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We suggest and analyze some implicit iterative methods for solving the extended
general nonconvex variational inequalities using the projection technique. We show that the convergence of these
iterative methods requires only the

Variational inequalities, which were introduced and studied in early sixties, contain wealth of new ideas. Variational inequalities can be considered as a natural extension of the variational principles. It is well known that the variational inequalities characterize the optimality conditions of the differentiable convex functions on the convex sets in normed spaces. In recent years, Noor [

Motivated and inspired by the recent activities in this dynamic field, we consider the extended general noncomvex variational inequalities on the prox-regular sets. We use the projection technique to establish the equivalence between the extended general nonconvex variational inequalities and the fixed point problems. We use this alternative formulation to some unified implicit and extragradient methods for solving the extended general nonconvex variational inequalities. These new methods include the modified projection method of Noor [

Let

The proximal normal cone of

The proximal normal cone

Let

The Clarke normal cone, denoted by

For a given

We now recall the well-known proposition which summarizes some important properties of the uniformly prox-regular sets

Let

For given nonlinear operators

We remark that if

We note that, if

If

We note that if

If

We now prove that the projection operator

Let

Let

We note that, if

An operator

It is known that the extended general nonconvex variational inequalities (

Lemma

For a given

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

For a given

For a given

To implement Algorithm

For a given

Algorithm

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 paper, we have introduced and considered a new class of general variational inequalities, which is called the general nonconvex variational inequalities. Some new characterizations of the nonconvex projection operator are proved. We have established the equivalent between the general nonconvex variational inequalities and fixed point problem using the technique of the projection operator. This equivalence is used to suggest and analyze some iterative methods for solving the nonconvex general variational inequalities. Several special cases are also discussed. Results proved in this paper can be extended for multivalued and system of general nonconvex variational inequalities using the technique of this paper. The comparison of the iterative method for solving nonconvex general variational inequalities is an interesting problem for future research. We hope that the ideas and technique of this paper may stimulate further research in this field.

This research is supported by the Visiting Professorship Program of King Saud University, Riyadh, Saudi Arabia, and Research Grant no. KSU.VPP.108. The research of Z. Huang is supported by National Natural Science Foundation of China (NSFC Grant no. 10871092), supported by the Fundamental Research Funds for the Central University of China (Grant no. 1113020301 and Grant no. 1116020301), and supported by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD Grant). The authors are also grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing the excellent research facilities.