Weak and Strong Convergence of an Implicit Iteration for Hemi-Contractive Semigroups in Banach Spaces
Abstract
We study an implicit iteration for approximating common fixed points of strongly continuous semigroups on closed convex subsets of real Banach spaces. Under a stepwise Lipschitz restriction and hemi-contractivity relative to the common fixed point set, we establish well-definedness and Fej% ér monotonicity of the iteration. In reflexive Banach spaces satisfying Opial's condition, we obtain a sufficient condition for weak convergence based on a normalized residual estimate, uniform continuity at zero along the generated sequence, and demiclosedness at fixed positive times. In general real Banach spaces, we prove strong convergence under a time-dependent directional contraction condition and a divergence condition coupling the iteration weights with the contraction strength. A linear example shows that vanishing time steps and a divergent sum of iteration weights alone do not ensure convergence to a common fixed point. A nonlinear planar evolution semigroup illustrates the applicability of the strong convergence theorem beyond the nonexpansive class.
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BibTeX
@misc{gurbuz2026weak,
title = {{Weak and Strong Convergence of an Implicit Iteration for Hemi-Contractive Semigroups in Banach Spaces}},
author = {Ferit Gürbüz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.25799},
eprint = {2606.25799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.25799},
}
RIS
TY - UNPB
TI - Weak and Strong Convergence of an Implicit Iteration for Hemi-Contractive Semigroups in Banach Spaces
AU - Ferit Gürbüz
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2606.25799
UR - https://arxiv.org/abs/2606.25799
AB - We study an implicit iteration for approximating common fixed points of strongly continuous semigroups on closed convex subsets of real Banach spaces. Under a stepwise Lipschitz restriction and hemi-contractivity relative to the common fixed point set, we establish well-definedness and Fej% ér monotonicity of the iteration. In reflexive Banach spaces satisfying Opial's condition, we obtain a sufficient condition for weak convergence based on a normalized residual estimate, uniform continuity at zero along the generated sequence, and demiclosedness at fixed positive times. In general real Banach spaces, we prove strong convergence under a time-dependent directional contraction condition and a divergence condition coupling the iteration weights with the contraction strength. A linear example shows that vanishing time steps and a divergent sum of iteration weights alone do not ensure convergence to a common fixed point. A nonlinear planar evolution semigroup illustrates the applicability of the strong convergence theorem beyond the nonexpansive class.
ER -