satisfies a quadratic inequality and is bounded above by M : = ⇒ ⇐⇒ = ⇒ α || x || 2 ≤ a ( x, x ) ≤ C ‖ x ‖‖ y 0 ‖ + ‖ ℓ ‖‖ x ‖ + ‖ ℓ ‖‖ y 0 ‖ α || x || 2 − ( C ‖ y 0 ‖ + ‖ ℓ ‖ ) ‖ x ‖ − ‖ ℓ ‖‖ y 0 ‖ ≤ 0 || x || 2 − β ‖ x ‖ − γ ≤ 0 ‖ x ‖ ≤ 1 $_{2}$( β + √ β 2 + 4 γ ) = M. Consequently, if x ∈ X, ‖ x ‖ > M, then f ( x, y 0 ) > 0 and the compactness condition in Theorem 14 is satisfied. Since f ( x, x ) = 0 for any x ∈ X, (A) of Theorem 14 is impossible, and (B) holds, i.e., f (¯ x, y ) = a (¯ x, ¯ x − y ) − ℓ (¯ x )+ ℓ ( y ) ≤ 0 for some ¯ x ∈ X and all y ∈ X and the proof of the existence is complete. The uniqueness follows at once from the bilinearity and the coercivity of the form a as follows: if a (¯ x$_{i}$, ¯ x$_{i}$ − y ) − ℓ (¯ x$_{i}$ ) + ℓ ( y ) ≤ 0 for two elements ¯ x$_{i}$ ∈ X, i = 1 , 2 , and all y ∈ X, then adding a (¯ x 1 , ¯ x 1 − ¯ x 2 ) ≤ ℓ (¯ x 1 ) − ℓ (¯ x 2 ) to a (¯ x 2 , ¯ x 2 − ¯ x 1 ) ≤ ℓ (¯ x 2 ) − ℓ (¯ x 1 ) gives 0 ≤ α ‖ ¯ x 1 − ¯ x 2 ‖ 2 ≤ a (¯ x 1 − ¯ x 2 , ¯ x 1 − ¯ x 2 ) ≤ 0 , i.e., ¯ x 1 = ¯ x 2 . The coincidence ( N , N − $^{1}$) (Theorem 12) can be expressed in analytical terms as a second alternative for nonlinear systems of inequalities as follows: Theorem 17 Let X and Y be two convex subsets of topological vector spaces and let f, g : X × Y −→ R be two functions satisfying: (i) f ( x, y ) ≤ g ( x, y ) for all ( x, y ) ∈ X × Y ; (ii) x ↦→ f ( x, y ) is quasiconcave on X, for each fixed y ∈ Y ; (iii) y ↦→ f ( x, y ) is l.s.c. and quasiconvex on Y, for each fixed x ∈ X ; (iv) x ↦→ g ( x, y ) is u.s.c. and quasiconcave on X, for each fixed x ∈ X ; (v) y ↦→ g ( x, y ) is quasiconvex on Y, for each fixed x ∈ X. (vi) Given λ ∈ R arbitrary, assume that either Y is compact, or X is com- pact, or there exist a compact subset K of X and a convex compact subset C of Y such that for any x ∈ X \ K there exists y ∈ C with g ( x, y ) < λ. 16