article

Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces

Authors:

Hafiz Fukhar-ud-din,

Abdul Rahim KhanAuthors Info & Claims

Computers & Mathematics with Applications, Volume 53, Issue 9

Pages 1349 - 1360

https://doi.org/10.1016/j.camwa.2007.01.008

Published: 01 May 2007 Publication History

Abstract

We introduce three-step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes. Mann-type and Ishikawa-type convergence results are included in the analysis of these new iteration schemes. The results presented in this paper substantially improve and extend the results due to [S.H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 8 (2005) 1295-1301], [N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005) 1031-1039], [W. Takahashi, T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1995) 45-58], [K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) 301-308] and [H.F. Senter, W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974) 375-380].

References

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Das, G. and Debata, J.P., Fixed points of quasi-nonexpansive mappings. Indian J. Pure Appl. Math. v17. 1263-1269.

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Ishikawa, S., Fixed points by a new iteration method. Proc. Amer. Math. Soc. v44. 147-150.

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Mann, W.R., Mean value methods in iteration. Proc. Amer. Math. Soc. v44. 506-510.

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Takahashi, W. and Kim, G.E., Approximating fixed points of nonexpansive mappings in Banach spaces. Math. Japon. v48. 1-9.

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Tan, K.K. and Xu, H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. v178. 301-308.

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Zhou, H.Y., Nonexpansive mappings and iterative methods in uniformly convex Banach spaces. Acta Math. Sinica. v20. 829-836.

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Takahashi, W. and Tamura, T., Convergence theorems for a pair of nonexpansive mappings. J. Convex Anal. v5. 45-58.

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Shahzad, N., Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Anal. v61. 1031-1039.

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Goebel, K. and Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. v35. 171-174.

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Bose, S.C., Weak convergence to the fixed point of an asymptotically nonexpansive map. Proc. Amer. Math. Soc. v68. 305-308.

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Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. v43. 153-159.

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Tan, K.K. and Xu, H.K., Fixed point iteration processes for asymptotically nonexpansive mapping. Proc. Amer. Math. Soc. v122. 733-739.

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Xu, B. and Noor, M.A., Fixed-points iteration for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. v267. 444-453.

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Cho, Y.J., Zhou, H.Y. and Guo, G., Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. v47. 707-717.

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Liu, Z.Q. and Kang, S.M., Weak and strong convergence for fixed points of asymptotically nonexpansive mappings. Acta Math. Sinica. v20. 1009-1018.

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Fukhar-ud-din, H. and Khan, S.H., Convergence of two-step iterative scheme with errors for two asymptotically nonexpansive mappings. Internat. J. Math. Math. Sci. v37. 1965-1971.

[17]

Khan, S.H. and Fukhar-ud-din, H., Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Anal. v8. 1295-1301.

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Khan, S.H. and Takahashi, W., Approximating common fixed points of two asymptotically nonexpansive mappings. Sci. Math. Japon. v53. 143-148.

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Takahashi, W. and Shimoji, K., Convergence theorems for nonexpansive mappings and feasibility problems. Math. Comput. Modelling. v32. 1463-1471.

Digital Library

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Xu, H.K. and Ori, R.G., An implicit iteration process for nonexpansive mappings. Numer. Funct. Anal. Optim. v22. 767-773.

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Kaczor, W., Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups. J. Math. Anal. Appl. v272. 565-574.

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Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. v73. 591-597.

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Bruck, R.E., A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J. Math. v32. 107-116.

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Senter, H.F. and Dotson, W.G., Approximating fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. v44. 375-380.

Cited By

Kettapun AKananthai ASuantai S(2018)A new approximation method for common fixed points of a finite family of asymptotically quasi-nonexpansive mappings in Banach spacesComputers & Mathematics with Applications10.1016/j.camwa.2010.06.02560:5(1430-1439)Online publication date: 31-Dec-2018
https://dl.acm.org/doi/10.1016/j.camwa.2010.06.025
Chang SWang GWang LTang YMa Z(2014)Δ -convergence theorems for multi-valued nonexpansive mappings in hyperbolic spacesApplied Mathematics and Computation10.1016/j.amc.2014.10.076249:C(535-540)Online publication date: 15-Dec-2014
https://dl.acm.org/doi/10.1016/j.amc.2014.10.076

Index Terms

Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations

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Published In

cover image Computers & Mathematics with Applications

Computers & Mathematics with Applications Volume 53, Issue 9

May, 2007

158 pages

ISSN:0898-1221

Issue’s Table of Contents

Copyright © Elsevier Ltd © 2007.

Publisher

Pergamon Press, Inc.

United States

Publication History

Published: 01 May 2007

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Cited By

Kettapun AKananthai ASuantai S(2018)A new approximation method for common fixed points of a finite family of asymptotically quasi-nonexpansive mappings in Banach spacesComputers & Mathematics with Applications10.1016/j.camwa.2010.06.02560:5(1430-1439)Online publication date: 31-Dec-2018
https://dl.acm.org/doi/10.1016/j.camwa.2010.06.025
Chang SWang GWang LTang YMa Z(2014)Δ -convergence theorems for multi-valued nonexpansive mappings in hyperbolic spacesApplied Mathematics and Computation10.1016/j.amc.2014.10.076249:C(535-540)Online publication date: 15-Dec-2014
https://dl.acm.org/doi/10.1016/j.amc.2014.10.076

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