Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces (original) (raw)
Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces
Fixed Point Theory and Applications, 2010
We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces.
Journal of Global Optimization, 2011
Mann-like iteration methods are significant to deal with convex feasibility problems in Banach spaces. We focus on a relaxed Mann implicit iteration method to solve a general system of accretive variational inequalities with an asymptotically nonexpansive mapping in the intermediate sense and a countable family of uniformly Lipschitzian pseudocontractive mappings. More convergence theorems are proved under some suitable weak condition in both 2-uniformly smooth and uniformly convex Banach spaces.
On Solutions of Variational Inequality Problems via Iterative Methods
Abstract and Applied Analysis, 2014
We investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of -inverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.
Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces
Acta Applicandae Mathematicae, 2010
In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed iterative method under some suitable conditions.
The Journal of Nonlinear Sciences and Applications, 2017
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.
Journal of Inequalities and Applications, 2013
In this paper, we propose and analyze some iterative algorithms by hybrid viscosity approximation methods for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex Banach space which has a uniformly Gâteaux differentiable norm, and we prove some strong convergence theorems under appropriate conditions. The results presented in this paper improve, extend, supplement and develop the corresponding results recently obtained in the literature. MSC: 49J30; 47H09; 47J20
Viscosity iterative method for a new general system of variational inequalities in Banach spaces
Journal of Inequalities and Applications, 2013
In this paper, we study a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative method without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others. MSC: 46B10; 46B20; 47H10; 49J40 Keywords: a new general system of variational inequalities; relaxed cocoercive mapping; strong convergence Ax-Ay ≤ L xy , ∀x, y ∈ C; (ii) accretive if there exists j(xy) ∈ J(xy) such that Ax-Ay, j(xy) ≥ , ∀x, y ∈ C; (iii) α-inverse strongly accretive if there exist j(xy) ∈ J(xy) and α > such that Ax-Ay, j(xy) ≥ α Ax-Ay , ∀x, y ∈ C;
On solutions of a system of variational inequalities and fixed point problems in Banach spaces
Fixed Point Theory and Applications, 2013
In this paper, considering the problem of solving a system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in Banach spaces, we propose a two-step relaxed extragradient method which is based on Korpelevich’s extragradient method and viscosity approximation method. Strong convergence results are established. MSC:49J30, 47H09, 47J20.
Strong Convergence Theorems of Iterative Algorithm for Nonconvex Variational Inequalities
Thai Journal of Mathematics, 2016
In this work, we suggest and analyze an iterative scheme for solving the system of nonconvex variational inequalities by using projection technique. We prove strong convergence of iterative scheme to the solution of the system of nonconvex variational inequalities requires to the modified mapping T which is Lipschitz continuous but not strongly monotone mapping. Our result can be viewed and improvement the result of N. Petrot [18]
General variational inequalities and nonexpansive mappings
Journal of Mathematical Analysis and Applications, 2007
In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.