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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 381980, 7 pages
Data Dependence Results for Multistep and CR Iterative Schemes in the Class of Contractive-Like Operators
1Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey
2Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey
Received 17 February 2013; Accepted 10 July 2013
Academic Editor: Adem Kilicman
Copyright © 2013 Vatan Karakaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- E. Picard, “Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives,” Journal de Mathématiques Pures et Appliquées, vol. 6, pp. 145–210, 1890.
- W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
- M. A. Krasnoselkij, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 1(63), pp. 123–127, 1955.
- S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.
- M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000.
- B. E. Rhoades and S. M. Soltuz, “The equivalence between Mann-Ishikawa iterations and multistep iteration,” Nonlinear Analysis, vol. 58, no. 1-2, pp. 219–228, 2004.
- R. Chugh, V. Kumar, and S. Kumar, “Strong converge of a new three step iterative scheme in Banach spaces,” American Journal of Computational Mathematics, vol. 2, pp. 345–357, 2012.
- R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968.
- S. K. Chatterjea, “Fixed-point theorems,” Doklady Bolgarskoĭ Akademii Nauk, vol. 25, pp. 727–730, 1972.
- T. Zamfirescu, “Fix point theorems in metric spaces,” Archiv der Mathematik, vol. 23, pp. 292–298, 1972.
- B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.
- V. Berinde, “On the convergence of the Ishikawa iteration in the class of quasi contractive operators,” Acta Mathematica Universitatis Comenianae, vol. 73, no. 1, pp. 119–126, 2004.
- M. O. Osilike and A. Udomene, “Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 12, pp. 1229–1234, 1999.
- C. O. Imoru and M. O. Olatinwo, “On the stability of Picard and Mann iteration processes,” Carpathian Journal of Mathematics, vol. 19, no. 2, pp. 155–160, 2003.
- F. Gursoy, V. Karakaya, and B. E. Rhoades, “Data dependence results of new multistep and S-iterative schemes for contractive-like operators,” Fixed Point Theory and Applications, vol. 2013, article 76, 2013.
- A. Rafiq, “On the convergence of the three-step iteration process in the class of quasi-contractive operators,” Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, vol. 22, no. 3, pp. 305–309, 2006.
- B. E. Rhoades, “Fixed point theorems and stability results for fixed point iteration procedures. II,” Indian Journal of Pure and Applied Mathematics, vol. 24, no. 11, pp. 691–703, 1993.
- I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001.
- İ. Yildirim, M. Özdemir, and H. Kiziltunç, “On the convergence of a new two-step iteration in the class of quasi-contractive operators,” International Journal of Mathematical Analysis, vol. 3, no. 37–40, pp. 1881–1892, 2009.
- J. A. Park, “Mann-iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces,” Journal of the Korean Mathematical Society, vol. 31, no. 3, pp. 333–337, 1994.
- M. Ertürk and V. Karakaya, “n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces,” Journal of Inequalities and Applications, vol. 2013, article 196, 2013.
- W. Phuengrattana and S. Suantai, “On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval,” Journal of Computational and Applied Mathematics, vol. 235, no. 9, pp. 3006–3014, 2011.
- R. Chugh and V. Kumar, “Strong convergence of SP iterative scheme for quasi-contractive operators in Banach spaces,” International Journal of Computer Applications, vol. 31, no. 5, pp. 21–27, 2011.
- S. M. Soltuz, “Data dependence for Mann iteration,” Octogon Mathematical Magazine, vol. 9, pp. 825–828, 2001.
- S. M. Soltuz, “Data dependence for Ishikawa iteration,” Lecturas Matemáticas, vol. 25, no. 2, pp. 149–155, 2004.
- S. M. Soltuz and T. Grosan, “Data dependence for Ishikawa iteration when dealing with contractive-like operators,” Fixed Point Theory and Applications, vol. 2008, Article ID 242916, 7 pages, 2008.
- S. M. Soltuz, “The equivalence between Krasnoselskij, Mann, Ishikawa, Noor and multistep iterations,” Mathematical Communications, vol. 12, no. 1, pp. 53–61, 2007.
- F. Gürsoy, V. Karakaya, and B. E. Rhoades, “The equivalence among new multistep iteration, S-iteration and some other iterative schemes,” http://arxiv.org/abs/1211.5701.
- N. Hussain, R. Chugh, V. Kumar, and A. Rafiq, “On the rate of convergence of Kirk-type iterative schemes,” Journal of Applied Mathematics, vol. 2012, Article ID 526503, 22 pages, 2012.
- V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, Germany, 2007.
- R. Chugh and V. Kumar, “Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators,” International Journal of Computer Applications, vol. 40, no. 15, pp. 41–46, 2011.