Graphical Cyclic $\mathcal{K}$-Quasi-Contractive Mappings and the Existence of their Best Proximity Points

The underlying aim of this paper is first to state the Cyclic version of $\mathcal{K}$-quasi-contractive mappings introduced by Fallahi and Aghanians [On quasi-contractions in metric spaces with a graph, Hacet. J. Math. Stat. 45 (4) (2016), 1033-1047]. Secondly, it seeks to show to show the existence of fixed point and best proximity points for such contractive mappings in a metric space with a graph, which can entail a large number of former fixed point and best proximity point results. One fundamental issue that can be distinguished between this work and previous studies is that it can also involve all of results stated by taking comparable and $\eta$-close elements.