On modified Halpern and Tikhonov-Mann iterations

Abstract

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. Leuştean 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. Leuştean and B. Dinis, P. Pinto respectively. A transformation in the converse direction is also possible in a slightly more restricted geodesic setting.

Linear rates of asymptotic regularity for Halpern-type iterations

“…In [6], the authors and Kohlenbach computed linear rates of (T −)asymptotic regularity, for a special choice of the parameter sequences, for the modified Halpern iteration, a generalization of the Halpern iteration. One gets immediately linear rates for this iteration too.…”

“…As an immediate consequence, linear rates for the Halpern iteration are obtained. The lemma of Sabach and Shtern has since been employed to obtain linear rates of asymptotic regularity for the Tikhonov-Mann and modified Halpern iterations in [6] and for the alternating Halpern-Mann iteration in [16]. The following lemma is a slight reformulation of [18,Lemma 3], proved in [16,Lemma 2.8].…”