Cayley-Dickson algebra-valued metric spaces and convergent fixed point theory for hypercomplex contractions

Document Type : Research Paper

Authors

1 Department of Mathematics, Raiganj Surendranath Mahavidyalaya

2 2Department of Mathematics, Raiganj Surendranath Mahavidyalaya, West Bengal, India

Abstract

This paper introduces a novel theoretical framework for metric spaces valued in Cayley-Dickson algebras, extending beyond the limitations of bicomplex-valued b metric spaces to encompass the entire hierarchy of hypercomplex number systems in cluding quaternions, octonions, and sedenions. We establish a comprehensive theory of Cayley-Dickson algebra-valued metric spaces (CD-metric spaces) with generalized triangle inequalities parameterized by coefficients that account for the non-associative and non-commutative properties inherent in higher-dimensional Cayley-Dickson con
structions. Our principal contributions include: (1) the formulation of a consistent partial ordering relation across all finite-dimensional Cayley-Dickson algebras, (2) the development of convergence theory and completeness criteria for CD-metric spaces, (3) the establishment of fixed point theorems for contractive mappings under rational type conditions involving hypercomplex control functions, and (4) novel applications to systems of hypercomplex integral equations. The results significantly generalize existing bicomplex metric space theory and provide new tools for analyzing fixed point
problems in non-associative algebraic structures.

Keywords

Main Subjects


[1] A. Azam, B. Fisher, and M. Khan, Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis and Optimization, 32(3) (2011), 243–253.
[2] M. A. Ahmed, F. M. Zeyada, and G. F. Hassan, Fixed point theorems in quaternion-valued metric spaces, Abstract and Applied Analysis, 2014 (2014), Article ID 258985.
[3] J. Choi, S. K. Datta, T. Biswas, and N. Islam, Some fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces, Honam Mathematical Journal, 39(1) (2017), 115–126.
[4] S. K. Datta, D. Pal, R. Sarkar, and A. Manna, A common fixed point theorem in bicomplex valued b-metric spaces, The Mathematics Student, 90(3–4) (2021), 1–17.
[5] X. L. Qiu, S. C¸ etin, ¨ O. Ki¸si, M. G¨urdal, and Q. B. Cai, Octonion-valued b-metric spaces and results on its application, AIMS Mathematics, 10(5) (2025), 10504–10527.
[6] S. C¸ etin, ¨ O. Ki¸si, and M. G¨urdal, Generalized statistical convergence via modulus function in octonionvalued metric spaces, Journal of Nonlinear Sciences and Applications, 15(4) (2024), 167–182.
[7] J. C. Baez, The octonions, Bulletin of the American Mathematical Society, 39(2) (2002), 145–205. 
[8] S. L. Hahn and K. M. Snopek, Complex and hypercomplex analytic signals: theory and applications, Artech House, 2020.
[9] C. Culbert, Cayley-Dickson algebras and loops, Commentationes Mathematicae Universitatis Carolinae, 48(1) (2007), 1–17.
[10] L. E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Annals of Mathematics, 20(3) (1919), 155–171.
[11] A. Hurwitz, ¨ Uber die Komposition der quadratischen Formen von beliebig vielen Variabeln, Nachrichten von der Gesellschaft der Wissenschaften zu G¨ottingen, (1898), 309–316.
[12] G. Frobenius, ¨ Uber lineare Substitutionen und bilineare Formen, Journal f¨ur die reine und angewandte Mathematik, 84 (1878), 1–63.
[13] G. B. Price, An introduction to multicomplex spaces and functions, Marcel Dekker, New York, 1991. 
[14] D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, Analele Universitatii
din Oradea, Fascicola Matematica, 11 (2004), 71–110.
[15] C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, Mathematische Annalen, 40 (1892), 413–467.
[16] N. Spampinato, Estensione nel campo bicomplesso di due teoremi del Levi-Civita e del Severi, Rendiconti della Reale Accademia Nazionale dei Lincei, 22 (1935), 38–43.
[17] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales, Fundamenta Mathematicae, 3 (1922), 133–181.
[18] L. E. J. Brouwer, ¨ Uber Abbildung von Mannigfaltigkeiten, Mathematische Annalen, 71(4) (1912), 497–515.
[19] J. Schauder, Der Fixpunktsatz in Funktionalr¨aumen, Studia Mathematica, 2(1) (1930), 171–180. 
[20] A. Tychonoff, Ein Fixpunktsatz, Mathematische Annalen, 111(1) (1935), 767–776.
[21] S. B. Nadler Jr., Multi-valued contraction mappings, Pacific Journal of Mathematics, 30(2) (1969), 475–488.
[22] S. Reich, Fixed points of contractive functions, Bollettino della Unione Matematica Italiana, 4(1) (1971), 26–42.
[23] R. Kannan, Some results on fixed points, Bulletin of the Calcutta Mathematical Society, 60 (1968), 71–76.
[24] S. K. Chatterjea, Fixed-point theorems, Comptes Rendus de l’Acad´emie Bulgare des Sciences, 25 (1972), 727–730.
[25] T. Zamfirescu, Fix point theorems in metric spaces, Archiv der Mathematik, 23(1) (1972), 292–298.