Hans-Peter A. Künzi | Searcharxiv

arXiv2019

Some remarks on "Convergence of Picard's iteration using projection algorithm for noncyclic contractions" [Indag. Math. 30 (2019) 227--239]

In this note, at first we prove that the existence of best proximity points for cyclic relatively nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that a main result of the paper "Proximal normal structure and relatively nonexpansive mappings", Studia Math., (171~(2005) 283--293) immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper published in Indag. Math., 30(1) (2019) 227--239 is obtained exactly from Picard's iteration sequence.

math.FA↗

arXiv2003

Maximal (sequentially) compact topologies

We revisit the known problem whether each compact topology is contained in a maximal compact topology and collect some partial answers to this question. For instance we show that each compact topology is contained in a compact topology in which convergent sequences have unique limits. We also answer a question of D.E. Cameron by showing that each sequentially compact topology is contained in a maximal sequentially compact topology. We finally observe that each sober compact T_1-topology is contained in a maximal compact topology and that each sober compact T_1-topology which is locally compact or sequential is the infimum of a family of maximal compact topologies.

math.GN↗