Shrinking projection method for solving inclusion problem and fixed point problem in reflexive Banach spaces

ABSTRACT

The purpose of this paper is to find a common element in the intersection of the set of zeros of the inclusion problem of sum of two monotone mappings and the set of fixed points of a Bregman quasi nonexpansive mapping in a reflexive Banach space by using Bregman distance and shrinking projection method. Under suitable conditions, some strong convergence theorems are proved. As applications, we utilize our results to study the convex minimization problem, variational inequality problem.

Keywords:

• Bregman distance
• Bregman quasi-nonexpansive mapping
• Bregman strongly nonexpansive mapping
• maximal monotone operator
• Bregman inverse strongly monotone mapping

2010 AMS Subject Classifications:

• 47H09
• 47H10
• 49J20
• 49J40

Acknowledgments

The authors would like to express their thanks to the Editor and the Reviewers for their helpful comments and advices.

This study was supported by the Natural Science Foundation of China Medical University, Taichung, Taiwan; and the grand from Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, Taiwan.

Disclosure statement

The authors declare that they have no competing interests.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This study was supported by the Natural Science Foundation of China Medical University, Taichung, Taiwan; and the grand from Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, Taiwan.