On stability and convergence of a novel iterative method for fixed point problems

Gaurav Aggarwal; Aynur Şahin; Izhar Uddin; Sabiya Khatoon; Gaurav Aggarwal; Aynur Şahin; Izhar Uddin; Sabiya Khatoon

doi:10.3934/math.20251089

AIMS Mathematics

2025, Volume 10, Issue 10: 24564-24579. doi: 10.3934/math.20251089

Previous Article Next Article

Research article Special Issues

On stability and convergence of a novel iterative method for fixed point problems

1.
Department of Mathematics, Jaypee Institute of Information Technology, Noida 201309, India
2.
Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Türkiye
3.
Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India

Received: 21 August 2025 Revised: 15 October 2025 Accepted: 17 October 2025 Published: 27 October 2025
MSC : 47H10, 54H25

Regarded as a cornerstone of mathematics, the fixed-point (fp) explores invariant outcomes under defined operators, thus offering powerful tools for problems that arise in mathematics, physics, engineering, computer science, and economics. This paper presents a novel iterative method to approximate the fps of non-expansive maps in Banach spaces (BSs). We investigate the stability of the proposed method and provide its convergence analysis. A numerical example further illustrates its performance in comparison to existing iterations. Consequently, we theoretically and numerically prove that our new iterative algorithm converges faster than some leading iterative algorithms in the literature for non-expansive maps. Hence, our results generalize and improve several well-known results in the existing literature.
- non-expansive map,
- iterative methods,
- fixed point problems,
- weak stability,
- convergence rate
Citation: Gaurav Aggarwal, Aynur Şahin, Izhar Uddin, Sabiya Khatoon. On stability and convergence of a novel iterative method for fixed point problems[J]. AIMS Mathematics, 2025, 10(10): 24564-24579. doi: 10.3934/math.20251089

Related Papers:

Abstract

Regarded as a cornerstone of mathematics, the fixed-point (fp) explores invariant outcomes under defined operators, thus offering powerful tools for problems that arise in mathematics, physics, engineering, computer science, and economics. This paper presents a novel iterative method to approximate the fps of non-expansive maps in Banach spaces (BSs). We investigate the stability of the proposed method and provide its convergence analysis. A numerical example further illustrates its performance in comparison to existing iterations. Consequently, we theoretically and numerically prove that our new iterative algorithm converges faster than some leading iterative algorithms in the literature for non-expansive maps. Hence, our results generalize and improve several well-known results in the existing literature.

