New Rational Contraction on b-Metric Spaces

This paper introduces a new rational contractive principle on b-metric spaces, termed the RMS-rational contraction, which unifies and strictly extends a broad spectrum of fixed point frameworks. The proposed condition incorporates multi-term interpoint distances in a rational structure and naturally adapts to the geometry of b-metrics through the sharp convergence threshold sθ < 1, where s is the b-metric coefficient and θ is determined by the contraction parameters. Within this setting we prove: (i) existence and uniqueness of fixed points, (ii) linear convergence of Picard iterations, (iii) explicit a priori and a posteriori error bounds, and (iv) a quantitative stability result controlling perturbations of fixed points under data variations. A cyclic extension on two closed subsets is also established, guaranteeing convergence to a unique point in their intersection whenever a forward orbit is bounded. Our framework recovers, as special or limiting cases, the classical principles of Banach, Kannan–Chatterjea, Hardy–Rogers, Meir–Keeler, Boyd–Wong, Geraghty,