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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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References

Bruck, R. E., Kuczumow, T. and Reich, S., ‘Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property’, Colloq. Math. 65(2) (1993), 169–179.CrossRefGoogle Scholar
Goebel, K. and Kirk, W. A., ‘A fixed point theorem for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 35(1) (1972), 171–174.CrossRefGoogle Scholar
Kim, G. E. and Takahashi, W., ‘Approximating common fixed points of nonexpansive semigroups in Banach spaces’, Sci. Math. Jpn. 63 (2006), 31–36.Google Scholar
Kozlowski, W. M., ‘Common fixed points for semigroups of pointwise Lipschitzian mappings in Banach spaces’, Bull. Aust. Math. Soc. 84 (2011), 353–361.CrossRefGoogle Scholar
Kozlowski, W. M., ‘On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces’, Comment. Math. 52(2) (2012), 113–136.Google Scholar
Kozlowski, W. M., ‘Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces’, Comment. Math. 54(2) (2014), 203–208.Google Scholar
Kozlowski, W. M., ‘On convergence of iteration processes for nonexpansive semigroups in uniformly convex and uniformly smooth Banach spaces’, J. Math. Anal. Appl. 426 (2015), 1182–1191.CrossRefGoogle Scholar
Kozlowski, W. M., ‘On stability of iteration processes convergent to stationary points of semigroups of nonlinear operators in metric spaces’, Optimization, to appear. Published online (7 October 2024), 14 pages.CrossRefGoogle Scholar
Kozlowski, W. M., ‘Convergence of implicit iterative processes for semigroups of nonlinear operators acting in regular modular spaces’, Mathematics 12(24) (2024), Article no. 4007.CrossRefGoogle Scholar
Kozlowski, W. M., ‘Stability of iteration processes weakly convergent to stationary points of semigroups of nonlinear operators’, Rend. Circ. Mat. Palermo II 74 (2025), Article no. 51.Google Scholar
Kozlowski, W. M. and Sims, B., ‘On the convergence of iteration processes for semigroups of nonlinear mappings in Banach spaces’, in: Computational and Analytical Mathematics, Springer Proceedings in Mathematics and Statistics, 50 (eds. Bailey, D. H., Bauschke, H. H., Borwein, P. B., Garvan, F., Théra, M., Vanderwerff, J. D. and Wolkowicz, H.) (Springer, New York, 2013), 463–484.CrossRefGoogle Scholar
Opial, Z., ‘Weak convergence of the sequence of successive approximations for nonexpansive mappings’, Bull. Amer. Math. Soc. (N.S.) 73 (1967), 591–597.CrossRefGoogle Scholar
Saejung, S., ‘Strong convergence theorems for nonexpansive semigroups without Bochner integrals’, Fixed Point Theory Appl. 2008 (2008), Article no. 745010.CrossRefGoogle Scholar
Schu, J., ‘Weak and strong convergence to fixed points of asymptotically nonexpansive mappings’, Bull. Aust. Math. Soc. 43 (1991), 153–159.CrossRefGoogle Scholar
Suzuki, T., ‘On strong convergence to common fixed points of nonexpansive mappings in Hilbert spaces’, Proc. Amer. Math. Soc. 131(7) (2002), 2133–2136.CrossRefGoogle Scholar
Suzuki, T., ‘Strong convergence of Krasnoselskii and Mann type sequences for one-parameter nonexpansive semigroups without Bochner integrals’, J. Math. Anal. Appl. 305(1) (2005), 227–239.CrossRefGoogle Scholar
Thong, D. V., ‘An implicit iteration process for nonexpansive semigroups’, Nonlinear Anal. 74 (2011), 6116–6120.CrossRefGoogle Scholar
Xu, H.-K., ‘A strong convergence theorem for contraction semigroups in Banach spaces’, Bull. Aust. Math. Soc. 72 (2005), 371–379.CrossRefGoogle Scholar
Zaslavski, A. J., ‘Two convergence results for inexact orbits of nonexpansive operators in metric spaces with graphs’, Axioms 12(10) (2023), Article no. 999.CrossRefGoogle Scholar
Zaslavski, A. J., Solutions of Fixed Point Problems with Computational Errors, Springer Optimization and Its Applications, 210 (Springer, Cham, 2024).CrossRefGoogle Scholar
Zeidler, E., Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems (Springer, New York, 1986).CrossRefGoogle Scholar