Perturbed mixed variational-like inequalities

1.   Introduction

Variational inequalities introduced by Stampacchia ^[1] have played an important role in studying various problems which arise in transportation, elasticity, economics, operational research, optimization and in the field of pure and applied sciences, see ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]} and references therein. The minimum of the differentiable convex function on convex set can be characterized by variational inequalities.

In recent years, the concept of convexity has been generalized and extended in many directions, which has potential and important applications in many fields. A significant generalization of convex functions is that of invex functions, which was introduced by Hanson ^[4]. In mathematical programming, Hanson^[4] showed many results involving convex functions actually hold for invex functions. A generalized class of convex functions, preinvex functions was introduced by Ben-Israel and Mond ^[2] and Weir and Jeyakumar ^[16]. In optimization problems, Weir and Mond ^[17] studied that the convex functions can be replaced by preinvex functions. Noor ^[8] and Yang and Chen ^[18] proved that the optimal conditions of differentiable preinvex functions on the invex set can be characterized by a class of variational inequalities, known as variational-like inequalities, which were introduced and studied by Parida and Sen ^[13].

In this paper, we introduce and consider a new class of variational-like inequalities which is known as perturbed mixed variational-like inequalities. The existence of the solution of perturbed mixed variational-like inequalities cannot be studied by resolvent method and projection method as these perturbed mixed variational-like inequalities involve the nonlinear terms and bifunction $\eta(., .)$. This fact motivated us to use the auxiliary principle technique to develop the existence theory for variational-like inequalities. Auxiliary principle technique was introduced and developed by Lions and Stampacchia ^[5] and Glowinski et al.^[3]. To study the existence of a solution of perturbed mixed variational-like inequalities, we use the auxiliary principle technique. Some special cases are also considered. Our results show a significant refinement and improvement of the already existing results.

2.   Preliminaries

Let H be a real Hilbert space. The inner product and norm are denoted by $\langle., .\rangle$ and $\parallel.\parallel$ respectively. Let $K_{\eta}$ be a nonempty invex set in H and $\eta(., .): H\times H\rightarrow R$ be a continuous bifunction.

First, we recall the well-known basic definitions and concepts.

Definition 2.1: ^[7] Let $K$ be a nonempty closed set in H. The set $K$ is said to be a convex set, if

Definition 2.2:^[7] A function $F: K \rightarrow R$ is said to be a convex function, if

Definition 2.3:^[15] A function $F: K \rightarrow R$is said to be a strongly convex function, if there exists a constant $\mu > 0,$ such that

Definition 2.4:^[16,17] A set $K_{\eta}$ is said to be an invex set, if there exists a bifunction $\eta(., .)$, such that

Definition 2.5:^[6,17] A function $F: K_{\eta} \rightarrow R$ is said to be a preinvex function, if there exists a bifunction $\eta(., .)$, such that

Definition 2.6:^[10,11] A function $F: K_{\eta} \rightarrow R$is said to be a strongly preinvex function with respect to a bifunction $\eta(., .)$, if there exists a constant $\mu > 0,$ such that

We note that, if $\eta (v, u) = v-u$, the invex set $K_{\eta}$ reduces to the convex set $K$, preinvex functions and strongly preinvex functions reduce to convex functions and strongly convex functions.

Now we consider the functional $I[v]$, defined as

where $F$ is a differentiable preinvex function and $\phi$ is a nondifferentiable strongly preinvex function. The minimum $u \in K_{\eta}$ of the functional $I[v]$, defined by (2.1), can be characterized by a class of variational-like inequalities.

The inequality (2.2) is called perturbed mixed variational-like inequality.

It is well-known that in many significant applications, variational-like inequalities do not arise as a result of minimization problem. This motivated us to consider a more general class of variational-like inequalities, of which (2.2) is a special cases.

Given an operator $T: H \rightarrow R$ and a function $\Phi: H \rightarrow R$, we consider the problem of finding $u \in H$ and constant $\mu > 0,$ such that

The inequality (2.3) is called perturbed mixed variational-like inequality (PMVLI).

If $T u = F^{\prime} (u),$ then PMVLI (2.3) includes inequality (2.2) as a special case. These perturbed mixed variational-like inequalities are more general and include classical mixed variational-like inequalities and related optimization problems as special cases. For the applications of PMVLI (2.3), see ^[2,3,5,9].

