The main purpose of this paper is to show that the Meir–Keeler contraction principle, as well as some of its generalizations, is, in general, not true in quasi-metric spaces. After that, we suggest a new Meir–Keeler type contraction that guaranties the existence and uniqueness of fixed points in quasi-metric spaces. Finally, to illustrate the wide usability of our findings, we discuss the existence and uniqueness of solutions for an integro-differential equation in Musielak–Orlicz spaces.

中文翻译：

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关于拟度量空间中的 Meir-Keeler 定理

本文的主要目的是表明 Meir-Keeler 收缩原理及其一些推广一般在拟度量空间中不成立。之后，我们提出了一种新的 Meir-Keeler 型收缩，它保证了拟度量空间中不动点的存在性和唯一性。最后，为了说明我们发现的广泛可用性，我们讨论了 Musielak-Orlicz 空间中积分微分方程解的存在性和唯一性。