A study of Wiener-Hopf dynamical systems for variational inequalities in the setting of fractional calculus

1.   Introduction

In modern analysis theory of inequalities has played a significant role due to its wide range of applications both in pure and applied sciences. Variational inequalities are one of the most significant and intensively studied inequalities. Recently these inequalities have appeared as an interesting and dynamic field of pure and applied sciences. Variational techniques are being used to study a wide class of problems with 'applications in industry, structural engineering, mathematical finance, economics, optimization, transportation, and optimization problems, see ^{[1,9,10,11,18,21,23,24,25,30]} and references therein. This has motivated researchers to introduce and study several classes of variational inequalities. Stampacchia ^[30] has studied that the minimum of differentiable convex functions on a convex set can be characterized by variational inequalities.

Using the fixed-point formulation of the variational inequalities, Dupuis and Nagurney ^[4] introduced and considered the projected dynamical systems in which the right hand side of the ordinary differential equation is a projection operator associated with variational inequalities. The innovative and novel feature of a projected dynamical system is that its set of stationary points corresponds to the set of solutions of the corresponding variational inequality problem. Hence, equilibrium and nonlinear problems arising in various branches in pure and applied sciences, which can be formulated in the setting of the variational inequalities, can be studied in the more general setting of dynamical systems. It has been shown ^{[5,6,10,11,14,16,17,32]} that the dynamical systems are useful in developing some efficient numerical techniques for solving variational inequalities and related optimization problems. In recent years, much attention has been given to study the globally asymptotic stability of these projected dynamical systems. Xia and Wang ^[32] have shown that the projected dynamical systems can be used effectively in designing neural network for solving variational inequalities.

Zeng et al. ^[31] have investigated the fractional dynamical systems associated with linear variational inequalities. They have investigated the criteria for the asymtotically stability of the equilibrium points. Noor et al. ^[22] have also suggested and analyzed fractional implicit dynamical systems for linear quasi variational inequalities by extending and modifying their techniques. In this paper, we propose and consider a fractional Wiener-Hopf dynamical system associated with the variational inequalities. We show that the fractional Wiener-Hopf dynamical system is exponentially stable and converges to its unique equilibrium point. Some special cases are discussed. Our results are more general than the results of Zeng et al. ^[31]. The ideas and techniques of this paper may inspire the interested readers for further research in this area.

2.   Formulation and basic results

Let $H$ be a real Hilbert space, whose norm and inner product are denoted by $\left\Vert \cdot \right\Vert$ and $\langle \cdot, \cdot \rangle,$ respectively. Let ${K}$ be any closed and convex set in $H$.

For given operator ${{\mathcal{T}}}:H\rightarrow H$, consider a problem of finding $u\in {K}$ such that

The inequality of type $\left(2.1\right)$ is called the variational inequality. This problem was introduced and studied by Stampacchia ^[30]. For the recent applications, numerical algorithms, sensitivity analysis, dynamical systems and formulations of variational inequalities, see ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,33]} and the references therein.

Definition 2.1. A nonlinear operator ${{\mathcal{T}}}:H\rightarrow H$ is said to be monotone, if

Definition 2.2. A nonlinear operator ${{\mathcal{T}}}:H\rightarrow H$ is said to be pseudomonotone, if

Definition 2.3. A nonlinear operator ${{\mathcal{T}}}:H\rightarrow H$ is said to be Lipschitz continuous, if there exists a constant $\beta > 0$ such that

It is well known that monotonicity implies pseudomonotonicity, but not conversely.

Lemma 2.4. ^[16] Let ${K}$ be a closed and convex set in $H$. Then for a given $z\in H$, $u\in {K}$ satisfies

if and only if

where ${P}_{{K}}$ is the projection of $H$ onto a closed and convex set ${K}$ in $H$.

It is well known that the projection operator ${P}_{{K}}$ is non-expensive, that is

Using Lemma 2.4, one can show that Problem $\left(2.1\right)$ is equivalent to the fixed point problem.

