A unified analytic approach to O-metric inequalities and applications to fixed point theory

The concept of O-metrics was recently introduced as a generalization of several metric-type structures by replacing the addition operation of the standard triangle inequality with a binary operation that may fail to be associative. This non-associativity naturally to generalized polygon inequalities and patterned compositions. In this work we develop an analytic framework for inequalities arising in such settings. By introducing generalized $\text{om}$-series governed by admissible control functions $\phi$, we establish a separation principle that resolves inequalities of the form $u\le v\text{om}\phi (k,u)$. This result provides convergence criteria for patterned compositions and determines intervals for the admissible contraction parameter $k$. As application, some fixed point theorems of '{C}iri'{c} type are obtained for mappings on O-metric spaces, as well as corresponding results for b-metric spaces. The approach provides a unified analytic framework for studying contractive conditions and iterative processes in generalized metric spaces.