Some new classes of general quasi variational inequalities

In this paper, we introduce and consider some new classes of general quasi variational inequalities, which provide us with unified, natural, novel and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We propose some new inertial projection methods for solving the general quasi variational inequalities and related problems. Convergence analysis is investigated under certain mild conditions. Since the general quasi variational inequalities include quasi variational inequalities, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inequalities for further research activities.


Introduction
Variational inequality theory, which was introduced by Stampacchia [1] in 1964 in potential theory, provides us with a simple, natural, unified, novel and general framework to study an extensive range of unilateral, obstacle, free, moving and equilibrium problems arising in fluid flow through porous media, elasticity, circuit analysis, transportation, oceanography, operations research, finance, economics, and optimization. It is worth mentioning that the variational inequalities can be viewed as a significant and novel generalization of the Riesz-Frechet representation theorem and Lax-Milgram lemma for linear continuous functionals and bifunction functions. For recent developments, see Noor and Noor [2] and the references therein. We would like to emphasize that the origin of the variational inequality theory can be traced back to Euler, Lagrange, Newton and Bernoulli's brothers. It is very simple fact that the minimum of a differentiable convex functions on the convex sets can be characterized by an inequality, some inertial projection methods for some classes of general quasi variational inequalities. Convergence analysis of these inertial type methods has been considered under some mild conditions. For more details see [31][32][33][34][35][36] and reference therein. Recently, Faraci et al. [37] and Jadamba et al. [38] have considered the theory of stochastic variational inequalities, which is another interesting aspect of variational inequalities. For more details, see [39][40][41][42][43][44] and the references therein.
Motivated and inspired by the recent research activities, we consider and study some new classes of quasi variational inequalities involving two arbitrary two operators, which are called the general quasi variational inequalities. We have shown that nonsymmetric and odd-order obstacle boundary vale problems can be studied in the framework of general quasi variational inequalities. Some special important cases are also discussed. We have shown that the general quasi variational inequalities are equivalent to the fixed point problems. This equivalence is used to propose some new inertial type methods for solving general quasi variational inequalities. These inertial methods include the extragradient method of Koperlevich [39]and double modified projections of Noor [7]. The convergence of the proposed inertial methods is considered. We have only considered theoretical aspects of the suggested methods. It is an interesting problem to implement these methods and to illustrate the efficiency. Comparison with other methods need further research efforts. The ideas and techniques of this paper may be extended for other classes of quasi variational inequalities and related optimization problems.

Basic definitions and results
Let K be a set in a real Hilbert space H with norm · and inner product ·, · . Let T, g : H −→ H be nonlinear operators in H. Let K : H −→ H be a set-valued mapping which, for any element µ ∈ H, associates a convex-valued and closed set K(µ) ⊂ H.
We also need the following concepts.
For the nonsymmetric and odd-order problems, many methods have developed by several authors to construct the energy functional of type (2.4) by introducing the'concept of g-symmetry and g-positivity of the operator g.
Definition 2.1. [45,46]. ∀u, v ∈ H, the operator T : H −→ H is said to be : (a). g-symmetric , if and only if, (c). g-coercive (g-elliptic, if there exists a constant α > 0 such that Note that g-coercivity implies g-positivity, but the converse is not true. It is also worth mentioning that there are operators which are not g-symmetric but g-positive. On the other hand, there are g-positive, but not g-symmetric operators. We consider the problem of finding µ ∈ H : g(µ) ∈ K(µ), such that which is called the general quasi variational inequality.

Applications
To convey an idea of the applications of the quasi variational inequality, we consider the third-order implicit obstacle boundary value problem of finding u such that where f (x) is a continuous function and M(u) is the cost (obstacle) function. The prototype encountered is In (2.3), k represents the switching cost. It is positive when the unit is turned on and equal to zero when the unit is turned off. Note that the operator M provides the coupling between the unknowns u = (u 1 , u 2 , . . . , u i ). We study the problem (2.2) in the framework of general quasi variational inequality approach. To do so, we first define the set K as , on Ω}, which is a closed convex-valued set in H 2 0 (Ω), where H 2 0 (Ω) is a Sobolev (Hilbert) space, see [4,5,45]. One can easily show that the energy functional associated with the problem (2.2) is It is clear that the operator T defined by (2.5) is linear, nonsymmetric and g-positive. Using the technique of Noor [7,47], one can show that the minimum of the functional I[v] defined by (2.4) associated with the problem (2.2) on the closed convex-valued set K(u) can be characterized by the inequality of type which is exactly the quasi variational inequality (2.1).

