Abstract In the current paper, we discuss sufficient and necessary conditions for the existence of best proximity points for cyclic f- contractive mappings in the setting of strictly convex Banach spaces. Extensions of Edelstein’s theorem are considered as well as an extension of a main result in Park [Park, S. (1978). Fixed points of f-contractive maps. Rocky Mountain J. Math. 8:743–750]. Another existence result of best proximity points will be obtained for asymptotically relatively nonexpansive mappings under different conditions with respect to the recent paper of Rajesh and Veeramani [Rajesh, S., Veeramani, P. (2016). Best Proximity point theorems for asymptotically relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 37:80–91].

中文翻译：

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严格凸 Banach 空间中循环收缩映射的 Edelstein 定理

摘要 在本文中，我们讨论了在严格凸 Banach 空间的设置中循环 f-收缩映射的最佳邻近点存在的充分必要条件。Edelstein 定理的扩展被认为是 Park [Park, S. (1978) 中主要结果的扩展。f-收缩贴图的固定点。落基山 J. 数学。8:743–750]。对于 Rajesh 和 Veeramani 最近的论文 [Rajesh, S., Veeramani, P. (2016)，在不同条件下，对于渐近相对非膨胀映射，将获得最佳邻近点的另一个存在结果。渐近相对非扩展映射的最佳邻近点定理。数字。功能。肛门。优化。37:80-91]。