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

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.