Generalizedg-quasivariational inequality

Suppose that X is a nonempty subset of a metric space E and Y is a nonempty subset of a topological vector space F. Let g:X→Y and È:X×Y→℠be two functions and let S:X→2Y and T:Y→2F∗ be two maps. Then the generalized g-quasivariational inequality problem (GgQVI) is to find a point x¯∈X and a point f∈T(g(x¯)) such that g(x¯)∈S(x¯) and supy∈S(x¯){Reâ¡〈f,y−g(x¯)〉+È(x¯,y)}=È(x¯,g(x¯)). In this paper, we prove the existence of a solution of (GgQVI)