Lemma 2.5. ^[16] Let ${K}$ be a closed and convex set in a real Hilbert space $H$. The function $u\in {K}$ is a solution of Problem $\left(2.1\right),$ if and only if, $u\in {K}$ satisfies the relation

where $\rho > 0$ is a constant.

From Lemma 2.5, it follows that Problem $\left(2.1\right)$ is equivalent to a fixed point Problem $\left(2.3\right)$. This equivalent formulation plays an important role in developing several iterative methods, see ^[17,19,20].

We now define a residue vector ${{\mathcal{R}}}\left(u\right)$ as:

It is clear from Lemma 2.5 that Problem $\left(2.1\right)$ has a solution $u\in {K}$, if and only if, $u\in {K}$ is a zero of the equation

It is well known that the variational inequalities are also equivalent to Wiener Hopf equations. The Wiener Hopf equations technique has been used to develop some efficient and powerful iterative methods for solving variational inequalities and complementarity problems. We now use the Wiener Hopf equations technique to suggest and analyze another dynamical system.

Let $Q_{{K}} = I-{P}_{{K}}$, where $I$ is the identity operator and ${P}_{{K}}$ is the projection of $H$ onto the closed and convex set ${K}.$ For given nonlinear operator ${{\mathcal{T}}}:H\rightarrow H$, consider a problem of finding $z\in H$ such that

Problem $\left(2.5\right)$ is called Wiener Hopf equations associated with the variational inequalities $\left(2.1\right).$ This problem was introduced and studied by Shi ^[29]. It is well known that Problems $\left(2.1\right)$ and $\left(2.5\right)$ are equivalent. For the sake of completeness, we recall this result without proof.

Lemma 2.6. ^[16] The Problem $\left(2.1\right)$ has a solution $u\in {K},$ if and only if, Problem $\left(2.5\right)$ has a solution $z\in H$, where

where $\rho > 0$ is a constant.

Using Lemma 2.6$,$ the Wiener-Hopf Eq $\left(2.5\right)$ can be written as

which is equivalent to

Thus it is clear from Lemma 2.6 that $u\in {K}$ is a solution of Problem $\left(2.1\right),$ if and only if, $u\in {K}$ satisfies the Eq $\left(2.9\right)$.

We now suggest a new dynamical system:

where $0 < \alpha < 1$ and $\gamma$ is a constant, associated with Problem $\left(2.1\right)$. The dynamical system of type $\left(2.10 \right)$ is called fractional Wiener-Hopf dynamical system related to the Problem $\left(2.1\right)$.

We now discuss some special cases of Problem $\left(2.10\right)$.

$(1)$ If $\alpha = 1$, then the fractional Wiener-Hopf dynamical System $\left(2.10\right)$ reduces to following dynamical system:

which was introduced and considered by Noor ^[16].

$(2)$ In the affine case, that is, if ${{\mathcal{T}}}u = Au+b,$ where $A = a_{ij}$ is a real $n\times n$ matrix and $b = b_{j}$ is an $n$-dimensional vector, then the System $\left(2.10\right)$ reduces to:

which is called linear fractional Wiener-Hopf dynamical system.

We also need the following well-known fundamental results and concepts.

Definition 2.7. ^[8] The fractional integral (Riemann-Liouville integral) with order $\alpha \in {{\mathcal{R}}}_{+}$ of continuous function $u\left({\omega}\right)$ is defined as:

where $\Gamma$ denotes the Euler gamma function.

Definition 2.8. ^[8] The Caputo derivative with order $\alpha \in {{\mathcal{R}}}_{+}$ of continuous function $u\left({\omega}\right) \in C^{n}\left(\left[{\omega}_{0}, +\infty \right], {{\mathcal{R}}}\right)$ is defined as:

where $n$ is positive integer such that $n-1 < \alpha < n$ and $\Gamma$ denotes the Euler gamma function.

