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Article:Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators

by Renu Chugh, Vivek Kumar
International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Volume 31 - Number 5
Year of Publication: 2011
Authors: Renu Chugh, Vivek Kumar
10.5120/3820-5294

Renu Chugh, Vivek Kumar . Article:Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators. International Journal of Computer Applications. 31, 5 ( October 2011), 21-27. DOI=10.5120/3820-5294

@article{ 10.5120/3820-5294,
author = { Renu Chugh, Vivek Kumar },
title = { Article:Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators },
journal = { International Journal of Computer Applications },
issue_date = { October 2011 },
volume = { 31 },
number = { 5 },
month = { October },
year = { 2011 },
issn = { 0975-8887 },
pages = { 21-27 },
numpages = {9},
url = { https://ijcaonline.org/archives/volume31/number5/3820-5294/ },
doi = { 10.5120/3820-5294 },
publisher = {Foundation of Computer Science (FCS), NY, USA},
address = {New York, USA}
}
%0 Journal Article
%1 2024-02-06T20:17:20.895248+05:30
%A Renu Chugh
%A Vivek Kumar
%T Article:Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators
%J International Journal of Computer Applications
%@ 0975-8887
%V 31
%N 5
%P 21-27
%D 2011
%I Foundation of Computer Science (FCS), NY, USA
Abstract

In this paper , we study the strong convergence of SP iterative scheme for quasi-contractive operators in Banach spaces. We show that Picard , Mann , Ishikawa , Noor, new two step and SP iterative schemes are equivalent for quasi-contractive operators. In addition, we show that the rate of convergence of SP iterative scheme is better than the other iterative schemes mentioned above for increasing and decreasing functions.

References
  1. Berinde , V., On the convergence of the Ishikawa iteration in the class of quasi-contractive operators, Acta Math. Univ. Comenianae LXXIII(2004), 119-126.
  2. Berinde, V., Iterative Approximation of Fixed Points, Springer-Verlag, Berlin Heidelberg, 2007.
  3. Berinde, V., Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory and Applications, volume 2(2004), 97-105.
  4. Ishikawa, S. , Fixed points by a new iteration method , Proc. Amer. Math. Soc. 44(1974), 147-150.
  5. Mann , W.R., Mean value methods in iteration, Proc. Amer. Math. Soc. 4(1953), 506-510.
  6. Noor, M. A., New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(2000), 217-229.
  7. Phuengrattana , Withunand , Suantai, Suthep , 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, 235(2011), 3006- 3014.
  8. Rafiq, A.,On the convergence of the three- step iteration process in the class of quasi-contractive operators , Acta Mathematica Academiae Paedagogicae Nyiregyhaziens 22(2006), 305-309.
  9. Rhoades , B. E., Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56(1976), 741-750.
  10. Rhoades , B. E. and Soltuz , S¸.M., On the equivalence of Mann and Ishikawa iteration methods, Int. J. Math. Math. Sci. 2003(2003), 451-459.
  11. Rhoades ,B. E. and Soltuz , S¸.M., The equivalence of the Mann and Ishikawa iteration for non-Lipschitzian operators, Int. J. Math. Math. Sci. 42(2003), 2645-2652.
  12. Rhoades ,B. E. and Soltuz , S¸.M., The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically pseudocontractive map, J. Math. Anal. Appl. 283(2003), 681-688.
  13. Rhoades , B. E. and Soltuz , S¸.M., The equivalence of Mann and Ishikawa iteration for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly accretive maps, Tamkang J. Math., 35(2004), 235-245.
  14. Rhoades , B. E. and S¸oltuz , S¸.M., The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289(2004), 266-278.
  15. Rhoades , B. E. and Soltuz , S¸.M., The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Analysis 58(2004), 219-228.
  16. Singh , S. L., A new approach in numerical Praxis, Progress Math.(Varansi) 32(2) (1998),75-89.
  17. Soltuz , S¸.M., The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators, Math. Comm. 10(2005), 81-89.
  18. Soltuz ,S¸. M., The equivalence between Krasnoselskij , Mann, Ishikawa ,Noor and multistep iterations, Mathematical Communications 12(2007),53-61.
  19. Thiainwan, S., Common fixed points of new iterations for two asymptotically nonexpansive nonself mappings in Banach spaces, Journal of Computational and Applied Mathematics,2009,688-695.
  20. Yildirim ,Isa, Ozdemir, Murat and Kiziltunc ,Hukmi, On the convergence of the new two-step iteration process in the class of quasi-contractive operators , Int. Journal of Math Analysis,Vol 3(2009),no. 38 ,1881-1892.
  21. Zamfirescu, T., Fixed point theorems in metric spaces, Arch. , Math., 23(1972), 292-298.
  22. Zhiqun, Xue , Remarks on equivalence among Picard, Mann and Ishikawa iterations in Normed spaces. Fixed Point Theory and Applications, volume 2007, 5 pages.
Index Terms

Computer Science
Information Sciences

Keywords

SP iteration Picard iteration Mann iteration Ishikawa iteration Noor iteration new two step iteration Strong convergence Quasi-contractive operators