Implicit and explicit iterative algorithms for hierarchical variational inequality in uniformly smooth Banach spaces

Lu-Chuan Ceng; Yung-Yih Lur; Ching-Feng Wen

doi:10.1186/s13660-017-1523-8

Implicit and explicit iterative algorithms for hierarchical variational inequality in uniformly smooth Banach spaces

J Inequal Appl. 2017;2017(1):247. doi: 10.1186/s13660-017-1523-8. Epub 2017 Oct 4.

Authors

Lu-Chuan Ceng¹, Yung-Yih Lur², Ching-Feng Wen^{3

4}

Affiliations

¹ Department of Mathematics, Shanghai Normal University, Shanghai, 200234 China.
² Department of Industrial Management, Vanung University, Taoyuan, Taiwan.
³ Center for Fundamental Science; and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 80702 Taiwan.
⁴ Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung, 80702 Taiwan.

Abstract

The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors.

Keywords: fixed point; global convergence; hierarchical variational inequality; implicit and explicit iterative algorithms; nonexpansive mapping; system of variational inequalities.