Notes on convergence and stability of iteration processes

Wojciech M. Kozlowski; Wojciech M. Kozlowski

doi:10.3934/math.20261156

AIMS Mathematics

2026, Volume 11, Issue 9: 29062-29105. doi: 10.3934/math.20261156

Previous Article Next Article

Research article Special Issues

Notes on convergence and stability of iteration processes

Wojciech M. Kozlowski ^,

School of Mathematics and Statistics, University of New South Wales, Sydney 2052, NSW, Australia

Received: 07 July 2026 Revised: 21 August 2026 Accepted: 01 September 2026 Published: 09 September 2026
MSC : 47H10, 47H20, 47J26, 46B20, 46A80

The theory developed in this paper defines a mathematical technology for automating residually stable convergence of iterative processes to common fixed points of semigroups of nonlinear operators. Our ultimate aim is to simplify future research on convergence and stability across a broad range of scenarios. By 'automation', we refer to the availability of a framework based on a single property of a given Banach space and a single theorem that enables conclusions to be drawn regarding stable convergence for a wide range of commonly used iterative processes. Such a novel framework is developed in this paper. This is exemplified by the direct application of our results to convergence in measure, convergence in Marcinkiewicz quasi-norm, and convergence in Lorentz quasi-norm within spaces such as Lebesgue spaces, Sobolev spaces, Orlicz spaces, Musielak-Orlicz spaces, variable exponent Lebesgue spaces, and other spaces constructed upon them. These specific outcomes, ready for use, are novel results in their own right.
- fixed point,
- semigroup of nonlinear operators,
- iteration process,
- process convergence,
- process stability
Citation: Wojciech M. Kozlowski. Notes on convergence and stability of iteration processes[J]. AIMS Mathematics, 2026, 11(9): 29062-29105. doi: 10.3934/math.20261156

Related Papers:

Abstract

The theory developed in this paper defines a mathematical technology for automating residually stable convergence of iterative processes to common fixed points of semigroups of nonlinear operators. Our ultimate aim is to simplify future research on convergence and stability across a broad range of scenarios. By 'automation', we refer to the availability of a framework based on a single property of a given Banach space and a single theorem that enables conclusions to be drawn regarding stable convergence for a wide range of commonly used iterative processes. Such a novel framework is developed in this paper. This is exemplified by the direct application of our results to convergence in measure, convergence in Marcinkiewicz quasi-norm, and convergence in Lorentz quasi-norm within spaces such as Lebesgue spaces, Sobolev spaces, Orlicz spaces, Musielak-Orlicz spaces, variable exponent Lebesgue spaces, and other spaces constructed upon them. These specific outcomes, ready for use, are novel results in their own right.

