Solution Of General Variational Inequalities Research Articles

Strong convergence theorems for fixed point problems for strict pseudo-contractions and variational inequalities for inverse-strongly accretive mappings in uniformly smooth Banach spaces

In this paper, we first study a simple viscosity iterative algorithm for finding a fixed point of a nonexpansive mapping in uniformly smooth Banach space and obtain a strong convergence theorem under suitable conditions. Then, we apply our iterative algorithm to find a common element of the set of fixed points for a strict pseudo-contraction and the set of fixed points for a nonexansive mapping in uniformly smooth Banach space. We also apply our main results to find a common element of the set of fixed points for strict pseudo-contraction and nonexpansive mapping, the set of solutions of general variational inequalities for two inverse-strongly accretive mappings and equilibrium problems in uniformly smooth Banach spaces or Hilbert spaces. Finally, two numerical examples are given in support of our main results.

Locally unique solutions of nonsmooth general variational inequalities

We study nonsmooth general variational inequalities where the underlying functions admit the H -differentiability but are not necessarily locally Lipschitzian nor directionally differentiable, and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We give some sufficient conditions to guarantee the local uniqueness of solutions to nonsmooth general variational inequalities. Also, we show how the solution of a linearized general variational inequality is related to the solution of the general variational inequality. When specialized to the classical variational inequality and nonlinear complementarity problems, our results extend/unify various similar results proved for C 1 and locally Lipschitzian.

General Wiener–Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces

In this paper, we show the general variational inequality problems are equivalent to solving the general Wiener–Hopf equations. By using the equivalence, we establish a general iterative algorithm for finding the solution of general variational inequalities, general Wiener–Hopf and the fixed point of nonexpansive mappings. Our results extend and improve the recent corresponding results announced by many others.

Merit functions for general variational inequalities

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.

Local Uniqueness of Solutions of General Variational Inequalities

We use the concept of Frechet approximate Jacobian matrices to establish the local uniqueness of solutions to general variational inequalities which involve continuous, not necessarily locally Lipschitz continuous data.