Abstract
In this paper, we first prove the weak convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Then we prove the weak convergence for the Moudafi’s iterative scheme of quasi-nonexpansive mapping and nonexpansive mapping. Finally, we prove the strong convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Our results generalize the recent results due to Iemoto and Takahashi.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)
Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)
Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa Iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)
Takahashi, W., Kim, G.E.: Approximating fixed points of nonexpansive mappings in Banach spaces. Math. Jpn. 48, 1–9 (1998)
Iemoto, S., Takahashi, W.: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71(12), 2082–2089 (2009)
Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Probl. 23, 1635–1640 (2007)
Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2005)
Browder, F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Am. Math. Soc. 74, 660–665 (1968)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)
Khan, S.H., Fukhar-ud-din, H.: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Anal. 61, 1295–1301 (2005)
Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974)
Groetsch, C.W.: A note on segmenting Mann iterates. J. Math. Anal. Appl. 40, 369–372 (1972)
Takahashi, W.: Nonlinear Functional Analysis. Kindaikagaku, Tokyo (1988) (Japanese)
Kim, G.E.: Convergence theorems for quasi-nonexpansive mappings in Banach spaces (submitted)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Viorel Barbu.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Kim, G.E. Weak and Strong Convergence Theorems of Quasi-Nonexpansive Mappings in a Hilbert Spaces. J Optim Theory Appl 152, 727–738 (2012). https://doi.org/10.1007/s10957-011-9924-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9924-1