SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2023 / Volume: 52 Issue: 6

Algorithm analysis of solving fixed point of nonexpansive mappings based on runge-kutta method

Xiangyao Li0000-0002-0877-7995
Li-jun Zhu(North Minzu University)0000-0001-8883-5034
Mihai Postolache(University Politehnica of Bucharest)0000-0003-0738-787X
Pages
—
Publication Date
2023-11-03
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We show that the second algorithm is bounded perturbations resilient, and it is strongly convergent.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In order to solve the fixed point of nonexpansive mappings, we propose two iterative algorithms based on runge-kutta method.
📊
Key Findings & Data:The first algorithm is focused on solving the fixed point problem of a single nonexpansive mapping, and weak convergence has been proved. we suggest the second algorithm by dynamic string-averaging rule.
💡
Conclusion & Impact:We show that the second algorithm is bounded perturbations resilient, and it is strongly convergent.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:nonexpansive mapping · common fixed point · runge-kutta method

Abstract

In order to solve the fixed point of nonexpansive mappings, we propose two iterative algorithms based on runge-kutta method. The first algorithm is focused on solving the fixed point problem of a single nonexpansive mapping, and weak convergence has been proved. we suggest the second algorithm by dynamic string-averaging rule. It can be used to find a common fixed point of a family of finite nonexpansive mappings. We show that the second algorithm is bounded perturbations resilient, and it is strongly convergent.

🔖

Cite This Article

Li, X., Zhu, L., Postolache, M. (2023). Algorithm analysis of solving fixed point of nonexpansive mappings based on runge-kutta method. Hacettepe Journal of Mathematics and Statistics, Volume: 52 Issue: 6. https://ajindex.com/articles/1402149

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (20)

  1. M. A. Alghamdi, M. A. Alghamdi, N. Shahzad and H. K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory and Applications, 96, 1-9, 2014.
  2. L. C. Ceng, D. R. Sahu and J. C. Yao, Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings, J. Comput. Appl. 11, 2902-2915, 2010.
  3. L. C. Ceng, N. C. Wong and J. C. Yao, Strong and weak convergence theorems for an infinite family of nonexpansive mappings and applications, Fixed Point Theory and Applications, 117, 2012.
  4. Y. Censor, R. Davidi and G. T. Herman, Perturbation resilience and superiorization of iterative algorithms, Inverse Probl. 26, 065008, 2010.
  5. Y. Censor and A. J. Zaslavski, Convergence and perturbation resilience of dynamic string-averaging projection methods, Comput. Optim. Appl. 54, 65-76, 2013.
  6. P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev. 27, 505-535, 1985.
  7. B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73, 957-961, 1967.
  8. L. He, L. J. Zhu and Y. M. Fu, An iterative algorithm based on simpson methods for solving fixed point problem of nonexpansive mappings, UPB Sci. Bull. A: Appl. Math. Phys. 83, 13-20, 2021.
  9. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67, 274-276, 1979.
  10. H. K. Xun and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22, 767-773, 2001.
  11. Y. H. Yao, Y. C. Liou, T. L. Lee and N. C. Wong, An iterative algorithm based on the implicit midpoint rule for nonexpansive mappings, J. Nonlinear Convex Anal. 17, 655-668, 2016.
  12. A. J. Zaslavski Approximate solutions of common fixed point problems, Commun. Appl. Nonlinear Anal. 22, 80-89, 2015.
  13. A. J. Zaslavski, Asymptotic behavior of two algorithms for solving common fixed point problems, Inverse Probl. 33, 044004, 2017.
  14. A. J. Zaslavski, Algorithms for Solving Common Fixed Point Problems, Springer Science and Business Media LLC. 2018.
  15. H. Y. Zhou, X. L. Qin, Fixed Points of Nonlinear Operators, Walter de Gruyter GmbH. 2020.
  16. L. J. Zhu, M. Postolache and Y. Y. She, Convergence of self-adaptive projection methods with linear search for pseudomonotone variational inequalities and fixed point problems, J. Nonlinear Convex Anal. 22, 1541-1554, 2021.
  17. L. J. Zhu, Y. H. Yao and M. Postolache, Projection methods with linesearch technique for pseudomonotone equilibrium problems and fixed point problems, UPB Sci. Bull. A: Appl. Math. Phys. 83, 3-14, 2021.
  18. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67, 274-276, 1979.
  19. H. K. Xun and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22, 767-773, 2001.
  20. L. J. Zhu, M. Postolache and Y. Y. She, Convergence of self-adaptive projection methods with linear search for pseudomonotone variational inequalities and fixed point problems, J. Nonlinear Convex Anal. 22, 1541-1554, 2021.