Metrization of Polygonal b-Metric Spaces and Some Fixed Point in Extended Polygonal b-Metric Spaces with Applications

We establish a metrization result for some bv(s)-metric spaces, extending a result recently established for rectangular b-metric spaces. Furthermore, we introduce the notion of an extended polygonal b-metric space (or bv(θ)-metric space), which unifies and generalizes several classes of spaces, including metric spaces, rectangular metric spaces, b-metric spaces, rectangular b-metric spaces, polygonal metric spaces, and bv(s)-metric spaces. Some fixed-point results in bv(θ)-metric spaces are established under the weak orbital completeness condition in the framework of the Banach contraction principle and for generalized expansive Hardy-Rogers-type mappings. An a priori error estimate for the iterative process is obtained in both bv(θ)-metric and bv(s)-metric spaces. We also establish the Ulam-Hyers stability of fixed-point equations in both bv(θ)-metric and bv(s)-metric spaces. Several examples are provided, and applications to certain types of integral equations and initial value problems are presented, illustrating the applicability and effectiveness of the obtained results.