Hardy–Rogers-Type Fixed-Point and Fixed-Disc Results on Quasi M-Metric Spaces

In this manuscript, we investigate original fixed-point (fp) and fixed-disc results in quasi M-metric spaces. First, we present a Hardy–Rogers-type fp result for self-mappings satisfying a specific contractive condition in mq-complete quasi M-metric spaces. Furthermore, we extend these results to an integral-type contraction, providing a more generalized version of the Hardy–Rogers fp theorem. In the second part of this study, we address the fixed-disc problem, which generalizes the fixed-circle problem to the setting of quasi M-metric spaces. We establish both forward and backward Hardy–Rogers-type fixed-disc theorems, identifying the conditions under which a self-mapping T fixes a disc Dζ0,β. The existence and uniqueness of such fps and discs are supported by detailed proofs and illustrative examples. Our results extend and unify several existing theorems in the literature, including Banach- and Kannan-type results on quasi M-metric spaces.