Fejér Monotone Sequences Revisited

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.