COMPATIBILITY OF SOME SYSTEMS OF INEQUALITIES

. In this paper, necessary and suﬃcient conditions for the compatibility of some systems of quasi-convex, or convex inequalities are established. Finally a new proof for a theorem of Shioji and Takahashi (1988) is given.


INTRODUCTION
Ky Fan studied in [4] the existence of solutions for some systems of convex inequalities involving lower semicontinuous functions defined on a compact convex set in a topological vector space (all the topological vector spaces considered in this paper are real and Hausdorff).Particularly, he proved the following theorem.
Theorem A. Let C be a nonempty compact convex subset of a topological vector space and let F be a family of real-valued lower semicontinuous convex functions defined on C. Then the following assertions are equivalent: (i) The system of convex inequalities (1) f (x) ≤ 0, f ∈ F, is compatible on C, i.e., there exists x ∈ C satisfying (1).(ii) For any n nonnegative numbers α i with n i=1 α i = 1 and for any f 1 , f 2 , . . ., f n ∈ F, there exists x ∈ C such that n i=1 α i f i (x) ≤ 0.
Shioji and Takahashi in [10, Th. 1] have established a Fan type theorem in the case when some function of two variables associated to the system of inequalities (1) is convex-like in one of the variables.This theorem receives a new proof in Section 3.
For any positive integer n we denote by S n the set The standard abbreviations conv A, cl A, card A are used to define the convex hull, closure and cardinality of a set A, respectively.

SYSTEMS OF QUASI-CONVEX INEQUALITIES
We recall that a real-valued function f defined on a convex set C is said to be quasi-convex if for every real number α, the set {x ∈ C : f (x) ≤ α} is convex.
In proving Theorem 2 we shall need the following lemma which is an analogous result of a classical Fan's section theorem.
Lemma 1. [6, Th. 2.2].Let C be a nonempty compact convex subset of a locally convex topological vector space X and K a nonempty closed convex subset of a topological vector space Y .Let A be a subset of C × K having the following properties: Then there exists Theorem 2. Let C be a nonempty compact convex subset of a locally convex topological vector space X and let F be a family of continuous quasi-convex functions f : C → R, satisfying the condition (2) any convex combination of functions in F is quasi-convex.
Then the following assertions are equivalent: In order to prove the reverse implication we shall apply Lemma 1 taking in the posture of Y , the vector space of all continuous functions f : C → R, endowed with the uniform norm f = max{|f (x)| : x ∈ C}.Also we take K = cl(conv F) (the closure being taken with respect to the uniform topology), ) and (f n ) n∈N a sequence in conv F uniformly converging to f (such a sequence there exists since Y is a normed space).By (ii) it follows that for each n ∈ N there exists According to (2), all the functions in conv F are quasi-convex.It is easily checked that the quasi-convexity is conserved by the pointwise convergence, hence by the uniform convergence too.
(c) For every x ∈ C, the set {f ∈ K; f (x) > 0} is obviously convex.So all the conditions of Lemma 1 are satisfied.Therefore there exists It should be mentioned that in the case when the topological vector space X is locally convex, Theorem 3 can be derived from Theorem 2. Indeed let C be a nonempty compact convex subset of a locally convex space and let F be a family of lower semicontinuous convex functions f : C → (−∞, ∞].It is clear that (i) implies (ii).In order to prove the reverse implication, for each f ∈ F let A f be the set of all continuous affine functions g : C → R satisfying g(x) ≤ f (x), for all x ∈ C, and denote by G = ∪{A f : f ∈ F}.
If g 1 , g 2 , . . ., g k ∈ A f 1 , g k+1 , g k+2 , . . ., g l ∈ A f 2 , . . ., g r+1 , g r+2 , . . ., g n ∈ A fm , then for every x ∈ C we have The sum in the right-hand side of the above inequality is a convex combination of the functions f 1 , f 2 , . . ., f m hence, according to (ii), it is ≤ 0 for at least one x ∈ C.This shows that G satisfies (ii).Theorem 2 applied to the family of functions G puts into evidence an x 0 ∈ C such that g(x 0 ) ≤ 0 for each g ∈ G.By (3) it follows immediately that f (x 0 ) ≤ 0, for each f ∈ F.
Remark 1. Observe that if the family of functions F is finite, having card F = n, the condition (ii) in each of Theorems A, 2, 3 can be replaced by (ii ) For each (α 1 , α 2 , . . ., α n ) ∈ S n and any f 1 , f 2 , . . ., f n ∈ F there exists x ∈ C such that n i=1 α i f i (x) ≤ 0. In Theorem 5 we shall give a set of sufficient conditions for the compatibility of systems of convex inequalities.The proof will be based on Theorem 3 and on the following intersectional result for convex sets (see [1] or [3]).

