ϵ -Compatible Map and New Approach for Common Fixed Point Theorems in Partial Metric Space Endowed with Graph

Abstract

In 2016, Muralisankar and Jeyabal introduced the concept of e-Compatible maps and studied the set of common fixed points. They generalized the Banach contraction, Kannan contraction, Reich contraction and Bianchini type contraction to obtain some common fixed point theorems for e-Compatible mappings which don't involve the suitable containment of the ranges for the given mappings in the setting of metric spaces. Motivated by this new concept of mappings, we establish a new approach for some common fixed point theorems via ϵ -compatible maps in context of complete partial metric space including a directed graph G=(V,E). By the remarkable work of Jachymski in 2008, we extend the results obtained by Muralisankar and Jeyabal in 2016. In 2008, Jachymski obtained some important fixed point results introduced by Ran and Reurings (2004) using the languages of graph theory instead of partial order and gave an interesting approach in this direction. After that, his work is considered as a reference in this domain. Sometimes, there are some mappings which do not satisfy the contractive nature on whole set M(say) but these can be made contractive on some subset of M and this can be done by including graph as shown in our Example 2.6 which is provided to substantiate the validity of our results.