Biannual Report - Fachbereich Mathematik - Technische Universität ...

Together with K. Schade we extended the work done in [1] to so-called modified Halpern
iterations in CAT(0)-spaces (see [4]). Ongoing work (with D. Körnlein) deals with metastability
bounds for the resolvent of nonexpansive and accretive operators in Hilbert and uniformly
smooth spaces. In particular, we aim at a quantitative treatment of so-called sunny
nonexpansive retracts.
Support: Kurt-Gödel-Society, John Templeton Foundation, DFG projects KO 1737/5-1
and KO 1737/5-2
Partner: L. Leuştean, Romanian Academy, Bucharest
Contact: U. Kohlenbach
References
[1] U. Kohlenbach and L. Leu¸stean. Effective metastability for Halpern iterates in CAT(0) space.
Advances in Mathematics, 231:2526–2556, 2012.
[2] U. Kohlenbach and L. Leu¸stean. On the computational content of convergence proofs via
Banach limits. Philosophical Transactions of the Royal Society A, 370:3449–3463, 2012.
[3] D. Körnlein and U. Kohlenbach. Effective rates of convergence for Lipschitzian pseudocontractive
mappings in general Banach spaces. Nonlinear Analysis, 74:5253–5267, 2011.
[4] K. Schade and U. Kohlenbach. Effective metastability for modified halpern iterations in CAT(0)
spaces. Fixed Point Theory and Applications, 2012:19pp., 2012.
Project: Effective metastability in nonlinear ergodic theory
In this project we extract explicit effective rates of metastability (in the sense of Tao) for
nonlinear generalizations of the von Neumann mean ergodic theorem due to Baillon and
Wittmann. In the absence of linearity the strong convergence of the ergodic mean fails to
hold in general while weak convergence is still true due to the famous Baillon nonlinear
ergodic theorem. In [3] we extract a rate of metastability for the weak Cauchy property
for Baillon’s theorem in the Hilbert space case (based on a computational analysis of weak
compactness from [2]). While strong convergence in general fails, there are important
cases where it is still true, e.g. for odd operators (Baillon) or even more general operators
satisfying a condition due to Wittmann. In this situation, an explicit primitive recursive
rate of the metastability of the strong convergence is extracted in [4]. For related results
obtained in this project, see [1]. Another important nonlinear generalization of the von
Neumann theorem is again due to Wittmann who proved that a so-called Halpern iteration
– which in the linear case coincides with the Cesàro mean from the mean ergodic theorem
– strongly converges in Hilbert space. This is discussed in the project “New Frontiers in
Proof Mining”.
Support: German Science Foundation (DFG) as part of project KO 1737/5-1
Contact: U. Kohlenbach, P. Safarik
References
[1] U. Kohlenbach. On the asymptotic behavior of odd operators. Journal of Mathematical Analysis
and Applications, 382:615–620, 2011.
[2] U. Kohlenbach. Gödel functional interpretation and weak compactness. Annals of Pure and
Applied Logic, 163:1560–1579, 2012.
[3] U. Kohlenbach. A uniform quantitative form of sequential weak compactness and Baillon’s
nonlinear ergodic theorem. Communications in Contemporary Mathematics, 14, 2012.
50 1 Research