A General Inequality Related to Variational Inequalities and Its Consequences

Abstract

Let X  be a Hausdorff topological vector space with dual X*  and K  a nonempty closed and convexsubset of X. Let the value of u E X*  at x E  be denoted by (u,x). Let g:K - R   be a map (possibly nonlinear). The classical minimization problem for the pair(g,K)  is to find xo E K   such that g(xo) = min g(y). If we define a function f : K X K-R as f (x,y) =g (y)-g(x) for all x,yEK , then the above problem reduces to the problem of finding xo E K  such that f (xo,y) > 0 for all y E K .