## 4.1 Alternatives for Systems of Nonlinear Inequalities and Applications 

Theorems 6 and 9 can be expressed in terms of an alternative for nonlinear systems of inequalities ` a la Ky Fan . 

Recall first the basic concepts of semicontinuity and quasiconvexity for real functions. 

Definition 13 A real function f : X −→ R defined on a subset X of a t.v.s is: (i) quasiconvex if ∀ λ ∈ R , the level set { x ∈ X ; f ( x ) ≤ λ } is a convex subset of X ; 

(ii) quasiconcave if − f is quasiconvex; 

(iii) lower semicontinuous (l.s.c.) if ∀ λ ∈ R , the level set { x ∈ X ; f ( x ) ≤ λ } is a closed subset of X ; 

(iv) upper semicontinuous (u.s.c.) if − f is l.s.c.. 

Naturally, every convex functional is quasiconvex and the converse is false. Also, a real function on a topological space is continuous if and only if it is both upper and lower semicontinuous. 

Theorem 14 Let X be a convex subset of a t.v.s. E , Y a non-empty subset of X, and f : X × Y −→ R a function satisfying: 

(i) x ↦→ f ( x, y ) is l.s.c. and quasiconvex on X, for each fixed y ∈ Y. 

(ii) y ↦→ f ( x, y ) quasiconcave on Y, for each fixed x ∈ X ; 

Assume that for a given λ ∈ R , there exist a compact subset K of X and a convex compact subset D of Y 

such that ∀ x ∈ X \ K, ∃ y ∈ D with f ( x, y ) > λ. 

Then the following alternative holds: 

(A) there exists x 0 ∈ X such that f ( x 0 , x 0 ) > λ, or 

