Best Proximity Point Theorems for $p$-Proximal $\alpha$-$\eta$-$\beta$-Quasi Contractions in Metric Spaces with $w_0$-Distance
Received:December 31, 2020  Revised:August 14, 2021
Key Word: best proximity point   $p$-proximal $\alpha$-$\eta$-$\beta$ quasi contraction   $w_0$-distance   $\alpha$-proximal admissible mapping with respect to $\eta$   $(\alpha,d)$ regular mapping with respect to $\eta$
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.12161056; 11701259; 11771198) and the Natural Science Foundation of Jiangxi Province (Grant No.20202BAB201001).
 Author Name Affiliation Mengdi LIU Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China Zhaoqi WU Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China Chuanxi ZHU Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China Chenggui YUAN Department of Mathematics, Swansea University, Singleton Park SA2 8PP, UK
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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.