-PROJECTIVE AND -INVOLUNTARY VARIATIONAL INEQUALITIES AND IMPROVED PROJECTION METHOD FOR PROJECTED DYNAMICAL SYSTEM

Purpose of study: To introduce the concept of projective and involuntary variational inequality problems of order and respectively. To study the equivalence theorem between these problems. To study the projected dynamical system using self involutory variational inequality problems. Methodology: Improved extra gradient method is used. Main Finding: Using a self-solvable improved extra gradient method we solve the variational inequalities. The algorithm of the projected dynamical system is provided using the RK-4 method whose equilibrium point solves the involutory variational inequality problems. Application of this study: Runge-Kutta type method of order 2 and 4 is used for the initial value problem with the given projected dynamical system with the help of self involutory variational inequality problems. The originality of this study: The concept of self involutory variational inequality problems, projective and involuntary variational inequality problems of order and respectively are newly defined.


INTRODUCTION
In a Banach space X with dual , we say that an operator A: X → is -projective if its minimal polynomials for some and an operator is -involutory if its minimal polynomial equation is for some . Now for any two Banach spaces X and Y, if f: and g:

LITERATURE REVIEW
In the recent decades, the theory of variational inequality is used to solve various types of inequality, and equilibrium problems arises in the branches of Engineering, Physical Sciences, Applied Mathematics, Finance, Medical, and so on. In fact, the problem are can be expressed in the form of variational inequality problems (VIP) which is introduced by Various authors have studied the theory of variational inequalities using the projection method. Solodov and Svaiter (1997) have developed the improved projection method to solve the variational inequalities. Nagurney, A., &Zhang, D. The variational inequality problems (VIP) is to find x K such that where the directed feasible set K + (x) of the solution of VIP is defined by The dual variational inequality problems (DVIP) is to find x K such that where directed feasible set K − (x) of the solution of DVIP is defined by

Projection operator and its application:
A map P X → X is a projection map if it satisfies P 2 = P, i.e., P(x) = 0 or (I − P)(x) = 0 for all x X. The zero space of P is defined by and the range space of P is defined by In fact, X = Z(P)⊕R(P), i.e., X = Z(P) ∪R(P) and Z(P) ∩ R(P) = . Let X be a Hilbert space and K be a subset of X.

Definition
The projection operator K → K is non-expansive, i.e.,∥ (x 1 ) − (x 2 )∥≤∥x 1 − x 2 ∥for all x 1 ,x 2 K, i.e., F, implying is continuous on K and has a fixed point in K. It is obvious that (x) = x for all x K and holds the following result. If K is a closed set in X, then for each x K, there exists an unique y K such that y = (x), i.e., ∥x − y∥ = ∥x − z∥, which can be written as a problem to find an unique y K such that ⟨y,z− y⟩ ≥ ⟨x,z− y⟩ for all z K. This is a particular representation of the variational inequality problem where F(x) = y − x.

f-Projection Method
Let f K → K be a Lipschiz continuous function on X. For our need, we define the concept of projection functional operator as follows: Definition: Let f X → X be any linear function. If for each x K, there exists an unique y K such that for all For f K has a fixed point, implying f has a fixed point on K. Let x = y be the fixed point of f, then f(x ) = x = y , we get ⟨y ,z− y ⟩≥ ⟨x + f(x ) -y ,z-y ⟩ = ⟨x ,z-y ⟩ = ⟨f(x ),z -y ⟩ for all z K where y is the unique point corresponds to x . This completes the proof.
Definition: For x K and v X, the f-projection of the vector v at x (with respect to K) is defined by = .

σ-Involutory Variational Inequality Problems and Projection Method
Let X be a Banach space. We consider a class of map A X → X satisfying the condition A 3 = σA for some σ R. In this case, we have either for all x X. For simplicity, we denote the operator = =σ −1 A 2 and = I − A 2 ( ;σ) = A −1 − σ −1 A. Thus for each x X, For σ = 1, we have A 1,2 = A 2 . Therefore A σ,2 = σ −1 A and A σ,2 = I − A σ,2 .
The zero space of A and the range space of A are defined by It is obvious that R(A) = σy= Z(A 1,2 ) but Z(A 1,2 ) Z(A σ,2 ). Thus X has a superclass partition as For σ <0, we say A is skew σ-involutory or idempotent and for ρ <0, we say B is skew ρ-projective.
Example: Let the function f R → R be any arbitrary function.
(i) The matrix A(x,y) = Inv(N 2 ;x 2 ) and the transpose of A(x,y) is also x 2 -involutory for all x X.
(ii) The matrix Inv(N 2 ; xy) and the transpose of B(x,y) is also xy-involutory for all x X.
(iii) The matrix C(x,y)= Inv(N 3 ;x 2 ) and the transpose of C(x,y) is also x 2 -involutory for all x X.
(iv) In general, the matrix D(x,y) = (d ij ) where for 1 ≤ i,j ≤ n and the transpose of D(x,y) is also x n -involutory for all x X.

RESULTS/FINDINGS
The Problems: Let X be a Hilbert space and K X, P K → be a continuous, invertible map and non-expansive map on K. The concepts of involutory variational inequality problem (IVIP) and Projective variational inequality problem (PVIP) are defined as follows: 1. The σ-involutory variational inequality problem (IVIP) is to find y K such that ⟨A 2 (y;σ),x − y⟩ ≥ 0 for all x K 2. The ρ-projective variational inequality problem (PVIP) is to find y K such that ⟨B(y;ρ),x − y⟩ ≥ 0 for all x K