K–R Contractions: A Unified Framework for Fixed Points, Stability, and Generalized Metric Structures

We introduce a new two-parameter framework for fixed-point analysis in metric spaces based on a pair of constants that simultaneously quantify the contraction strength and the stability radius of a self-mapping. A mapping is called a K–R contraction if where and . This formulation extends the classical Banach contraction principle () and unifies several well-known contraction types, including Kannan- and Chatterjea-type mappings, under a single analytic structure. We prove that if satisfies , then admits a unique fixed point, and the associated Picard iteration converges to it with an explicit error estimate. In addition, we establish a K–R stability theorem, providing quantitative Hyers–Ulam stability bounds and showing robustness of fixed points under perturbations of the operator. Several counterexamples demonstrate that the condition is sharp. Further extensions to partial metric spaces, b-metric spaces, and multivalued mappings are also presented. The proposed K–R framework generalizes classical fixed-point results, offers improved flexibility for nonlinear operator analysis, and provides a unified basis for new applications in iterative methods, integral equations, and nonlinear dynamical systems.