We now discuss some special cases of PMVLI (2.3).

(Ⅰ) For $\eta (v, u) = v-u,$ problem (2.3) reduces to

which is known as strongly mixed variational inequality, introduced and studied by Noor ^[8].

(Ⅱ) For $\mu = 0,$ problem (2.3) is equivalent to finding $u \in H,$ such that

which is known as the mixed variational-like inequality (MVLI), introduced by Noor ^[9]. For the formulation of numerical methods and applications of (2.5), see ^[9,10,11,12].

(Ⅲ) For $\eta(v, u) = v-u,$ problem (2.5) is equivalent to finding $u \in H,$ such that

which is known as mixed variational inequality (MVI), introduced and studied by Lions and Stampacchia ^[4]. Mixed variational inequalities have applications in elasticity, structural engineering and electronic network, see ^[3,9,10,11] and the references therein.

(Ⅳ) If $\Phi (.)$ is an indicator function of a closed invex set $K_{\eta}$ in H, that is

then problem (2.5) is equivalent to finding $u\in K_{\eta},$ such that

which is known as the variational-like inequality (VLI), introduced by Parida and Sen ^[13]. For the applications of variational-like inequalities and optimization problems, see ^[9,10,12].

(Ⅴ) For $\eta(v, u) = v-u,$ problem (2.7) reduces to

which is known as variational inequality, proposed and studied by Stampacchia ^[1] and Lions and Stampacchia ^[5]. Several equilibrium and obstacle problems in finance, economics, mathematical and engineering sciences can be studied by variational inequality of type (2.8), see ^{[3,5,8,13,16,18]} and the references therein.

For a proper and suitable choice of $T, \eta, \Phi$ and the invex set, one can get a number of known and unknown variational inequalities and complementary problems. It is evident that the problems (2.4)–(2.8) are special cases of the PMVLI (2.3). In fact, PMVLI (2.3) is the most general and unifying one, which is the inspirational force of our work.

We also need the following assumption regarding the bifunction $\eta(., .)$ to obtain the main results.

Assumption 2.1: The bifunction $\eta(., .): H\times H\rightarrow H$ satisfies the following condition

We assume that $\eta (v, u)$ is skew symmetric, that is, $\eta (v, u) = - \eta (u, v), \forall u, v \in H,$ unless otherwise.

The assumption 2.1 played an important part in the study of the existence of a solution of variational-like inequalities, see ^[9,10].

Definition 2.6: ^[9] An operator $T: H \rightarrow H$ is said to be:

(ⅰ) strongly $\eta-$monotone, if there exists a constant $\alpha > 0$ such that

(ⅱ) $\eta-$ monotone, if

(ⅲ) partially relaxed strongly $\eta-$monotone, if there exists a constant $\alpha > 0$ such that

(ⅳ) Lipschitz continuous, if there exists a constant $\beta > 0$ such that

We note that for $z = u$, partially relaxed strongly monotonicity reduces to monotonicity.

Definition 2.7: ^[11] The bifunction $\eta(., .):H \times H\rightarrow H$ is said to be:

(a) Strongly monotone: if there exists a constant $\sigma > 0,$ such that

(b) Lipschitz continuous: if there exists a constant $\delta > 0,$ such that

From (a) and (b), it is observed that $\sigma \leq \delta.$

If $\eta(v, u) = Tv-Tu$, then definition (2.7) reduces to the strongly monotonicity and Lipschitz continuity of the nonlinear operator $T.$

3.   Existence results

In this section, we consider the conditions under which the perturbed mixed variational-like inequality (2.3) has an unique solution. We use the auxiliary principle technique of Glowinski et al.^[3] to study the existence of the solution of perturbed mixed variational-like inequality (2.3).

Theorem 3.1: Let the operator $T: H \rightarrow H$ be strongly monotone and Lipschitz continuous with constants $\alpha > 0$, $\beta > 0$ and the bifunction $\eta(., .):H \times H\rightarrow H$ be strongly monotone and Lipschitz continuous with constants $\sigma > 0$, $\delta > 0,$ respectively. If assumption 2.1 holds and there is a constant $\rho > 0,$ such that

where

then there exists a unique solution of (2.3).

