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A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system is suggested and analyzed. Also the convergence of the suggested iterative algorithm under some suitable conditions is proved.

Variational inequality theory, introduced by Stampacchia [

Projection method and its variant forms represent important tool for finding the approximate solution of various types of variational and quasi-variational inequalities, the origin of which can be traced back to Lions and Stampacchia [

It should be pointed that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting. Consequently, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. It is known that the uniformly prox-regular sets are nonconvex and include the convex sets as special cases. For more details, see, for example, [

In this paper, we introduce and consider a new system of extended general nonlinear regularized nonconvex set-valued variational inequalities. We establish the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems, and then, by this equivalent formulation, we suggest and analyze a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system. We also prove the convergence of the suggested iterative algorithm under some suitable conditions.

Throughout this paper, we will let

Let

The proximal normal cone of

Let

The above inequality is called the

Let

Let

The generalized directional derivative defined earlier can be used to develop a notion of tangency that does not require

The tangent cone

Having defined a tangent cone, the likely candidate for the normal cone is the one obtained from

In 1995, Clarke et al. [

For any

This means that, for all

Obviously, the class of normalized uniformly prox-regular sets is sufficiently large to include the class of convex sets,

A closed set

If

The following proposition summarizes some important consequences of the uniform prox-regularity needed in the sequel. The proof of this results can be found in [

Let

For all

For all

The proximal normal cone is closed as a set-valued mapping.

As a direct consequent of part (c) of Proposition

In order to make clear the concept of

The single-valued operator

Let

A two-variable set-valued operator

It should be pointed that if the domain of

In this section, we introduce a new system of extended general nonlinear regularized nonconvex set-valued variational inequalities and a new system of extended general nonlinear set-valued variational inequalities in Hilbert spaces and investigate their relations.

Let

The problem (

If

Let

The problem (

Some special cases of the system (

If

Find

If

If

If

If

If

If

If

Find

In this section, by using the projection operator technique, we first verify the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities (

Let

Let

The equality (

The fixed point formulation (

Let

Let

It should be pointed that

when

if

In this section, we establish the strongly convergence of the sequence generated by the perturbed projection iterative Algorithms

Let

If the constants

It follows from (

Let

If the constants

Using the method presented in this paper, one can extend Theorems

The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2011-0021821).