On general implicit hybrid iteration method for triple hierarchical variational inequalities with hierarchical variational inequality constraints

Abstract

In this paper, we study the triple hierarchical variational inequality problem (THVIP, for short) with the hierarchical variational inequality constraint for finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Based on Korpelevich's extragradient method, hybrid steepest-descent method, Mann's implicit iteration method and Halpern's iteration method, a general implicit hybrid iterative algorithm for solving this THVIP is proposed. Under mild conditions, the strong convergence of the iterative sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.

Keywords:

• General implicit hybrid iterative algorithm
• triple hierarchical variational inequality
• Mann's implicit iteration method
• Korpelevich's extragradient method
• nonexpansive mapping

2010 Mathematics Subject Classifications:

• 49J30
• 47H09
• 47J20
• 49M05

Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments and suggestions, which helped to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

In this research, Lu-Chuan Ceng was partially supported by the Innovation Program of Shanghai Municipal Education Commission [grant number 15ZZ068], PhD Program Foundation of Ministry of Education of China [grant number 20123127110002] and Program for Outstanding Academic Leaders in Shanghai City [grant number 15XD1503100]; Xiaopeng Zhao was partially supported by the National Natural Science Foundation of China [grant number 11801411], Natural Science Foundation of Tianjin [grant number 18JCQNJC01100], and The Science & Technology Development Fund of Tianjin Education Commission for Higher Education [grant number 2018KJ224].