A Unified Framework for Contractive Mappings: The Theory of p-Contractions and their Fixed Points in Metric Spaces

Abstract

Metric fixed point theory is based on the Banach Contraction Principle (BCP). Piles of generalizations of BCP is being proposed in the literature. Current unifying models, including implicit relations and F-contractions are popular yet require complicated boundary conditions or auxiliary functions. In the current study, a new class of contrac-tive mappings, p-contractions is introduced and discussed. This class of mapping is driven by the need for a more efficient analytical tool. The proposed approach uses an intrinsic tri-variable control function. Instead of depending on multi-layered admissibility require-ments these are controlled by minimum axioms to encode contractive behavior producing more robust generalization. A single continuous function p(u, v, w) controls the con-tractive behavior of these mappings. The present work explicitly show the mathematical benefits of this framework by demonstrating both extremely non-linear contractions and traditional linear models (Banach, Reich, Kannan). Apart from the primary theorem the present work also discusses its applicability in solving nonlinear integral equations, convergence rates, and makes clear comparisons with current frameworks.