Accelerated hybrid iterative algorithm for common fixed points of a finite families of countable Bregman quasi-Lipschitz mappings and solutions of generalized equilibrium problem with application

Volume 11, Issue 1, pp 108--130 http://dx.doi.org/10.22436/jnsa.011.01.09

Publication Date: December 31, 2017 Submission Date: July 11, 2017 Revision Date: September 06, 2017 Accteptance Date: September 09, 2017

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Authors

Jingling Zhang - Department of Mathematics, Tianjin University, Tianjin 300350, P. R. China. Ravi P. Agarwal - Department of Mathematics, Texas A&M University-Kingsville, Texas 78363, U. S. A.. Nan Jiang - Department of Mathematics, Tianjin University, Tianjin 300350, P. R. China. - School of Mechanical Engineering, Tianjin University, Tianjin 300350, P. R. China.

Abstract

The purpose of this paper is to introduce and consider a new accelerated hybrid shrinking projection method for finding a common element of the set \(EP \cap F\) in reflexive Banach spaces, where \(EP\) is the set of all solutions of a generalized equilibrium problem, and \(F\) is the common fixed point set of finite uniformly closed families of countable Bregman quasi-Lipschitz mappings. It is proved that the sequence generated by the accelerated hybrid shrinking projection iteration, converges strongly to the point in \(EP \cap F,\) under some conditions. This result is also applied to find the fixed point of Bregman asymptotically quasi-nonexpansive mappings. It is worth mentioning that, there are multiple projection points from the multiple points in the projection algorithm. Therefore the new projection method in this paper can accelerate the convergence speed of iterative sequence. The new results improve and extend the previously known ones in the literature.

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Keywords

• Bregman distance
• Bregman quasi-Lipschitz mapping
• accelerated hybrid algorithm
• Bregman asymptotically quasi-nonexpansive mappings
• equilibrium problem

MSC

•  47H05
•  47H09
•  47H10

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