Partial extended b-metric and some fixed point theorem

In this paper, we introduce the concept of partial extended b-metric spaces (PEBMS) as a unification and generalization of extended b-metric spaces and partial b-metric spaces. This new structure incorporates a point-dependent control function together with the possibility of non-zero self-distance, providing a more flexible framework for the study of generalized metric spaces. We establish several fundamental properties of PEBMS, including convergence, Cauchy sequences, and 0-completeness. By introducing the notion of 0-Cauchy sequences, we extend various results from extended b-metric spaces to the PEBMS setting. In particular, we prove fixed point theorems for contractive mappings and show the existence and uniqueness of fixed points under suitable conditions. Furthermore, we demonstrate that every extended b-metric space can be viewed as a special case of a PEBMS. As an application, we study the stability of discrete dynamical systems within this framework. The results presented here generalize and enrich existing theories in metric-type spaces and open new directions for further research.