An extension of high-order Kou’s method for solving nonlinear systems and its stability analysis

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, Kou’s method is extended to solve nonlinear systems. The convergence order of the iterative method is proved. Using fractal theory, we study the theoretical operators related to the iterative method, and analyze the stability of the iterative method. Properties related to strange fixed points and critical points are explored. The fractal results indicate that the iterative method is most stable when the parameter γ equals zero. The extended iterative method is applied to solve the Hammerstein equation and some nonlinear systems. The dynamic plane and numerical experiments show that the extended iterative method can solve the nonlinear system of equations with good convergence and stability.

Keywords

nonlinear systems / iterative method / stability

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] C. Li, H. Zhang, X. Yang, A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK, Comput. Math. Appl., 176 (2024), 349-370. https://doi.org/10.1016/j.camwa.2024.11.006

[2] H. Zhang, X. Jiang, F. Wang, X. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., 70 (2024), 1127-1151. https://doi.org/10.1007/s12190-024-02000-y

[3] X. Yang, Z. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972

[4] Z. Chen, H. Zhang, H. Chen, Adi compact difference scheme for the two-dimensional integro-differential equation with two fractional Riemann-Liouville integral kernels, Fractal Fract., 8 (2024), 707. https://doi.org/10.3390/fractalfract8120707

[5] J. Wang, X. Jiang, X. Yang, H. Zhang, A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto-Sivashinsky equation, Comput. Appl. Math., 43 (2024), 381. https://doi.org/10.1007/s40314-024-02883-4

[6] X. Shen, X. Yang, H. Zhang, The high-order ADI difference method and extrapolation method for solving the two-dimensional nonlinear parabolic evolution equations, Mathematics, 12 (2024), 3469. https://doi.org/10.3390/math12223469

[7] X. Yang, W. Wang, Z. Zhou, H. Zhang, An efficient compact difference method for the fourth-order nonlocal subdiffusion problem, Taiwan. J. Math., 29 (2025), 35-66. https://doi.org/10.11650/tjm/240906

[8] X. Yang, Z. Zhang, Superconvergence analysis of a robust orthogonal Gauss collocation method for 2D fourth-order subdiffusion equations, J. Sci. Comput., 100 (2024). https://doi.org/10.1007/s10915-024-02616-z

[9] N. Sidorov, Solvability of Nonlinear equations with parameters: Branching, regularization, group symmetry and solutions blow-up, Symmetry, 14 (2022), 226. https://doi.org/10.3390/sym14020226

[10] Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Levy processes, IEEE Trans. Autom. Control, 70 (2025), 1176-1183. http://doi.org/10.1109/TAC.2024.3448128

[11] X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7

[12] J. Wang, X. Jiang, X. Yang, H. Zhang, A compact difference scheme for mixed-type time-fractional black-scholes equation in European option pricing, Math. Methods Appl. Sci., 2025. https://doi.org/10.1002/mma.10717

[13] K. R. Nilay, A new semi-explicit atomistic molecular dynamics simulation method for membrane proteins, J. Comput. Methods Sci. Eng., 19 (2019), 259-286. https://doi.org/10.3233/jcm-180851

[14] X. Wang, N. Shang, Local convergence analysis of a novel derivative-free method with and without memory for solving nonlinear systems, Int. J. Comput. Math., (2025), 1-18. http://doi.org/10.1080/00207160.2025.2464701

[15] D. Ruan, X. Wang, Y. Wang, Local convergence of seventh-order iterative method under weak conditions and its applications, Eng. Comput., 2025. http://doi.org/10.1108/EC-08-2024-0775

[16] D. Ruan, X. Wang, A high-order Chebyshev-type method for solving nonlinear equations: local convergence and applications, Electron. Res. Arch., 33 (2025), 1398-1413. https://doi.org/10.3934/era.2025065

[17] X. Wang, W. Li, Fractal behavior of King’s optimal eighth-order iterative method and its numerical application, Math. Comm., 29 (2024), 217-236.

