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An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces

Abstract

Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator with constants , be nonexpansive mappings with where denotes the fixed-point set of , and be a -contraction with coefficient . Let and , where . For each , let be a unique solution of the fixed-point equation . We derive the following conclusions on the behavior of the net along the curve : (i) if , as , then strongly, which is the unique solution of the variational inequality of finding such that and (ii) if , as , then strongly, which is the unique solution of some hierarchical variational inequality problem.

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Correspondence to Ching-Feng Wen.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zeng, L., Wen, CF. & Yao, J. An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces. Fixed Point Theory Appl 2011, 748918 (2011). https://doi.org/10.1155/2011/748918

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  • DOI: https://doi.org/10.1155/2011/748918

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