A Wiener-Hopf Dynamical System for Mixed Equilibrium Problems

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.


Introduction
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, [1][2][3][4][5][6].In 2002, Moudafi [5] considered a class of mixed equilibrium problems which includes variational inequalities as well as complementarity problems, convex optimization, saddle point problems, problems of finding a zero of a maximal monotone operator, and Nash equilibria problems as special cases.He studied sensitivity analysis and developed some iterative methods for mixed equilibrium problems.In recent years, much attention has been given to consider and analyze the projected dynamical systems associated with variational inequalities and nonlinear programming problems, in which the right-hand side of the ordinary differential equation is a projection operator.Such types of the projected dynamical system were introduced and studied by Dupuis and Nagurney [7].Projected dynamical systems are characterized by a discontinuous right-hand side.The discontinuity arises from the constraint governing the question.The innovative and novel feature of a projected dynamical systems is that the set of stationary points of dynamical system correspond to the set of solution of the variational inequality problems.It has been shown in [8][9][10][11][12][13][14] that the dynamical systems are useful in developing efficient and powerful numerical technique for solving variational inequalities and related optimization problems.Xia and Wang [13], Zhang and Nagurney [14], and Nagurney and Zhang [11] have studied the globally asymptotic stability of these projected dynamical systems.Noor [15][16][17] has also suggested and analyzed similar resolvent dynamical systems for variational inequalities.It is worth mentioning that there is no such type of the dynamical systems for mixed equilibrium problems.
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, [6,13,[15][16][17].
The following definitions and theorem will be needed in the sequel.
Theorem 2 (see [14]).If the following conditions hold true for  :  ×  → R: (i)  is monotone and upper hemicontinuous, (ii) (, ⋅) is convex and lower semicontinuous for each  ∈ , (iii) there exists a compact subset  of R  and there exists  0 ∈  ∩  such that (,  0 ) < 0 for each  ∈  \ , then the set of solutions to the equilibrium problem is nonempty convex and compact.Moreover, if  is strictly monotone, then the solution of equilibrium problem is unique.
Let us recall the extension of the Yosida approximation notion introduced in [5].Let  > 0, for a given bifunction ; the associated Yosida approximation,   , over  and the corresponding regularized operator,    , are defined as follows: in which    () ∈  is the unique solution of Remark 3 (see [5]).(i) The existence and uniqueness of the solution of problem ( 9) follow by invoking Theorem 2.
(ii) If (, ) = sup ∈ ⟨,  − ⟩ and  = R  ,  being a maximal monotone operator, it directly yields where ( Now combining (11) with  =    ()+   () and  =    ()+    (), we obtain On the other hand The announced result follows by noticing that which can be written as where  > 0 is a constant.Thus, for all  ∈ , we have which is equivalent to by Lemma 4. This completes the proof.
We now define the residue vector () by the relation Invoking Lemma 5, one can observe that  ∈  is a solution of MEP (1) if and only if  ∈  is a zero of Now related to MEP (1), we consider the following Wiener-Hopf equation (in short, WHE): find  ∈ R  such that, for  ∈ ,  (, ) +    () = 0,  =    () , for  > 0. ( where  is a constant.The system of type (29) is called the resolvent dynamical system associated with mixed equilibrium problem (29) (in short, RDS-MEP).Here the right-hand side is associated with resolvent and hence is discontinuous on the boundary of .It is clear from the definitions that the solution to (29) belongs to the constraints set .This implies that the results such as the existence, uniqueness, and continuous dependence on the given data can be studied.It is worth mentioning that RDS-MEP (29) is different from one considered and studied in [15][16][17].
The following concepts and results are useful in the sequel.
Definition 7. The dynamical system is said to converge to the solution set  * of MEP (1) if and only if, irrespective of the initial point, the trajectory of the dynamical system satisfies lim where It is easy to see that if the set  * has a unique point  * , then (30) implies that lim  → ∞ () =  * .If the dynamical system is still stable at  * in the Lyapunov sense, then the dynamical system is globally asymptotically stable at  * .Definition 8.The dynamical system is said to be globally exponentially stable with degree  at  * if and only if, irrespective of the initial point, the trajectory of the system () satisfies where  1 and  are positive constants independent of the initial point.It is clear that globally exponential stability is necessarily globally asymptotical stability and the dynamical system converges arbitrarily fast.

𝑦 (𝑠) 𝑑𝑠}) . (34)
In the sequel, one assumes that the bifunction  involved in MEP (1) satisfies conditions of Theorem 2. Further, from now onward one assumes that  * is nonempty and is bounded, unless otherwise specified.Furthermore, assume that, for all  ∈ , there exists a constant  > 0 such that ‖ (, )‖ ≤  (‖‖ + ‖‖) . ( We study some properties of RDS-MEP (29) and analyze the global stability of the system.First of all, we discuss the existence and uniqueness of RDS-MEP (29).

Existence and Uniqueness of Solution
First, we define the following concepts.
where  is a constant.For all ,  ∈ R  , we have Therefore, using Lemma 9, we have Hence, the solution ‖()‖ is bounded on [ 0 , ).So  = ∞.This completes the proof.

Stability Analysis
We now study the stability of RDS-MEP (29).The analysis is in the spirit of Xia and Wang [13].
It is clear that lim  → ∞ (  ) = +∞, whenever the sequence {  } ⊂  and lim  → ∞   = +∞.Consequently, we conclude that the level sets of  are bounded.Let  * ∈  be a solution of MEP ( 1  (55) This implies that () is a global Lyapunov function for RDS-MEP (29) which is stable in the sense of Lyapunov.Since {() :  ≥  0 } ⊂  0 where  0 = { ∈  : () ≤ ( 0 )} and the function () is continuously differentiable on the bounded and closed set , it follows from LaSalle's invariance principle [9] that the trajectories () will converge to Ω, the largest invariant subset of the following set:

Theorem 12 .
Let the mappings , , and  be the same asTheorem 11.Let the function  be -pseudomonotone with respect to , where  is defined as Proof.Since the mappings , , and  are Lipschitz continuous, it follows from Theorem 11 that RDS-MEP (29) has a unique continuous solution () over [ 0 , ) for any fixed  0 ∈ .Let () = (,  0 ;  0 ) be the solution of the initial-value problem (29).For a given * ∈ , consider the following Lyapunov function: