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


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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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.


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

Mathematics Subject Classification (2010)

  • 47H10
  • 47H09
  • 47E10