We introduce a concept of complete proximal normal structure and used to investigate the existence of a best proximity point for an arbitrary family of cyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces. We also prove that every bounded, closed and convex pair in uniformly convex Banach spaces as well as every compact and convex pairs in Banach spaces has complete proximal normal structure. Furthermore, we consider a class of cyclic relatively nonexpansive mappings in the sense of Suzuki and establish a new best proximity point theorem in the setting of Hilbert spaces.

中文翻译：

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通过完全近端法向结构的概念共同的最佳邻近对

我们引入了完全近端法向结构的概念，并用于研究在严格凸 Banach 空间的设置中，任意循环相对非扩展映射族的最佳邻近点的存在。我们还证明了一致凸 Banach 空间中的每个有界、闭和凸对以及 Banach 空间中的每个紧和凸对都具有完整的近端正规结构。此外，我们考虑了铃木意义上的一类循环相对非膨胀映射，并在希尔伯特空间的设置中建立了一个新的最佳邻近点定理。