In this paper, we study a class of self-mappings that are not necessarily contractions but admit a contractive iterate at a point. Working in the framework of strong $b$-metric spaces, we establish existence and uniqueness results for fixed points of such mappings. Moreover, we obtain an upper bound for the distance between an arbitrary initial point and the corresponding fixed point. Additionally, we investigate the behavior of fixed points under perturbations of the operator and prove the iterative Ulam–Hyers stability result for the associated fixed point problem.