Mathematics and Computing for Scientific Innovation Vol. 1

Fixed-point theory plays a fundamental role in nonlinear analysis and has been extensively extended from classical metric spaces to various generalised distance spaces. Among these, b₂-metric spaces provide a unified framework that combines the characteristics of b-metric and 2-metric spaces. However, the behaviour of contractive-type mappings in complete b₂-metric spaces, particularly without assuming continuity of the b₂-metric function, has received limited attention. This study examines fixed-point results for contractive-type self-mappings in complete b₂-metric spaces, a framework that extends b-metric and 2-metric spaces through a rectangular inequality involving three variables and a coefficient. After presenting the relevant definitions of 2-metric spaces, b₂-metric spaces, convergence, Cauchy sequences, and completeness, two existence and uniqueness results are developed. The first result considers a pair of self-mappings satisfying a generalized contractive condition. Beginning with an arbitrary point, an iterative sequence is constructed, and successive estimates are used to establish that the sequence is Cauchy. Completeness then guarantees convergence to a limit in the space. The contractive condition is subsequently applied to show that this limit is a common fixed point of the two mappings, while a contradiction argument establishes uniqueness. The second result concerns a single self-mapping governed by a contractive condition involving a monotone, upper semicontinuous comparison function. A corresponding iterative procedure proves convergence to a fixed point, and the same contractive framework yields uniqueness. Corollaries are stated for continuous mappings under the respective assumptions. The analysis demonstrates that standard iterative fixed-point techniques can be formulated in the b₂-metric setting without requiring continuity of the b₂-metric function itself. The validity of the conclusions depends on completeness and on the specified restrictions imposed on the contraction parameters and comparison function.