International Journal of Nonlinear Analysis and Applications

A new faster iteration process to fixed points of generalized α-nonexpansive mappings in Banach spaces

Document Type : Research Paper

Authors

Department of Mathematics, University of Maragheh, Maragheh, Iran

Abstract

In this paper, we introduce a new iterative scheme to approximate the fixed point of generalized α-nonexpansive mappings. we first prove that the proposed iteration process is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas and Thakur processes for contractive mappings. We also obtain some weak and strong convergence theorems for generalized α-nonexpansive mappings. Using the example presented in [R. Pant and R. Shukla, Approximating fixed point of generalized α-nonexpansive mappings in Banach spaces, J. Numer. Funct. Anal. Optim. 38(2017) 248-266.], we compare the convergence behavior of the new iterative process with other iterative processes.

Keywords

[1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mate. Vesnik 66 (2014), no. 2, 223–234.
[2] R.P. Agarwal, D. O’Regan, and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), no. 1, 61–79.
[3] K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal. 74 (2011), 4387–4391.
[4] D. Ariza-Ruiz, C. Hermandez Linares, E. Llorens-Fuster, and E. Moreno-Galvez, On α-nonexpansive mappings in Banach spaces, Carpath. J. Math. 32 (2016), 13–28.
[5] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl. 2 (2004), 97–105.
[6] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
[7] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.
[8] M.A. Noor, New approximation schemes for general variational inequalities, J. Math.Anal. Appl. 251 (2000), no. 1, 217–229.
[9] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Mathe. Soc. 73 (1967), 591–597.
[10] R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim.: Int. J. 38 (2017), no. 2, 248–266.
[11] D.R. Sahu, Applications of the S-iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory 12 (2011), no. 1, 187–204.
[12] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153–159.
[13] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. −340- (2008), 1088–1095.
[14] W. Takahashi, Nonlinear Functional Analysis, Yokohoma Publishers, Yokohoma, 2000.
[15] B.S. Thakur, D. Thakur and M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat 30 (2016), 2711–2720.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 5
May 2024
Pages 1-10
  • Receive Date: 01 April 2021
  • Revise Date: 06 August 2021
  • Accept Date: 21 August 2021