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Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces ()

In this paper, we suggest a new type of three step iterative scheme called the CR iterative scheme** **and study the strong convergence of this iterative scheme for a certain class of quasi-contractive operators in Banach spaces. We show that for the aforementioned class of operators, the CR iterative scheme is equivalent to and faster than Picard, Mann, Ishikawa, Agarwal *et al**.*, Noor and SP iterative schemes. Moreover, we also present various numerical examples using computer programming in C++ for the CR iterative scheme to compare it with the other above mentioned iterative schemes. Our results show that as far as the rate of convergence is concerned 1) for increasing functions the CR iterative scheme is best, while for decreasing functions the SP iterative scheme is best; 2) CR iterative scheme is best for a certain class of quasi-contractive operators.

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*American Journal of Computational Mathematics*, Vol. 2 No. 4, 2012, pp. 345-357. doi: 10.4236/ajcm.2012.24048.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | W. R. Mann, “Mean Value Methods in Iteration,” Proceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3 |

[2] | S. Ishikawa, “Fixed Points by a New Iteration Method,” Proceedings of the American Mathematical Society, Vol. 44, 1974, pp. 147-150. doi:10.1090/S0002-9939-1974-0336469-5 |

[3] | R. P. Agarwal, D. O’Regan and D. R. Sahu, “Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpasive Mappings,” Journal of Nonlinear and Convex Analysis, Vol. 8, No. 1, 2007, pp. 61-79. |

[4] | M. A. Noor, “New Approximation Schemes for General Variational Inequalities,” Journal of Mathematical Analysis and Applications, Vol. 251, No. 1, 2000, pp. 217-229. doi:10.1006/jmaa.2000.7042 |

[5] | W. Pheungrattana and S. Suantai, “On the Rate of Convergence of Mann, Ishikawa, Noor and SP Iterations for Continuous on an Arbitrary Interval,” Journal of Computational and Applied Mathematics, Vol. 235, No. 9, 2011, pp. 3006-3914. doi:10.1016/j.cam.2010.12.022 |

[6] | T. Zamfirescu, “Fixed Point Theorems in Metric Spaces,” Archiv der Mathematik, Vol. 23, No. 1, 1972, pp. 292-298. doi:10.1007/BF01304884 |

[7] | V. Berinde, “On the Convergence of the Ishikawa Iteration in the Class of Quasi Contractive Operators,” Acta Mathematica Universitatis Comenianae, Vol. 73, No. 1, 2004, pp. 119-126. |

[8] | Z. Q. Xue, “Remarks on Equivalence among Picard, Mann and Ishikawa Iterations in Normed Spaces. Fixed Point Theory and Applications, Vol. 2007, 2007, Article ID: 61434. |

[9] | B. E. Rhoades and S. M. Soltuz, “On the Equivalence of Mann and Ishikawa Iteration Methods,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, 2003, pp. 451-459. doi:10.1155/S0161171203110198 |

[10] | B. E. Rhoades, and S. M. Soltuz, “The Equivalence of the Mann and Ishikawa Iteration for Non-Lipschitzian Operators,” International Journal of Mathematics and Mathematical Sciences, Vol. 42, 2003, pp. 2645-2652. doi:10.1155/S0161171203211418 |

[11] | B. E. Rhoades and S. M. Soltuz, “The Equivalence between the Convergences of Ishikawa and Mann Iterations for Asymptotically Pseudo-Contractive Map,” Journal of Mathematical Analysis and Applications, Vol. 283, 2003, pp. 681-688. doi:10.1016/S0022-247X(03)00338-X |

[12] | B. E. Rhoades and S. M. Soltuz, “The Equivalence of Mann and Ishikawa Iteration for a Lipschitzian Psi-Uniformly Pseudocontractive and Psi-Uniformly Accretive Maps, Tamkang Journal of Mathematics, Vol. 35, 2004, pp. 235-245. |

[13] | B. E. Rhoades and S. M. Soltuz, “The Equivalence between the Convergences of Ishikawa and Mann Iterations for Asymptotically Nonexpansive Maps in the Intermediate Sense and Strongly Successively Pseudocontractive Maps,” Journal of Mathematical Analysis and Applications, Vol. 289, No. 1, 2004, pp. 266-278. doi:10.1016/j.jmaa.2003.09.057 |

[14] | B. E. Rhoades and S. M. Soltuz, “The Equivalence between Mann-Ishikawa Iterations and Multistep Iteration,” Nonlinear Analysis, Vol. 58, No. 1-2, 2004, pp. 219-228. doi:10.1016/j.na.2003.11.013 |

[15] | S. M. Soltuz, “The Equivalence of Picard, Mann and Ishikawa Iterations Dealing with Quasi-Contractive Operators,” Mathematical Communications, Vol. 10, 2005, pp. 81-89. |

[16] | S. M. Soltuz, “The Equivalence between Krasnoselskij, Mann, Ishikawa, Noor and Multistep Iterations,” Mathematical Communications, Vol. 12, 2007, pp. 53-61. |

[17] | R. Chugh and V. Kumar, “Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators,” International Journal of Computer Applications, Vol. 31, No. 5, 2011, pp. 21-27. |

[18] | B. E. Rhoades, “Comments on Two Fixed Point Iteration Methods,” Journal of Mathematical Analysis and Applications, Vol. 56, 1976, pp. 741-750. doi:10.1016/0022-247X(76)90038-X |

[19] | S. L. Singh, “A New Approach in Numerical Praxis,” Progress of Mathematics (Varanasi), Vol. 32, No. 2, 1998, pp. 75-89. |

[20] | V. Berinde, “Iterative Approximation of Fixed Points,” Springer-Verlag, Berlin, 2007. |

[21] | V. Berinde, “Picard Iteration Converges Faster than Mann Iteration Iteration for a Class of Quasi-Contractive Operators,” Fixed Point Theory and Applications, Vol. 2, 2004, pp. 97-105. doi:10.1155/S1687182004311058 |

[22] | Y. Qing and B. E. Rhoades, “Comments on the Rate of Convergence between Mann and Ishikawa Iterations Applied to Zamfirescu Operators,” Fixed Point Theory and Applications, Vol. 2008, 2008, Article ID: 387504. |

[23] | L. B. Ciric, B. S. Lee and A. Rafiq, “Faster Noor Iterations,” Indian Journal of Mathematics, Vol. 52, No. 3, 2010, pp. 429-436. |

[24] | N. Hussian, A. Rafiq, D. Bosko and L. Rade, “On Rate of Convergence of Various Iterative Schemes,” Fixed Point Theory and Applications, Vol. 45, 2011, p. 1-6. |

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