On Partially Ordered -Dislocated Metric Spaces

Abstract

     Dislocated metrics play an important role in logical programming. Moreover, they are pivotal in fields such as topology and electronic engineering. These spaces are significant because they have the property of modifying self-distance property. This feature made many authors undertake extensive investigations of this metric, where they examined the fixed points of maps that exist in this space that meet specific requirements and studied the characteristics that set it apart. Additionally, they looked into whether -contractions have the best proximity in the framework of dislocated metric spaces, and specific requirements be established to ensure that a best proximity point for these contractions is unique. Researchers worked on generalizing the concept of dislocated metric space to include b-dislocated metric space, S-dislocated metric space, and other forms of metric structures. This paper introduces the concept of Ps-dislocated metric spaces and the concepts of Ps-proximal contraction map and generalized Ps-proximity contraction mapping. Also, we study the existence of a best proximity point of non-self-map via generalized -proximity contraction  map and given the condition to be unique. After that, we used these results to derive the theorems' best proximity point on -dislocated metric spaces endowed with a partial order, and we gave an example that cleared these concepts in this space.