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From graphical metric spaces to fixed point theory in binary related distance spaces

Roldán López de Hierroa Antonio Francisco (University of Granada, Department of Quantitative Methods for Economics and Business, Granada, Spain + PAIDI Research Group FQM-, University of Ja´en, Ja´en, Spain)
Shahzad Naseer (King Abdulaziz University, Department of Mathematics, Jeddah, Saudi Arabia)

Very recently, many fixed point results have been introduced in the setting of graphical metric spaces. Due to their intimate links, such works also deal with metric spaces endowed with partial orders. As the reachability relationship in any directed graph (containing all cycles) is a reflexive transitive relation (that is, a preorder), but it is not necessarily a partial order, results on graphical metric spaces are independent from statements on ordered metric spaces. The main aim of this paper is to show that fixed point theorems in the setting of graphical metric spaces can be directly deduced from their corresponding results on measurable spaces endowed with a binary relation. Finally, we also describe the main advantages of involving this last class of spaces.

Keywords: Partially ordered metric space, Graphical metric space, Fixed point, Contractive mapping