Iterative Schemes for a Class of Mixed Trifunction Variational Inequalities

Muhammad Aslam Noor,Khalida Inayat Noor,Eisa Al-Said

doi:10.1155/2012/125373

Abstract

We use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of variational inequalities, which is called the mixed trifunction variational inequality. The mixed trifunction variational inequality includes the trifunction variational inequalities and the classical variational inequalities as special cases. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of variational inequalities and its variant forms.

Highlights

In recent years, variational inequalities have appeared an interesting and dynamic field of pure and applied sciences
It is well known that the minimum of the differentiable convex functions on the convex set can be characterized by the variational inequalities
We have used the auxiliary principle technique to suggest and analyze several explicit and inertial proximal point algorithms for solving the trifunction equvariational inequality problem

Summary

Introduction

Variational inequalities have appeared an interesting and dynamic field of pure and applied sciences. Inspired and motivated by the ongoing research in this dynamic and fascinating field, we consider and analyze a new class of variational inequalities, called the mixed trifunction variational inequality. This new class of trifunction variational inequalities includes the trifunction bifunction variational inequality and the classical variational inequality as special cases. Due to the nature of the trifunction variational inequality problem, projection methods and its variant form such as WienerHopf equations cannot be used for solving the trifunction variational inequality This fact motivated us to use the auxiliary principle technique of Glowinski et al as developed by Aslam Noor and, Noor et al. This fact motivated us to use the auxiliary principle technique of Glowinski et al as developed by Aslam Noor and, Noor et al This technique is quite flexible and general one. The ideas and techniques of this paper stimulate further research in this area of pure and applied sciences

Preliminaries

Main Results

Conclusion