An iterative process for a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems in reflexive Banach spaces

18 pages•Published: January 16, 2022

Abstract

The equilibrium problem and its generalizations had a great influence in the development of some branches of pure and applied sciences. The equilibrium problems theory provides a natural and novel approach for some problems arising in nonlinear analysis, physics and engineering, image reconstruction, economics, finance, game theory and optimization. In recent times, there were many methods in order to solve the equilibrium problem and its generalizations. Some authors proposed many iterative methods and studied the convergence of such iterative methods for equilibrium problems and nonexpansive mappings in the setting of Hilbert spaces and Banach spaces. Note that a generalized mixed equilibrium problem is a generalization of an equilibrium problem and a Bregman totally quasi-asymptotically nonexpansive mapping is a generalization of a nonexpansive mapping in reflexive Banach spaces. The purpose of this paper is to combine the parallel method with the Bregman distance and the Bregman projection in order to introduce a new parallel hybrid iterative process which is to find common solutions of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems. After that, we prove that the proposed iteration strongly converges to the Bregman projection of initial element on the intersection of common fixed point set of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and the solution set of a system of generalized mixed equilibrium problems in reflexive Banach spaces. As application, we obatin some strong convergence results for a Bregman totally quasi-asymptotically nonexpansive mapping and a generalized mixed equilibrium problem in reflexive Banach spaces. These results are extensions and improvements to the main results in [7, 8]. In addition, a numerical example is provided to illustrate for the obtained result.

Keyphrases: bregman totally quasi asymptotically nonexpansive mapping, generalized mixed equilibrium problem, iterative method, reflexive banach space, strong convergence

In: Tich Thien Truong, Trung Nghia Tran, Thanh Nha Nguyen and Quoc Khai Le (editors). Proceedings of International Symposium on Applied Science 2021, vol 4, pages 32-49.

Links:	https://easychair.org/publications/paper/M9ml
	https://doi.org/10.29007/2mn6

BibTeX entry

@inproceedings{ISAS2021:iterative_process_finite_family,
  author    = {Trung Hieu Nguyen},
  title     = {An iterative process for a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of  generalized mixed equilibrium problems in reflexive Banach spaces},
  booktitle = {Proceedings of International Symposium on Applied Science 2021},
  editor    = {Tich Thien Truong and Trung Nghia Tran and Thanh Nha Nguyen and Quoc Khai Le},
  series    = {Kalpa Publications in Engineering},
  volume    = {4},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2515-1770},
  url       = {/publications/paper/M9ml},
  doi       = {10.29007/2mn6},
  pages     = {32-49},
  year      = {2022}}

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