Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities
- Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China.
- 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.
- Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan.
- Center for Big Data Analytics \& Intelligent Healthcare, Kaohsiung Medical University, Kaohsiung 807, Taiwan.
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.
- System of variational inequalities
- nonexpansive mapping
- fixed point
- hybrid steepest-descent method
- global convergence.
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