Polynomiograph Comparison and Stability of a New Iteration Process

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-22-2024-136
Published: Aug 15, 2024
Cite as:
Bashir Nawaz, Kifayat Ullah, Maha Noorwali, Polynomiograph Comparison and Stability of a New Iteration Process, Int. J. Anal. Appl., 22 (2024), 136.

Main Article Content

Bashir Nawaz, Kifayat Ullah, Maha Noorwali

Abstract

In this paper, we introduce a geometric version of the F iteration process. We establish some strong and weak convergence results for our proposed iteration process in the setting of generalized contractive mappings. We also prove stability of our proposed iteration process. Additionally, we support our analysis with polynomiographs generated by our proposed iteration process, compared with those from established iteration processes in the literature, showcasing the superiority and innovation of our approach.

Article Details

References

  1. T. Zamfirescu, Fix Point Theorems in Metric Spaces, Arch. Math. 23 (1972), 292–298. https://doi.org/10.1007/bf01304884.
  2. V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007. https://doi.org/10.1007/978-3-540-72234-2.
  3. M. Osilike, Stability Results for Fixed Point Iteration Procedures, J. Nigerian Math. Soc. 14 (1995), 17–29.
  4. C.O. Imoru, M.O. Olatinwo, On the Stability of Picard and Mann Iteration Processes, Carpathian J. Math. 19 (2003), 155–160. https://www.jstor.org/stable/43996782.
  5. S. Banach, Sur les Opérations dans les Ensembles Abstaits et leur Application aux Équations Intégrales, Fund. Math. 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181.
  6. M. Abbas, T. Nazir, A New Faster Iteration Process Applied to Constrained Minimization and Feasibility Problems, Mat. Vesnik 66 (2014), 223–234. http://hdl.handle.net/2263/43663.
  7. R.P. Agarwal, D. O’Regan, D.R. Sahu, Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpasive Mappings, J. Nonlinear Convex Anal. 8 (2007), 61–79.
  8. S. Ishikawa, Fixed Points by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147–150. https://doi.org/10.1090/s0002-9939-1974-0336469-5.
  9. K. Ullah, M. Arshad, Numerical Reckoning Fixed Points for Suzuki’s Generalized Nonexpansive Mappings via New Iteration Process, Filomat 32 (2018), 187–196. https://doi.org/10.2298/fil1801187u.
  10. E. Picard, Mémoire sur la Théorie des équations aux Dérivées Partielles et la Méthode des Approximations Successives, J. Math. Pures Appl. 6 (1890), 145–210.
  11. W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. https://doi.org/10.1090/s0002-9939-1953-0054846-3.
  12. V. Kanwar, P. Sharma, I.K. Argyros, R. Behl, C. Argyros, A. Ahmadian, M. Salimi, Geometrically Constructed Family of the Simple Fixed Point Iteration Method, Mathematics 9 (2021), 694. https://doi.org/10.3390/math9060694.
  13. P. Sharma, H. Ramos, R. Behl, V. Kanwar, A New Three-Step Fixed Point Iteration Scheme With Strong Convergence and Applications, J. Comput. Appl. Math. 430 (2023), 115242. https://doi.org/10.1016/j.cam.2023.115242.
  14. J. Ali, M. Jubair, F. Ali, Stability and Convergence of F Iterative Scheme With an Application to the Fractional Differential Equation, Eng. Comput. 38 (2020), 693–702. https://doi.org/10.1007/s00366-020-01172-y.
  15. A.m. Harder, T.L. Hicks, Stability Results for Fixed Point Iteration Procedures, Math. Japon. 33 (1988), 693–706.
  16. Vasile Berinde, On the Stability of Some Fixed Point Procedures, Bul. ¸St. Univ. Baia Mare Ser. B Fasc. Math. Inf. 18 (2002), 7–14. https://www.jstor.org/stable/44001825.
  17. X. Weng, Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping, Proc. Amer. Math. Soc. 113 (1991), 727–727. https://doi.org/10.1090/s0002-9939-1991-1086345-8.
  18. V. Berinde, Picard Iteration Converges Faster Than Mann Iteration for a Class of Quasi-Contractive Operators, Fixed Point Theory Appl. 2004 (2004), 716359. https://doi.org/10.1155/s1687182004311058.
  19. B. Kalantari, Polynomiography: From the Fundamental Theorem of Algebra to Art, Leonardo 38 (2005), 233–238. https://doi.org/10.1162/0024094054029010.
  20. B. Kalantari, Art and Math via Cubic Polynomials, Polynomiography and Modulus Visualization, LASER J. 2 (2024), 1–35.
  21. I. Go´sciniak, K. Gdawiec, Control of Dynamics of the Modified Newton-Raphson Algorithm, Commun. Nonlinear Sci. Numer. Simul. 67 (2019), 76–99. https://doi.org/10.1016/j.cnsns.2018.07.010.
  22. B. Nawaz, K. Ullah, K. Gdawiec, Convergence Analysis of Picard-SP Iteration Process for Generalized αNonexpansive Mappings, Numer. Algor. (2024). https://doi.org/10.1007/s11075-024-01859-z.
  23. B. Kalantari, Polynomial Root-Finding and Polynomiography, World Scientific, Singapore, 2009. https://doi.org/10.1142/6265.
  24. K. Gdawiec, W. Kotarski, A. Lisowska, On the Robust Newton’s Method With the Mann Iteration and the Artistic Patterns From Its Dynamics, Nonlinear Dyn. 104 (2021), 297–331. https://doi.org/10.1007/s11071-021-06306-5.