The concept of quasivariational inequality problem on proximally smooth sets is studied. Some sufficient conditions for solving the existence of solutions of such a problem are provided; also some interesting cases are discussed. Of course, due to the significance of proximally smooth sets, the results which are presented in this paper improve and extend many important results in the literature.

Variational inequality theory is a branch of the mathematics which is important, and it was also the inspiration for researchers to find new works, both in terms of mathematics and applications such as in economics, physical, biological, and engineering science, and other applied sciences. In 1973, Bensoussan et al. [

In the early period of the research, it should be pointed out that almost all the results regarding the existence and iterative schemes for solving those variational inequalities problems are being considered in the convexity setting. This is because they need the convexity assumption for guaranteeing the well definedness of the proposed iterative algorithm which depends on the projection mapping. However, in fact, the convexity assumption may not be required because it may be well defined even if the considered set is nonconvexs (e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of a Hilbert space, etc.). However, it may be from the practical point of view one may see that the nonconvex problems are more useful than convex case. Consequently, now many researchers are paying attention to many nonconvex cases.

Let

Let

For each

The following is called

Let

We recall also [

In 1995, Clarke et al. [

For a given

For the case of

From now on, we will denote

If

The following lemma which summarizes some important consequences of the uniformly prox-regularity sets is needed in the sequel. The proof of this result can be found in [

Let

For all

For all

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

If

The following definition and lemma are also needed, in order to obtain our main results.

A set-valued mapping

Let

In this paper, we are interested in the following classes of mappings.

Let

there is

Let

The following remark is very useful in order to prove our results. Before seeing that, for the sake of simplicity, let us make a notation: for each

Let

Next, for a fixed positive real number

Now we are in a position to present our main results.

Let

If there is

Firstly, we will define a sequence

Further, let us consider

Next, we will show that

Now, by using the Assumption (

We now finish the proof by showing that

Next we show that

A condition which has been proposed in the assumptions of Theorem

The function

Assume that all assumptions of Theorem

Recall that a set-valued mapping

Let

It is well known that if

Let

Finally, in view of Remark

Let

In this paper, we provide some conditions for the existence theorems of the quasivariational inequality problem on a class of nonconvex sets. In fact, there are two constraints on the assumptions of considered mapping

the range of mapping

This work is supported by the Centre of Excellence in Mathematics, Commission on Higher Education, Thailand.

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