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We suggest and analyze dynamical systems associated with mixed equilibrium problems by using the resolvent operator technique. We show that these systems have globally asymptotic property. The concepts and results presented in this paper extend and unify a number of previously known corresponding concepts and results in the literature.

Equilibrium problems theory has emerged as an interesting and fascinating branch of applicable mathematics. This theory has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, optimization, and operation research in a general and unified way. There are a substantial number of papers on existence results for solving equilibrium problems based on different-relaxed monotonicity notions and various compactness assumptions; see, for example, [

In this paper, we show that such type of dynamical systems can be suggested for the mixed equilibrium problems. We consider a mixed equilibrium problem and give its related Wiener-Hopf equation and fixed point formulation. Using this fixed point formulation and Wiener-Hopf equation, we suggest dynamical systems associated with mixed equilibrium problems. We use these dynamical systems to prove the uniqueness of a solution of mixed equilibrium problems. Further, we show that the dynamical systems have globally asymptotic stability property. Our results can be viewed as significant and unified extensions of the known results in this area; see, for example, [

Let

This problem has potential and useful applications in nonlinear analysis and mathematical economics. For example, if we set

The basic case of variational inclusions corresponds to

Moreover, if

In particular if

Another example corresponds to Nash equilibria in noncooperative games. Let

The following definitions and theorem will be needed in the sequel.

Let

If the following conditions hold true for

there exists a compact subset

then the set of solutions to the equilibrium problem

Let us recall the extension of the Yosida approximation notion introduced in [

(i) The existence and uniqueness of the solution of problem (

(ii) If

(iii) The operator

Assume that conditions of Theorem

From the relation (

Equation (

MEP (

Let

We now define the residue vector

Invoking Lemma

Now related to MEP (

MEP (

Using (

Thus it is clear from Lemma

Using this equivalence, we suggest a new dynamical system associated with MEP (

The following concepts and results are useful in the sequel.

The dynamical system is said to

It is easy to see that if the set

If the dynamical system is still stable at

The dynamical system is said to be

Let

In the sequel, one assumes that the bifunction

We study some properties of RDS-MEP (

First, we define the following concepts.

Let

Let the mappings

Let

Therefore, using Lemma

Hence, the solution

We now study the stability of RDS-MEP (

Let the mappings

Since the mappings

It is clear that

Since

Taking

Setting

From (

Thus, from (

Conversly, if

Thus, we conclude that

Therefore RDS-MEP (

Hence RDS-MEP (

Let the mappings

It follows from Theorem

Now, we estimate

From (

Thus, for

The authors declare that there is no conflict of interests regarding the publication of this paper.

Farhat Suhel is thankful to NBHM, Department of Atomic Energy, India, Grant no. NBHM/PDF.2/2013/291, for supporting this research work.