Extended Extragradient Methods for Generalized Variational Inequalities

We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed

Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

We consider a general variational inequality and fixed point problem, which is to find a point x* with the property that (GVF): x*∈GVI(C,A) and g(x*)∈Fix(S) where GVI(C,A) is the solution set of some variational inequality Fix(S) is the fixed points set of nonexpansive mapping S, and g is a nonlinear operator. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following method g(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)], n≥0. It is shown that the sequence {xn} converges strongly to x*∈Ω which is the unique solution of the variational inequality 〈F(x*)-g(x*),g(x)-g(x*)〉≤0, for all x∈Ω

The Meir-Keeler Type for Solving Variational Inequalities and Fixed Points of Nonexpansive Semigroups in Banach Spaces

The aim of this paper is to introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for nonexpansive semigroups by using the modified viscosity approximation method associate with Meir-Keeler type mappings and obtain some strong convergence theorem in a Banach spaces under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu (2010), Wangkeeree and Preechasilp (2012), Yao and Maruster (2011), and many others