References

[1]	W. M. Kozlowski, Fixed points of semigroups of pointwise Lipschitzian operators: A nonexpansive and asymptotic approach, Springer Cham, 2026. https://doi.org/10.1007/978-3-032-08869-7
[2]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597.
[3]	M. Budzyńska, T. Kuczumow, M. Michalska, The $\Gamma$-opial property, Bull. Aust. Math. Soc., 73 (2006), 473–476. https://doi.org/10.1017/S0004972700035474 doi: 10.1017/S0004972700035474
[4]	M. A. Japon, Some geometric properties in modular spaces and application to fixed point theory, J. Math. Anal. Appl., 295 (2004), 576–594. https://doi.org/10.1016/j.jmaa.2004.02.047 doi: 10.1016/j.jmaa.2004.02.047
[5]	T. D. Benavides, M. A. Japon, Opial modulus, moduli of noncompact convexity and fixed points for asymptotically regular mappings, Nonlinear Anal.-Theor., 41 (2000), 617–630. https://doi.org/10.1016/S0362-546X(98)00300-9 doi: 10.1016/S0362-546X(98)00300-9
[6]	T. D. Benavides, M. A. Japon, Fixed point theorems for asymptotically regular mappings in Orlicz function spaces, Nonlinear Anal.-Theor., 44 (2001), 829–842. https://doi.org/10.1016/S0362-546X(99)00313-2 doi: 10.1016/S0362-546X(99)00313-2
[7]	T. D. Benavides, M. A Khamsi, S. Samadi, Asymptotically nonexpansive mappings in modular function spaces, J. Math. Anal. Appl., 265 (2002), 249–263. https://doi.org/10.1006/jmaa.2000.7275 doi: 10.1006/jmaa.2000.7275
[8]	H. Hudzik, M. Mastylo, Strongly extreme points in Köthe–Bochner spaces, Rocky Mountain J. Math. Anal., 23 (1993), 899–909. https://doi.org/10.1216/rmjm/1181072531 doi: 10.1216/rmjm/1181072531
[9]	T. D. Benavides, , H. Hudzik, G. Lopez, M. Mastylo, B. Sims, Complete characterizations of Kadec-Klee properties in Orlicz spaces, Houston J. Math., 29 (2003), 1027–1044.
[10]	G. Sadeghi, R. Saadat, On geometrical properties of noncommutative modular function spaces, New York J. Math., 21 (2015), 759–781.
[11]	W. M. Kozlowski, Modular function spaces, New York/Basel: Dekker, 1988.
[12]	J. Musielak, Orlicz spaces and modular spaces, Berlin, Heidelberg: Springer, 1983. https://doi.org/10.1007/BFb0072210
[13]	M. A. Khamsi, W. M. Kozlowski, Fixed point theory in modular function spaces, Birkhäuser Cham, 2015. https://doi.org/10.1007/978-3-319-14051-3
[14]	S. Reich, A. J. Zaslavski, Convergence of inexact iterates of uniformly locally nonexpansive mappings with summable errors, JP J. Fixed Point Theory Appl., 18 (2022), 1–11. http://doi.org/10.17654/0973422822003 doi: 10.17654/0973422822003
[15]	W. M. Kozlowski, On stability of iteration processes convergent to stationary points of semigroups of nonlinear operators in metric spaces, Optimization, 72 (2025), 4613–4626. https://doi.org/10.1080/02331934.2024.2410259 doi: 10.1080/02331934.2024.2410259
[16]	W. M. Kozlowski, Stability of iteration processes weakly convergent to stationary points of semigroups of nonlinear operators, Rend. Circ. Mat. Palermo, II. Ser, 74 (2025), 51.
[17]	K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. Math. Soc., 35 (1972), 171–174. https://doi.org/10.2307/2038462 doi: 10.2307/2038462
[18]	W. A. Kirk, H. K. Xu, Asymptotic pointwise contractions, Nonlinear Anal.-Theor., 69 (2008), 4706–4712. https://doi.org/10.1016/j.na.2007.11.023 doi: 10.1016/j.na.2007.11.023
[19]	W. M. Kozlowski, Common fixed points for semigroups of pointwise Lipschitzian mappings in Banach spaces, Bull. Aust. Math. Soc., 84 (2011), 353–361. https://doi.org/10.1017/S0004972711002668 doi: 10.1017/S0004972711002668
[20]	S. T. Chen, Geometry of Orlicz spaces, Warszawa: Polish Scientific Publisher, 1996.
