Improved Convergence Theorems for Monotone α-Nonexpansive Mappings in Uniformly Convex Banach Spaces
Mots-clés :
Monotone α-nonexpansive mapping, Mann iteration, Halpern iteration, inertial hybrid algorithm, strong convergence, duality mapping, uniformly smooth Banach space, counterexamplesRésumé
This paper investigates convergence properties of monotone α-nonexpansive
mappings in uniformly convex Banach spaces and introduces new inertial and hybrid schemes that extend and unify classical Mann and Halpern iterations. We
first establish weak and strong convergence theorems for the Mann and Halpern
processes under natural control conditions on {βn} and {αn}. The strong convergence results are then extended beyond Opial’s condition to Banach spaces that
are uniformly smooth or possess Gateaux/Fr´echet differentiable norms, using the
continuity of the normalized duality mapping.
An accelerated inertial Halpern–Hybrid algorithm is proposed, incorporating
an explicit inertial term {µn} to enhance convergence speed. Strong convergence
and asymptotic regularity of the new scheme are rigorously proven. Furthermore,
several analytical counterexamples are constructed to demonstrate that the boundedness of {βn}, the non-summability of {αn}, and the decay of {µn} are essential
for stability. These results collectively provide a comprehensive framework for understanding and improving iterative fixed-point algorithms in smooth, convex, and
non-Hilbert settings.
