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A NOTE ON IMPLICIT ITERATION PROCESSES

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  16 May 2025

WOJCIECH M. KOZLOWSKI Open the ORCID record for WOJCIECH M. KOZLOWSKI [Opens in a new window]
WOJCIECH M. KOZLOWSKI*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
*
e-mail: w.m.kozlowski@unsw.edu.au
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Abstract

Let C be a closed, bounded, convex subset of a uniformly convex Banach space, and let $\{T_s\}$ be an asymptotic nonexpansive semigroup of nonlinear mappings acting within C. Consider the implicit iteration process defined by the sequence of equations:

$$ \begin{align*} x_{k+1} = c_k T_{s_{k+1}}(x_{k+1}) + (1 - c_k) x_k,\end{align*} $$

where each $c_k \in (0,1)$ and the initial point $x_0 \in C$ is arbitrarily chosen. In this context, we investigate the conditions under which the sequence $\{x_k\}$ converges, either weakly or strongly, to a common fixed point of the semigroup $\{T_s\}$. We also touch upon the question of the stability of such processes.

Keywords

semigroups of nonlinear operatorsfixed point theoryimplicit iteration processfixed point construction process convergence

MSC classification

Primary: 47H20: Semigroups of nonlinear operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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