Proof. Denote by A the family of all sets
, where f i ∈ F. Since the functions f i ∈ F are lower semicontinuous and convex, the corresponding sets A i are closed in C and convex.The proof of Theorem 5 will be achieved whenever we verify the conditions (i) and (ii) in Lemma 4 for the family A.
If A does not satisfy the condition (i) then there exists k functions, pairwise distinct, f 1 , f 2 , . . ., f k in F and x in C such that f j (x) > 0, for all j ∈ {1, 2, . . ., k}.But in this case for any (α 1 , α 2 , . . ., α k ) ∈ S k we have k j=1 α j f j (x) > 0, which contradicts condition (a).Now given a subfamily {A 1 , A 2 , . . ., A l } of l members of A, i.e., A j = {x ∈ C : f j (x) ≤ 0}, f j ∈ F, then condition (b) together with Theorem 3, via Remark 1, yield an x ∈ C such that f j (x) ≤ 0, for all j ∈ {1, 2, . . ., k}, that is, The following result can be proved by applying the same argument as in the previous proof, using Theorem 2 instead of Theorem 3. Theorem 6.Let C be a nonempty compact convex subset of a locally convex topological vector space, F a family of continuous quasi-convex functions f : C → R satisfying condition (2) in Theorem 2, and k, l two positive integers with k ≤ l + 1 ≤ card F. If conditions (a) and (b) in Theorem 5 hold, then there exists x ∈ C such that f (x) ≤ 0, for all f ∈ F.

THE SHIOJI-TAKAHASHI THEOREM
In [10, Th. 1] Shioji and Takahashi have extended Fan's theorem to functions more general than the convex ones.The goal of this section is to give a new proof of this result, using a minimax theorem.
Before going to this result, we first recollect the following definitions (see [2, p. 161]).
Remark 2. It is clear from condition (i) that the following property results (i ) for every x 1 , x 2 , . . ., x n ∈ A and (α 1 , α 2 , . . ., α n ) ∈ S n , there exists A similar statement for condition (ii) holds.The above lemma has been formulated in [2,Th. 3.5], in the case when A and B are compact convex sets, each in a topological vector space, but the proof given there holds too in the conditions imposed by us.
The following theorem was obtained by Shioji and Takahashi in [10, Th. 1].We present another proof relied on Lemma 7.
Theorem 8. Let C be a nonempty compact space (not necessarily Hausdorff).Let F be a family of lower semicontinuous functions f : C → R such that the function F : F ×C → R defined by F (f, x) = f (x), for each f ∈ F and x ∈ C, is convex-like in its second variable.Then the assertions (i) and (ii) in Theorem 2 are equivalent.
Proof.We have only to prove the implication (ii) ⇒ (i).The set C being compact and the functions f ∈ F being lower semicontinuous, it follows immediately that an infinite system (1) is compatible if and only if every finite subsystem is compatible.So, we may assume that the family F is finite, namely Clearly L is linear, hence continuous in its first variable.On the other side, from hypothesis it follows that L is lower semicontinuous convex-like in its second variable.
Our assumption (ii) can be written as a minimax inequality, namely

In [ 10 ,
Th. 2], Shioji and Takahashi extend Fan's theorem to families of lower semicontinuous convex functions with values in (−∞, ∞].More exactly they have established Theorem 3. Let C be a nonempty compact convex subset of a topological vector space X and let F be a family of lower semicontinuous convex functions f : C → (−∞, ∞].Then the assertions (i) and (ii) in Theorem 2 are equivalent.

Lemma 4 .
Let C be a compact convex subset of a topological vector space, A a family of closed convex subsets of C, and k, l two positive integers with k ≤ l + 1 ≤ card A. Suppose that (i) ∪A = C, for any subfamily A of A with card A = k; (ii) ∩A = ∅, for any subfamily A of A with card A = l.Then ∩A = ∅.Theorem 5. Let C be a nonempty compact convex subset of a topological vector space, F be a family of lower semicontinuous convex functions f : C → (−∞, ∞], and k, l two positive integers with k ≤ l + 1 ≤ card F. Suppose that (a) for each k functions, pairwise distinct, f 1 , f 2 , . . ., f k ∈ F and any x ∈ C there exists (α 1 , α 2 , . . ., α k ) ∈ S k such that k j=1 α j f j (x) ≤ 0; (b) for each l functions f 1 , f 2 , . . ., f l ∈ F and any (α 1 , α 2 , . . ., α l ) ∈ S l there exists x ∈ C such that f (x) ≤ 0, for all f ∈ F.

Lemma 7 .FF
Let A and B be compact topological spaces and let F : A×B → R be an upper-lower semicontinuous concave-convex-like function.Then max x∈A min y∈B (x, y) = min y∈B max x∈A (x, y).

α
max α∈Sn min x∈C L(α, x) ≤ 0. The existence of an x ∈ C satisfying all n inequalities f i (x) ≤ 0, 1 ≤ i ≤ n, is equivalent to the truth of the relation min i f i (x) = min x∈C max α∈Sn L(α, x) ≤ 0. This relation can be obtained by Lemma 7 and relation (4) as follows min x∈C max α∈Sn L(α, x) = max α∈Sn min x∈C L(α, x) ≤ 0.