spaces", preprint, 2024.
  • J. S. Connor, "On strong matrix summability with respect to a modulus and statistical convergence", Canadian Mathematical Bulletin, 32, 194-198, 1989.
  • P. Das, E. Savaş and S. Kr. Ghosal, "On generalized of certain summability methods using ideals", Applied Mathematic Letters, 36, 1509-1514, 2011.
  • B. G. Dhage, "Generalized metric spaces and mapping with fixed point", Bulletin of the Calcutta Mathematical Society, 84, 329-336, 1992.
  • T. Dray and C. Manogue, "The Geometry of the Octonions", World Scientific, 2015.
  • Ö. Ege, "Complex valued rectangular b-metric spaces and an application to linear equations", Journal of Nonlinear Sciences and Applications, 6, 1014-1021, 2015.
  • H. Fast, "Sur la convergence statistique", Colloquium Mathematicum, 10, 142-149, 1951.
  • A. R. Freedman, J. Sember and M. Raphael, "Some Cesàro-type summability spaces", Proceeding London of Mathematic Society, 37, 508-520, 1978.
  • J. A. Fridy and C. Orhan, "Lacunary statistical convergence", Pacific Journal of Mathematics, 160, 43-51, 1993.
  • D. Fiorenza, H. Sati and U. Schreiber, "Super-exceptional embedding construction of the heterotic M5: Emergence of SU(2)-flavor sector", Journal of Geometry and Physics, 170, 104349, 2021.
  • M. M. Fréchet, "Sur quelques points du calcul fonctionnel", Rendiconti del Circolo Matematico di Palermo, 22, 1-72, 1906.
  • J. A. Fridy, "On statistical convergence", Analysis, 5, 301-313, 1985.
  • S. Gähler "2-metrische Raume und ihre topologische struktur", Mathematische Nachrichten, 26, 115-148, 1963.
  • M. Gürdal, "Some types of convergence", Suleyman Demirel University, Diss. Doctoral Dissertation, Isparta, 2004.
  • M. Gürdal and M. O. Özgür, A generalized statistical convergence via moduli, Electronic Journal of Mathematical Analysis and Applications, 3, 173-178, 2015.
  • M. Gürdal and U. Yamancı, "Statistical convergence and some questions of operator theory", Dynamic Systems and Applications, 24, 305-311, 2015.
  • H. Ş. Kandemir, M. Et, and N. D. Aral, "Strongly λ-convergence of order α in neutrosophic normed spaces", Dera Natung Government College Research Journal, 7, 1--9, 2022.
  • M. Kansu, E.M. Tanşl and S. Demir, "Octonion form of duality-invariant field equations for dyons", Turkish Journal of Physicsi 44, 10-23, 2020.
  • M. A. Khamsi, "Generalized metric spaces: A survey", Journal of Fixed Point Theory and Applications, 17, 455-475, 2015.
  • Ö. Kişi, S. Çetin and M. Gürdal, "Octonion-valued b-metric spaces and ideal convergence", preprint, 2025.
  • P. Kostyrko, T. alá t and W. Wilczyński, "I-convergence", Real Analysis Exchange, 26, 669-686, 2000-2001.
  • I. J. Maddox, "Sequence spaces defined by a modulus, Mathematical Proceedings of the Cambridge Philosophical Society, 100, 161-166, 1986.
  • A. Nabiev, S. Pehlivan and M. Gürdal, "On I-Cauchy sequences", Taiwanese Journal of Mathematics, 11, 569-576, 2007.
  • A. A. Nabiev, E. Savaş and M. Gürdal, "Statistically localized sequences in metric spaces", Journal of Applied Analysis and Computation, 9, 739-746, 2019.
  • A. A. Nabiev, E. Savaş and M. Gürdal, "I-localized sequences in metric spaces", Facta Universitatis, Series: Mathematics and Informatics, 35, 459-469, 2020.