Proof (a) Uniqueness: Let $u_1 \neq u_2 \in H,$ be two solutions of problem (2.3). Then, we have

Replacing $v$ by $u_2$ in (3.3) and $v$ by $u_1$ in (3.4), adding the resultants and using the Assumption 2.1, we have

which can be written as

Since $T$ is a strongly monotone and Lipschitz continuous with constants $\alpha > 0$ and $\beta > 0,$ we have

Since $\eta (., .)$ is strongly monotone and Lipschitz continuous with constants $\sigma > 0$ and $\delta > 0$, we have

From (3.2), (3.6) and (3.7), we get

It follows that

Thus, for $\alpha > \gamma+2\mu\delta^{2}$, we get

which implies that $u_1 = u_2$ is the uniqueness of the solution of (2.3).

Existence: We use the auxiliary principle technique to prove the existence of a solution of problem (2.3), which is mainly due to Glowinski et al.^[3] as developed by Noor ^[10].

For a given $u \in H$ satisfying (2.3), we consider the problem of finding $w \in H,$ such that

where $\rho > 0$ is the constant.

The inequality (3.9) is called auxiliary perturbed mixed variational-like inequality. The relation (3.9) defines a mapping $w = w(u)$ between the problem (2.3) and (3.9). In order to prove the existence of a solution of problem (2.3), it is enough to show that the connecting mapping is a contraction mapping and consequently has a unique fixed point.

For $w_1 \neq w_2 \in H$ (corresponding to $u_1 \neq u_2 \in H$), consider two solutions of (3.9). Then

and

By taking $v = w_2$ in (3.10) and $v = w_1$ in (3.11) and by adding the resultants and using the Assumption 2.1, we get

Thus, we have

Since $T$ is strongly monotone and Lipschitz continuous with constants $\alpha > 0$ and $\beta > 0,$ we have

By combining (3.2), (3.7), (3.12) and (3.13) and using the Lipschitz continuity of $T,$ we have

where

From above, it follows that

where

From (3.1), it follows that $\theta < 1,$ so the mapping $w$ defined by (3.9) is a contraction mapping and consequently, it has a fixed point $w(u) = u \in H$ which satisfy the perturbed mixed variational-like inequality (2.3). $\square$

Special cases:

(Ⅰ) If $\eta (v, u) = v-u,$ Theorem (3.1) reduces to:

Theorem 3.2: ^[8] Let the operator $T: H \rightarrow H$ be strongly monotone and Lipschitz continuous with constants $\alpha > 0$, $\beta > 0$ respectively. If there is a constant $\rho > 0,$ such that

then there exists a unique solution $u \in H$ satisfying strongly mixed variational inequality (2.4).

(Ⅱ) If $\mu = 0,$ Theorem (3.1) reduces to:

Theorem 3.3: ^[9] Let the operator $T: H \rightarrow H$ be strongly monotone and Lipschitz continuous with constants $\alpha > 0$, $\beta > 0$ and the bifunction $\eta(., .):H \times H\rightarrow H$ be strongly monotone and Lipschitz continuous with constants $\sigma > 0$, $\delta > 0,$ respectively. If assumption 2.1 holds and there is a constant $\rho > 0,$ such that

then there exists a unique solution $u \in H$ satisfying mixed variational-like inequality (2.5).

For different choices of bifunction $\eta (., .)$ and invex set $K_{\eta}$, one can obtain various known and new results as special cases of Theorem (3.1). This shows that our results are more general and unifying one.

It can be shown that the solution of the auxiliary perturbed mixed variational-like inequality (3.9) can be characterized as a minimum of the functional $I[w]$ on the invex set $K_{\eta}$ in $H$, where

The functional $I[w]$ can be used to construct the class of gap (merit) functions for variational-like inequality (2.3). This functional can be used to propose and analyze many iterative methods for solving perturbed mixed variational-like inequalities. For more details, see ^[9,10,13] and the references therein.

4.   Conclusions

In this paper, we have introduced and considered a new class of variational-like inequalities, which is called the perturbed mixed variational-like inequalities. We have studied the existence of the solution of PMVLI by using the auxiliary principle technique. Some special cases have been also considered under some suitable conditions. Further research is required to develop some numerical schemes to find the approximate solution of these inequalities, which will be quite interesting problem for new researchers.

Acknowledgements

The authors would like to thank the Rector, COMSATS University Islamaabd, Islamabad, Pakistan, for providing excellent research and academic environments. Authors are grateful to the referees for their valuable suggestions and comments.

Conflict of interest

The authors declare no conflict of interest.