A Wiener-Hopf Dynamical System for Mixed Equilibrium Problems

Abstract

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.

1. 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, [1–6]. 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 [8–14] 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 [15–17] 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, 15–17].

2. Formulation and Basic Facts

Let be an Euclidean space, whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex set in , let be nonlinear mappings, and let be a nonlinear mapping, if is a given bifunction satisfying for all . Consider the following mixed equilibrium problem (for short MEP): find such that

This problem has potential and useful applications in nonlinear analysis and mathematical economics. For example, if we set , for all , , a real-valued function, and , then MEP (1) reduces to the following minimization problem subject to implicit constraints:

The basic case of variational inclusions corresponds to with , a set-valued maximal monotone operator. Actually, MEP (1) is equivalent to the following: find such that

Moreover, if , then inclusion (3) reduces to find such that

In particular if , for all , where , and is a closed and convex cone, then inequality (4) can be written as where , for all is the polar cone to . The problem of finding such is an important instance of well-known complementarity problem of mathematical programming.

Another example corresponds to Nash equilibria in noncooperative games. Let (the set of players) be a finite index set. For every , let (the strategy set of ith player) be a given set, (the loss function of the ith player, defending on the strategies of all players) a given function with . For , we define , . The point is called Nash equilibrium if and only if for all the following inequalities hold true: (i.e., no player can reduce his loss by varying his strategy alone). Let and define by . Then is a Nash equilibrium if and only if solves MEP (1).

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

Definition 1 (see [14]). Let be a real-valued function. Then is said to be (a)monotone if , for each ;(b)strictly monotone if , for each , with ;(c)upper hemicontinuous, if, for all , .

Theorem 2 (see [14]). If the following conditions hold true for : (i) is monotone and upper hemicontinuous,(ii) is convex and lower semicontinuous for each ,(iii)there exists a compact subset of and there exists such that 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 , 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 and , being a maximal monotone operator, it directly yields where is the Yosida approximation of , and one recovers classical concepts.
(iii) The operator is cocoercive and nonexpansive.

Lemma 4. Assume that conditions of Theorem 2 are fulfilled; then the operator is cocoercive with modulus 1; that is, is 1-firmly nonexpansive, that is, and is cocoercive with modulus , that is,

Proof. From the relation (9), we can write By adding the last two inequalities and using the monotonicity of , we obtain the desired result.
Equation (12) follows from (11); indeed we have successively Now combining (11) with and , we obtain On the other hand The announced result follows by noticing that

Lemma 5. MEP (1) has a solution if and only if satisfies

Proof. Let be a solution of MEP (1); then which can be written as where 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 such that, for ,

Lemma 6. MEP (1) has a solution if and only if WHE (26) has a solution where
Using (27), WHE (26) can be written as
Thus it is clear from Lemma 6 that is a solution of MEP (1) if and only if satisfies (28).
Using this equivalence, we suggest a new dynamical system associated with MEP (1) as 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 [15–17].

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 where
It is easy to see that if the set has a unique point , then (30) implies that .
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 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.

Lemma 9 (Gronwall; see [9]). Let and be real-valued nonnegative continuous function with domain and let , where is a monotone increasing function. If, for , then
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 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).

3. Existence and Uniqueness of Solution

First, we define the following concepts.

Definition 10. Let , , and be nonlinear mappings. Then, for all , (a) is -Lipschitz continuous if there exists a constant such that (b) is -Lipschitz continuous if there exist constants such that (c) is mixed monotone with respect to and , if (d) is said to be -pseudomonotone, where is a real-valued multivariate function, if

Theorem 11. Let the mappings , , and be -Lipschitz continuous, -Lipschitz continuous, and -Lipschitz continuous, respectively. For each , there exists a unique continuous solution of RDS-MEP (29) with over .

Proof. Let where is a constant. For all , we have This implies that the mapping is Lipchitz continuous in . So, for each , there exists a unique and continuous solution of RDS-MEP (29), defined in an interval with initial condition . Let be its maximal interval of existence; we show that . We estimate for any ; then where
Therefore, using Lemma 9, we have
Hence, the solution is bounded on . So . This completes the proof.

4. Stability Analysis

We now study the stability of RDS-MEP (29). The analysis is in the spirit of Xia and Wang [13].

Theorem 12. Let the mappings , , and be the same as Theorem 11. Let the function be -pseudomonotone with respect to , where is defined as and let be mixed monotone with respect to and . If , then RDS-MEP (29) is stable in the sense of Lyapunov and globally converges to the solution subset of MEP (1).

Proof. Since the mappings , , and are Lipschitz continuous, it follows from Theorem 11 that RDS-MEP (29) has a unique continuous solution over for any fixed . Let be the solution of the initial-value problem (29). For a given , consider the following Lyapunov function:
It is clear that , whenever the sequence and . Consequently, we conclude that the level sets of are bounded. Let be a solution of MEP (1); then
Since is -pseudomonotone and is mixed monotone then (48) implies that That is,
Taking in (50), we have
Setting , , and in (9), we have
From (51), (52), and (24), we have which implies that
Thus, from (29), (47), and (54), we have This implies that is a global Lyapunov function for RDS-MEP (29) which is stable in the sense of Lyapunov. Since where 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 , the largest invariant subset of the following set: Note that if , then and hence is an equilibrium point of RDS-MEP (29); that is, .
Conversly, if  , then it  follows that .
Thus, we conclude that , which is nonempty, convex, and invariant set contained in the solution set . So, .
Therefore RDS-MEP (29) converges globally to the solution set of MEP (1). In particular, if we set , then
Hence RDS-MEP (29) is globally asymptotically stable. This completes the proof.

Theorem 13. Let the mappings , , and be the same as in Theorem 11. If , then RDS-MEP (29) converges globally exponentially to the unique solution of MEP (1).

Proof. It follows from Theorem 11 that there exists a unique continuously differentiable solution of RDS-MEP (29) over [). Then where is the solution of MEP (1). Thus
Now, we estimate
From (59) and (61), we have where .
Thus, for , where is a positive constant, we have which shows that the trajectory of the solution of RDS-MEP (29) converges globally exponentially to the unique solution of MEP (1). This completes the proof.

Conflict of Interests

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

Acknowledgment

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