On a class of fixed points for set contractions on partial metric spaces with a digraph

We investigate the existence of fixed point problems on a partial metric space. The results obtained are for set contractions in the domain of sets and the pattern for the partial metric space is constructed on a directed graph. Essentially, our main strategy is to employ generalized $\varphi$-contractions in order to prove our results, where the fixed points are investigated with a graph structure. Moreover, we state and prove the well-posedness of fixed point based problems of the generalized $\varphi$-contractive operator in the framework of a partial metric space. We illustrate the main results in this manuscript by providing several examples.