Common Fixed Point Theorems for (β, α)-Generalized Enriched Contractions in Banach Spaces

Abstract

This paper investigates common fixed point theorems for (β,α)-generalized enriched contractions in Banach spaces. We provide a corrected proof of an existing theorem on enriched contractions, thereby strengthening the reliability of results in this area. Our analysis further shows that a previously published illustrative example does not satisfy the proposed enriched contraction condition, since the condition fails for x=0, y=18, and b=45. We then introduce a generalized pair of mappings and prove two common fixed point theorems for single-valued (β,α)-generalized enriched contractions under weak commutativity and compatibility conditions. Strong convergence to the unique common fixed point is established through a Mann-type iteration process, and an example is provided to validate the proposed generalizations.