A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings

2019 ◽  
Vol 83 (3) ◽  
pp. 867-883
Author(s):  
Faik Gürsoy ◽  
Müzeyyen Ertürk ◽  
Mujahid Abbas
Keyword(s):  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
General Variational Inequalities

scholarly journals Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints

2015 ◽  
Vol 46 (4) ◽  
pp. 453-503
Author(s):  
Lu-Chuan Ceng
Keyword(s):  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Steepest Descent Method ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Special Cases ◽  
Mixed Equilibrium ◽  
Split Feasibility

In this paper, we introduce a hybrid extragradient iterative algorithm with regularization for solving the triple hierarchical variational inequality problem (THVIP) (defined over the common fixed point set of finitely many nonexpansive mappings and a strictly pseudocontraction) with constraints of a general mixed equilibrium problem (GMEP), a split feasibility problem (SFP) and a general system of variational inequalities (GSVI). The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest descent method and gradient-projection method (GPM) with regularization. It is proven that, under very mild conditions, the sequences generated by the proposed algorithm converge strongly to a unique solution of the THVIP. We also give the applications of our results for solving some special cases of the THVIP. The results presented in this paper improve and extend some corresponding ones in the earlier and recent literature.

scholarly journals Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Adrian Petrusel ◽  
Mu-Ming Wong ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Finite Family ◽  
Steepest Descent Method ◽  
Common Element ◽  
Monotone Mappings ◽  
Mixed Equilibria

We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method.

Sign in / Sign up

Export Citation Format

Share Document