[21]	A. Kamińska, On uniform convexity of Orlicz spaces, Indagat. Math., 85 (1982), 27–36. https://doi.org/10.1016/1385-7258(82)90005-1 doi: 10.1016/1385-7258(82)90005-1
[22]	W. M. Kozlowski, Notes on modular projections, Appl. Set-Valued Anal. Optim., 4 (2022), 337–348. https://doi.org/10.23952/asvao.4.2022.3.07 doi: 10.23952/asvao.4.2022.3.07
[23]	J. Kisyński, Convergence du type L, Colloq. Math., 7 (1960), 205–211.
[24]	R. M. Dudley, On sequential convergence, T. Am. Math. Soc., 112 (1964), 483–507. https://doi.org/10.2307/1994157
[25]	W. M. Kozlowski, On modular approximants in sequential convergence spaces, J. Approx. Theory, 264 (2021), 105535. https://doi.org/10.1016/j.jat.2020.105535 doi: 10.1016/j.jat.2020.105535
[26]	M. A. Krasnosel'skii, Y. B. Rutickii, Convex functions and Orlicz spaces, Groningen, Holland: P. Nordhoff Ltd, 1961.
[27]	M. M. Rao, Z. D. Ren, Theory of Orlicz spaces, New York/Basel: Dekker, 1991.
[28]	S. Goldstein, L. Labuschange, Noncommutative measures and $L^p$ and Orlicz spaces, with applications to quantum physics, Oxford, UK: Oxford University Press, 2025. https://doi.org/10.1093/oso/9780198950202.001.0001
[29]	D. V. Cruz-Uribe, A. Fiorenza, Convergence in variable Lebesgue spaces, Publ. Mat., 54 (2010), 441–459. https://doi.org/10.5565/PUBLMAT_54210_08 doi: 10.5565/PUBLMAT_54210_08
[30]	D. V. Cruz-Uribe, A. Fiorenza, Variable Lebesgu spaces: Foundations and harmonic analysis, Basel: Birkhäuser, 2013. https://doi.org/10.1007/978-3-0348-0548-3
[31]	D. Cruz-Uribe, A. Fiorenza, M. Ruzhansky, J. Wirth, Variable Lebesgue spaces and hyperbolic systems, Birkhäuser Basel, 2014. https://doi.org/10.1007/978-3-0348-0840-8
[32]	H. Hudzik, The problems of separability, duality, reflexivity and of comparison for generalized Orlicz-Sobolev spaces $W^k_M(\Omega$), Comment. Math. Prace Mat., 21 (1980), 315–324.
[33]	A. Kamińska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr., 147 (1990), 29–38. https://doi.org/10.1002/mana.19901470104 doi: 10.1002/mana.19901470104
[34]	H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal.-Theor., 16 (1991) 1127–1138. https://doi.org/10.1016/0362-546X(91)90200-K
[35]	W. M. Kozlowski, B. Sims, On the convergence of iteration processes for semigroups of nonlinear mappings in Banach spaces, In: Computational and analytical mathematics, New York: Springer, 2013,463–484. https://doi.org/10.1007/978-1-4614-7621-4_22
[36]	W. M. Kozlowski, On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces, Comment. Math., 52 (2012), 113–136.
[37]	H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983) 486–490. https://dx.doi.org/10.1090/s0002-9939-1983-0699419-3
[38]	A. M. Ostrowski, The round-off stability of iterations, Z. Angew. Math. Mech., 47 (1967), 77–81. https://doi.org/10.1002/zamm.19670470202 doi: 10.1002/zamm.19670470202
[39]	D. Butnariu, S. Reich, A. J. Zaslavski, Convergence to fixed points of inexact orbits of Bregman-Monotone and of nonexpansive operators in Banach spaces, Fixed Point Theor. Appl., 2006, 11–32.
[40]	D. Butnariu, S. Reich, and A.J. Zaslavski, Asymptotic behavior of inexact orbits for a class of operators in complete metric spaces, J. Appl. Anal., 13 (2007), 1–11.
[41]	S. Reich, A. J. Zaslavski, Three convergence results for inexact iterates of uniformly locally nonexpansive mappings, Symmetry, 15 (2023), 1084. https://doi.org/10.3390/sym15051084 doi: 10.3390/sym15051084
[42]	A. J. Zaslavski, Two convergence results for inexact orbits of nonexpansive operators in metric spaces with graphs, Axioms, 12 (2023), 999. https://doi.org/10.3390/axioms12100999 doi: 10.3390/axioms12100999
