Fixed Point Results for -Rational Contraction in Complete Metric Spaces

This paper introduces and investigates a new family of nonlinear contraction-type mappings, termed -rational contractions, defined on complete metric spaces. This class is formulated through a function satisfying appropriate monotonicity and asymptotic regularity conditions, and incorporates both rational-type expressions and power-type nonlinearities. The proposed framework unifies and extends several well-known contractive conditions, including the Jleli-Samet -contractions and the rational-type contractions of Dass-Gupta. A general fixed point theorem is established, ensuring that a fixed point exist and is unique for -rational contractions, together with the convergence of the associated Picard iteration. A corollary shows that, by choosing , the contractive condition reduces to a strict -contraction, thereby generalizing and strengthening the -contraction introduced by Jleli and Samet. Illustrative examples on both finite and continuous metric spaces are included to show the applicability of the established findings. Additionally, a common fixed point result is established for commuting pairs of -rational contractions. These results highlight the flexibility of the -rational framework and offer a unified approach to a broad class of nonlinear contractive conditions.