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We introduce and study some new classes of variational inequalities and the Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational in equalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed.

Variational inequality problems were initially studied by Stampacchia in 1964. Variational inequalities have applications in diverse disciplines such as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance, see [

Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to consider a unified approach to these two different problems. Noor and Huang [

Let

We now discuss several special cases.

(A) If

(B) If

If

(C) If

In the sequel, we need the following well-known lemma.

For a given

By using Lemma

Related to the general multivalued variational inequality (

Using essentially the technique of Noor [

If

Let

Let

Using Remark

For a given

Note that, if

For a given

If

For a given

If

For a given

We recall the well-known concepts. The multivalued mapping

A mapping

Now we state and prove our main result.

Let

Let

From (

At the same time, we note that

One of the most difficult and important problems in variational inequalities is the development of an efficient numerical methods. One of the technique is called the projection method and its variant forms. Projection method represent an important tool for finding the approximate solution of various types of variational inequalities. The projection type methods were developed in 1970s. The main idea in this techniques is to establish the equivalence between the variational inequalities and the fixed point problem using the concept of projection. These methods have been extended and modified in various ways. Shi [

The authors thank the referees for useful comments and suggestions. The research of Professor M. Aslam Noor is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia, and Research Grant: KSU.VPP.108.