arXiv subjects

Horatiu Cheval

Publications and source records attributed to Horatiu Cheval.

3 recordsLinked to original sources

arXiv2024

Quantitative metastability of the Tikhonov-Mann iteration for countable families of mappings

In this paper, we obtain rates of metastability for the Tikhonov-Mann iteration for countable families of mappings in CAT(0) spaces. This iteration was recently defined by the author in the setting of W-hyperbolic spaces as a generalization of the strongly convergent version of the Krasnoselskii-Mann iteration introduced by Bot and Meier for finding common fixed points of families of nonexpansive mappings in Hilbert spaces, and as an extension of the Tikhonov-Mann iteration for single mappings, for which Leustean and the author obtained rates of asymptotic regularity in W-hyperbolic spaces.

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arXiv2023

On modified Halpern and Tikhonov-Mann iterations

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 Boţ et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Boţ 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. We also obtain rates of asymptotic regularity of order $O(1/n)$ for both the modified Halpern (and so in particular for the Halpern iteration) and the Tikhonov-Mann iteration in a general geodesic setting for a special choice of scalars.

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arXiv2022

Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration

In this paper, we compute quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration in W-hyperbolic spaces. This iteration is an extension to a nonlinear setting of the modified Mann iteration defined recently by Bot, Csetnek and Meier in Hilbert spaces. Furthermore, we show that the Douglas-Rachfors and forward-backward algorithms with Tikhonov regularization terms are special cases, in Hilbert spaces, of our Tikhonov-Mann iteration.

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