(B) is true: there exists ¯ x ∈ X such that f (¯ x, y ) = ϕ (¯ x ) − ϕ ( y ) ≤ 0 , for all y ∈ X. We now derive from the nonlinear alternative in Theorem 14 the celebrated theorem of Stampacchia for variational inequalities. Recall that given a normed space E, a form a : E × E −→ R is said to be (i) bilinear if it linear in each of its arguments; (ii) continuous if there exists a constant C > 0 with | a ( x, y ) | ≤ C || x |||| y || for all x, y ∈ E ; and (iii) coercive if there exists a constant α > 0 with a ( x, x ) ≥ α || x || 2 for all x ∈ E. Corollary 16 (Stampacchia Theorem) Let E be a reflexive Banach space, a : E × E −→ R be a continuous and coercive bilinear form, and let ℓ : E −→ R be a bounded linear functional. Given a non-empty closed and convex subset X in E, there exists a unique ¯ x ∈ X such that a (¯ x, ¯ x − y ) ≤ ℓ (¯ x ) − ℓ ( y ) for all y ∈ X. Proof. For the existence, we apply Theorem 14 to f : X × X −→ R defined by f ( x, y ) := a ( x, x − y ) − ℓ ( x − y ) , ( x, y ) ∈ X × X, λ = 0 , D = { y 0 } with 0 = y 0 ∈ X arbitrary, and K := { x ∈ X : ‖ x ‖ ≤ M } where β = ( C || y 0 || + || ℓ || ) /α and γ = || ℓ |||| y 0 || /α. Indeed, first note that if E is equipped with the weak topology, then f ( x, y ) is l.s.c. and quasiconvex in x and quasiconcave in y (it is in fact linear and continuous for the norm topology in both arguments). Since X is closed and convex, it follows that K is a closed, convex and bounded, hence weakly com- pact, subset of X. D is obviously a weakly compact subset of X. Note now that if f ( x, y 0 ) ≤ 0 for any given x ∈ X, i.e., a ( x, x ) ≤ a ( x, y 0 ) + ℓ ( x − y 0 ) , then ‖ x ‖ 15 M := 1 2 ( β + √ β 2 + 4 γ ) ,