Journal of Convex Analysis 33 (2026), No. 1&2, 075--090
Copyright Heldermann Verlag 2026
Fejér Monotone Sequences Revisited
Ulrich Kohlenbach
Department of Mathematics, Technische Universitaet, Darmstadt, Germany
kohlenbach@mathematik.tu-darmstadt.de
Pedro Pinto
Department of Mathematics, Technische Universitaet, Darmstadt, Germany
pinto@mathematik.tu-darmstadt.de
We introduce a localized and relativized generalization of the usual concept of Fejer monotonicity together with uniform and quantitative versions thereof and show that the main quantitative results obtained by the first author together with Nicolae and Leustean in 2018 and with Lopez-Acedo and Nicolae in 2019 respectively, extend to this generalization. Our framework, in particular, covers the sequence generated by the Dykstra algorithm while the latter is not Fejer-monotone in the ordinary sense. This gives a theoretical explanation why under a metric regularity assumption one obtains an explicit rate of convergence for Dykstra's algorithm which was proved recently by the second author.
Keywords: Fejer monotonicity, rates of convergence, metastability, Dykstra's algorithm.
MSC: 47J25, 41A25, 03F10.
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