Definition 2.9. ^[31] The point $u^{\ast }$ is said to be an equilibrium point of the fractional projected dynamical System $\left(2.4\right)$, if $u^{\ast }$ satisfies the following

Definition 2.10. ^[33] The dynamical system $\left(2.4\right)$ is said to be $\alpha$-exponentially stable with degree $\lambda$ if, for any two solutions $u\left({\omega}\right)$ and $v\left({\omega}\right)$ of $\left(2.4 \right)$ with different initial values by $u_{0}$ and $v_{0}$ satisfies

where $\eta > 0$ is a constant.

Lemma 2.11. (Gronwall's Lemma ^[17]) Let $u$ and $v$ be real valued non-negative continuous functions with domain $\left\{ {\omega}:{\omega}\geq {\omega}_{0}\right\}$ and let $\alpha \left({\omega}\right) = \alpha _{0}\left\vert {\omega}-{\omega}_{0}\right\vert$, where $\alpha _{0}$ is a monotone increasing function. If, for ${\omega}\geq {\omega}_{0}$,

then

Lemma 2.12. ^[8] Let $n$ is a positive integer such that $n-1 < \alpha < n$. If $u\left({\omega}\right) \in C^{n}\left[a, b\right]$, then

In particular, if $0 < \alpha \leq 1$ and $u\left({\omega}\right) \in C^{1}\left[a, b \right]$

Lemma 2.13. ^[33] Let $u\left({\omega}\right)$ be a continuous function on $\left[0, +\infty \right)$ and satisfies

where $0 < \alpha < 1$ and $\theta$ is a constant. Then

Proof. For a nonnegative continuous function $h\left({\omega}\right)$, relation $\left(2.14\right)$ can be written as:

By taking the fractional integral of order $\alpha$ of $\left(2.15 \right)$, we have

Using Lemma $2.12$, we have

Since $h\left({\omega}\right)$ is a nonnegative continuous function, therefore by the definition of fractional integral we have

and

Combining $\left(2.16\right)$–$\left(2.19\right)$, we have

which implies

where we have used Lemma $2.11$. This is the desired result.

Lemma 2.14. ^[12,31] Consider the system

with initial condition $u\left({\omega}_{0}\right)$, where $0 < \alpha \leq 1$ and $g:\left[{\omega}_{0}, \infty \right) \times \Omega \rightarrow H,$ $\Omega \subset H$. If $g\left({\omega}, u\left({\omega}\right) \right)$ satisfies the locally Lipschitz condition with respect to $u$, then there exists a unique solution of $\left(2.20\right)$ on $\left[{\omega}_{0}, \infty \right) \times \Omega$.

Lemma 2.15. ^[12] For the real valued continuous function $g\left({\omega}, u\left({\omega}\right) \right)$, defined in $\left(2.20\right)$, we have $\left\Vert I_{{\omega}}^{\alpha }g\left({\omega}, u\left({\omega}\right) \right) \right\Vert \leq I_{{\omega}}^{\alpha }\left\Vert g\left({\omega}, u\left({\omega}\right) \right) \right\Vert$, where $\alpha \geq 0$ and $\left\Vert \cdot \right\Vert$ denotes an arbitrary norm.

3.   Main results

In this section we study the main properties of the dynamical system $\left(2.10\right)$ and analyze the global stability of the systems.

Theorem 3.1. Let the operator ${{\mathcal{T}}}$ is Lipschitz continuous with constant $\beta > 0.$ If $\gamma > 0$, then there exists a unique solution $u\left({\omega}\right) \in H$ of Problem $\left(2.10\right)$ with $u\left({\omega}_{0}\right) = u_{0}$, that is defined for all ${\omega}\in \left[{\omega}_{0}, \infty \right)$.

