Fixed Point Theorem for Path-Averaged Contractions in Complete b-Metric Spaces

Abstract

We extend the fixed point result for Path-Averaged Contractions (PA-contractions) from complete metric spaces to complete b-metric spaces. We prove that every PA-contraction on a complete b-metric space has a unique fixed point, provided the contraction constant $ \alpha $ satisfies $s \alpha^{1/N} < 1$, where $ s \geq 1 $ is the b-metric coefficient and $N$ the averaging parameter. Moreover, we establish that every PA-contraction is automatically continuous. The proof relies on geometric decay of successive distances and the generalized triangle inequality. This result paves the way for extending averaged contraction principles to other classical types, such as Kannan, Chatterjea, and \'Ciri\'c-type mappings, as well as Wardowski's F-contractions, in generalized metric settings.