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On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. (English) Zbl 1215.65095
The authors introduce a new iterative method to find a fixed point of a continuous function on an interval. Its convergence is characterized and shown to be faster than that of three other known methods.

65H05Single nonlinear equations (numerical methods)
Full Text: DOI
[1] Mann, W. R.: Mean value methods in iteration, Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603 · doi:10.2307/2032162
[2] Ishikawa, S.: Fixed points by a new iteration method, Proc. amer. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036 · doi:10.2307/2039245
[3] Noor, M. A.: New approximation schemes for general variational inequalities, J. math. Anal. appl. 251, 217-229 (2000) · Zbl 0964.49007 · doi:10.1006/jmaa.2000.7042
[4] Rhoades, B. E.: Comments on two fixed point iteration methods, J. math. Anal. appl. 56, 741-750 (1976) · Zbl 0353.47029 · doi:10.1016/0022-247X(76)90038-X
[5] Borwein, D.; Borwein, J.: Fixed point iterations for real functions, J. math. Anal. appl. 157, 112-126 (1991) · Zbl 0742.26006 · doi:10.1016/0022-247X(91)90139-Q
[6] Qing, Y.; Qihou, L.: The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval, J. math. Anal. appl. 323, 1383-1386 (2006) · Zbl 1101.47056 · doi:10.1016/j.jmaa.2005.11.058
[7] Soltuz, S. M.: The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators, Math. commun. 10, 81-88 (2005) · Zbl 1089.47051
[8] Babu, G. V.; Prasad, K. N.: Mann iteration converges faster than Ishikawa iteration for the class of zamfirescu operators, Fixed point theory appl. 49615, 1-6 (2006) · Zbl 1106.47053 · doi:10.1155/FPTA/2006/49615
[9] Qing, Y.; Rhoades, B. E.: Comments on the rate of convergence between Mann and Ishikawa iterations applied to zamfirescu operators, Fixed point theory appl. 387504, 1-3 (2008) · Zbl 1203.47076 · doi:10.1155/2008/387504