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

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.

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References

  1. 1.

    Banach, S.: Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales. Fund. Math. 3, 133–181 (1922)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Kelisky, R.P., Rivlin, T.J.: Iterates of Bernstein polynomials. Pac. J. Math. 21, 511–520 (1967)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Glowinski, R., Tallec, P.L.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanic. SIAM, Philadelphia (1989)

    Google 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)

    MathSciNet  Article  Google 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)

  14. 14.

    Opial, Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Google Scholar 

  20. 20.

    Sangago, M.G.: Convergence of iterative schemes for nonexpansive mappings. Asian-European J. Math. 4(4), 671–682 (2011)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Wajtaszczyk, P.: Banach Spaces for Analysts. Cambridge Univ. Press (1991)

  22. 22.

    Rhoades, B.E.: Comments on two fixed point iteration method. J. Math. Anal. Appl. 56(2), 741–750 (1976)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Google 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

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 20 (2004)

  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)

    Article  Google Scholar 

  30. 30.

    Zeidler, E.: Nonlinear Functional Analysis and its Applications: Variational Methods and Applications. Springer, New York (1985)

    Google Scholar 

  31. 31.

    Aubin, J.P., Cellina, A.: Diffierential Inclusions: Set-valued Maps and Viability Theory. Springer, Berlin (1984)

    Google Scholar 

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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

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

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Correspondence to Tanakit Thianwan.

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Yambangwai, D., Aunruean, S. & Thianwan, T. A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph. Numer Algor 84, 537–565 (2020). https://doi.org/10.1007/s11075-019-00768-w

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Keywords

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

Mathematics Subject Classification (2010)

  • 47H10
  • 47H09
  • 47E10