Some best proximity point results via a new family of F-contraction and an application to homotopy theory

In this paper, taking into account the lack of symmetry property of quasi-metric, we first revise some concepts and notations related to best proximity point theory. Next, we define a new family of F -contraction, larger than the family that is widely used for multivalued mappings in the literature. Then, we obtain some best proximity point results which improve many results in the literature. Hence, we give a positive answer to the question in Altun et al. (J Nonlinear Convex Anal 16(4):659–666, 2015) on quasi-metric spaces. Also, some comparative examples are provided to support our main results. Finally, we give some applications of our new best proximity point theorems to homotopy theory. Therefore, we prove some best proximity point results for homotopic mappings.