Analytical Meir-Keeler type contraction mappings and equivalent characterizations

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.

Automated summary

The paper presents a new fixed point theorem which offers a solution to Rhoades' problem on the existence of contractive mappings with discontinuity at the fixed point, providing the first Meir-Keeler type solution to this issue. The theorem is shown to characterize the completeness of the metric space, and the structure of complete subspaces of the real line in which contractive mappings do not have discontinuity at the fixed point is also given. This resolves Rhoades' question completely in the context of the real line.