Best Proximity Point Theorems for $p$-Proximal $\alpha$-$\eta$-$\beta$-Quasi Contractions in Metric Spaces with $w_0$-Distance

Abstract:

      In this paper, we propose a new class of non-self mappings called $p$-proximal $\alpha$-$\eta$-$\beta$-quasi contraction, and introduce the concepts of $\alpha$-proximal admissible mapping with respect to $\eta$ and $(\alpha,d)$ regular mapping with respect to $\eta$. Based on these new notions, we study the existence and uniqueness of best proximity point for this kind of new contractions in metric spaces with $w_0$-distance and obtain a new theorem, which generalize and complement the results in [Ayari, M. I. et al. Fixed Point Theory Appl., 2017, 2017: 16] and [Ayari, M. I. et al. Fixed Point Theory Appl., 2019, 2019: 7]. We give an example to show the validity of our main result. Moreover, we obtain several consequences concerning about best proximity point and common fixed point results for two mappings, and we present an application of a corollary to discuss the solutions to a class of systems of Volterra type integral equations.