Path--Averaged Contractions: A New Generalization of the Banach Contraction Principle

Abstract

We introduce a novel class of self-mappings on metric spaces, called \textbf{PA-contractions} (Path-Averaged Contractions), defined by an averaging condition over iterated distances. We prove that every continuous PA-contraction on a complete metric space has a unique fixed point, and the Picard iterates converge to it. This condition strictly generalizes the classical Banach contraction principle. We provide examples showing that PA-contractions are independent of F-contractions, Kannan, Chatterjea, and \'Ciri\'c contractions. A comparison table highlights the distinctions. The PA-condition captures long-term contractive behavior even when pointwise contraction fails.