Abstract:

In this paper we introduce a new orbit-based contractive framework in the setting of \( G \)-metric spaces, called \( (m,\alpha) \) \( G \)-path-averaged (\( G \)-PA) contractions with \( m \geq 2 \). This notion extends Fabião’s path-averaged contractions to the triadic geometry of Mustafa–Sims \( G \)-metrics and is designed to avoid collapse to pointwise contractivity. For a \( G \)-continuous self-map on a complete \( G \)-metric space, we establish existence and uniqueness of a fixed point and prove that the Picard iteration converges to it in the sense of \( G \)-convergence. Moreover, we derive explicit quantitative estimates, including a posteriori and a priori geometric error bounds for the iterates. We also relate the new class to the induced metric \( d_G \), showing that every \( G \)-PA contraction yields a path-averaged contraction on \( (X, d_G) \), and we provide examples demonstrating that the \( G \)-PA class can be strictly larger than the Banach-type contraction class. Finally, we obtain multi-step (\( t \)-point) fixed point and convergence results by embedding the recursion into a shift map on the product space \( (X^t, \sigma^t) \) and applying the single-valued theory.