Noor (“Extended general variational inequalities,” 2009, “Auxiliary principle technique for extended general variational inequalities,” 2008, “Sensitivity analysis of extended general variational inequalities,” 2009, “Projection iterative methods for extended general variational inequalities,” 2010) introduced and studied a new class of variational inequalities, which is called the extended general variational inequality involving three different operators. This class of variational inequalities includes several classes of variational inequalities and optimization problems. The main motivation of this paper is to review some aspects of these variational inequalities including the iterative methods and sensitivity analysis. We expect that this paper may stimulate future research in this field along with novel applications.

Variational inequalities, which were introduced and studied in early sixties, contain a wealth of new ideas. Variational inequalities can be considered as a natural extension of the variational principles. It is now well known that the variational inequalities enable us to study a wide class of problems such as free, moving, obstacle, unilateral, equilibrium, and fixed points in a unified and simple framework. Variational inequalities are closely connected with the convexity optimization problem. We would like to point out that the minimum of a differentiable convex function on a convex set in a normed space can be characterized by the variational inequalities. This shows that the variational inequalities are closely related to the convexity. In recent years, the concept of the convexity has been extended and generalized in several direction using some novel and innovative techniques. We emphasize that these generalizations of the convexity have played a fundamental and basic part in the introduction of a new class of variational inequalities. Motivated by these developments, Noor [

Motivated and inspired by the research activities going on in this dynamic field, Noor [

Several numerical techniques have been developed for solving variational inequalities using different technique and ideas. Using the projection technique, one can establish the equivalence between the variational inequalities and the fixed point problem. This alternative equivalent form has been used to study the existence of a solution of the variational inequalities and related problems. This technique and its variant forms have been used to develop several iterative methods for solving the extended general variational inequalities and optimization problems.

Theory of extended general variational inequalities is quite a new one. We shall content ourselves to give the main flavour of the ideas and techniques involved. The technique used to analyze the various iterative methods and other results for extended general variational inequalities are a beautiful blend of ideas of pure and applied sciences. In this paper, we have presented the main results regarding the various iterative methods, their convergence analysis, and other aspects. The language used is necessary to be that of functional analysis, convex analysis, and some knowledge of elementary Hilbert space theory. The framework chosen should be seen as a model setting for more general results for other classes of variational inclusions. One of the main purposes of this paper is to demonstrate the close connection among various classes of iterative methods for solving the extended general variational inequalities. We would like to emphasize that the results obtained and discussed in this paper may motivate and bring a large number of novel, innovative, and important applications, extensions, and generalizations in other fields.

Let

For given nonlinear operators

For this purpose, we recall the following well-known concepts, see [

Let

From now onward, we assume that

The function

We now show that the minimum of a differentiable

Let

Let

Conversely, let

Lemma

We now list some special cases of the extended general variational inequality (

(i) If

(ii) For

(iii) For

(iv) For

(v) If

From the above discussion, it is clear that the extended general variational inequalities (

We would like to emphasize that problem (

If

We also need the following concepts and results.

Let

For all

It follows from the strongly monotonicity of the operator

The operator

An operator

It is known that the extended general variational inequality (

We rewrite the the relation (

We now study those conditions under which the extended general variational inequality (

Let the operators

From Lemma

Since the operator

From (

Using the fixed point formulation (

For a given

We again use the fixed point formulation to suggest and analyze the following iterative method for solving (

For a given

For a given

To implement Algorithm

For a given

We now consider the convergence analysis of Algorithm

Let

Let

Let

Let

We again use the fixed point formulation (

For a given

For a given

For a given step size

For a given

In this section, we use the auxiliary principle technique to study the existence of a solution of the extended general variational inequality (

Let

We use the auxiliary principle technique to prove the existence of a solution of (

The inequality of type (

We note that, if

For a given

For a given

We now use the auxiliary principle technique to suggest the implicit iterative method for solving the extended general variational inequality (

It is clear that, if

For a given

For a given

The auxiliary principle technique can be used to develop several two-step, three-step, and alternating direction methods for solving the extended general variational inequalities. This is an interesting problem for further research.

We now define the residue vector

For a given

It is worth mentioning that one can suggest and analyze a wide class of iterative methods for solving the extended general variational inequality and its variant forms by using the technique of Noor [

In this paper, we have introduced and considered a new class of variational inequalities, which is called the extended general variational inequalities. We have established the equivalent between the extended general variational inequalities and fixed point problem using the technique of the projection operator. This equivalence is used to study the existence of a solution of the extended general variational inequalities as well as to suggest and analyze some iterative methods for solving the extended general variational inequalities. Several special cases are also discussed. Results proved in this paper can be extended for multivalued and system of extended general variational inequalities using the technique of this paper. The comparison of the iterative method for solving extended general variational inequalities is an interesting problem for future research. Using the technique of Noor [

The author is grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing the excellent research facilities.