A General Inequality Related to Variational Inequalities and Its Consequences

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 .