Common Fixed Point for Finite Sets of Mappings in Generalized Metric Spaces

Most classical results in common fixed point theory focus on the case of two mappings sharing a fixed point. In contrast, this article adopts a broader perspective by investigating the existence of a common fixed point for any finite number of mappings. Our main result demonstrates that a finite family of mappings f1,f2,…,fk admits a unique common fixed point under certain well-defined conditions. By leveraging the distinctive structure of the MP-metric, particularly its ability to accommodate multiple entries, we establish a contraction condition that simultaneously involves several mappings and employ it as the central tool to achieve our result. In addition, we introduce two general frameworks that ensure the existence and uniqueness of a fixed point for a single mapping and we propose new criteria for determining when two mappings share a common fixed point. Beyond these contributions, this work lays the groundwork for future research, including extending the theory under alternative assumptions and exploring potential applications in solving systems of equations arising from such fixed point formulations.