A new type of $\mathcal{R}$-contraction and its best proximity points

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.