Peyman Salimi

In this paper, we introduce the concepts of an $\alpha$ -admissible nonself-mapping, an $\alpha$ -F-contractive nonself-mapping, a weak $\alpha$ -F-contractive nonself-mapping, and a generalized $\alpha$ -F-contractive nonself-mapping and prove some $PPF$ (past-present-future)-dependent fixed point theorems for the proposed contractive nonself-mappings in certain Razumikhin classes. By using our results, we derive some $PPF$ -dependent fixed point theorems for an $\alpha$ -F-contractive nonself-mapping endowed with a graph or a partial order. Finally, we give some applications to illustrate the main results.

G-metric spaces proved to be a rich source for fixed point theory; however, the best proximity point problem has not been considered in such spaces. The aim of this paper is to introduce certain new classes of proximal contraction mappings and establish the best proximity point theorems for such kind of mappings inG-metric spaces. As a consequence of these results, we deduce certain new best proximity and fixed point results. Moreover, we present an example to illustrate the usability of the obtained results.