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

DOI：10.3770/j.issn:2095-2651.2022.01.009

 作者 单位 刘孟递 南昌大学数学系, 江西 南昌 330031 吴照奇 南昌大学数学系, 江西 南昌 330031 朱传喜 南昌大学数学系, 江西 南昌 330031 袁成桂 英国斯旺西大学数学系, 斯旺西 SA2 8PP

本文提出了一类称为$p$-逼近$\alpha$-$\eta$-$\beta$-拟压缩的新的非自映射,并引进了关于$\eta$的$\alpha$-逼近可容许映射和关于$\eta$的$(\alpha,d)$正则映射的概念.基于这些新概念,在$w_0$-距离度量空间中研究了此类新压缩最佳逼近点的存在唯一性,并给出了一个新的定理,推广和补充了文[Ayari, M. I. et al. Fixed Point Theory Appl., 2017, 2017: 16]和[Ayari, M. I. et al. Fixed Point Theory Appl., 2019, 2019: 7]中的结果.给出了一个例子来说明主要结果的有效性.进一步地,作为推论得到关于两个映射的最佳逼近点和公共不动点定理.作为其中一个推论的应用,讨论了一类Volterra型积分方程组的求解问题.

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.