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Mathematics 2019, 7(2), 142; https://doi.org/10.3390/math7020142

Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems

1,†
and
2,*,†
1
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2
School of Mathematics and Statistics, Linyi University, Linyi 276000, China
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Received: 22 December 2018 / Revised: 26 January 2019 / Accepted: 29 January 2019 / Published: 2 February 2019
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
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Abstract

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces. View Full-Text
Keywords: hybrid Mann viscosity implicit iteration method; triple hierarchical constrained variational inequality; general system of variational inequalities; fixed point; asymptotically nonexpansive mapping; pseudocontractive mapping; strong convergence; Hilbert spaces hybrid Mann viscosity implicit iteration method; triple hierarchical constrained variational inequality; general system of variational inequalities; fixed point; asymptotically nonexpansive mapping; pseudocontractive mapping; strong convergence; Hilbert spaces
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Ceng, L.-C.; Yuan, Q. Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems. Mathematics 2019, 7, 142.

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