  • A. A. Nabiev, S. Pehlivan and M. Gürdal, "On I-Cauchy sequences", Taiwanese Journal of Mathematics, 11, 569-576, 2007.
  • H. Nakano, "Concave modular", Journal of the Mathematical Society of Japan, 5, 29-49, 1953.
  • F. Nuray and W. H. Ruckle, "Generalized statistical convergence and convergence free spaces", Journal of Mathematical Analysis and Applications, 245, 513-527, 2000.
  • S. Okubo, "Introduction to Octonion and Other Non-Associative Algebras in Physics", Cambridge University Press, Cambridge, 1995.
  • W. H. Ruckle, FK Spaces in which the sequence of coordinate vectors is bounded, Canadian Journal of Mathematics, 25, 973-978, 1973.
  • A. Şahiner, M. Gürdal and T. Yigit, "Ideal convergence characterization of the completion of linear -normed spaces", Computers & Mathematics with Applications, 61, 683-689, 2011.
  • T. Salat, "On statistically convergent sequences of real numbers", Mathematica Slovaca, 30, 139-150, 1980.
  • E. Savaş, S. Ç etin, Ö. Kişi and M. Gürdal, "On some convergence type in octonion-valued b-metric spaces", preprint, 2025.
  • E. Savaş and M. Gürdal, "Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces, Journal of Intelligent & Fuzzy Systems, 27, 2067-2075, 2014.
  • E. Savaş and M. Gürdal, "Ideal convergent function sequences in random 2-normed spaces", Filomat, 30, 557-567, 2016.
  • E. Savaş, Ö. Kişi and M. Gürdal, "On statistical convergence in credibility space", Numerical Functional Analysis and Optimization, 43, 987-1008, 2022.
  • E. Savaş, U. Yamancı and M. Gürdal, "I-lacunary statistical convergence of weighted g via modulus functions in 2-normed spaces, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68, 2324-2332, 2019.
  • I. J. Schoenberg, "The integrability of certain functions and related summability methods", American Mathematical Monthly, 66, 361-375, 1959.
  • H. Steinhaus, "Sur la convergence ordiniaire et la convergence asymptotique", Colloquium Mathematicum, 2, 73-84, 1951.
  • K. Takahashi, M. Fujita and M. Hashimoto, "Remarks on Octonion-valued Neural Networks with Application to Robot Manipulator Control", IEEE International Conference on Mechatronics (ICM), Kashiwa, Japan, 2021, pp. 1-6.
  • J. Wu, L. Xu, F. Wu, Y. Kong, L. Senhadji and H. Shu, "Deep octonion networks", Neurocomputing, 397, 179-191, 2020.
  • J. J. Quan, S. Çetin, Ö. Kişi and M. Gürdal, "On certain results of statistical convergence in octonion-valued metric spaces", preprint, 2024.
  • U. Yamancı and M. Gürdal, "Statistical convergence and operators on Fock space", New York Journal of Mathematics, 22, 199-207, 2016.
  • U. Yamancı and M. Gürdal, "On lacunary ideal convergence in random n-normed space", Journal of Mathematics, 2013, 1-8, 2013.
  • U. Yamancı and M. Gürdal, "I-statistical convergence in 2-normed space", Arab Journal of Mathematical Sciences, 20, 41-47, 2014.
  • T. Yaying, “On Λ-Fibonacci difference sequence spaces of fractional order”, Dera Natung Government College Research Journal, 6, 92--102, 2021.
  • T. Yaying, “Arithmetic continuity in cone metric space”, Dera Natung Government College Research Journal, 5, 55-62, 2020.
  • A. Zygmund, Trigonometric Series. Cambridge Univ. Press, UK, 1979.

