Octonion-valued $b$-metric spaces and results on its application

This study introduces octonion-valued b-metric spaces as a natural extension of the octonion-valued metric spaces developed by establishing a partial ordering relation on octonions. Octonion-valued $b$-metric spaces are constructed by modifying the triangle inequality of a semi-metric space, where one side of the inequality is multiplied by a positive scalar $b\ge 1$. On the other hand, octonion-valued metric spaces generalize the concept of classical metric spaces by employing octonions, which provide a higher-dimensional and non-associative algebraic framework. Two key reasons make this novel generalization of metric spaces very interesting: First, octonions are not even a ring since they do not have the associative feature in multiplication; second, the spaces do not meet the standard triangle inequality. In addition to explanations on sequences, convergence, Cauchy characteristics, boundedness, theorems, and associated conclusions, examples are given to help visualize this recently formed metric space. Lastly, the building of a fixed point finds extensive applications in a variety of mathematical analytic subjects as well as applied mathematics domains like differential equations and dynamical systems. Because of this, octonion-valued $b$-metric spaces have been used to study the Banach fixed-point theorem and a few additional fixed-point theorems.