[43]	A. Taiwo, S. Reich, Bounded perturbation resilience of a regularized forward-reflected-backward splitting method for solving variational inclusion problems with applications, Optimization, 73 (2024), 2089–2122. https://doi.org/10.1080/02331934.2023.2187664 doi: 10.1080/02331934.2023.2187664
[44]	A. J. Zaslavski, Approximate solutions of common fixed point problems, Springer Cham, 2016. https://doi.org/10.1007/978-3-319-33255-0
[45]	A. J. Zaslavski, Algorithms for solving common fixed point problems, Springer Cham, 2018. https://doi.org/10.1007/978-3-319-77437-4
[46]	A. J. Zaslavski, Solutions of fixed point problems with computational errors, Springer Cham, 2024. https://doi.org/10.1007/978-3-031-50879-0
[47]	A. J. Zaslavski, The Krasnoselskii-Mann method for common fixed point problems, Springer Cham, 2025. https://doi.org/10.1007/978-3-031-85841-3
[48]	W. M. Kozlowski, Convergence of implicit iterative processes for semigroups of nonlinear operators acting in regular modular space, Mathematics, 12 (2024), 4007. https://doi.org/10.3390/math12244007 doi: 10.3390/math12244007
[49]	W. M. Kozlowski, A note on implicit iteration processes, Bull. Aust. Math. Soc., 113 (2026), 161–172. https://doi.org/10.1017/S0004972725000292 doi: 10.1017/S0004972725000292
[50]	Y. Li, Z. Rui, B. Hu, Monotone iterative and quasilinearization method for a nonlinear integral impulsive differential equation, AIMS Mathematics, 10 (2025), 21–37. https://doi.org/10.3934/math.2025002 doi: 10.3934/math.2025002
[51]	B. Hu, Y. Liao, Convergence conditions for extreme solutions of an impulsive differential system, AIMS Mathematics, 10 (2025), 10591–10604. https://doi.org/10.3934/math.2025481 doi: 10.3934/math.2025481
[52]	H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. U.S.A, 56 (1966), 1907–1911. https://doi.org/10.1073/pnas.56.6.1907 doi: 10.1073/pnas.56.6.1907
[53]	V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, Cambridge, UK, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511760303
[54]	V. N. Kolokoltsov, The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups, Probab. Theory Relat. Fields, 151 (2011), 95–123. https://doi.org/10.1007/s00440-010-0293-8 doi: 10.1007/s00440-010-0293-8
[55]	R. Carmona, F. Delarue, Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics, Ann. Probab., 43 (2015), 2647–2700. https://doi.org/10.1214/14-AOP946 doi: 10.1214/14-AOP946
[56]	H. Pham, X. Wei, Dynamic programming for optimal control of stochastic McKean–Vlasov dynamics, SIAM J. Control Optim., 55 (2017), 1069–1101. https://doi.org/10.1137/16M1071390 doi: 10.1137/16M1071390
[57]	B. A. Neumann, Nonlinear Markov chains with finite state space: Invariant distributions and long-term behaviour, J. Appl. Probab., 60 (2023), 30–44. 10.1017/jpr.2022.23 doi: 10.1017/jpr.2022.23
[58]	W. A. Majewski, L.E. Labuschange, On applications of Orlicz spaces to Statistical Physics, Ann. Henri Poincare, 15 (2014), 1197–1221. https://doi.org/10.1007/s00023-013-0267-3 doi: 10.1007/s00023-013-0267-3
[59]	L. E. Labuschange, W. A. Majewski, Dynamics on noncommutative Orlicz spaces, Acta Math. Sci., 40 (2020), 1249–1270. https://doi.org/10.1007/s10473-020-0507-9 doi: 10.1007/s10473-020-0507-9
[60]	L. Arens, M. Gritl, On local well-posedness for the nonlinear Schrödinger equation with general power nonlinearity, 2025, arXiv: 2506.10595. https://doi.org/10.48550/arXiv.2506.10595

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)