In this paper, we first introduce the concepts of $d$- and $d^{-1}$-proximal Ćirić contraction mappings. Also, we present new definitions and notations by taking into account the lack of symmetry property of quasi-metric spaces. Moreover, we give some examples to support our definitions and notations. Then, we prove some right and left best proximity point results for these mappings on best orbitally complete quasi-metric spaces. Hence, we obtain some generalizations of famous results in the literature.

In this paper, we aim to overcome the problem given by Abkar et al. [Abstr. Appl. Anal., 2013 (2013), 189567], and so to obtain real generalizations of fixed point results in the literature. In this direction, we introduce a new class of functions, which include $\mathcal{R}$-functions. Thus, we present a new type of $\mathcal{R}$ -contraction and weaken $\mathcal{R}$-contractions that have often been studied recently. We also give a new definition of the $P$-property. Hence, we obtain some best proximity point results, including fixed point results for the new kind of $\mathcal{R}$-contractions. Then, we provide an example to show the effectiveness of our results. Finally, inspired by a nice and interesting technique, we investigate the existence of a best proximity point of the homotopic mappings with the help of our main result.