Fixed point results for mappings of asymptotically Hölder nonexpansive type
Abstract
We introduce asymptotically Hölder nonexpansive mappings and mappings of asymptotically Hölder nonexpansive type, in which the Hölder exponents converge to one along the iterates. We prove that if K is a nonempty closed bounded convex subset of a Banach space X with characteristic of convexity ε0(X)<1, then every self-mapping of K of asymptotically Hölder nonexpansive type admits a point x∈ K such that Tn x→ x. Consequently, a fixed point exists whenever some positive iterate of T is continuous; in particular, this applies to asymptotically Hölder nonexpansive mappings. We also obtain fixed-point-free constructions in spaces containing copies of c0, and a related construction on a bounded convex, not necessarily closed, subset of every Banach space containing an isomorphic copy of ℓ1. Further examples show that the new classes properly extend their classical counterparts and may contain mappings with no continuous positive iterate. Finally, we examine Lin's renorming of ℓ1, identify an obstruction related to shift-type constructions, and establish a shrinking-diagonal criterion that reduces the remaining closed-set problem to the construction of a suitably controlled uniformly Lipschitzian mapping.
Related
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BibTeX
@misc{barroso2026fixed,
title = {{Fixed point results for mappings of asymptotically Hölder nonexpansive type}},
author = {Cleon S. Barroso and Carlos Sérgio R. da Silva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.30678},
eprint = {2605.30678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.30678},
}
RIS
TY - UNPB
TI - Fixed point results for mappings of asymptotically Hölder nonexpansive type
AU - Cleon S. Barroso
AU - Carlos Sérgio R. da Silva
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2605.30678
UR - https://arxiv.org/abs/2605.30678
AB - We introduce asymptotically Hölder nonexpansive mappings and mappings of asymptotically Hölder nonexpansive type, in which the Hölder exponents converge to one along the iterates. We prove that if K is a nonempty closed bounded convex subset of a Banach space X with characteristic of convexity ε_0(X)<1, then every self-mapping of K of asymptotically Hölder nonexpansive type admits a point x∈ K such that T^n x→ x. Consequently, a fixed point exists whenever some positive iterate of T is continuous; in particular, this applies to asymptotically Hölder nonexpansive mappings. We also obtain fixed-point-free constructions in spaces containing copies of c_0, and a related construction on a bounded convex, not necessarily closed, subset of every Banach space containing an isomorphic copy of ℓ_1. Further examples show that the new classes properly extend their classical counterparts and may contain mappings with no continuous positive iterate. Finally, we examine Lin's renorming of ℓ_1, identify an obstruction related to shift-type constructions, and establish a shrinking-diagonal criterion that reduces the remaining closed-set problem to the construction of a suitably controlled uniformly Lipschitzian mapping.
ER -