A New Fixed‐Point Framework for Nonexpansive and Averaged Mappings in Normed GE‐Algebras

In this paper, we develop a systematic framework for studying fixed‐point theory in the setting of normed GE‐algebras. Building on the GE‐norm, we introduce and analyze nonexpansive mappings, α ‐averaged mappings, and enriched contractions with respect to the quasimetric induced by the GE‐norm. Several new fixed point results are established, including the nonexpansiveness of averaged operators, a demiclosedness principle, and the sequential closedness of fixed‐point sets. We further prove convergence theorems for Picard‐type iterations associated with α ‐averaged and enriched nonexpansive mappings, thereby extending classical fixed point principles from normed linear and metric spaces to the broader, nonsymmetric, and algebraic setting of GE‐algebras. The results presented here demonstrate that normed GE‐algebras provide a natural and robust framework for nonlinear operator theory and open new directions for fixed point analysis within algebraic systems.