Journal of Advances in Mathematics and Computer Science

This paper introduces the notion of multiplicative hemi-metric spaces as a higher-order extension of multiplicative metric structures. The proposed distance function is defined on ((m+1))-tuples and takes values in ([1,∞)), with the usual additive simplex inequality of an (m)-hemi-metric replaced by a multiplicative simplex inequality. Basic concepts associated with this setting are developed, including multiplicative convergence, multiplicative Cauchy sequences, completeness, and uniqueness of limits. A central feature of the study is the logarithmic correspondence between multiplicative hemi-metric spaces and classical (m)-hemi-metric spaces. It is shown that every multiplicative hemi-metric induces an (m)-hemi-metric through the logarithmic transformation, while every (m)-hemi-metric generates a multiplicative hemi-metric through the exponential transformation. This correspondence also yields an equivalence of completeness between the two settings. Using these observations, a Banach-type fixed point theorem is established for multiplicative contractive mappings on complete multiplicative hemi-metric spaces. The result is further expressed through the associated logarithmic formulation, showing how fixed point problems in the multiplicative framework may be treated by means of the corresponding additive structure. Several examples are included to illustrate the definitions and the applicability of the main theorem. Finally, the developed fixed point result is applied to a nonlinear integral equation on the space of continuous real-valued functions on ([0,1]), where sufficient conditions are given for the existence and uniqueness of a continuous solution.