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

Hafiz Fukhar-ud-din; Abdul Rahim Khan

doi:10.1016/j.camwa.2007.01.008

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

Hafiz Fukhar-ud-din
, Abdul Rahim Khan^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

32 Scopus citations

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].

Original language	English
Pages (from-to)	1349-1360
Number of pages	12
Journal	Computers and Mathematics with Applications
Volume	53
Issue number	9
DOIs	https://doi.org/10.1016/j.camwa.2007.01.008
State	Published - May 2007

Bibliographical note

Funding Information:
The authors thank the referees for their comments and suggestions, which improved the presentation of this paper. The author A.R. Khan gratefully acknowledges support provided by King Fahd University of Petroleum and Minerals during this research.

Keywords

Common fixed point
Fréchet differentiable norm
Kadec-Klee property
Nonexpansive map
Opial property
Strong convergence
Three-step iteration process with errors
Weak convergence

ASJC Scopus subject areas

Modeling and Simulation
Computational Theory and Mathematics
Computational Mathematics

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title = "Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces",

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].",

keywords = "Common fixed point, Fr{\'e}chet differentiable norm, Kadec-Klee property, Nonexpansive map, Opial property, Strong convergence, Three-step iteration process with errors, Weak convergence",

author = "Hafiz Fukhar-ud-din and Khan, \{Abdul Rahim\}",

note = "Funding Information: The authors thank the referees for their comments and suggestions, which improved the presentation of this paper. The author A.R. Khan gratefully acknowledges support provided by King Fahd University of Petroleum and Minerals during this research.",

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AU - Fukhar-ud-din, Hafiz

AU - Khan, Abdul Rahim

N1 - Funding Information: The authors thank the referees for their comments and suggestions, which improved the presentation of this paper. The author A.R. Khan gratefully acknowledges support provided by King Fahd University of Petroleum and Minerals during this research.

PY - 2007/5

Y1 - 2007/5

N2 - 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].

AB - 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].

KW - Common fixed point

KW - Fréchet differentiable norm

KW - Kadec-Klee property

KW - Nonexpansive map

KW - Opial property

KW - Strong convergence

KW - Three-step iteration process with errors

KW - Weak convergence

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JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 9

ER -

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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