References

[1]	S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fundam. Math., 3 (1922), 133–181.
[2]	F. E. Browder, Fixed-point theorem for noncompact mappings in Hilbert space, Proc. Net. Acad. Sci. USA, 53 (1965), 1272–1276. https://doi.org/10.1073/pnas.53.6.1272 doi: 10.1073/pnas.53.6.1272
[3]	F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Net. Acad. Sci. USA, 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
[4]	D. Göhde, Zum prinzip der kontraktiven abbildung, Math. Nachr., 30 (1965), 251–258. https://doi.org/10.1002/mana.19650300312 doi: 10.1002/mana.19650300312
[5]	W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Am. Math. Mon., 72 (1965), 1004–1006. https://doi.org/10.2307/2313345 doi: 10.2307/2313345
[6]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.2307/2032162 doi: 10.2307/2032162
[7]	S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. https://doi.org/10.2307/2039245 doi: 10.2307/2039245
[8]	M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042 doi: 10.1006/jmaa.2000.7042
[9]	R. P. Agarwal, D. O. Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61–79.
[10]	M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik, 66 (2014), 223–234.
[11]	B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for approximating fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147–155. https://doi.org/10.1016/j.amc.2015.11.065 doi: 10.1016/j.amc.2015.11.065
[12]	K. Ullah, M. Arshad, New iteration process and numerical reckoning fixed points in Banach spaces, UPB Sci. Bull. Ser. A, 79 (2017), 113–122.
[13]	K. Ullah, M. Arshad, Numerical reckoning fixed points for Suzuki's generalized nonexpansive mappings via new iteration process, Filomat, 32 (2018), 187–196.
[14]	N. Hussain, K. Ullah, M. Arshad, Fixed point approximation for Suzuki generalized nonexpansive mappings via new iteration process, J. Nonlinear Convex Anal., 19 (2018), 1383–1393.
[15]	K. Ullah, M. Arshad, New three-step iteration process and fixed point approximation in Banach spaces, J. Linear Topol. Algebra, 7 (2018), 87–100.
[16]	H. Piri, B. Daraby, S. Rahrovi, M. Ghasemi, Approximating fixed points of generalized $\alpha$-nonexpansive mappings in Banach spaces by new faster iteration process, Numer. Algor., 81 (2019), 1129–1148. https://doi.org/10.1007/s11075-018-0588-x doi: 10.1007/s11075-018-0588-x
[17]	C. Garodia, I. Uddin, A new iterative method for solving split feasibility problem, J. Appl. Anal. Comput., 10 (2020), 986–1004. https://doi.org/10.11948/20190179 doi: 10.11948/20190179
[18]	A. Şahin, E. Öztürk, G. Aggarwal, Some fixed-point results for the $KF$-iteration process in hyperbolic metric spaces, Symmetry, 15 (2023), 1360. https://doi.org/10.3390/sym15071360 doi: 10.3390/sym15071360
[19]	A. Şahin, Z. Kalkan, The $AA$-iterative algorithm in hyperbolic spaces with applications to integral equations on time scales, AIMS Mathematics, 9 (2024), 24480–24506. https://doi.org/10.3934/math.20241192 doi: 10.3934/math.20241192
[20]	Y. Saad, Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics, 2003. https://doi.org/10.1137/1.9780898718003
[21]	A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89–97. https://doi.org/10.1023/B:JMIV.0000011325.36760.1e doi: 10.1023/B:JMIV.0000011325.36760.1e
[22]	D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, In: The 3rd international conference for learning representations, San Diego, 2015.
[23]	Ç. Dinçkal, New predictor-corrector type iterative methods for solving nonlinear equations, Sakarya Univ. J. Sci., 21 (2017), 463–468. https://doi.org/10.16984/saufenbilder.275466 doi: 10.16984/saufenbilder.275466
[24]	A. Şahin, Z. Kalkan, H. Arısoy, On the solution of a nonlinear Volterra integral equation with delay, Sakarya Univ. J. Sci., 21 (2017), 1367–1376. https://doi.org/10.16984/saufenbilder.305632 doi: 10.16984/saufenbilder.305632
[25]	E. Köse, A. Mühürcü, G. Mühürcü, M. Özdemir, PI parameter optimization by fairly algorithm for optimal controlling of a buck converter's output state variable, Sakarya Univ. J. Sci., 22 (2018), 1267–1273. https://doi.org/10.16984/saufenbilder.327296 doi: 10.16984/saufenbilder.327296
[26]	M. Gürcan, N. Halisdemir, Y. Güral, Modelling the effect size of microbial fuel cells using Bernstein polynomial approach via iterative method, Sakarya Univ. J. Sci., 25 (2021), 28–35. https://doi.org/10.16984/saufenbilder.641591 doi: 10.16984/saufenbilder.641591
[27]	N. G. Adar, Real time control application of the robotic arm using neural network based inverse kinematics solution, Sakarya Univ. J. Sci., 25 (2021), 849–857. https://doi.org/10.16984/saufenbilder.907312 doi: 10.16984/saufenbilder.907312
[28]	M. Saif, B. Almarri, M. Aljuaid, I. Uddin, Numerical solutions of boundary value problems via fixed point iteration, Comp. Appl. Math., 43 (2024), 440. https://doi.org/10.1007/s40314-024-02933-x doi: 10.1007/s40314-024-02933-x
[29]	B. Almarri, M. Saif, M. Akram, I. Uddin, An effective fixed point approach based on Green's function for solving BVPs, J. Nonlinear Convex Anal., 25 (2024), 2881–2891.
[30]	J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396–414. https://doi.org/10.2307/1989630
[31]	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 doi: 10.1016/0362-546X(91)90200-K
[32]	J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 43 (1991), 153–159. https://doi.org/10.1017/S0004972700028884 doi: 10.1017/S0004972700028884
[33]	R. P. Agarwal, D. O'Regan, D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, New York: Springer, 2009. http://dx.doi.org/10.1007/978-0-387-75818-3
[34]	T. M. Gallagher, The demiclosedness principle for mean nonexpansive mappings, J. Math. Anal. Appl., 439 (2016), 832–842. http://dx.doi.org/10.1016/j.jmaa.2016.03.029 doi: 10.1016/j.jmaa.2016.03.029
[35]	Y. Dong, Comments on 'the proximal point algorithm revisited', J. Optim. Theory Appl., 166 (2015), 343–349. http://dx.doi.org/10.1007/s10957-014-0685-5 doi: 10.1007/s10957-014-0685-5
[36]	A. M. Harder, T. L. Hicks, Stability result for fixed point iteration procedures, Math. Japonica, 33 (1988), 693–706.
[37]	T. Cardinali, P. Rubbioni, A generalization of the Cariti fixed point theorem in metric space, Fixed Point Theory, 11 (2010), 3–10.
[38]	I. Timiş, On the weak stability of Picard iteration for some contractive type mappings, Annal. Univ. Craiova Mat., 37 (2010), 106–114.
[39]	B. S. Thakur, D. Thakur, M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat, 30 (2016), 2711–2720. https://doi.org/10.2298/FIL1610711T doi: 10.2298/FIL1610711T
[40]	H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380. https://doi.org/10.2307/2040440 doi: 10.2307/2040440
[41]	S. Khatoon, I. Uddin, M. Başarır, A modified proximal point algorithm for a nearly asymptotically quasi-nonexpansive mapping with an application, Comp. Appl. Math., 40 (2021), 250. https://doi.org/10.1007/s40314-021-01646-9 doi: 10.1007/s40314-021-01646-9
[42]	S. Temir, Ö. Korkut, Approximating fixed points of generalized $\alpha$-nonexpansive mappings by the new iteration process, J. Math. Sci. Model., 5 (2022), 35–39. https://doi.org/10.33187/jmsm.993823 doi: 10.33187/jmsm.993823
[43]	Z. Kalkan, A. Şahin, A. Aloqaily, N. Mlaiki, Some fixed point and stability results in $b$-metric-like spaces with an application to integral equations on time scales, AIMS Mathematics, 9 (2024), 11335–11351. https://doi.org/10.3934/math.2024556 doi: 10.3934/math.2024556
[44]	A. Arslanhan, M. Başarır, A. Şahin, A. Arslanhan, On the $F$-iterative method for the class of maps satisfying enriched condition $(C\gamma)$ in Banach spaces, J. Nonlinear Convex Anal., 26 (2025), 767–780.
[45]	U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Am. Math. Soc., 357 (2004), 89–128.

Reader Comments

Your name:*

Email:*
© 2025 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)