Proof. Let

To prove that $G\left(u\left({\omega}\right) \right)$ is Lipschitz continuous for all $u\left({\omega}\right), v\left({\omega}\right) \in H$, we have to consider

$\left\Vert G\left(u\left({\omega}\right) \right) -G\left(v\left({\omega}\right) \right) \right\Vert$

where we have used Lipschitz continuity of operator ${{\mathcal{T}}}$ with constant $\beta > 0$. This implies that operator $G\left(u\right)$ is a Lipschitz continuous in $H$. Thus from Lemma 2.14, it is clear that there exists a unique solution $u\left({\omega}\right)$ of Problem $\left(2.10 \right).$ Let $\left[{\omega}_{0}, \infty \right)$ be its maximal interval of existence; we show that ${{\mathcal{T}}}_{1} = \infty$. Consider

where we have used the Lipschitz continuity of operator ${{\mathcal{T}}},$ with constant $\beta > 0.$

Taking the fractional integral of $\left(3.1\right)$, we have

using Lemma 2.12 and Lemma 2.15, we have

From which by using Lemma 2.11, we obtain

where

Thus from $\left(3.2\right),$ it follows that the solution is bounded on $\left[{\omega}_{0}, \infty \right)$.

Theorem 3.2. Let the operator ${{\mathcal{T}}}$ be pseudomonotone and Lipschitz continuous. If $\gamma > 0,$ then the dynamical system $\left(2.10 \right)$ is stable in the sense of Lyapunov and globally converges to the solution of variational inequality $\left(2.1\right)$.

Proof. Since the operator is Lipschitz continuous, it follows from Theorem 3.1 that the dynamical system $\left(2.10\right)$ has a unique continuous solution $u({\omega})$ over $\left[{\omega}, {{\mathcal{T}}}_{1}\right)$ for any fixed $u_{0}\in {K}$. Let $u({\omega}, {\omega}_{0};u_{0})$ be he solution of the initial value Problem $\left(2.1\right)$. For a given $u^{\ast }\in {K}$, consider the following Lyapunov function

It is clear that $\underset{n\rightarrow \infty }{\lim }L\left(u_{n}\right) = \infty$, whenever the sequence $\left\{ u_{n}\right\} \subset {K}$ and $\underset{n\rightarrow \infty }{\lim }u_{n} = \infty$. Consequently, we conclude that the level sets of $L$ are bounded. Let $u^{\ast }\in {K}$ be a solution of the variational inequality $\left(2.1\right)$. Then

which implies, using pseudomonotonicity of operator ${{\mathcal{T}}},$

Setting $v = {P}_{{K}}\left[u-\rho {{\mathcal{T}}}u\right]$ in $\left(3.4\right)$, we have

Now, setting $v = u^{\ast }$, $u = {P}_{{K}}\left[u-\rho {{\mathcal{T}}}u\right]$ and $z = u-\rho {{\mathcal{T}}}u$ in $\left(2.2\right)$, we have

Adding $\left(3.5\right)$ and $\left(3.6\right)$, we have

using $\left(2.4\right)$, $\left(3.7\right)$ can be written as

which implies that

$\langle u-u^{\ast }, {{\mathcal{R}}}\left(u\right) -\rho {{\mathcal{T}}}u+\rho {{\mathcal{T}}}{P}_{{K}}\left[u-\rho {{\mathcal{T}}}u\right] \rangle$

where we have used the fact that the operator ${{\mathcal{T}}}$ is Lipschitz continuous with constant $\delta > 0$.

Thus from $\left(2.10\right)$, $\left(3.3\right)$ and $\left(3.8\right)$, we have

This implies that $L(u)$ is a global Lyapunov function for the System $\left(2.10\right)$ and the system is stable in the sense of Lyapunov. Since $\left\{ u\left({\omega}\right) :{\omega}\geq {\omega}_{0}\right\} \subset {K}_{0}$, where ${K}_{0} = \left\{ u\in {K}:L\left(u\right) \leq L\left(u_{0}\right) \right\}$ and the function is differentiable on the bounded and closed set ${K},$ then it follows from LaSalle's invariance principle that the trajectory will converge to $\Omega$, the largest invariant subset of the following subset:

Note that, if $D_{{\omega}}^{\alpha }L\left(u\left({\omega}\right) \right) = 0$, then

and hence $u$ is the equilibrium point of the dynamical system $\left(2.4\right)$, that is,