Generalized Statistical Convergence via Modulus Function in Octonion Valued b-Metric Spaces

Year 2025, Volume: 20 Issue: 1, 103 - 125, 25.05.2025

Ömer Kişi , Selim Çetin , Mehmet Gürdal

https://doi.org/10.29233/sdufeffd.1643711

Abstract

The key ideas of summability theory have been the subject of extensive investigation in recent years in a variety of metric space extensions. Octonion-valued metric spaces are based on modifying the triangle inequality of a semi-metric space by multiplying one side of the inequality by a scalar b. This new generalisation of metric spaces is very interesting since octonions are not even a ring since they do not have the associative property of multiplication and the spaces do not satisfy the standard triangle inequality. We are prompted by this to study the notions of strong I-Cesàro summability, I-statistical convergence, I-lacunary statistical convergence, and similar notions that respect the modulus function in octonion valued b-metric spaces, an extension of metric spaces. We also examine the connections among these ideas.

Keywords

Ideal convergence Statistical convergence Prime numbers Octonion valued -metric space Modulus function

References

  • A. El-Sayed Ahmed, S. Omran and A. J. Asad, "Fixed point theorems in quaternion valued metric spaces", Abstract Applied Analysis, Article ID 258958, 1-9, 2014.
  • A. Aizpuru, M. C. Listan-Garcia and F. Rambla-Barreno, "Density by moduli and statistical convergence", Quaestiones Mathematicae, 1-6, 2014.
  • A. Azam, B. Fisher and M. Khan, "Common fixed point theorems in complex valued metric spaces", Numerical Functional Analysis and Optimization, 32, 243-253, 2011.
  • J. C. Baez, "The octonions", Bulletin of the American Mathematical Society, 39, 145-205, 2002.
  • Q. B. Cai, S. Çetin, Ö. Kişi and M. Gürdal, "Octonion-valued b-metric spaces and results on its application", preprint, 2025.
  • J. H. Conway and D. A. Smith, "On quaternions and octonions: their geometry, arithmetic, and symmetry", Bulletin of the American Mathematical Society, 42, 229-243, 2005.
  • S. Çetin, M. Mursaleen, Ö. Kişi and M. Gürdal, "Octonion-valued metric spaces and the concept of convergence in these spaces", preprint, 2024.
  • J. S. Connor, "On strong matrix summability with respect to a modulus and statistical convergence", Canadian Mathematical Bulletin, 32, 194-198, 1989.
  • P. Das, E. Savaş and S. Kr. Ghosal, "On generalized of certain summability methods using ideals", Applied Mathematic Letters, 36, 1509-1514, 2011.
  • B. G. Dhage, "Generalized metric spaces and mapping with fixed point", Bulletin of the Calcutta Mathematical Society, 84, 329-336, 1992.
  • T. Dray and C. Manogue, "The Geometry of the Octonions", World Scientific, 2015.
  • Ö. Ege, "Complex valued rectangular b-metric spaces and an application to linear equations", Journal of Nonlinear Sciences and Applications, 6, 1014-1021, 2015.
  • H. Fast, "Sur la convergence statistique", Colloquium Mathematicum, 10, 142-149, 1951.
  • A. R. Freedman, J. Sember and M. Raphael, "Some Cesàro-type summability spaces", Proceeding London of Mathematic Society, 37, 508-520, 1978.
  • J. A. Fridy and C. Orhan, "Lacunary statistical convergence", Pacific Journal of Mathematics, 160, 43-51, 1993.
  • D. Fiorenza, H. Sati and U. Schreiber, "Super-exceptional embedding construction of the heterotic M5: Emergence of SU(2)-flavor sector", Journal of Geometry and Physics, 170, 104349, 2021.
  • M. M. Fréchet, "Sur quelques points du calcul fonctionnel", Rendiconti del Circolo Matematico di Palermo, 22, 1-72, 1906.
  • J. A. Fridy, "On statistical convergence", Analysis, 5, 301-313, 1985.
  • S. Gähler "2-metrische Raume und ihre topologische struktur", Mathematische Nachrichten, 26, 115-148, 1963.
  • M. Gürdal, "Some types of convergence", Suleyman Demirel University, Diss. Doctoral Dissertation, Isparta, 2004.
  • M. Gürdal and M. O. Özgür, A generalized statistical convergence via moduli, Electronic Journal of Mathematical Analysis and Applications, 3, 173-178, 2015.
  • M. Gürdal and U. Yamancı, "Statistical convergence and some questions of operator theory", Dynamic Systems and Applications, 24, 305-311, 2015.
  • H. Ş. Kandemir, M. Et, and N. D. Aral, "Strongly λ-convergence of order α in neutrosophic normed spaces", Dera Natung Government College Research Journal, 7, 1--9, 2022.
  • M. Kansu, E.M. Tanşl and S. Demir, "Octonion form of duality-invariant field equations for dyons", Turkish Journal of Physicsi 44, 10-23, 2020.
  • M. A. Khamsi, "Generalized metric spaces: A survey", Journal of Fixed Point Theory and Applications, 17, 455-475, 2015.
