COMPLEMENTARITY PROBLEM FOR η-MONOTONE-TYPE MAPS

The theory of complementarity problem (CP) has a unique place in optimization theory due to its wide range of applications to various problems like equilibrium problems in economics and certain nonlinear optimization problems. Variational inequality problems are also very important due to their wide range of applications to various problems, that is, constrained mechanics problems, fluid flow through porus media, transportation problems, various engineering applications, and many others. In 1964, R. W. Cottle mentioned the complementarity problem in his Ph.D. thesis. Stampacchia discovered variational inequality in the year 1966. Both the theories of variational inequality and complementarity problem are related, and these two problems are essentially the same as was shown by Karamardian [2] in the year 1971. Generally, people working in applied mathematics or engineering sciences use variational inequality and their problem is infinite-dimensional, whereas people working in economic and operations research use complementarity problem and their problem is finite-dimensional. Many people have worked on various aspects like existence, uniqueness, and algorithmic approach of the solution under different conditions on the operator as in [1, 2]. In this paper, we prove the existence of solution of complementarity problems in Banach space. We define the complementarity problem for η-monotonicity and establish some results. Before going to the results we mention the preliminaries.


Introduction
The theory of complementarity problem (CP) has a unique place in optimization theory due to its wide range of applications to various problems like equilibrium problems in economics and certain nonlinear optimization problems.Variational inequality problems are also very important due to their wide range of applications to various problems, that is, constrained mechanics problems, fluid flow through porus media, transportation problems, various engineering applications, and many others.In 1964, R. W. Cottle mentioned the complementarity problem in his Ph.D. thesis.Stampacchia discovered variational inequality in the year 1966.Both the theories of variational inequality and complementarity problem are related, and these two problems are essentially the same as was shown by Karamardian [2] in the year 1971.Generally, people working in applied mathematics or engineering sciences use variational inequality and their problem is infinite-dimensional, whereas people working in economic and operations research use complementarity problem and their problem is finite-dimensional.Many people have worked on various aspects like existence, uniqueness, and algorithmic approach of the solution under different conditions on the operator as in [1,2].In this paper, we prove the existence of solution of complementarity problems in Banach space.We define the complementarity problem for η-monotonicity and establish some results.
Before going to the results we mention the preliminaries.

Preliminaries
Let X be a reflexive real Banach space and let X * be its dual.In particular, X could be R n ; in that case X * = R n .The value of f ∈ X * at x ∈ X is denoted by ( f ,x).Let K be a closed convex cone in X.
The nonlinear variational inequality (NVI) problem with respect to η is defined as follows.
Find x such that Another NVI can be stated as follows.Find x such that We refer to the above variational inequalities as NVI(1) and NVI(2), and their solution sets as NVIS(1) and NVIS(2), respectively.If η(y,x) = y − x, then the above variational inequalities reduce to usual variational inequalities.
Let K be a closed convex cone in X.Let K * be the subset of X * defined by will be called the generalized complementarity problem (GCP).Let C denote the set of all solutions of GCP.
Case 2. Let x u ∈ S u for all u ∈ K. Then by our assumption, we have a solution for the complementarity problems.
Note that the result holds if η is linear.Certainly, the result is not true if η is any continuous convex function.
For example, if X = R, K = [0,1], Tx = −1, and η(y,x) = |y − x|.Then for any x 0 ∈ K, if we choose y ∈ K such that y = x 0 , we have (Tx 0 ,η(y,x)) = −|y − x|.But for the result to be true, there would exist x 0 ∈ K such that −|y − x 0 | ≥ 0 for all y ∈ K and this cannot happen.