Expansive mappings endowed with a directed graph in dislocated metric space and some new fixed-point results

In this paper, we introduce the concept of $Gr$-expansive and $ \theta$-$Gr$-expansive mapping and derive some new fixed-point results endowed with a directed graph considering such mappings in a complete dislocated metric space. A coupled fixed-point theorem is also established by applying $Gr$-expansive condition. Examples are provided in support of the derived results. The solution to a Fredholm integral equation is also formulated to show the applicability of the results.