Special cases
We now discuss some special cases of general quasi variational inequalities (2.1) which is called the general variational inequality, introduced and studied by Noor [6,7,21,23]. It has been shown a wide class of nonsymmetric and odd-order obstacle boundary value and initial value problems can be studied in the general framework of general variational inequalities (2.6).
which is known as the general quasi complementarity problem and appears to be a new one. If g = I, then problem 2.8 is called the implicit complementarity problem, introduced and studied by Noor [13]. For g(u) = m(u) + K, where m is a point-to-point mapping, the problem(2.8) is called the implicit (quasi) complementarity problem. If g ≡ I, then problem (2.8) is known as the generalized complementarity problems. Such problems have been studied extensively in recent years. 3. For g = I, problem (2.1) is equivalent to finding µ ∈ K(µ) such that is known as quasi variational inequality introduced and studied by Bensoussan and Lions [10]. 4. If we take K(µ) = K, and g = I, the identity operator, then problem (2.1) reduces to the variational inequality: That is, finding µ ∈ K, such that is classical variational inequality. It was introduced and studied by Stampacchia [1].
For a different and appropriate choice of the operators and spaces, one can obtain several known and new classes of variational inequalities and related problems. This clearly shows that the problem (2.1) considered in this paper is more general and unifying one.
We need the following well-known definitions and results in obtaining our results. i. The mapping T is called r-strongly monotone (r ≥ 0), if For ξ = 0, T is r-strongly monotone. The class of relaxed (ξ , r)−cocoercive mapping is the generalized class than the r−strongly monotone mapping and ξ−cocoercive. iv. The mapping T is called η−Lipschitz continuous (η > 0), if The following projection result plays an indispensable role in achieving our results.
if and only if The implicit projection operator Π K(µ) is nonexpansive and has the following characterization.
In many important applications, the convex-valued set K(u) is of the form where m is a point-to-point mapping and K is a closed convex set. In this case, If m is a Lipschitz continuous with constant ν, then This show that the Assumption 2.1 holds.

Inertial projection methods
In this section, we suggest some new inertial-type approximation schemes for solving the general quasi variational inequality (2.1). One can prove that the general quasi variational inequality (2.1) is equivalent to fixed point problem by using Lemma 2.1.
where ρ > 0 is a constant and Π K(µ) is the projection of H into K(µ).
Lemma 3.1 implies that the problem (2.1) is equivalent to a fixed point problem (3.1). This alternate form is very useful from both numerical and theoretical point of views.
Using the result (3.1), we can propose some iterative approximation schemes for solving the general quasi variational inequality (2.1).
Such type of inertial projection methods for solving general variational inequalities have been considered by Noor [7] and Noor et al. [7][8][9].
Using this technique, we can suggest the following inertial type methods for solving general quasi variational inequalities (2.1).
For a different and suitable choice of operators and spaces in Algorithm (3.9), one can obtain numerous new and previous iterative schemes for solving inequality (2.1) and related problems. This shows that the Algorithm (3.9) is quite flexible and unifying ones.
(I.) If K(µ) = K, then the following result can be obtained from Theorem 4.1.
(II.) If K(µ) = K and g = I, then we get the following result from Theorem 4.1.

Conclusions
In this paper, several new inertial projection methods have been suggested and analyzed for solving general quasi variational inequalities involving two operators. It is shown that odd-order implicit obstacle problems can be studied in the unified frame work of general quasi variational inequalities. We have proved that the general quasi variational inequalities are equivalent to the implicit fixed point problems. This alternative equivalent formulation is used to suggest a wide class of inertial type iterative methods for solving general quasi variational inequalities. Several important special cases are discussed. Convergence analysis of these proposed inertial projection methods is investigated. It is an interesting problem to compare the efficiency of the proposed methods with other known methods. Similar methods can be suggested for stochastic variational inequalities, which is an interesting and challenging problem. We expect that the ideas and techniques of this paper will motivate and inspire the interested readers to explore its applications in various fields.