An enriched iterative scheme for common fixed points of enriched nearly asymptotically nonexpansive mappings in Banach spaces

Abstract

In this paper, we introduce and analyze a new two-step iterative scheme for approximating common fixed points of two enriched nearly asymptotically nonexpansive self-mappings defined on a closed convex subset of a uniformly convex Banach space. The proposed iteration combines averaged mappings with a carefully selected convex parameter to achieve an optimal contraction rate. We first establish that for enriched contractions, the new scheme converges with a rate factor equal to the contraction constant \(\kappa_t\), which is strictly faster than the factor \(k + \alpha_n(1-k)\) obtained for the existing Yao–-Chen iteration and for the classical Mann iteration. Under more general enriched nearly asymptotically nonexpansive conditions, we prove asymptotic regularity, provided the sequence parameters lie in a closed interval within \((0,1)\) and certain conditions hold on the nearly asymptotically nonexpansive sequences. Using this regularity, we establish the demiclosedness principle for \(I-\gamma\) at zero and, under Opial’s condition, Fréchet differentiability of the norm, or the Kadec–-Klee property of the dual space, we prove weak convergence of the iterates to a common fixed point. Furthermore, under condition \((A')\), we obtain strong convergence. A concrete example illustrates the applicability of the considered mapping class. The results extend and unify several recent findings in fixed point theory, particularly in the context of enriched and nearly asymptotically nonexpansive mappings.