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.