4.1 Alternatives for Systems of Nonlinear Inequalities and Applications 13 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