Fixed Point Theorems for Contravariant Maps in Bipolar b-Metric Spaces with Integration Application

As a natural extension of the metric and the bipolar metric, this article introduces the new abstract bipolar $b-$ metric. The bipolar $b-$metric is a novel technique addressed in this article; it is explained by combining the well-known $b-$metric in the theory of metric spaces, as defined by Mutlu and G\"{u}rdal (2016) {mg1}, with the description of the bipolar metric. In this new definition, well-known mathematical terms such as Cauchy and convergent sequences are utilized. In the bipolar $b-$metric, fundamental topological concepts are also defined to investigate the existence of fixed points implicated in such mappings under different contraction conditions. An example is provided to demonstrate the presented results.