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.