On the convergence and data dependence results for multistep Picard-Mann iteration process in the class of contractive-like operators

Volume 9, Issue 6, pp 3773--3786 http://dx.doi.org/10.22436/jnsa.009.06.27

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)

• 1553 Downloads
• 2672 Views

Authors

Isa Yildirim - Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey. Mujahid Abbas - Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan. Nazli Karaca - Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey.

Abstract

In this paper, we introduce a new iteration process and prove the convergence of this iteration process to a fixed point of contractive-like operators. We also present a data dependence result for such mappings. Our results unify and extend various results in the existing literature.

Share and Cite

Keywords

• Fixed point
• contractive-like operators
• convergence and data dependence.

MSC

•  47H10
•  54H25

References

• [1] M. Abbas, P. Vetro, S. H. Khan, On fixed points of Berinde's contractive mappings in cone metric spaces, Carpathian J. Math., 26 (2010), 121-133.


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


• [3] V. Berinde , On the convergence of the Ishikawa iteration in the class of quasi contractive operators, Acta Math. Univ. Comenianae, 73 (2004), 119-126.


• [4] V. Berinde, Iterative approximation of fixed points, Springer Berlin Heidelberg, New York (2007)


• [5] C. E. Chidume, S. A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc., 129 (2001), 2359-2363.


• [6] R. Chugh, V. Kumar, Strong convergence of SP iterative scheme for quasi-contractive operators in Banach spaces, Int. J. Comput. Appl., 31 (2011), 21-27.


• [7] R. Chugh, V. Kumar, S. Kumar, Stronge converge of a new three step iterative scheme in Banach spaces, Amer. J. Comput. Math., 2 (2012), 345-357.


• [8] C. O. Imoru, M. O. Olantiwo, On the stability of Picard and Mann iteration processes, Carpathian J. Math., 19 (2003), 155-160.


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


• [10] V. Karakaya, F. Gürsoy, K. Doğan, M. Ertürk, Data dependence results for multistep and CR iterative schemes in the class of contractive-like operators, Abstr. Appl. Anal., 2013 (2013), 7 pages.


• [11] S. H. Khan, Common fixed points of quasi-contractive type operators by a generalized iterative process, IAENG Int. J. Appl. Math., 41 (2011), 260-264.


• [12] S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 2013 (2013), 10 pages.


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


• [14] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229.


• [15] M. O. Osilike, A. Udomene, Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian J. Pure Appl. Math., 30 (1999), 1229-1234.


• [16] W. Phuengrattana, S. Suantai , On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval , J. Comput. Appl. Math., 235 (2011), 3006-3014.


• [17] E. Picard , Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl., 6 (1890), 145-210.


• [18] B. E. Rhoades, S. M. Şoltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map, J. Math. Anal. Appl., 283 (2003), 681-688.


• [19] B. E. Rhoades, S. M. Şoltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Int. J. Math. Math. Sci., 42 (2003), 2645-2651.


• [20] B. E. Rhoades, S. M. Şoltuz, The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Anal., 58 (2004), 219-228.


• [21] S. M. Şoltuz, The equivalence between Krasnoselskij, Mann, Ishikawa, Noor and multistep iterations, Math. Commun, 12 (2007), 53-61.


• [22] S. M. Şoltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl., 2008 (2008), 7 pages.


• [23] S. Thianwan, Common fixed points of new iterations for two asymptotically nonexpansive nonself mappings in a Banach space, J. Comput. Appl. Math., 224 (2009), 688-695.


• [24] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., 113 (1991), 727-731.


• [25] I. Yildirim, M. Ozdemir, A new iterative process for common fixed points of finite families of non-self- asymptotically non-expansive mappings, Nonlinear Anal., 71 (2009), 991-999.


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