Advertisement

A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph

  • Damrongsak Yambangwai
  • Sukanya Aunruean
  • Tanakit ThianwanEmail author
Original Paper
  • 39 Downloads

Abstract

In the present article, we establish weak and strong convergence theorems of a new modified three-step iteration method for three G-nonexpansive mappings in a uniformly convex Banach space with a directed graph. Moreover, weak convergence theorem without making use of the Opial’s condition is proved. We also show the numerical experiment for supporting our main results and comparing rate of convergence of the new modified three-step iteration with the three-step Noor iteration and the SP iteration. We also provide some numerical examples to illustrate the convergence behavior and advantages of the proposed method. Furthermore, we apply our results to find solutions of constrained minimization problems and split feasibility problems.

Keywords

G-nonexpansive mapping Three-step Noor iteration SP iteration Uniformly convex Banach space Directed graph 

Mathematics Subject Classification (2010)

47H10 47H09 47E10 

Notes

Acknowledgments

The authors would like to thank the editor and anonymous referees for their valuable comments and suggestions which improved the original version of this article.

Funding information

This study was supported by University of Phayao, Phayao, Thailand (Grant No. UoE62001).

References

  1. 1.
    Banach, S.: Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales. Fund. Math. 3, 133–181 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc. 136(4), 1359–1373 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kelisky, R.P., Rivlin, T.J.: Iterates of Bernstein polynomials. Pac. J. Math. 21, 511–520 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aleomraninejad, S.M.A., Rezapour, S., Shahzad, N.: Some fixed point result on a metric space with a graph. Topol. Appl. 159, 659–663 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alfuraidan, M.R., Khamsi, M.A.: Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph. Fixed Point Theory Appl.  https://doi.org/10.1186/s13663-015-0294-5 (2015)
  6. 6.
    Alfuraidan, M.R.: Fixed points of monotone nonexpansive mappings with a graph. Fixed Point Theory Appl.  https://doi.org/10.1186/s13663-015-0299-0 (2015)
  7. 7.
    Tiammee, J., Kaewkhao, A., Suantai, S.: On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory Appl.  https://doi.org/10.1186/s13663-015-0436-9 (2015)
  8. 8.
    Tripak, O.: Common fixed points of G-nonexpansive mappings on Banach spaces with a graph. Fixed Point Theory Appl.  https://doi.org/10.1186/s13663-016-0578-4 (2016)
  9. 9.
    Noor, M.A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251(1), 217–229 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Glowinski, R., Tallec, P.L.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanic. SIAM, Philadelphia (1989)CrossRefzbMATHGoogle Scholar
  11. 11.
    Haubruge, S., Nguyen, V.H., Strodiot, J.J.: Convergence analysis and applications of the Glowinski Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 97, 645–673 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sridarat, P., Suparaturatorn, R., Suantai, S., Cho, Y.J.: Covergence analysis of SP-iteration for G-nonexpansive mappings with directed graphs. Bull. Malays. Math. Sci Soc.  https://doi.org/10.1007/s40840-018-0606-0 (2017)
  13. 13.
    Johnsonbaugh, R.: Discrete Mathematics. New Jersey (1997)Google Scholar
  14. 14.
    Opial, Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shahzad, S., Al-Dubiban, R.: Approximating common fixed points of nonexpansive mappings in Banach spaces. Georgian Math. J. 13(3), 529–537 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43(1), 153–159 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Suantai, S.: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 331, 506–517 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Springer, New York (2009)zbMATHGoogle Scholar
  20. 20.
    Sangago, M.G.: Convergence of iterative schemes for nonexpansive mappings. Asian-European J. Math. 4(4), 671–682 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wajtaszczyk, P.: Banach Spaces for Analysts. Cambridge Univ. Press (1991)Google Scholar
  22. 22.
    Rhoades, B.E.: Comments on two fixed point iteration method. J. Math. Anal. Appl. 56(2), 741–750 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Phuengrattana, W., Suantai, S.: On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 235, 3006–3014 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Brooks/Cole Cengage Learning, Boston (2010)Google Scholar
  25. 25.
    Berinde, V.: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare (2002)zbMATHGoogle Scholar
  26. 26.
    Suparatulatorn, R., Cholamjiak, W., Suantai, S.: A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs. Numer Algorithm.  https://doi.org/10.1007/s11075-017-0324-y (2017)
  27. 27.
    Thianwan, T., Yambangwai, D.: Convergence analysis for a new two-step iteration process for G-nonexpansive mappings with directed graphs. J. Fixed Point Theory Appl. 21, 44 (2019).  https://doi.org/10.1007/s11784-019-0681-3 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 20 (2004)Google Scholar
  29. 29.
    Podilchuk, C.I., Mammone, R.J.: Image recovery by convex projections using a least squares constraint. J. Optical Soc. Am. A 7, 517–521 (1990)CrossRefGoogle Scholar
  30. 30.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications: Variational Methods and Applications. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  31. 31.
    Aubin, J.P., Cellina, A.: Diffierential Inclusions: Set-valued Maps and Viability Theory. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Damrongsak Yambangwai
    • 1
  • Sukanya Aunruean
    • 1
  • Tanakit Thianwan
    • 1
    Email author
  1. 1.Department of Mathematics, School of ScienceUniversity of PhayaoPhayaoThailand

Personalised recommendations