Viscosity iterative method for a new general system of variational inequalities in Banach spaces

Abstract

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(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; © 2013 Imnang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In particular, if q = , the mapping J  is called the normalized duality mapping (or duality mapping), and usually we write J  = J. If X is a Hilbert space, then J = I. Further, we have the following properties of the generalized duality mapping J q .() J q (x) = x q- J  (x) for all x ∈ X with x = . () J q (tx) = t q- J q (x) for all x ∈ X and t ∈ [, ∞).() J q (-x) = -J q (x) for all x ∈ X.

Viscosity Approximation Methods for a General Variational Inequality System and Fixed Point Problems in Banach Spaces

“…We study this viscosity approximation method to find a common element of the fixed point set of an asymptotically nonexpansive mapping and the set of solutions of the general variational inequality system in Banach spaces. Our results presented in this paper generalize and complement many recent ones [3,5,6,9,10,[12][13][14]17].…”

“…Some strong convergence theorems are obtained and the numerical experiments can be guaranteed by Theorem 1. We give an extension to the general variational inequality system in Banach spaces and we generalize the Hilbert spaces to Banach spaces, and the nonexpansive mapping to the asymptotically nonexpansive mappings of Imnang [5] and Cai et al [14], for the fixed point problem and variational inequality problem. In Theorem 1, if t = 0 δ n = 0 in Hilbert spaces, this is the main results of Ceng et al [3].…”

“…Variational inequality theory has played a significant role in nonlinear analysis and the optimization problem. Many iterative methods have been used to solve variational inequality problems due to the applications in some branches of applied science, convex optimization, mathematical physics, and operator studies, see [1][2][3][4][5][6][7][8][9] and the references therein. In fact, the classical variational inequality problem in Banach spaces is to find q ∈ E such that Aq, j(x − q) ≥ 0, ∀x ∈ E.…”

A Simple proof for Imnang's algorithms

“…3. A simple proof for the theorem S. Imnang [2] considered an iterative algorithm for finding a common element of the set of fixed points of nonexpansive mappings and the set of solutions of a variational inequality. First, it is needed to prove a lemma as follows: Lemma 3.1.…”

A New Algorithm for the Common Solutions of a Generalized Variational Inequality System and a Nonlinear Operator Equation in Banach Spaces

“…The set of the solutions of the variational inequality (1) is denoted by V I(C, A). Lots of problems in physics, optimization, differential equation (inclusion), finance and minimax problem reduce to find an element of (1) and relevant numerical analysis methods can be considered to solve the problems, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”