On Halpern-type sequences with applications in variational inequality problems

Abstract

We introduce a Halpern-type sequence and give a necessary and sufficient condition for a strong convergence of this sequence. In particular, we obtain two strong convergence theorems for approximation of a fixed point of nonexpansive mappings or of quasi-nonexpansive ones. We also apply our result for various iterative methods in variational inequality problem. For the Lipschitz continuous mappings, we deal with the extragradient method of Korpelevič, the subgradient extragradient method of Censor et al. and the extragradient of Tseng where the step size rule is priorly or posteriorly chosen. For the non-Lipschitz continuous mappings, we use our results to deduce the convergence results of Shehu and Iyiola and of Thong and Gibali. Our approach allows us to conclude many new results with some new assumptions.

Keywords:

• Strong convergence approximation
• fixed point
• variational inequality
• extragradient method
• iterative method with line search technique

2010 Mathematics Subject Classifications:

• 47H10
• 47J20
• 47J25

Acknowledgments

The authors thank the referees and professor Christiane Tammer for their comments and suggestions on the manuscript. The first author is supported by the Department of Applied Mathematics and Statistics, Faculty of Sciences and Liberal Arts, Rajamangala University of Technology Isan. The second author is supported by the Thailand Research Fund and Khon Kaen University under grant RSA6280002.

Disclosure statement

No potential conflict of interest was reported by the author(s).

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 work was supported by Thailand Research Fund [ RSA6280002].