Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities

Volume 10, Issue 9, pp 4574--4596 http://dx.doi.org/10.22436/jnsa.010.09.03

Publication Date: September 06, 2017 Submission Date: June 13, 2017

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Authors

Lu-Chuan Ceng - Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China. Yeong-Cheng Liou - Department of Healthcare Administration and Medical Informatics, Center for Big Data Analytics \& Intelligent Healthcare, and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan. - Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan. Ching-Feng Wen - Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan. Ching-Hua Lo - Center for Big Data Analytics \& Intelligent Healthcare, Kaohsiung Medical University, Kaohsiung 807, Taiwan.

Abstract

The purpose of this paper is to find a solution of a general system of variational inequalities (for short, GSVI), which is also a unique solution of a hierarchical variational inequality (for short, HVI) for an infinite family of nonexpansive mappings in Banach spaces. We introduce general implicit and explicit iterative algorithms, which are based on the hybrid steepest-descent method and the Mann iteration method. Under some appropriate conditions, we prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the GSVI, which is also a unique solution of the HVI.

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Keywords

• System of variational inequalities
• nonexpansive mapping
• fixed point
• hybrid steepest-descent method
• global convergence.

MSC

•  49J30
•  47H09
•  47J20
•  49M05

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