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

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1567 - 1577, 03.11.2023

Xiangyao Li , Li-jun Zhu , Mihai Postolache

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.

Keywords

nonexpansive mapping, common fixed point, runge-kutta method

Supporting Institution

National Natural Science Foundation of China

Project Number

11861003

References

• [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.