Conversely, if $D_{{\omega}}^{\alpha }u\left({\omega}\right) = 0$, then it follows that $D_{{\omega}}^{\alpha }L\left(u\left({\omega}\right) \right) = 0$. Thus, we conclude that

which is nonempty, convex and invariant set containing the solution set ${K}^{\ast }$. So

Therefore the dynamical system $\left(2.10\right)$ converges globally to the solution set of the variational inequalities $\left(2.1\right)$. In particular, if the set $E = \left\{ u^{\ast }\right\}$, then

Hence the dynamical system $\left(2.10\right)$ is globally asymptotically stable.

We now discuss the stability and existence of the equilibrium point for the dynamical System $\left(2.10\right)$ under some suitable conditions.

Theorem 3.3. Let the operator ${{\mathcal{T}}}$ be Lipschitz continuous with constant $\beta > 0.$ If $\gamma < 0$, then fractional dynamical System $\left(2.10 \right)$ is $\alpha$-exponentially stable.

Proof. Let $u\left({\omega}\right)$ and $v\left({\omega}\right)$ be any two solutions of dynamical System $\left(2.10\right)$ with initial values $u\left(0\right) = u_{0}$ and $v\left(0\right) = v_{0}$ respectively.

Let

then $e\left(0\right) \neq 0$ and taking the fractional derivative of above equation, we have

where $0 < \alpha < 1$, ${\omega}\geq 0$. From Theorem 3.1, $u\left({\omega}\right)$ and $v\left({\omega}\right)$ are uniquely determined solutions. Therefore $e\left({\omega}\right)$ is the uniquely determined solution of error System $\left(3.9\right)$ with initial value $e\left(0\right) = e_{0}$.

We claim that if $e\left(0\right) > 0$, then $e\left({\omega}\right) \geq 0$ for ${\omega}\geq 0$, if $e\left(0\right) < 0$, then $e\left({\omega}\right) \leq 0$ for ${\omega}\geq 0$. In fact, if $e\left(0\right) > 0$, there exists ${\omega}_{1}$, such that $e\left({\omega}\right) < 0$ for ${\omega}\geq {\omega}_{1}$, so there must be $0 < {\omega}_{0} < {\omega}_{1}$ such that $e\left({\omega}_{0}\right) = 0$. It means that dynamical System $\left(2.10\right)$ has two different solutions with initial value ${\omega}_{0}$ to $e\left({\omega}_{0}\right) \leq 0$ for ${\omega}\geq {\omega}_{0}$, which contradicts to Theorem 3.1. In a similar way, we can prove that if $e\left(0\right) < 0$, then $e\left({\omega}_{0}\right) \leq 0$ for ${\omega}\geq 0$.

So, we can have

which implies

thus by using Lemma 2.13, we have

where

Let $\theta _{1} = -\theta _{2},$ where $\theta _{2}$ is a positive constant. Then

which shows that the dynamical system $\left(2.10\right)$ is $\alpha$ -exponentially stable.

Remark 3.4. We would like to mention that, for $\alpha = 1,$ Theorems 3.1–3.3 are also valid under the same conditions. This gives us a new technique to study dynamical systems associated with variational inequalities.

4.   Conclutions

In this paper, we have introduced and studied fractional Wiener-Hopf dynamical systems for classical variational inequalities. These fractional dynamical systems related to the variational inequalities are proposed and analyzed using the projection technique. We have proved that these dynamical systems converge exponentially to the unique solution of variational inequality problems under some suitable conditions. The suggested dynamical systems can be used in designing recurrent neural networks for solving variational inequalities and related optimization problems.

Acknowledgments

The authors are thankful to the editor and the anonymous reviewers for their valuable comments and suggestions. This research was supported by the Department of Mathematics, Faculty of Science, Khon Kaen University, Fiscal Year 2022.

Conflict of interest

The authors declare that they have no conflicts of interest.