# A strongly convergent Krasnosel'skivi-Mann-type algorithm for finding a common fixed point of a countably infinite family of nonexpansive operators in Hilbert spaces

arXiv: https://arxiv.org/abs/1911.11656 · HTML: https://mlsanity.com/paper/1911.11656/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Nov 2019.

## Built on by (most cited first, 1 of 1)

- [HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration](https://mlsanity.com/paper/2407.13120/) (2024)

## Abstract (arXiv)

In this article, we propose a Krasnosel'skivi-Mann-type algorithm for finding a common fixed point of a countably infinite family of nonexpansive operators (Tn)n ≥ 0 in Hilbert spaces. We formulate an asymptotic property which the family (Tn)n ≥ 0 has to fulfill such that the sequence generated by the algorithm converges strongly to the element in bigcapn ≥ 0 operatornameFix Tn with minimum norm. Based on this, we derive a forward-backward algorithm that allows variable step sizes and generates a sequence of iterates that converge strongly to the zero with minimum norm of the sum of a maximally monotone operator and a cocoercive one. We demonstrate the superiority of the forward-backward algorithm with variable step sizes over the one with constant step size by means of numerical experiments on variational image reconstruction and split feasibility problems in infinite dimensional Hilbert spaces.