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 ψ : X × Y → ℝ be two functions and let S : X → 2 Y and T : Y → 2 F ∗ be two maps. Then the generalized g -quasivariational inequality problem (G g QVI) is to find a point x ¯ ∈ X and a point f ∈ T ( g ( x ¯ ) ) such that g ( x ¯ ) ∈ S ( x ¯ ) and sup y ∈ S ( x ¯ ) { Re ⁡ 〈 f , y − g ( x ¯ ) 〉 + ψ ( x ¯ , y ) } = ψ ( x ¯ , g ( x ¯ ) ) . In this paper, we prove the existence of a solution of (G g QVI).