[18] X. Wang, T. Zhang, Y. Qin, Efficient two-step derivative-free iterative methods with memory and their dynamics, Int. J. Comput. Math., 93 (2015), 1423-1446. http://doi.org/10.1080/00207160.2015.1056168

[19] A. Cordero, J. R. Torregrosa, M. P. Vassileva, Three-step iterative methods with optimal eighth-order convergence, J. Comput. Appl. Math., 235 (2011), 3189-3194. http://doi.org/10.1016/j.cam.2011.01.004

[20] A. Cordero, M. Moscoso-Martinez, J. R. Torregrosa, Chaos and stability in a new iterative family for solving nonlinear equations, Algorithms, 14 (2021), 101. http://doi.org/10.3390/a14040101

[21] J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM, 52 (2000), 521-572. https://doi.org/10.1137/1.9780898719468.bm

[22] F. A. Potra, V. Ptak, Nondiscrete induction and iterative processes. SIAM Rev., 1984. https://doi.org/10.1137/1029105

[23] A. Cordero, J. R. Torregrosa, On interpolation variants of Newton’s method for functions of several variables, J. Comput. Appl. Math., 234 (2010), 34-43. https://doi.org/10.1016/j.cam.2009.12.002

[24] J. Kou, Y. Li, X. Wang, A composite fourth-order iterative method for solving non-linear equations, Appl. Math. Comput., 184 (2007), 471-475. http://doi.org/10.1016/j.amc.2006.05.181

[25] K. Falconer, Fractal geometry: mathematical foundations and applications, Wiley, 2013.

[26] B. B. Mandelbrot, Fractal geometry: what is it, and what does it do?, Proc. R. Soc. Lond. A Math. Phys. Sci., 423 (1989), 3-16. https://doi.org/10.1098/rspa.1989.0038

[27] M. Y. Lee, Y. I. Kim, The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere, Numer. Algorithms, 83 (2020), 1063-1090. https://doi.org/10.1007/s11075-019-00716-8

[28] X. Wang, W. Li, Choosing the best members of the optimal eighth-order Petkovic’s family by its fractal behavior, Fractal Fract., 6 (2022), 749. http://doi.org/10.3390/fractalfract6120749

[29] S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl., 366 (2010), 24-32. https://doi.org/10.1016/j.jmaa.2010.01.047

[30] A. Cordero, T. Lotfi, K. Mahdiani, J. R. Torregrosa, A stable family with high order of convergence for solving nonlinear equations, Appl. Numer. Math., 254 (2015), 240-251. https://doi.org/10.1016/j.amc.2014.12.141

[31] H. H. H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math., 176 (2005), 425-432. https://doi.org/10.1016/j.cam.2004.07.027

[32] C. Chun, Some third-order families of iterative methods for solving nonlinear equations, Appl. Math. Comput., 188 (2007), 924-933. https://doi.org/10.1016/j.amc.2006.10.043

[33] A. Cordero, J. R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas, Appl. Math. Comput., 190 (2007), 686-698. https://doi.org/10.1016/j.amc.2007.01.062

[34] J. Kou, On Chebyshev-Halley methods with sixth-order convergence for solving non-linear equations, Appl. Math. Comput., 190 (2007), 126-131. https://doi.org/10.1016/j.amc.2007.01.011

[35] Y. I. Kim, R. Behl, S. S. Motsa, Higher-order efficient class of Chebyshev-Halley-type methods, Appl. Math. Comput., 273 (2016), 1148-1159. https://doi.org/10.1016/j.amc.2015.09.013

[36] S. Qureshi, A. Soomro, A. A. Shaikh, E. Hincal, N. Gokbulut, A novel multistep iterative technique for models in medical sciences with complex dynamics, Comput. Math. Methods Med., 2022 (2022), 7656451. https://doi.org/10.1155/2022%2F7656451

[37] J. R. Sharma, S. Kumar, H. Singh, A new class of derivative-free root solvers with increasing optimal convergence order and their complex dynamics, SeMA J., 80 (2023), 333-352. https://doi.org/10.1007/s40324-022-00288-z

[38] O. S. Solaiman, I. Hashim, Two new efficient sixth order iterative methods for solving nonlinear equations, J. King Saud Univ. Sci., 31 (2019), 701-705. https://doi.org/10.1016/j.jksus.2018.03.021

[39] J. Kou, Y. Li, X. Wang, A modification of Newton method with third-order convergence, Appl. Numer. Math., 181 (2019), 1106-1111. https://doi.org/10.1016/j.amc.2006.01.076