Kamaleldin AbodayehSyed Khayyam ShahMuhammad SarwarVaraporn WattanakejornThanin Sitthiwirattham2025-03-102025-03-102024Fixed Point Theory and Algorithms for Sciences and Engineering1687182010.1186/s13663-024-00767-02-s2.0-85197791737https://repository.dusit.ac.th//handle/123456789/4470This article focuses on studying some fixed-point results via _-contraction of HardyÐRogers type in the context of supermetric space and ordered supermetric space. We also introduced rational-type z-contraction on supermetric space. For authenticity, some illustrative examples and applications have been included. © The Author(s) 2024.All Open Access; Gold Open AccessF-contractionFixed pointHardyÐRogers typeIntegral contractionOrdered supermetric spaceSupermetric spaceArticleScopus