Journal
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
Volume 235, Issue 9, Pages 3006-3014Publisher
ELSEVIER
DOI: 10.1016/j.cam.2010.12.022
Keywords
Continuous functions; Convergence theorem; Fixed point; Nondecreasing functions; Rate of convergence
Categories
Funding
- Center of Excellence in Mathematics
- Commission on Higher Education
- Thailand Research Fund
- Graduate School of Chiang Mai University
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In this paper, we propose a new iteration, called the SP-iteration, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the SP-iteration of continuous functions on an arbitrary interval is given. We also compare the convergence speed of Mann, Ishikawa, Noor and SP-iterations. It is proved that the SP-iteration is equivalent to and converges faster than the others. Our results extend and improve the corresponding results of Borwein and Borwein [D. Borwein, J. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991) 112-126], Qing and Qihou [Y. Qing, L Qihou, The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval, J. Math. Anal. Appl. 323 (2006) 1383-1386], Rhoades [B.E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976) 741-750], and many others. Moreover, we also present numerical examples for the SP-iteration to compare with the Mann. Ishikawa and Noor iterations. (C) 2011 Elsevier B.V. All rights reserved.
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