Abstract
In this paper, we introduce a new iterative scheme to approximate fixed point of generalized α-nonexpansive mappings and then, we 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. At the end, by using an example for generalized α-nonexpansive mappings, we compare the convergence behavior of new iterative process with other iterative processes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal Appl. 340, 1088–1095 (2008)
Aoyama, K., Kohsaka, F.: Fixed point theorem for α-nonexpansive mappings in Banach spaces. Nonlinear Anal. 74, 4387–4391 (2011)
Ariza-Ruiz, D., Hermandez Linares, C., Llorens-Fuster, E., Moreno-Galvez, E.: On α-nonexpansive mappings in Banach spaces. Carpath. J. Math. 32, 13–28 (2016)
Pant, R., Shukla, R.: Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim. 38, 248–266 (2017)
Mann, W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)
Ishikawa, S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44, 147–150 (1974)
Noor, M. A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000)
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61–79 (2007)
Abbas, M., Nazir, T.: A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesn. 66, 223–234 (2014)
Shahin, F., Adrian, G., Mihai, P., Shahram, R.: A comparative study on the convergence rate of some iteration methods involving contractive mappings. Fixed Point Theory Appl, pp. 24 (2015)
Thakur, B.S., Thakur, D., Postolache, M.: A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. App. Math. Comp. 275, 147–155 (2016)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)
Takahashi, W.: Nonlinear Functional Analysis. Yokohoma Publishers, Yokohoma (2000)
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications Series: Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York (2009)
Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43, 153–159 (1991)
Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12, 187–204 (2011)
Berinde, V.: Picard iteration converges faster than Mann iteration for a class of quasi contractive operators. Fixed Point Theory Appl. 2, 97–105 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Piri, H., Daraby, B., Rahrovi, S. et al. Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces by new faster iteration process. Numer Algor 81, 1129–1148 (2019). https://doi.org/10.1007/s11075-018-0588-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-018-0588-x