Mathematics

CLC number: O177.91

Fixed Point Results for Extended Contractions with Various Weak Conditions in Generalized Metric Spaces

广义度量空间中各类弱条件下压缩映射的不动点定理

School of Mathematics and Statistics, Zhaotong University, Zhaotong 657000, Yunnan, China

Received: 15 September 2024

Abstract

In this work, we investigate several fixed point theorems in double controlled metric spaces by providing some new definitions, which serve as a natural extension of the existing notions. Moreover, we establish its various characterizations by some typical examples. Specifically, these examples hold in double-controlled metric-like spaces instead of usual b-metric spaces. The fixed point theorems are obtained under these new concepts, without appealing to the completeness of the spaces or the continuity of the mappings. Furthermore, we prove that the completeness in double-controlled metric-like spaces is necessary if the extended Kannan-type contractions have a fixed point in the spaces. The results contribute to a deeper understanding of Kannan-type contractions and their applicability in the context of fixed point theory in double-controlled metric spaces.

摘要

本文通过提出一些新的定义来研究双控制度量类空间中的几个不动点定理，这些定义是作为现有概念的推广。同时，通过一些典型例子说明了主要结果的应用。具体而言，这些实例适用于双控制度量类空间，而非通常的 b-度量空间。我们在这些新概念下获得了不动点定理，不要求空间的完备性或映射的连续性。此外，证明了在双控制度量类空间中，如果推广的 Kannan 型压缩映射在空间中有不动点，则该空间的完备性是必要的。本文的结果有助于更深入地理解 Kannan 型压缩映射及其在双控制度量类空间不动点理论中的应用。