  • Ö. Kişi, S. Çetin and M. Gürdal, "Octonion-valued b-metric spaces and ideal convergence", preprint, 2025.
  • P. Kostyrko, T. alá t and W. Wilczyński, "I-convergence", Real Analysis Exchange, 26, 669-686, 2000-2001.
  • I. J. Maddox, "Sequence spaces defined by a modulus, Mathematical Proceedings of the Cambridge Philosophical Society, 100, 161-166, 1986.
  • A. Nabiev, S. Pehlivan and M. Gürdal, "On I-Cauchy sequences", Taiwanese Journal of Mathematics, 11, 569-576, 2007.
  • A. A. Nabiev, E. Savaş and M. Gürdal, "Statistically localized sequences in metric spaces", Journal of Applied Analysis and Computation, 9, 739-746, 2019.
  • A. A. Nabiev, E. Savaş and M. Gürdal, "I-localized sequences in metric spaces", Facta Universitatis, Series: Mathematics and Informatics, 35, 459-469, 2020.
  • A. A. Nabiev, S. Pehlivan and M. Gürdal, "On I-Cauchy sequences", Taiwanese Journal of Mathematics, 11, 569-576, 2007.
  • H. Nakano, "Concave modular", Journal of the Mathematical Society of Japan, 5, 29-49, 1953.
  • F. Nuray and W. H. Ruckle, "Generalized statistical convergence and convergence free spaces", Journal of Mathematical Analysis and Applications, 245, 513-527, 2000.
  • S. Okubo, "Introduction to Octonion and Other Non-Associative Algebras in Physics", Cambridge University Press, Cambridge, 1995.
  • W. H. Ruckle, FK Spaces in which the sequence of coordinate vectors is bounded, Canadian Journal of Mathematics, 25, 973-978, 1973.
  • A. Şahiner, M. Gürdal and T. Yigit, "Ideal convergence characterization of the completion of linear -normed spaces", Computers & Mathematics with Applications, 61, 683-689, 2011.
  • T. Salat, "On statistically convergent sequences of real numbers", Mathematica Slovaca, 30, 139-150, 1980.
  • E. Savaş, S. Ç etin, Ö. Kişi and M. Gürdal, "On some convergence type in octonion-valued b-metric spaces", preprint, 2025.
  • E. Savaş and M. Gürdal, "Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces, Journal of Intelligent & Fuzzy Systems, 27, 2067-2075, 2014.
  • E. Savaş and M. Gürdal, "Ideal convergent function sequences in random 2-normed spaces", Filomat, 30, 557-567, 2016.
  • E. Savaş, Ö. Kişi and M. Gürdal, "On statistical convergence in credibility space", Numerical Functional Analysis and Optimization, 43, 987-1008, 2022.
  • E. Savaş, U. Yamancı and M. Gürdal, "I-lacunary statistical convergence of weighted g via modulus functions in 2-normed spaces, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68, 2324-2332, 2019.
  • I. J. Schoenberg, "The integrability of certain functions and related summability methods", American Mathematical Monthly, 66, 361-375, 1959.
  • H. Steinhaus, "Sur la convergence ordiniaire et la convergence asymptotique", Colloquium Mathematicum, 2, 73-84, 1951.
  • K. Takahashi, M. Fujita and M. Hashimoto, "Remarks on Octonion-valued Neural Networks with Application to Robot Manipulator Control", IEEE International Conference on Mechatronics (ICM), Kashiwa, Japan, 2021, pp. 1-6.
  • J. Wu, L. Xu, F. Wu, Y. Kong, L. Senhadji and H. Shu, "Deep octonion networks", Neurocomputing, 397, 179-191, 2020.
  • J. J. Quan, S. Çetin, Ö. Kişi and M. Gürdal, "On certain results of statistical convergence in octonion-valued metric spaces", preprint, 2024.
  • U. Yamancı and M. Gürdal, "Statistical convergence and operators on Fock space", New York Journal of Mathematics, 22, 199-207, 2016.
  • U. Yamancı and M. Gürdal, "On lacunary ideal convergence in random n-normed space", Journal of Mathematics, 2013, 1-8, 2013.
  • U. Yamancı and M. Gürdal, "I-statistical convergence in 2-normed space", Arab Journal of Mathematical Sciences, 20, 41-47, 2014.
  • T. Yaying, “On Λ-Fibonacci difference sequence spaces of fractional order”, Dera Natung Government College Research Journal, 6, 92--102, 2021.
  • T. Yaying, “Arithmetic continuity in cone metric space”, Dera Natung Government College Research Journal, 5, 55-62, 2020.
  • A. Zygmund, Trigonometric Series. Cambridge Univ. Press, UK, 1979.
There are 54 citations in total.

Details

Primary Language English
Subjects Operator Algebras and Functional Analysis
Journal Section Makaleler
Authors

Ömer Kişi 0000-0001-6844-3092

Selim Çetin 0000-0002-9017-1465

Mehmet Gürdal 0000-0003-0866-1869

Publication Date May 25, 2025
Submission Date February 20, 2025
Acceptance Date April 7, 2025
Published in Issue Year 2025 Volume: 20 Issue: 1

Cite

IEEE Ö. Kişi, S. Çetin, and M. Gürdal, “Generalized Statistical Convergence via Modulus Function in Octonion Valued b-Metric Spaces”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 20, no. 1, pp. 103–125, 2025, doi: 10.29233/sdufeffd.1643711.
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