On modified Halpern and Tikhonov-Mann iterations

Horatiu Cheval, Ulrich Kohlenbach, Laurentiu Leustean

Published 2022-03-21Version 1

We show that the asymptotic regularity and the strong convergence of the modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by A. Cuntavenapit and B. Panyanak and the Tikhonov-Mann iteration introduced by H. Cheval and L. Leustean as a generalization of an iteration due to Y. Yao et al. that has recently been studied by Bot et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Bot et al. together with a generalization from Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity and metastability due to K. Schade and U. Kohlenbach can be adapted and transformed into rates for the Tikhonov-Mann iteration corresponding to recent quantitative results on the latter of H. Cheval, L. Leustean and B. Dinis, P. Pinto respectively. A transformation in the converse direction is also possible in a slightly more restricted geodesic setting