The aim of this paper is to obtain a fixed point theorem which gives a new solution to the Rhoades’ problem on the existence of contractive mappings that admit discontinuity at the fixed point; and it is the first Meir–Keeler type solution of this problem. We prove that our theorem characterizes the completeness of the metric space. We also give the structure of complete subspaces of the real line in which contractive mappings do not admit discontinuity at the fixed point and, thus, in the setting of the real line we completely resolve the Rhoades’ question.

中文翻译：

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分析 Meir-Keeler 型收缩映射和等效特征

本文的目的是获得一个不动点定理，该定理为 Rhoades 问题的一个新的解决方案，即承认在不动点处不连续的收缩映射的存在性；它是该问题的第一个 Meir-Keeler 型解。我们证明了我们的定理表征了度量空间的完备性。我们还给出了实线的完全子空间的结构，其中收缩映射不允许在不动点处不连续，因此，在实线的设置中，我们完全解决了罗兹问题。