On a new iteration scheme for numerical reckoning fixed points of Berinde mappings with convergence analysis

Volume 9, Issue 5, pp 2553--2562 http://dx.doi.org/10.22436/jnsa.009.05.53

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1582 Downloads
• 2710 Views

Authors

Wutiphol Sintunavarat - Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand. Ariana Pitea - Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania.

Abstract

The aim of this work is to introduce a new three step iteration scheme for approximating fixed points of the nonlinear self mappings on a normed linear spaces satisfying Berinde contractive condition. We also study the sufficient condition to prove that our iteration process is faster than the iteration processes of Mann, Ishikawa and Agarwal, et al. Furthermore, we give two numerical examples which fixed points are approximated by using MATLAB.

Share and Cite

Keywords

• Picard iteration process
• Mann iteration process
• Ishikawa iteration process
• rate of convergence
• mean valued theorem.

MSC

•  47H09
•  47H10

References

• [1] M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik, 66 (2014), 223-234.


• [2] R. P. Agarwal, D. O'Regan, D. R. Sahu , Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61-79.


• [3] S. Banach , Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3 (1922), 133-181.


• [4] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (2004), 45-53.


• [5] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl., 2 (2004), 97-105.


• [6] V. Berinde, A convergence theorem for some mean value fixed point iteration procedures, Demonstratio Math., 38 (2005), 177-184.


• [7] S. K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727-730.


• [8] R. Dewangan, B. S. Thakur, M. Postolache, A hybrid iteration for asymptotically strictly pseudocontractive mappings, J. Inequal. Appl., 2014 (2014), 11 pages.


• [9] R. Dewangan, B. S. Thakur, M. Postolache, Strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration, Appl. Math. Comput., 248 (2014), 160-168.


• [10] R. H. Haghi, M. Postolache, Sh. Rezapour, On T-stability of the Picard iteration for generalized phi-contraction mappings, Abstr. Appl. Anal., 2012 (2012), 7 pages.


• [11] N. Hussain, A. Rafiq, B. Damjanovic, R. Lazovic, On rate of convergence of various iterative schemes, Fixed Point Theory Appl., 2011 (2011), 6 pages.


• [12] S. Ishikawa , Fixed point by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.


• [13] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.


• [14] M. O. Olatinwo, M. Postolache, Stability results for Jungck-type iterative processes in convex metric spaces, Appl. Math. Comput., 218 (2012), 6727-6732.


• [15] B. E. Rhoades, A comparison of various deffnitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257-290.


• [16] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math., 80 (1975), 325-330.


• [17] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.


• [18] B. S. Thakur, R. Dewangan, M. Postolache, Strong convergence of new iteration process for a strongly continuous semigroup of asymptotically pseudocontractive mappings, Numer. Funct. Anal. Optim., 34 (2013), 1418-1431.


• [19] B. S. Thakur, R. Dewangan, M. Postolache , General composite implicit iteration process for a finite family of asymptotically pseudo-contractive mappings, Fixed Point Theory Appl., 2014 (2014), 15 pages.


• [20] D. Thakur, B. S. Thakur, M. Postolache, New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 15 pages.


• [21] Y. Yao, M. Postolache, S. M. Kang, Strong convergence of approximated iterations for asymptotically pseudocontractive mappings, Fixed Point Theory Appl., 2014 (2014), 13 pages.


• [22] Y. Yao, M. Postolache, Y. C. Liou , Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings, Fixed Point Theory Appl., 2013 (2013), 8 pages.


• [23] T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., 23 (1972), 292-298.