A Classes of Variational Inequality Problems Involving Multivalued Mappings

The main objective of the Variational inequality problem is to study some functional analytic tools, projection method and fixed point theorems and then exploiting these to study the existence of solutions and convergence analysis of iterative algorithms developed for some classes of Variational inequality problem. The main objective of this paper is to study the existence of solutions of some classes of Variational inequalities using fixed point theorems for multivalued and using Banach contraction theorem we prove the existence of a unique solution of multi value Variational inequality problem.


Introduction
Variational inequalities and complementarity problem play equally important roles in applied mathematics, physics, control theory and optimization, equilibrium theory of transportation and economics, mechanics, and engineering sciences. We study the existence and convergence of solutions of some classes of Variational inequalities using fixed point theorem for multivalued mappings we develop an iterative algorithm for prove the approximate solution converges to solution of multi value Variational inequality problem.
Definition (1)(2): If and project any point , on the axis then the projection , 0 . Theorem (1-3): Let be a real Hilbert space, let ⊂ be a nonempty closed convex set and let due the projection mapping on . Then is non expansive, monotone, but not strictly monotone and strongly continuous.
Proof: Let us first note that the characterization of the projection # $ , # $ % 0, # ∈ can be written To show that is monotone, let us fix & '& and write And therefore Which in particular implies the monotonicity of P -, further if K H then P -1, an identity mapping, and one has strict monotonicity, but in general for K 0 H then is not injective and hence not strictly monotone, If ∉ then To show that Pis non expansive it is enough to apply the Schwartz inequality to (1.5) and obtain thus Or, dividing by ‖ − ‖ = 0 then ‖ − ‖ ≤ ‖ − ‖. the strong continuity follows immediately form (5).
Definition (2-2): let 7: → is said to be: A point is said to be fixed point 4 if = .
Fixed Point Problem: Let 4 be a mapping defined on a metric space , into itself, find ∈ such that 4 = .
And there exists a positive integer \ such that  We prove the existence of a unique solution of multivalued Variational inequality problem  Theorem (3-2): Let 4: → 2 6 be a % − Lipschitz continuous and x -strongly monotone multivalued mapping and let 5: → 2 6 be J −Lipschitz continuous multivalued mapping. then multivalued Variational inequality problem (3-1) has a solution.
Proof: By Lemma (3-1), it is enough to prove that multivalued mapping is contraction mapping.
Let P ∈ B & P ∈ B , we have Since { < 1. we have B W_ converges to B strongly in, we have L W_ → L strongly in and t W_ → t strongly in . This completes the proof #

Conclusions
We study the existence of solutions of some classes of Variational inequalities using fixed point theorems for multivalued and using Banach contraction theorem we prove the existence of a unique solution of multi value Variational inequality problem discussed in the article research.