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 Recent Issues Online First Volume 1, Issue 1

### Generalized Extragradient Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Fixed Point Problems and Variational Inclusions

Pages: 1 – 22

Lu-Chuan Ceng

#### Abstract

In this paper, let $X$ be a uniformly convex and $q$-uniformly smooth Banach space where $1< q \leq 2$. We introduce and consider a generalized extragradient implicit rule for solving a general system of variational inequalities (GSVI) with the constraints of a variational inclusion (VI) and a common fixed point problem (CFPP) of a countable family of nonexpansive mappings in $X$. Here the generalized extragradient implicit rule is based on Korpelevich's extragradient method, composite implicit viscosity approximation method and Mann's iteration method. We then prove the strong convergence of the sequences generated by the generalized extragradient implicit rule to a solution of the GSVI with the VI and CFPP constraints, which solves a hierarchical variational inequality (HVI). Our results improve and extend the corresponding results in the very recent literature.

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Received: 28 October 2019; Accepted: 20 January 2020.