Weak \(w^{2}\)-stability and data dependence of Mann iteration method in Hilbert spaces

Abstract

We study a weaker and more natural notion of stability called weak \(w^{2}\)-stability to get an insight in the corresponding results obtained by Măruşter and Măruşter (J Comput Appl Math 276:110–116, 2015) and Wang (J Comput Appl Math 285:226–229, 2015). A data dependence result for fixed points of strongly demicontractive operators is also established. Some illustrative examples are given to validate results obtained herein.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Abbas, M., Ali, B., Butt, A.R.: Existence and data dependence of the fixed points of generalized contraction mappings with applications. Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Math. RACSAM 109, 603–621 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Alfuraidan, M.R., Khamsi, M.A.: Fibonacci–Mann iteration for monotone asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. (2017). doi:10.1017/S0004972717000120

  3. 3.

    Berinde, V.: Iterative Approximation of Fixed Points. Springer, Berlin (2007)

    Google Scholar 

  4. 4.

    Berinde, V.: On the stability of some fixed point procedures. Bul. Ştiinţ. Univ. Baia Mare Ser. B, Matematică-Informatică 18, 7–14 (2002)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Berinde, V.: Summable almost stability of fixed point iteration procedures. Carpathian J. Math. 19, 81–88 (2003)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cardinali, T., Rubbioni, P.: A generalization of the Caristi fixed point theorem in metric spaces. Fixed Point Theory 11, 3–10 (2010)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Dehaish, B.I., Khamsi, M.A., Khan, A.R.: Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces. J. Math. Anal. Appl. 397, 861–868 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Espínola, R., Petruşel, A.: Existence and data dependence of fixed points for multivalued operators on gauge spaces. J. Math. Anal. Appl. 309, 420–432 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Gürsoy, F., Karakaya, V., Rhoades, B.E.: Data dependence results of new multi-step and S-iterative schemes for contractive-like operators. Fixed Point Theory Appl. 2013(1), 1–12 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Harder, A.M., Hicks, T.L.: Stability results for fixed point iteration procedures. Math. Jpn. 33, 693–706 (1988)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Ivan, D., Leuştean, L.: A rate of asymptotic regularity for the Mann iteration of \(\kappa -\)strict pseudo-contractions. Numer. Funct. Anal. Optim. 36, 792–798 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Karakaya, V., Gürsoy, F., Ertürk, M.: Some convergence and data dependence results for various fixed point iterative methods. Kuwait J. Sci. 43, 112–128 (2016)

    MathSciNet  Google Scholar 

  14. 14.

    Khan, A.R., Gürsoy, F., Karakaya, V.: Jungck–Khan iterative scheme and higher convergence rate. Int. J. Comput. Math. 93, 2092–2105 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Khan, A.R., Gürsoy, F., Kumar, V.: Stability and data dependence results for Jungck–Khan iterative scheme. Turkish J. Math. 40, 631–640 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Khan, A.R., Kumar, V., Hussain, N.: Analytical and numerical treatment of Jungck-type iterative schemes. Appl. Math. Comput. 231, 521–535 (2014)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Liu, Z., Kang, S.M., Shim, S.H.: Almost stability of the Mann iteration method with errors for strictly hemi-contractive operators in smooth Banach spaces. J. Korean Math. Soc. 40, 29–40 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Mann, W.R.: Mean value methods in iterations. Proc. Am. Math. Soc. 4, 506–510 (1953)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Măruşter, L., Măruşter, Ş.: On the error estimation and T-stability of the Mann iteration. J. Comput. Appl. Math. 276, 110–116 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Măruşter, L., Măruşter, Ş.: Strong convergence of the Mann iteration for \(\alpha -\)demicontractive mappings. Math. Comput. Model. 54, 2486–2492 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Măruşter, Ş., Rus, I.A.: Kannan contractions and strongly demicontractive mappings. Creat. Math. Inform. 24, 173–182 (2015)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Ostrowski, A.M.: The round-off stability of iterations. ZAMM Z. Angew. Math. Mech. 47, 77–81 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Rus, I.A., Petruşel, A., Sîntamarian, A.: Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Anal. 52, 1947–1959 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Soltuz, S.M., Grosan, T.: Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl. 2008, 1–7 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Timis, I.: On the weak stability of Picard iteration for some contractive type mappings and coincidence theorems. Int. J. Comput. Appl. 37, 9–13 (2012)

    MathSciNet  Google Scholar 

  26. 26.

    Urabe, M.: Convergence of numerical iteration in solution of equations. J. Sci. Hiroshima Univ. A 19, 479–489 (1956)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Wang, C.: A note on the error estimation of the Mann iteration. J. Comput. Appl. Math. 285, 226–229 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Wang, C., Zhang, C.Z.: Approximating common fixed points for a pair of generalized nonlinear mappings in convex metric space. J. Nonlinear Sci. Appl. 9, 1–7 (2016)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their constructive comments to improve quality and presentation of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Faik Gürsoy.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gürsoy, F., Khan, A.R., Ertürk, M. et al. Weak \(w^{2}\)-stability and data dependence of Mann iteration method in Hilbert spaces. RACSAM 113, 11–20 (2019). https://doi.org/10.1007/s13398-017-0447-y

Download citation

Keywords

  • Mann iteration
  • Strongly demicontractive operator
  • Data dependency
  • Stability

Mathematics Subject Classification

  • 47H09
  • 47H10
  • 54H25
  • 65J15